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+1
+Two-Dimensional Golay Complementary Array
+Sets With Arbitrary Lengths for
+Omnidirectional MIMO Transmission
+You-Qi Zhao, Cheng-Yu Pai, Zhen-Ming Huang, Zilong Liu, Senior
+Member, IEEE, and Chao-Yu Chen, Member, IEEE
+Abstract
+This paper presents a coding approach for achieving omnidirectional transmission of certain common
+signals in massive multi-input multi-output (MIMO) networks such that the received power at any
+direction in a cell remains constant for any given distance. Specifically, two-dimensional (2D) Golay
+complementary array set (GCAS) can be used to design optimal massive MIMO precoding matrix so as
+to achieve omnidirectional transmission due to its complementary autocorrelation property. In this paper,
+novel constructions of new 2D GCASs with arbitrary array lengths are proposed. Our key idea is to
+carefully truncate the columns of certain larger arrays generated by 2D generalized Boolean functions.
+Finally, the power radiation patterns and numerical results are provided to verify the omnidirectional
+property of the GCAS-based precoding. The error performances of the proposed precoding scheme are
+presented to validate its superiority over the existing alternatives.
+Index Terms
+This work was supported by the Ministry of Science and Technology, Taiwan, R.O.C., under Grant MOST 109–2628–E–006–
+008–MY3 and MOST 111–2218–E–305–002.
+You-Qi Zhao and C.-Y. Pai are with the Department of Engineering Science, National Cheng Kung University, Tainan 701,
+Taiwan, R.O.C. (e-mail: n98081505@gs.ncku.edu.tw).
+Z.-M. Huang is with the Institute of Computer and Communication Engineering, National Cheng Kung University, Tainan
+701, Taiwan, R.O.C. (e-mail: n98101012@gs.ncku.edu.tw).
+Zilong Liu is with the School of Computer Science and Electronic Engineering, University of Essex, United Kingdom (e-mail:
+zilong.liu@essex.ac.uk).
+C.-Y. Chen is with the Department of Electrical Engineering and the Institute of Computer and Communication Engineering,
+National Cheng Kung University, Tainan 701, Taiwan, R.O.C. (e-mail: super@mail.ncku.edu.tw).
+January 4, 2023
+DRAFT
+arXiv:2301.01225v1  [cs.IT]  3 Jan 2023
+
+2
+Generalized Boolean function (GBF), Golay complementary array pair (GCAP), Golay comple-
+mentary array set (GCAS), omnidirectional precoding (OP), uniform rectangular array (URA).
+I. INTRODUCTION
+Complementary pairs/sets of sequences have attracted a sustained research interest owing to
+their zero aperiodic correlation sums properties. To be specific, a Golay complementary pair
+(GCP) refers to a pair of equal-length sequences whose summation of aperiodic autocorrelations
+is zero except at the zero time-shift [1]. Such a concept was extended to Golay complementary
+set (GCS) with constituent sequences of more than 2 by Tseng and Liu in [2]. Furthermore, a
+maximum collection of GCSs is called a set of complete complementary code (CCC) [3] if any
+two different GCSs have zero aperiodic cross-correlation sums for all time-shifts. In the literature,
+GCSs and CCCs have been widely used for radar sensing [4], channel estimation [5], precoding
+for massive multi-input multi-output (MIMO) [6], peak-to-average power ratio (PAPR) reduction
+in orthogonal frequency division multiplexing (OFDM) [7]–[13], interference-free multicarrier
+code division multiple access [14]–[17], and many other applications [18], [19].
+Recently, there is a surge of research attention to study two-dimensional (2D) Golay com-
+plementary array sets (GCASs) [18]-[23], each having zero aperiodic autocorrelation sums
+property for two directions of shifts (compared to conventional GCSs and CCCs with time-
+shifts only). An important application of the 2D GCASs is for omnidirectional transmission in
+MIMO communication systems with a uniform rectangular array (URA) configuration [20], [21].
+In massive MIMO systems, some common messages (e.g., reference signals, synchronization
+signals, control signals, etc.) need to be power-uniformly broadcasted to all the angles within
+the whole cell. In this paper, we consider space-time block code (STBC) for the harvesting
+of the diversity gain. At the base station (BS), the STBC encoded symbols are assigned to
+several streams and then mapped onto the antenna arrays in URA by certain 2D GCASs assisted
+precoding matrices to achieve uniform power radiation at any angle.
+On the other hand, since a large number of antennas are considered in massive MIMO
+systems, a huge pilot overhead may be needed to acquire the channel state information (CSI). As
+pointed out in [22], this can be alleviated by omnidirectional precoding (OP) based transmission.
+For uniform linear arrays (ULAs), Zadoff-Chu (ZC) sequences were adopted to satisfy the
+requirements of the omnidirectional property. However, [22] only considered the omnidirectional
+January 4, 2023
+DRAFT
+
+3
+transmission in certain directions. Later in [6], GCSs and CCCs based OP matrices were proposed
+to meet the requirement of omnidirectional transmission across all directions.
+In [20], [21], [23], [24], 2D GCASs were employed for precoding matrices in URAs by
+applying interleaving and Kronecker-product to existing 1D sequences or 2D arrays. As a result,
+the array sizes of 2D GCASs are only feasible for certain lengths. A construction of 2D GCASs
+of array size pn × pm was proposed in [25] by using permutation ploynomials (PPs) functions
+and 2-level autocorrelation sequences, where p is a prime number, m, n are two positive integers,
+and p, m, n > 0. Furthermore, a unifying construction framework for 2D GCASs was developed
+in [26] by a multivariate polynomial matrix from certain seed para-unitary (PU) matrices. In
+[27], [28], Pai and Chen proposed direct constructions of 2D Golay complementary array pairs
+(GCAPs) and GCASs with array size 2n × 2m from 2D generliazed Boolean functions (GBFs)
+[29] where n, m are integers and n, m ≥ 2. 2D GCAP can be regarded as a case of 2D GCAS
+when the set size is equal to 2. Moreover, Pai et al. [30] proposed a direct construction of 2D
+CCCs with array size 2n×2m, which have ideal autocorrelations and cross-correlations. Later, Liu
+et al. [31] proposed a construction of GCASs with array size pn ×pm by using 2D multivariable
+functions, where p is a prime number, n, m are integers, and n, m ≥ 2. Based on [27], [32]
+developed a direct construction of GCASs with set size 4 and array size 2n × (2m−1 + 2v) by
+using 2D GBFs, where n, m, v are positive number with n, m ≥ 2, and 0 ≤ v ≤ m − 1.
+The aforementioned research efforts are generally driven by the need of highly flexible array
+sizes of 2D GCASs. Motivated by this, we aim for generating new GCASs with arbitrary array
+lengths. The key idea of our proposed constructions is to carefully truncate some columns of
+the certain larger arrays generated by 2D GBFs. Thus, our proposed GCASs can be applied to
+URAs with various array sizes. In addition, the proposed GCASs can be directly generated from
+2D GBFs without the requirements of any specific sequences or tedious sequence operations. In
+Table I, we compare the existing parameters of 2D GCASs with our proposed ones.
+The remainder of this paper is defined as follows. Section II discusses notations, definitions,
+system models, and the omnidirectional transmission in MIMO systems. Section III describes
+our proposed constructions of 2D GCASs. Section IV shows the power radiation pattern and bit
+error rate (BER) performance based on our proposed 2D GCASs precoding. Finally, Section V
+presents the conclusion.
+January 4, 2023
+DRAFT
+
+4
+TABLE I
+A COMPARISON OF CONSTRUCTIONS FOR 2D GCASS
+Construction
+Parameters
+Approaches
+[26, Th. 5]
+(N, N n, N m), N, n, m > 0
+Seed PU matrices
+[26, Th. 7]
+(2k, 2kn, 2km), n, m, k > 0
+[25, Th. 4]
+(p, pn, pm), prime p, n, m > 0
+PPs and 2-level
+autocorrelation sequences
+[25, Th. 6]
+(pk, pkn, pkm), prime p, k, n, m > 0
+[31, Th. 1]
+(pk1
+1 pk2
+2 , pn
+1 , pm
+2 ), primes p1, p2
+2D multivariable functions
+[31, Th. 2]
+(pk, pn, pm), prime p, n + m ≥ k > 0
+[27], [28], [30]
+(2k, 2n, 2m), n, m ≥ k > 0, and k > 0
+2D GBFs
+[32]
+(4, 2n, 2m−1 + 2v), n, m ≥ 2, and k > 0
+Th. 1
+(2k+1, 2n, 2m−1 + �k−1
+α=1 dα2m−k+α−1 + d02v),
+k < m, 0 ≤ v ≤ m − k, dα ∈ {0, 1}
+Th. 2
+(2k+1, 2n, 2m−1 + �k−1
+α=1 dα2π1(m−k+α)−1 + d02v),
+k < m, 0 ≤ v ≤ m − k, dα ∈ {0, 1}
+II. PRELIMINARIES AND DEFINITIONS
+A. Notations
+Throughout this paper, we present the notations in the following:
+• (a)i refers to the i-th element of the vector a.
+• (A)i,j denotes the (i, j)-th element of the array A.
+• (·)H refers to the conjugate transpose.
+• diag(A) refers to the column vector composed of the main diagonal of A.
+• (·)∗ refers to the complex conjugation of an element.
+• (·)T refers to the transpose.
+• vec(·) express stacking one column of the matrix into one another column.
+• 1 is a vector whose elements are all 1.
+• Let ξ = e2π√−1/q.
+• In this paper, q is an even number.
+Let X and Y be two arrays of size L1 × L2. Then X and Y can be stated as
+X = (Xg,i), Y = (Yg,i),
+(1)
+where g = 0, 1, · · · , L1 − 1 and i = 0, 1, · · · , L2 − 1.
+January 4, 2023
+DRAFT
+
+5
+Definition 1: Given two arrays X and Y of size L1 × L2, the 2D aperiodic cross-correlation
+function (AACF) is defined by
+ρ (X, Y; u1, u2) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+L1−1−u1
+�
+g=0
+L2−1−u2
+�
+i=0
+Yg+u1,i+u2X∗
+g,i, 0 ≤ u1 < L1,
+0 ≤ u2 < L2;
+L1−1−u1
+�
+g=0
+L2−1−u2
+�
+i=0
+Yg+u1,iX∗
+g,i−u2, 0 < u1 < L1,
+−L2 < u2 < 0;
+L1−1−u1
+�
+g=0
+L2−1−u2
+�
+i=0
+Yg,iX∗
+g−u1,i−u2, −L1 < u1 < 0,
+−L2 < u2 < 0;
+L1−1+u1
+�
+g=0
+L2−1−u2
+�
+i=0
+Yg,i+u2X∗
+g−u1,i, −L1 < u1 < 0,
+0 < u2 < L2.
+(2)
+When X = Y , then it is called 2D aperiodic autocorrelation function (AACF) and denoted
+by ρ(X; u1, u2). If taking L1 = 1, two 2D arrays X and Y are degraded as a 1-D sequence
+X = Xi for i = 0, 1, · · · , L2 − 1 and Y = Yi for i = 0, 1, · · · , L2 − 1, respectively. Then the
+1-D AACF of 1-D sequence X is related by
+ρ(X; u) =
+�
+�
+�
+�
+�
+�
+�
+L2−1−u
+�
+i=0
+Xi+uX∗
+i ,
+0 ≤ u ≤ L2 − 1;
+L2−1+u
+�
+i=0
+XiX∗
+i−u,
+−L2 + 1 ≤ u < 0.
+(3)
+In this paper, q-PSK modulation is employed. Thus, x and y denote q-ary arrays and (1) is
+expressed as
+X = (Xg,i) = (ξxg,i) = ξx;
+Y = (Yg,i) = (ξyg,i) = ξy,
+(4)
+where x = (xg,i), y = (yg,i), and xg,i, yg,i ∈ Zq = {0, 1, · · · , q−1} for 0 ≤ g < L1, 0 ≤ i < L2.
+Consider a set of N L-length sequences can be represented as
+C = {X0, X1, · · · , XN−1}
+where
+Xn = (Xn,0, Xn,1, · · · , Xn,L−1)
+for n = 0, 1, · · · , N − 1.
+January 4, 2023
+DRAFT
+
+6
+Definition 2: [19] If a set C consisting of N sequences of length L satisfies
+N−1
+�
+k=0
+ρ(Xk; u) =
+�
+�
+�
+�
+�
+NL,
+u = 0;
+0,
+u ̸= 0,
+(5)
+then the set C is called a Golay complementary set of size N, denoted by (N, L)-GCS. The
+GCP can be regarded as a special case of the GCS by setting N = 2.
+Definition 3: For a GCP (X0, X1), if another GCP (Y0, Y1) meets the following condition:
+ρ(X0, Y0; u) + ρ(X1, Y1; u) = 0, for all u,
+(6)
+then the two GCPs are called the Golay complementary mate of each other.
+Definition 4: A pair of arrays X and Y of array size L1 × L2 is called a 2D Golay
+complementary array pair if
+ρ(X; u1, u2) + ρ(Y ; u1, u2) =
+�
+�
+�
+�
+�
+2L1L2,
+u1 = u2 = 0;
+0,
+u1 ̸= 0 or u2 ̸= 0.
+(7)
+Definition 5: Let the array set G = {X0, X1, · · · , XN−1} where each array in set G is of
+size L1 × L2. If the array set G satisfies
+N−1
+�
+k=0
+ρ(Xk; u1, u2) =
+�
+�
+�
+�
+�
+NL1L2,
+u1 = u2 = 0;
+0,
+u1 ̸= 0 or u2 ̸= 0,
+(8)
+the set G is called the Golay complementary array set of set size N denoted by (N, L1, L2)-
+GCAS where L2 is defined as the length of the GCAS. If N = 2, the GCAS G is degraded as
+a GCAP.
+B. Generalized Boolean Functions
+A 2D generalized Boolean function (GBF) f in n + m binary variables y1, y2, · · · , yn,
+x1, x2, · · · , xm, is a function mapping: Zn
+2 ×Zm
+2 → Zq, where xi, yg ∈ {0, 1} for i = 1, 2, · · · , m
+and g = 1, 2, · · · , n. A monomial of degree r is given by any product of r distinct variables
+among y1, y2, · · · , yn, x1, x2, · · · , xm. For instance, x1x3y1y2 is a monomial of degree 4. Next,
+the variables z1, z2, · · · , zn+m are defined as
+zl =
+�
+�
+�
+�
+�
+yl
+if 1 ≤ l ≤ n;
+xl−n
+if n < l ≤ m + n,
+(9)
+January 4, 2023
+DRAFT
+
+7
+which are useful for our proposed constructions. For a 2D GBF with n + m variables, the 2D
+Zq-valued array
+f =
+�
+�
+�
+�
+�
+�
+�
+f0,0
+f0,1
+· · ·
+f0,2m−1
+f1,0
+f1,1
+· · ·
+f1,2m−1
+...
+...
+...
+...
+f2n−1,0
+f2n−1,1
+· · ·
+f2n−1,2m−1
+�
+�
+�
+�
+�
+�
+�
+(10)
+of size 2n×2m is given by letting fg,i = f((g1, g2, · · · , gn), (i1, i2, · · · , im)), where (g1, g2, · · · , gn)
+and (i1, i2, · · · , im) are binary vector representations of integers g = �n
+h=1 gh2h−1 and i =
+�n
+j=1 ij2j−1, respectively.
+Example 1: Taking q = 4, n = 2, and m = 3 for example, the 2D GBF is given as f =
+3z5z4 + z2z3 + 2z2. Then the array f of size 4 × 8 corresponding to f can be obtained, i.e.,
+f =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+0
+0
+0
+3
+3
+0
+0
+0
+2
+1
+1
+3
+3
+2
+3
+2
+3
+2
+3
+1
+2
+2
+3
+2
+3
+2
+3
+1
+2
+�
+�
+�
+�
+�
+�
+�
+.
+(11)
+The GBF f can be rewritten as f = 3x3x2 + y2x1 + 2y2. In this paper, we consider the array
+size ̸= 2n × 2m. Hence, we define the truncated array f (L) corresponding to the 2D GBF f by
+ignoring the last 2m − L columns of the corresponding array f.
+Example 2: Following the same notations given in Example 1, the truncated array f (6) is
+given by
+f (6) =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+0
+0
+0
+0
+0
+0
+2
+1
+1
+2
+3
+2
+3
+2
+3
+2
+3
+2
+3
+2
+3
+�
+�
+�
+�
+�
+�
+�
+.
+(12)
+For simplicity, we use f to stand for f (L) when L is known.
+C. System Model
+Considering downlink transmission from a BS to UEs where each has one single antenna,
+we suppose that the number of antennas at the BS is M = L1 × L2, i.e., the URA consists of
+L1 rows and L2 columns. Fig. 1 illustrates the diagram of data downlink transmission. For an
+January 4, 2023
+DRAFT
+
+8
+Fig. 1. Diagram of data transmission through STBC encoding and omnidirectional precoding.
+L1 × L2 URA, the steering matrix A(ϕ, θ) at the direction (ϕ, θ) with the (g, i)-th entry can be
+expressed as
+(A(ϕ, θ))g,i =e−j 2π
+λ gdy sin ϕ sin θ−j 2π
+λ idx sin ϕ cos θ,
+for g = 0, 1, . . . , L1 − 1, i = 0, 1, . . . , L2 − 1,
+θ ∈ [0, 2π], ϕ ∈ [0, π/2],
+(13)
+where dx and dy denote the vertical antenna and horizontal antenna inter-element spacings of
+the URA, respectively, and λ denotes the carrier wavelength. To enhance the spatial diversity
+and communication reliability, the STBC signal transmission scheme is used. The N ×M STBC
+is given by
+S ≜
+�
+�
+�
+�
+�
+�
+�
+s0(0)
+s0(1)
+· · ·
+s0(M − 1)
+s1(0)
+s1(1)
+· · ·
+s1(M − 1)
+...
+...
+...
+...
+sN−1(0)
+sN−1(1)
+· · ·
+sN−1(M − 1)
+�
+�
+�
+�
+�
+�
+�
+∈ CN×M
+(14)
+January 4, 2023
+DRAFT
+
+So(t)
+Xo(t) 
+7
+Data
+Omni-
+Space-time
+UE1
+Si(t)
+directional
+xi(t)
+block coding
+y(t)
+...
+precoding
+V
+SN-1(t)
+XL;L2
+-1(t)
+y
+d
+d
+(t)
+xo(t)
+x
+Omnidirectional
+precoding
+UE19
+where CN×M refers to the N-by-M complex space and sn(t) denotes the (n, t)-th element of
+the STBC at time instant t for t = 0, 1, · · · , M − 1. We define the precoding matrix Wn of size
+L1 × L2. The encoded symbols is given by
+x(t) = (x0(t), x1(t), · · · , xL1L2−1(t))T = vec
+�N−1
+�
+n=0
+Wn · sn(t)
+�
+, for t = 0, 1, · · · , M − 1,
+(15)
+which are transmitted by the L1L2 antennas of the URA. In the light-of-sight (LOS) channel
+without multipaths, the received signal at the direction (ϕ, θ) can be written as
+y(t) =
+N−1
+�
+n=0
+�
+vec(A(ϕ, θ))Tvec(Wn)
+�
+· sn(t) + η(t), t = 0, . . ., M − 1,
+(16)
+where η(t) is the additive Gaussian white noise (AWGN) at time instant t.
+D. Omnidirectional Precoding Matrices Based on 2D Arrays
+In this subsection, we list two necessary requirements for the design of OP matrices. Then,
+we will connect these two requirements with the conditions of 2D arrays.
+Requirement 1 (R1): Omnidirectional transmission.
+We consider the MIMO system with URA. Following (16), the received power E at the angle
+(ϕ, θ) is represented as
+E =
+N−1
+�
+n=0
+��[vec(A(ϕ, θ))Tvec(Wn)]
+��2 .
+(17)
+Therefore, to satisfy the omnidirectional transmission in the whole cell, (17) must be constant
+for all ϕ and θ.
+Requirement 2 (R2): Equal average power on each antenna.
+To enhance the efficiency of the power amplifier, the average transmission power on all L1×L2
+antennas is required to be equal. We define
+W = (vec(W0), vec(W1), · · · , vec(WN−1)) ,
+(18)
+where the array size of W is L1L2 × N. Hence, (15) can be rewritten as
+X = (x(0), x(1), · · · , x(M − 1)) = W S.
+(19)
+January 4, 2023
+DRAFT
+
+10
+Let s(t) be the t-th column of S. Throughout this paper, we assume E
+�
+s(t)s(t)H�
+=IN. The
+transmitted signal on the (l1, l2)-th antenna is (W s)l2L1+l1. The average power on the (l1, l2)-th
+antenna can be expressed as
+E
+�
+|(W s)l2L1+l1|2�
+=
+�
+W E
+�
+s(t)s(t)H�
+W H�
+l2L1+l1,l2L1+l1
+= (W W H)l2L1+l1,l2L1+l1.
+(20)
+Therefore, the condition to guarantee equal power on each antenna is equivalent to
+diag(W W H) = N1.
+(21)
+Next, we will derive two sufficient conditions on the precoding matrices to fulfill requirements
+R1 and R2.
+Lemma 1: [21] For an L1 × L2 URA, if the precoding matrices W0, W1, · · · , WN−1 of size
+L1 × L2 form an (N, L1, L2)-GCAS, then the omnidirectional transmission is achieved.
+Lemma 2: For an L1×L2 URA, if the precoding matrices W0, W1, · · · , WN−1 of size L1×L2
+are unimodular, then the average power on each antenna is equal.
+Proof: In order to meet the requirement for equal average power on each antenna, the
+precoding matrix W must satisfy (21). We let wi = vec(Wi), for i = 0, 1, · · · , N − 1. Then,
+diag
+�
+W W H�
+=
+�N−1
+�
+i=0
+|(wi)0|2 ,
+N−1
+�
+i=0
+|(wi)1|2 , · · · ,
+N−1
+�
+i=0
+|(wi)L1L2−1|2
+�T
+= N1
+(22)
+since we have
+|(wi)n|2 = 1,
+for i = 0, 1, · · · , N − 1 and n = 0, 1, · · · , L1L2 − 1.
+(23)
+According to (21), the requirement (R2) is fulfilled.
+In the sequel, the design of OP matrices W0, W1, · · · , WN−1 are based on Lemma 1 and
+Lemma 2. That is, our goal is to construct unimodular GCASs with flexible sizes.
+III. GCASS WITH FLEXIBLE ARRAY SIZE
+In this section, two constructions of 2D GCASs with arbitrary array lengths based on 2D
+GBFs will be proposed. By recalling the function mapping in (9), we present our first theorem
+in the following.
+January 4, 2023
+DRAFT
+
+11
+Theorem 1: For any integers q, m, n ≥ 2, and k < m, v is an integer satisfies 0 ≤ v ≤ m−k
+and let π be a permutation of {1, 2, · · · m + n − k} satisfying {zπ(1), zπ(2), · · · , zπ(v+n)} =
+{z1, z2, · · · , zv+n}. The 2D generalized Boolean function can be written as
+f = q
+2
+�m+n−k−1
+�
+l=1
+zπ(l)zπ(l+1)
+�
++
+m+n
+�
+s=1
+pszs + p0
+(24)
+where ps ∈ Zq. The array set
+G =
+�
+f + q
+2
+k
+�
+α=1
+λαzm+n−k+α + q
+2λk+1zπ(1) : λα ∈ {0, 1}
+�
+(25)
+is a q-ary (2k+1, 2n, 2m−1 + �k−1
+α=1 dα2m−k+α−1 + d02v)-GCAS where dα ∈ {0, 1}.
+Proof: Without loss of generality, we consider L1 = 2n and L2 = 2m−1+�k−1
+α=1 2m−k+α−1+ 2v.
+We need to show that
+�
+c∈G
+L1−1−u1
+�
+g=0
+L2−1−u2
+�
+i=0
+�
+ξcg+u1,i+u2−cg,i�
+= 0
+(26)
+for 0 ≤ u1 < 2n, 0 ≤ u2 < 2m−1 + �k−1
+α=1 2m−k+α−1 + 2v and (u1, u2) ̸= (0, 0). Then we let
+h = g + u1 and j = i + u2 for any integers g and i. We also let (g1, g2, · · · , gn),(i1, i2, · · · , im),
+(h1, h2, · · · , hn), and (j1, j2, · · · , jm) be the binary representations of g, i, h, and j, respectively.
+For the ease of presentation, we denote
+al =
+�
+�
+�
+�
+�
+gl
+for 1 ≤ l ≤ n;
+il−n for n < l ≤ n + m;
+bl =
+�
+�
+�
+�
+�
+hl
+for 1 ≤ l ≤ n;
+jl−n for n < l ≤ n + m;
+(27)
+In what follows, we consider four cases to show that the above formula holds.
+Case 1: If aπ(1) ̸= bπ(1), we can find that c′ = c + (q/2)zπ(1) for any arrayc ∈ G satisfying
+ch,j − cg,i − c′
+h,j+c′
+g,i = q
+2(aπ(1) − bπ(1)) ≡ q
+2
+(mod q).
+(28)
+Therefore, we have
+ξch,j−cg,i + ξc′
+h,j−c′
+g,i = 0.
+(29)
+Case 2: If am+n−k+α ̸= bm+n−k+α, we can find that c′ = c + (q/2)zm+n−k+α for any array
+c ∈ G. Similar to Case 1, we have
+ξch,j−cg,i + ξc′
+h,j−c′
+g,i = 0.
+(30)
+January 4, 2023
+DRAFT
+
+12
+Case 3: If aπ(1) = bπ(1) and am+n−k+α = bm+n−k+α for α = 1, 2, · · · , k. Suppose that α′
+is the largest integer satisfying am+n−k+α′ = bm+n−k+α′ = 0 for α′ ≤ k. Then we assume β
+is the smallest integer which satisfies aπ(β) ̸= bπ(β). Let a′ and b′ be integers distinct from a
+and b, respectively, only in one position π(β − 1). In other words, a′
+π(β−1) = 1 − aπ(β−1) and
+b′
+π(β−1) = 1 − bπ(β−1). If 1 ≤ π(β − 1) ≤ n, by using the above definition, we have
+cg′,i − cg,i
+= q
+2
+�
+aπ(β−2)g′
+π(β−1) − aπ(β−2)gπ(β−1) + g′
+π(β−1)aπ(β)
+−gπ(β−1)aπ(β)
+�
++ pπ(β−1)g′
+π2(β−1) − pπ(β−1)gπ(β−1)
+≡ q
+2(aπ(β−2) + aπ(β)) + pπ(β−1)(1 − 2gπ(β−1))
+(mod q).
+(31)
+where a′
+π(β−1) = g′
+π(β−1) and aπ(β−1) = gπ(β−1). Since aπ(β−2) = bπ(β−2) and aπ(β−1) = bπ(β−1),
+we have
+ch,j − cg,i − ch′,j + cg′,i
+≡ q
+2(aπ(β−2) − bπ(β−2) + aπ(β) − bπ(β))
++ pπ(β−1)(2hπ(β−1) − 2gπ(β−1))
+≡ q
+2(aπ(β) − bπ(β)) ≡ q
+2
+(mod q)
+(32)
+implying ξch,j−cg,i/ξch′,j−cg′,i = −1. We can also obtain
+ξch,j−cg,i + ξch′,j−cg′,i = 0.
+(33)
+If n < π(β − 1) ≤ n + m, note that a′
+π(β−1) = i′
+π(β−1)−n and aπ(β−1) = iπ(β − 1) − n according
+to (27). Following the similar argument as given above, we can get ξch,j−cg,i + ξch,j′−cg,i′ = 0.
+Case 4: If aπ(1) = bπ(1) and am+n−k+α = bm+n−k+α = 1 for α = 1, 2, · · · , k. We assume β is
+the smallest integer such that aπ(β) ̸= bπ(β). Since as = bs = 0 for s = v+n+1, v+n+2, · · · , m+
+n−k, we can obtain π(β) ≤ v+n implying π(β −1) ≤ v+n. If 1 ≤ π(β−1) ≤ n, by following
+the similar argument as given above, we have ξch,j−cg,i +ξch′,j−cg′,i = 0. If n < π(β −1) ≤ v+n,
+we have ξch,j−cg,i + ξch,j′−cg,i′ = 0. From Cases 1 to 4, the theorem can be proved.
+Remark 1: The parameter 2m−1+�k−1
+α=1 dα2m−k+α−1+d02v of the proposed GCASs in Theorem
+1 can be any arbitrary length since m, k, v are flexible and dα ∈ {0, 1}.
+Example 3: Taking q = 2, m = 6, n = 2, k = 1, and v = 0, we let π = (1, 2, 3, 4, 5, 6, 7).
+The generalized Boolean function is f = z1z2 + z2z3 + z3z4 + z4z5 + z5z6 + z6z7 = x1x2 +
+x2x3 + x3x4 + x4x5 + y1y2 + y2x1 by setting pk = 0 for k = 0, 1, . . . , m + n. The array set
+January 4, 2023
+DRAFT
+
+13
+TABLE II
+THE CONSTRUCTED (4, 4, 33)-GCAS IN EXAMPLE 3
+c0 =
+�
+�
+�
+�
+�
+�
+�
+0
+1
+1
+1
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+1
+0
+1
+1
+1
+1
+0
+0
+0
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+0
+1
+0
+0
+1
+1
+0
+0
+0
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+0
+1
+0
+0
+1
+�
+�
+�
+�
+�
+�
+�
+c1 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+1
+0
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+0
+0
+1
+1
+1
+1
+0
+1
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+0
+0
+1
+0
+1
+1
+0
+1
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+0
+0
+1
+1
+�
+�
+�
+�
+�
+�
+�
+c2 =
+�
+�
+�
+�
+�
+�
+�
+0
+1
+1
+1
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+1
+0
+1
+1
+1
+1
+0
+0
+0
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+0
+1
+0
+0
+1
+0
+1
+1
+1
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+1
+0
+1
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+1
+0
+1
+1
+0
+�
+�
+�
+�
+�
+�
+�
+c3 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+1
+0
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+0
+0
+1
+1
+0
+0
+1
+0
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+1
+1
+1
+0
+1
+0
+0
+1
+0
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+1
+1
+1
+0
+0
+�
+�
+�
+�
+�
+�
+�
+0
+200
+2
+400
+40
+600
+20
+800
+0
+1000
+0
+-20
+-2
+-40
+Fig. 2. The summation of autocorrelations of constituent arrays in the GCAS in Example 3.
+G = {f, f + x8, f + y1, f + x8 + y1} is a GCAS of size 4 and the array size is 4 × 33. We let
+G = {c0, c1, c2, c3} and list the constituent arrays in Table II. Fig. 2 shows the AACF sum of
+set G is zero at shift u1 ̸= 0 or u2 ̸= 0. Thus, we can find that array set G is a (4, 4, 33)-GCAS.
+January 4, 2023
+DRAFT
+
+14
+Next, we introduce a lemma which illustrates a construction of (4, 2n, 2m−1 +2v)-GCAS from
+2D GBFs.
+Lemma 3: [32, Th. 1] For nonnegative integers m, n, and v with 0 ≤ v < m − 1, let π1 be
+a permutation of {1, 2, · · · , m − 1} and π2 be a permutation of {1, 2, · · · , n}. The 2D GBF is
+given by
+f =q
+2
+�m−2
+�
+k=1
+xπ1(k)xπ1(k+1) +
+n−1
+�
+k=1
+yπ2(k)yπ2(k+1) + xπ1(m−1)xm + xmyπ2(1)
+�
++
+m
+�
+l=1
+plxl +
+n
+�
+s=1
+κsys + p0
+(34)
+where pl, κs ∈ Zq. Then the array set
+G =
+�
+f, f + q
+2xπ1(1), f + q
+2yπ2(n), f + q
+2xπ1(1) + q
+2yπ2(n)
+�
+is a (4, 2n, 2m−1 + 2v)-GCAS.
+Since the set size of the GCAS from Lemma 3 is limited to 4, we propose a general
+construction of 2D GCASs with more flexible array sizes and set sizes which can include Lemma
+3 as a special case.
+Theorem 2: For any integers q, m, n ≥ 2, and k < m, v is an integer satisfies 0 ≤ v ≤ m−k.
+Assume that π1 is a permutation of {1, 2, · · · m} and π2 is a permutation of {1, 2, · · · n}. The
+2D generalized Boolean function can be written as
+f =q
+2
+�m−k−1
+�
+l=1
+xπ1(l)xπ1(l+1) +
+n−1
+�
+s=1
+yπ2(s)yπ2(s+1) + xπ1(m)yπ2(n)
+�
++
+m−k
+�
+l=1
+µlxπ1(l)xπ1(m) +
+m
+�
+l=1
+plxk +
+n
+�
+s=1
+κsys + p0
+(35)
+where µl, pl, κs, ∈ Zq. The array set
+G =
+�
+f + q
+2
+k−1
+�
+α=1
+λαxπ1(m−k+α) + q
+2λkyπ2(1) + q
+2λk+1xπ1(1) : λα ∈ {0, 1}
+�
+(36)
+is a q-ary (2k+1, 2n, 2m−1 + �k−1
+α=1 dα2π1(m−k+α)−1 + d02v)-GCAS where dα ∈ {0, 1} if the
+following three conditions hold.
+(C1) {π1(1), π1(2), · · · , π1(v)} = {1, 2, · · · , v} if v > 0;
+(C2) π1(m − k + α) < π1(m − k + α + 1) for 1 ≤ α ≤ k − 1 where π1(m) = m;
+(C3) For 1 ≤ α ≤ k − 1 and 2 ≤ β ≤ m − k, if π1(β) < π1(m − k + α), then π1(β − 1) <
+π1(m − k + α).
+January 4, 2023
+DRAFT
+
+15
+Proof: Similarly, we consider L1 = 2n and L2 = 2m−1 + �k−1
+α=1 2π1(m−k+α)−1 + 2v. Then
+we would like to prove that
+�
+C
+ρ(C; u1, u2) =
+�
+c∈G
+L1−1−u1
+�
+g=0
+L2−1−u2
+�
+i=0
+�
+ξcg+u1,i+u2−cg,i�
+= 0
+(37)
+for 0 ≤ u1 < 2n, 0 ≤ u2 < 2m−1 + �k−1
+α=1 2π1(m−k+α)−1 + 2v and (u1, u2) ̸= (0, 0). From (4)
+we can find that
+c = q
+2
+�m−k−1
+�
+l=1
+xπ1(l)xπ1(l+1) +
+n−1
+�
+s=1
+yπ2(s)yπ2(s+1) + xπ1(m)yπ2(n)
+�
++
+m−k
+�
+l=1
+µlxπ1(l)xπ1(m) +
+m
+�
+l=1
+plxl +
+n
+�
+s=1
+κsys + p0 · 1.
+(38)
+Then we let h = g + u1 and j = i + u2 for any integers g and i. Next, we discuss seven cases
+to complete the proof.
+Case 1: Assuming u1 > 0, u2 ≥ 0, and gπ2(1) ̸= hπ2(1), we can find an array c′ = c +
+(q/2)yπ2(1) ∈ G for any array c ∈ G. Therefore, we can obtain
+ch,j − cg,i − c′
+h,j+c′
+g,i = q
+2(gπ2(1) − hπ2(1)) ≡ q
+2
+(mod q)
+(39)
+Since gπ2(1) ̸= hπ2(1), we have
+ξch,j−cg,i/ξc′
+h,j−c′
+g,i = ξ
+q
+2 = −1.
+(40)
+Thus,
+ξch,j−cg,i + ξc′
+h,j−c′
+g,i = 0.
+(41)
+Case 2: If u1 > 0, u2 ≥ 0, and gπ2(1) = hπ2(1). Let β be the smallest integer such that
+gπ2(β) ̸= hπ2(β). We define g′ and h′ are two integers which are distinct from g and h only in
+one position π2(β − 1), respectively. Then, similar to Case 2 of Theorem 1, we have
+ξch,j−cg,i + ξch′,j−cg′,i = 0.
+(42)
+Case 3: We suppose im ̸= jm, u1 = 0 and u2 > 0. We let g′ be an integer distinct from
+i only in one position, i.e., g′
+π2(n) = 1 − gπ2(n). Similar to Case 3 of Theorem 1, we have
+ξcg,j−cg,i + ξcg′,j−cg′,i = 0.
+Case 4: If u1 = 0, u2 > 0, and iπ1(1) ̸= jπ1(1) or iπ1(m−k+α) ̸= jπ1(m−k+α), we can find an
+array c′ = c + (q/2)xπ1(1) ∈ G or c′ = c + (q/2)xπ1(m−k+α) for any array c ∈ G. Similar to
+Case 1, we can obtain ξcg,j−cg,i + ξc′
+g,j−c′
+g,i = 0.
+January 4, 2023
+DRAFT
+
+16
+Case 5: Suppose u1 = 0, u2 > 0, iπ1(1) = jπ1(1), and iπ1(m−k+α) = jπ1(m−k+α) for all
+α = 1, 2, · · · , k. Suppose that α′ is the largest non-negative integer satisfying iπ1(m−k+α′) =
+jπ1(m−k+α′) = 0. Then we assume β is the smallest integer which satisfies iπ1(β) ̸= jπ1(β).
+Here, we have is = js = 0 for s = π1(m − k + α′) + 1, π1(m − k + α′) + 2, . . . , m − 1, and
+s ̸= π1(m − k + α) for α = α′ + 1, α′ + 2, . . . , k. Hence, it implies π1(β) < π1(m − k + α′)
+and π1(β − 1) < π1(m − k + α′) according to the condition (C-3). Let i′ and j′ be integers that
+differ from i and j, respectively, in the position π1(β − 1). Similar to Case 2, we have
+ξcg,j−cg,i + ξcg,j′−cg,i′ = 0.
+(43)
+Case 6: Suppose u1 = 0, u2 > 0, iπ1(1) = jπ1(1), and iπ1(m−k+α) = jπ1(m−k+α) = 1 for all
+α = 1, 2, · · · , k. Then we assume β is the smallest integer which satisfies iπ1(β) ̸= jπ1(β). Since
+is = js = 0 for s = v + 1, v + 2, · · · , m − k and s ̸= π1(m − k + α) for α = 1, 2, . . . , k − 1, we
+can obtain π1(β) ≤ v implying π1(β − 1) ≤ v. Similar to Case 2, we have
+ξcg,j−cg,i + ξcg,j′−cg,i′ = 0.
+(44)
+From Cases 1 to 6, the theorem can be proved.
+Remark 2: Taking σ2(l) = π2(n − l + 1) for l = 1, 2, . . . , n and π1(m − k + α) = m − k + α
+for α = 1, 2, . . . , k in Theorem 2, (34) can be represented as
+f =q
+2
+�m−k−1
+�
+k=1
+xπ1(k)xπ1(k+1) +
+n−1
+�
+k=1
+yσ2(k)yσ2(k+1) + xmyσ2(1)
+�
++
+m−k
+�
+l=1
+µlxπ1(l)xm
++
+m
+�
+l=1
+plxl +
+n
+�
+s=1
+κsys + p0
+(45)
+where pl, κs ∈ Zq. We can find that the result of Lemma 3 is a special case of Theorem 2 by
+simply setting k = 1, µm−1 = q
+2, and µl = 0 for l = 1, · · · , m − 2.
+Example 4: Taking q = 2, m = 5, n = 2, k = 2, and v = 0, we let π1 = (1, 2, 4, 3, 5) and
+π2 = (1, 2). The generalized Boolean function is f = x1x2 + x2x4 + y1y2 + x5y1 by setting
+pl, κs = 0. The array set G is a GCAS of size 8 when the truncated size L1 = 4 L2 = 21. We
+let G = {c0, c1, · · · , c7} and list the constituent arrays in Table III. Also, their AACF sum is
+shown as Fig. 3.
+IV. SIMULATION RESULTS
+In this section, we present the numerical results including the power radiation pattern and
+BER performance by using our proposed 2D GCASs for massive MIMO systems with URA.
+January 4, 2023
+DRAFT
+
+17
+TABLE III
+THE CONSTRUCTED (8, 4, 21)-GCAS IN EXAMPLE 4
+c0 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+1
+0
+1
+0
+1
+1
+0
+1
+1
+1
+0
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+1
+0
+1
+0
+1
+1
+0
+1
+1
+1
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+1
+1
+1
+1
+1
+0
+0
+0
+1
+0
+0
+1
+0
+1
+1
+1
+1
+0
+1
+1
+0
+0
+0
+0
+0
+0
+1
+1
+1
+0
+1
+1
+�
+�
+�
+�
+�
+�
+�
+c1 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+1
+1
+1
+1
+0
+0
+0
+1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+1
+1
+1
+1
+0
+1
+0
+0
+1
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+1
+0
+1
+1
+0
+1
+1
+0
+0
+1
+0
+1
+0
+0
+0
+1
+1
+1
+0
+1
+1
+0
+1
+0
+�
+�
+�
+�
+�
+�
+�
+c2 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+1
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+0
+1
+1
+1
+1
+0
+1
+1
+0
+0
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+�
+�
+�
+�
+�
+�
+�
+c3 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+1
+1
+0
+1
+0
+1
+0
+1
+1
+0
+0
+1
+0
+1
+0
+0
+0
+1
+1
+1
+0
+0
+0
+1
+0
+1
+�
+�
+�
+�
+�
+�
+�
+c4 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+1
+0
+1
+0
+1
+1
+0
+1
+1
+1
+0
+1
+1
+1
+0
+1
+0
+0
+0
+1
+1
+0
+1
+0
+1
+0
+0
+1
+0
+0
+0
+1
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+1
+1
+1
+1
+1
+0
+0
+0
+1
+0
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+1
+1
+1
+1
+1
+0
+0
+0
+1
+0
+0
+�
+�
+�
+�
+�
+�
+�
+c5 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+0
+1
+1
+1
+1
+1
+1
+1
+0
+0
+1
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+1
+0
+1
+0
+1
+0
+0
+1
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+0
+0
+1
+0
+1
+�
+�
+�
+�
+�
+�
+�
+c6 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+0
+1
+1
+1
+0
+0
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+1
+1
+1
+0
+1
+0
+0
+0
+1
+1
+0
+1
+0
+1
+0
+0
+0
+1
+1
+1
+0
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+1
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+0
+0
+0
+0
+1
+0
+0
+1
+1
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+�
+�
+�
+�
+�
+�
+�
+c7 =
+�
+�
+�
+�
+�
+�
+�
+0
+0
+0
+1
+1
+0
+0
+0
+0
+0
+1
+0
+0
+1
+0
+0
+1
+0
+0
+0
+0
+1
+1
+1
+0
+0
+1
+1
+1
+1
+1
+0
+1
+1
+0
+1
+1
+0
+1
+1
+1
+1
+0
+1
+0
+0
+1
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+1
+1
+0
+1
+0
+0
+1
+0
+0
+1
+1
+0
+1
+0
+1
+1
+1
+0
+0
+0
+1
+1
+1
+0
+1
+0
+�
+�
+�
+�
+�
+�
+�
+A. Power Radiation Pattern
+According to (16), the power radiation pattern �N−1
+n=0
+��[vec(A(ϕ, θ))Tvec(Wn)]
+��2 can be ob-
+tained. We first consider the massive MIMO system equipped with a URA of size 4 × 33,
+January 4, 2023
+DRAFT
+
+18
+0
+200
+2
+400
+40
+600
+20
+800
+0
+1000
+0
+-20
+-2
+-40
+Fig. 3. The summation of autocorrelations of constituent arrays in the GCAS in Example 4.
+i.e., L1 = 4 and L2 = 33. We take the GCS G = {c0, c1, c2, c3} listed in Table II to
+generate the precoding matrices {W0, W1, W2, W3} = {(−1)c0, (−1)c1, (−1)c2, (−1)c3} with
+the omnidirectional property. The power radiation pattern of the GCAS-based scheme with array
+size 4 × 33 is perfectly omnidirectional as illustrated in Fig 4(a).
+For the purpose of comparison, we also show the power radiation patterns of the precoding
+matrices based on Zadoff-Chu sequences and random-matrices whose elements are randomly
+generated from “+1” and “−1”. The ZC-based precoder consists of four 4 × 33 precoding
+matrices, which are obtained based on a ZC sequence of length 4 and a ZC sequence of length
+33 [21]. Fig. 4(b) illustrates the power radiation pattern of the ZC-based precoder. We can
+find that its power radiation pattern is not omnidirectional. The random-matrix-based precoder
+consists of four 4 × 33 precoding matrices. The elements in the random-matrix-based precoding
+matrices are generated by selecting the elements from {1, −1} with equal probability. Fig. 4(c)
+describes the power radiation pattern of the random matrix-based precoder. We can observe that
+the power radiation pattern is not omnidirectional.
+Next, we consider the massive MIMO system equipped with a URA of size 4 × 21, i.e.,
+January 4, 2023
+DRAFT
+
+19
+(a) GCAS-based precoding.
+(b) ZC-based precoding.
+(c) Random-matrix-based precoding.
+Fig. 4. Power radiation pattern with 4 × 33 URA and 4 × 4 STBC.
+L1 = 4 and L2 = 21. We use the GCS G = {c0, c1, · · · , c7} listed in Table III for the precoding
+matrix {W0, W1, · · · , W7} = {(−1)c0, (−1)c1, · · · , (−1)c7}. The power radiation pattern of the
+GCAS-based scheme with array size 4×21 is described in Fig. 5(a). The perfect omnidirectional
+property can be observed. We also see that the power radiation patterns of the ZC-based precoder
+and the random-matrix precoder shown in Fig. 5(b) and Fig. 5(c) are not omnidirectional. The
+ZC-based precoding matrices are obtained by a ZC sequence of length 4 and ZC sequence of
+21 [21].
+B. Bit Error Rate Performance
+In this subsection, we present the BER performance of our proposed 2D GCAS-based schemes.
+We first consider the massive MIMO system equipped with a URA of size 4×33. We let N = 4
+January 4, 2023
+DRAFT
+
+X-axis-0.500.51Sy-ax1sX-axis550.50.51Sy-ax.
+1S0.3X-axis00.50.5Sy-ax.
+1S20
+(a) GCAS-based precoding.
+(b) ZC-based precoding.
+(c) Random-matrix-based precoding.
+Fig. 5. Power radiation pattern with 4 × 21 URA and 8 × 8 STBC.
+and then the 4 × 4 orthogonal real STBC be presented as
+S =
+�
+�
+�
+�
+�
+�
+�
+s0
+−s1
+−s2
+−s3
+s1
+s0
+s3
+−s2
+s2
+−s3
+s0
+s1
+s3
+s2
+−s1
+s0
+�
+�
+�
+�
+�
+�
+�
+,
+(46)
+where s0, s1, s2, s3 are binary phase shift keying (BPSK) modulated symbols. We employ the
+maximum likelihood (ML) decoding here. For each realization, the elevation and the azimuth
+angles are uniformly distributed at random between [0, π/2] and [0, 2π], respectively. For com-
+parison, the ZC-based precoder and random-matrix-based precoder are the same as mentioned in
+Section IV-A. The BER performances of three different schemes are depicted in Fig. 6. We can
+find that the 2D GCAS-based scheme outperform the others. At BER of 10−4, there are 1.6 dB
+and 3.6 dB gains over the ZC-based scheme and the random-matrix-based scheme, respectively.
+January 4, 2023
+DRAFT
+
+X-axis-0.500.51Sy-ax1sX-axis-0.500.51Sy-ax1s0.8X-axis-0.500.510.2Sy-06.ax1s21
+-2
+0
+2
+4
+6
+8
+10
+12
+14
+SNR (dB)
+10-8
+10-6
+10-4
+10-2
+100
+BER
+GCAS-based precoding (N=4)
+ZC-based precoding (N=4)
+Random-matrix-based precoding (N=4)
+Fig. 6. BER performance of the different schemes for a 4 × 33 URA.
+Next, we consider the massive MIMO system equipped with a URA of size 4 × 21. We
+consider 8 × 8 STBC and the 8 × 8 orthogonal real STBC is given by
+S =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+s0
+s1
+s2
+s3
+s4
+s5
+s6
+s7
+−s1
+s0
+s3
+−s2
+s5
+−s4
+−s7
+s6
+−s2
+−s3
+s0
+s1
+s6
+s7
+−s4
+−s5
+−s3
+s2
+−s1
+s0
+s7
+−s6
+s5
+−s4
+−s4
+−s5
+−s6
+−s7
+s0
+s1
+s2
+s3
+−s5
+s4
+−s7
+s6
+−s1
+s0
+−s3
+s2
+−s6
+s7
+s4
+−s5
+−s2
+s3
+s0
+−s1
+−s7
+−s6
+s5
+−s4
+−s3
+s2
+s1
+s0
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(47)
+where s0, s1, · · · , s7 are BPSK modulated symbols. We also take the ZC-based precoding and
+random-matrix-based precoding for comparison. The BER performance comparison for these
+three different schemes is depicted in Fig. 7. At BER of 10−4, there are 0.2 dB and 1.8 dB
+gains over the ZC-based scheme and the random-matrix-based scheme, respectively. As a result,
+the 2D GCASs are good candidates as precoding matrices for omnidirectional transmission in
+January 4, 2023
+DRAFT
+
+22
+-2
+0
+2
+4
+6
+8
+10
+12
+14
+SNR (dB)
+10-8
+10-6
+10-4
+10-2
+100
+BER
+GCAS-based precoding (N=8)
+ZC-based precoding (N=8)
+Random-matrix-based precoding (N=8)
+Fig. 7. BER performance of the different schemes for a 4 × 21 URA.
+massive MIMO systems.
+V. CONCLUSION
+In this paper, constructions of 2D GCASs with flexible array sizes have been proposed in
+Theorems 1 and 2. Our constructions can be obtained directly from 2D GBFs without the aid
+of special sequences. Besides, our proposed GCASs have flexible array sizes which can fit
+more antenna configuration. Furthermore, Theorem 2 can include the results in [32] as a special
+case. Simulation results showed that the omnidirectional transmission can be achieved when
+the precoding matrices are based on the proposed GCASs. The BER performance due to their
+omnidirectional power radiation patterns, the ZC-based scheme and random-matix-based have
+inferior performances because their power radiation patterns both are not ideally omnidirectional.
+Although Theorems 1 and 2 can provide direct constructions of 2D GCASs, the first dimension
+has size L1 limited to 2n. Therefore, the future work includes the extension of constructions of
+2D GCASs of which both dimensions have non-power-of-two sizes.
+January 4, 2023
+DRAFT
+
+23
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+January 4, 2023
+DRAFT
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+24
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+Commun., vol. 64, no. 1, pp. 174–186, Jan. 2016.
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+complementary codes,” IEEE Trans. Inf. Theory, vol. 67, no. 7, pp. 4902–4928, Jul. 2021.
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+Boolean functions,” in Proc. IEEE Int. Symp. Inf. Theory, Los Angeles, California, USA, Jun. 2020, pp. 2931–2935.
+[28] C.-Y. Pai and C.-Y. Chen, “Two-dimensional Golay complementary array pairs/sets with bounded row and column sequence
+PAPRs,” IEEE Trans. Commun., vol. 70, no. 6, pp. 3695–3707, Jun. 2022.
+[29] Z. Liu, Y. L. Guan, and U. Parampalli, “New complete complementary codes for peak-to-mean power control in multi-
+carrier CDMA,” IEEE Trans. Commun., vol. 62, pp. 1105–1113, Mar. 2014.
+[30] C.-Y. Pai, Z. Liu, Y.-Q. Zhao, Z.-M. Hunag, and C.-Y. Chen, “Designing two-dimensional complete complementary codes
+for omnidirectional transmission in massive MIMO systems,” in Proc. IEEE Int. Symp. Inf. Theory, Espoo, Finland, Jun.
+2022, pp. 1699–1704.
+[31] T. Liu, X. Men, Y. Li, and X. Chen, “Constructions of 2-D Golay complementary array sets for MIMO omnidirectional
+transmission,” IEEE Commun. Lett., pp. 1459 – 1463, Jul. 2022.
+[32] B. Shen, Y. Yang, and R. Ren, “Three constructions of Golay complementary array sets,” Adv. Math. Commun., Oct. 2022.
+January 4, 2023
+DRAFT
+
diff --git a/3dAzT4oBgHgl3EQfR_t3/content/tmp_files/load_file.txt b/3dAzT4oBgHgl3EQfR_t3/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1ce969983dc0fc9a8e315879a7d2e523b3b01d12
--- /dev/null
+++ b/3dAzT4oBgHgl3EQfR_t3/content/tmp_files/load_file.txt
@@ -0,0 +1,2171 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf,len=2170
+page_content='1  Two-Dimensional Golay Complementary Array  Sets With Arbitrary Lengths for  Omnidirectional MIMO Transmission  You-Qi Zhao, Cheng-Yu Pai, Zhen-Ming Huang, Zilong Liu, Senior  Member, IEEE, and Chao-Yu Chen, Member, IEEE  Abstract  This paper presents a coding approach for achieving omnidirectional transmission of certain common  signals in massive multi-input multi-output (MIMO) networks such that the received power at any  direction in a cell remains constant for any given distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Specifically, two-dimensional (2D) Golay  complementary array set (GCAS) can be used to design optimal massive MIMO precoding matrix so as  to achieve omnidirectional transmission due to its complementary autocorrelation property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In this paper,  novel constructions of new 2D GCASs with arbitrary array lengths are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Our key idea is to  carefully truncate the columns of certain larger arrays generated by 2D generalized Boolean functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Finally, the power radiation patterns and numerical results are provided to verify the omnidirectional  property of the GCAS-based precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The error performances of the proposed precoding scheme are  presented to validate its superiority over the existing alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Index Terms  This work was supported by the Ministry of Science and Technology, Taiwan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=', under Grant MOST 109–2628–E–006–  008–MY3 and MOST 111–2218–E–305–002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  You-Qi Zhao and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Pai are with the Department of Engineering Science, National Cheng Kung University, Tainan 701,  Taiwan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' (e-mail: n98081505@gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='ncku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='tw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Huang is with the Institute of Computer and Communication Engineering, National Cheng Kung University, Tainan  701, Taiwan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' (e-mail: n98101012@gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='ncku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='tw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Zilong Liu is with the School of Computer Science and Electronic Engineering, University of Essex, United Kingdom (e-mail:  zilong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='liu@essex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Chen is with the Department of Electrical Engineering and the Institute of Computer and Communication Engineering,  National Cheng Kung University, Tainan 701, Taiwan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' (e-mail: super@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='ncku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='tw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='01225v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='IT]  3 Jan 2023  2  Generalized Boolean function (GBF), Golay complementary array pair (GCAP), Golay comple-  mentary array set (GCAS), omnidirectional precoding (OP), uniform rectangular array (URA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' INTRODUCTION  Complementary pairs/sets of sequences have attracted a sustained research interest owing to  their zero aperiodic correlation sums properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' To be specific, a Golay complementary pair  (GCP) refers to a pair of equal-length sequences whose summation of aperiodic autocorrelations  is zero except at the zero time-shift [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Such a concept was extended to Golay complementary  set (GCS) with constituent sequences of more than 2 by Tseng and Liu in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Furthermore, a  maximum collection of GCSs is called a set of complete complementary code (CCC) [3] if any  two different GCSs have zero aperiodic cross-correlation sums for all time-shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In the literature,  GCSs and CCCs have been widely used for radar sensing [4], channel estimation [5], precoding  for massive multi-input multi-output (MIMO) [6], peak-to-average power ratio (PAPR) reduction  in orthogonal frequency division multiplexing (OFDM) [7]–[13], interference-free multicarrier  code division multiple access [14]–[17], and many other applications [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Recently, there is a surge of research attention to study two-dimensional (2D) Golay com-  plementary array sets (GCASs) [18]-[23], each having zero aperiodic autocorrelation sums  property for two directions of shifts (compared to conventional GCSs and CCCs with time-  shifts only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' An important application of the 2D GCASs is for omnidirectional transmission in  MIMO communication systems with a uniform rectangular array (URA) configuration [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  In massive MIMO systems, some common messages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=', reference signals, synchronization  signals, control signals, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=') need to be power-uniformly broadcasted to all the angles within  the whole cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In this paper, we consider space-time block code (STBC) for the harvesting  of the diversity gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' At the base station (BS), the STBC encoded symbols are assigned to  several streams and then mapped onto the antenna arrays in URA by certain 2D GCASs assisted  precoding matrices to achieve uniform power radiation at any angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  On the other hand, since a large number of antennas are considered in massive MIMO  systems, a huge pilot overhead may be needed to acquire the channel state information (CSI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' As  pointed out in [22], this can be alleviated by omnidirectional precoding (OP) based transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  For uniform linear arrays (ULAs), Zadoff-Chu (ZC) sequences were adopted to satisfy the  requirements of the omnidirectional property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' However, [22] only considered the omnidirectional  January 4, 2023  DRAFT  3  transmission in certain directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Later in [6], GCSs and CCCs based OP matrices were proposed  to meet the requirement of omnidirectional transmission across all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  In [20], [21], [23], [24], 2D GCASs were employed for precoding matrices in URAs by  applying interleaving and Kronecker-product to existing 1D sequences or 2D arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' As a result,  the array sizes of 2D GCASs are only feasible for certain lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' A construction of 2D GCASs  of array size pn × pm was proposed in [25] by using permutation ploynomials (PPs) functions  and 2-level autocorrelation sequences, where p is a prime number, m, n are two positive integers,  and p, m, n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Furthermore, a unifying construction framework for 2D GCASs was developed  in [26] by a multivariate polynomial matrix from certain seed para-unitary (PU) matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In  [27], [28], Pai and Chen proposed direct constructions of 2D Golay complementary array pairs  (GCAPs) and GCASs with array size 2n × 2m from 2D generliazed Boolean functions (GBFs)  [29] where n, m are integers and n, m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 2D GCAP can be regarded as a case of 2D GCAS  when the set size is equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Moreover, Pai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' [30] proposed a direct construction of 2D  CCCs with array size 2n×2m, which have ideal autocorrelations and cross-correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Later, Liu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' [31] proposed a construction of GCASs with array size pn ×pm by using 2D multivariable  functions, where p is a prime number, n, m are integers, and n, m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Based on [27], [32]  developed a direct construction of GCASs with set size 4 and array size 2n × (2m−1 + 2v) by  using 2D GBFs, where n, m, v are positive number with n, m ≥ 2, and 0 ≤ v ≤ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  The aforementioned research efforts are generally driven by the need of highly flexible array  sizes of 2D GCASs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Motivated by this, we aim for generating new GCASs with arbitrary array  lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The key idea of our proposed constructions is to carefully truncate some columns of  the certain larger arrays generated by 2D GBFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Thus, our proposed GCASs can be applied to  URAs with various array sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In addition, the proposed GCASs can be directly generated from  2D GBFs without the requirements of any specific sequences or tedious sequence operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In  Table I, we compare the existing parameters of 2D GCASs with our proposed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  The remainder of this paper is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Section II discusses notations, definitions,  system models, and the omnidirectional transmission in MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Section III describes  our proposed constructions of 2D GCASs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Section IV shows the power radiation pattern and bit  error rate (BER) performance based on our proposed 2D GCASs precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Finally, Section V  presents the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  4  TABLE I  A COMPARISON OF CONSTRUCTIONS FOR 2D GCASS  Construction  Parameters  Approaches  [26, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 5]  (N, N n, N m), N, n, m > 0  Seed PU matrices  [26, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 7]  (2k, 2kn, 2km), n, m, k > 0  [25, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 4]  (p, pn, pm), prime p, n, m > 0  PPs and 2-level  autocorrelation sequences  [25, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 6]  (pk, pkn, pkm), prime p, k, n, m > 0  [31, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 1]  (pk1  1 pk2  2 , pn  1 , pm  2 ), primes p1, p2  2D multivariable functions  [31, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 2]  (pk, pn, pm), prime p, n + m ≥ k > 0  [27], [28], [30]  (2k, 2n, 2m), n, m ≥ k > 0, and k > 0  2D GBFs  [32]  (4, 2n, 2m−1 + 2v), n, m ≥ 2, and k > 0  Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 1  (2k+1, 2n, 2m−1 + �k−1  α=1 dα2m−k+α−1 + d02v),  k < m, 0 ≤ v ≤ m − k, dα ∈ {0, 1}  Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 2  (2k+1, 2n, 2m−1 + �k−1  α=1 dα2π1(m−k+α)−1 + d02v),  k < m, 0 ≤ v ≤ m − k, dα ∈ {0, 1}  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' PRELIMINARIES AND DEFINITIONS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Notations  Throughout this paper, we present the notations in the following:  (a)i refers to the i-th element of the vector a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (A)i,j denotes the (i, j)-th element of the array A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (·)H refers to the conjugate transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  diag(A) refers to the column vector composed of the main diagonal of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (·)∗ refers to the complex conjugation of an element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (·)T refers to the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  vec(·) express stacking one column of the matrix into one another column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  1 is a vector whose elements are all 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Let ξ = e2π√−1/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  In this paper, q is an even number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Let X and Y be two arrays of size L1 × L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then X and Y can be stated as  X = (Xg,i), Y = (Yg,i),  (1)  where g = 0, 1, · · · , L1 − 1 and i = 0, 1, · · · , L2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  5  Definition 1: Given two arrays X and Y of size L1 × L2, the 2D aperiodic cross-correlation  function (AACF) is defined by  ρ (X, Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u1, u2) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  L1−1−u1  �  g=0  L2−1−u2  �  i=0  Yg+u1,i+u2X∗  g,i, 0 ≤ u1 < L1,  0 ≤ u2 < L2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  L1−1−u1  �  g=0  L2−1−u2  �  i=0  Yg+u1,iX∗  g,i−u2, 0 < u1 < L1,  −L2 < u2 < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  L1−1−u1  �  g=0  L2−1−u2  �  i=0  Yg,iX∗  g−u1,i−u2, −L1 < u1 < 0,  −L2 < u2 < 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  L1−1+u1  �  g=0  L2−1−u2  �  i=0  Yg,i+u2X∗  g−u1,i, −L1 < u1 < 0,  0 < u2 < L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (2)  When X = Y , then it is called 2D aperiodic autocorrelation function (AACF) and denoted  by ρ(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u1, u2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' If taking L1 = 1, two 2D arrays X and Y are degraded as a 1-D sequence  X = Xi for i = 0, 1, · · · , L2 − 1 and Y = Yi for i = 0, 1, · · · , L2 − 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then the  1-D AACF of 1-D sequence X is related by  ρ(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u) =  �  �  �  �  �  �  �  L2−1−u  �  i=0  Xi+uX∗  i ,  0 ≤ u ≤ L2 − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  L2−1+u  �  i=0  XiX∗  i−u,  −L2 + 1 ≤ u < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (3)  In this paper, q-PSK modulation is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Thus, x and y denote q-ary arrays and (1) is  expressed as  X = (Xg,i) = (ξxg,i) = ξx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Y = (Yg,i) = (ξyg,i) = ξy,  (4)  where x = (xg,i), y = (yg,i), and xg,i, yg,i ∈ Zq = {0, 1, · · · , q−1} for 0 ≤ g < L1, 0 ≤ i < L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Consider a set of N L-length sequences can be represented as  C = {X0, X1, · · · , XN−1}  where  Xn = (Xn,0, Xn,1, · · · , Xn,L−1)  for n = 0, 1, · · · , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  6  Definition 2: [19] If a set C consisting of N sequences of length L satisfies  N−1  �  k=0  ρ(Xk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u) =  �  �  �  �  �  NL,  u = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  0,  u ̸= 0,  (5)  then the set C is called a Golay complementary set of size N, denoted by (N, L)-GCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The  GCP can be regarded as a special case of the GCS by setting N = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Definition 3: For a GCP (X0, X1), if another GCP (Y0, Y1) meets the following condition:  ρ(X0, Y0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u) + ρ(X1, Y1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u) = 0, for all u,  (6)  then the two GCPs are called the Golay complementary mate of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Definition 4: A pair of arrays X and Y of array size L1 × L2 is called a 2D Golay  complementary array pair if  ρ(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u1, u2) + ρ(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u1, u2) =  �  �  �  �  �  2L1L2,  u1 = u2 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  0,  u1 ̸= 0 or u2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (7)  Definition 5: Let the array set G = {X0, X1, · · · , XN−1} where each array in set G is of  size L1 × L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' If the array set G satisfies  N−1  �  k=0  ρ(Xk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u1, u2) =  �  �  �  �  �  NL1L2,  u1 = u2 = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  0,  u1 ̸= 0 or u2 ̸= 0,  (8)  the set G is called the Golay complementary array set of set size N denoted by (N, L1, L2)-  GCAS where L2 is defined as the length of the GCAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' If N = 2, the GCAS G is degraded as  a GCAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Generalized Boolean Functions  A 2D generalized Boolean function (GBF) f in n + m binary variables y1, y2, · · · , yn,  x1, x2, · · · , xm, is a function mapping: Zn  2 ×Zm  2 → Zq, where xi, yg ∈ {0, 1} for i = 1, 2, · · · , m  and g = 1, 2, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' A monomial of degree r is given by any product of r distinct variables  among y1, y2, · · · , yn, x1, x2, · · · , xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' For instance, x1x3y1y2 is a monomial of degree 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Next,  the variables z1, z2, · · · , zn+m are defined as  zl =  �  �  �  �  �  yl  if 1 ≤ l ≤ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  xl−n  if n < l ≤ m + n,  (9)  January 4, 2023  DRAFT  7  which are useful for our proposed constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' For a 2D GBF with n + m variables, the 2D  Zq-valued array  f =  �  �  �  �  �  �  �  f0,0  f0,1  · ·  f0,2m−1  f1,0  f1,1  · ·  f1,2m−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  f2n−1,0  f2n−1,1  · ·  f2n−1,2m−1  �  �  �  �  �  �  �  (10)  of size 2n×2m is given by letting fg,i = f((g1, g2, · · · , gn), (i1, i2, · · · , im)), where (g1, g2, · · · , gn)  and (i1, i2, · · · , im) are binary vector representations of integers g = �n  h=1 gh2h−1 and i =  �n  j=1 ij2j−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Example 1: Taking q = 4, n = 2, and m = 3 for example, the 2D GBF is given as f =  3z5z4 + z2z3 + 2z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then the array f of size 4 × 8 corresponding to f can be obtained, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=',  f =  �  �  �  �  �  �  �  0  0  0  0  0  0  3  3  0  0  0  2  1  1  3  3  2  3  2  3  2  3  1  2  2  3  2  3  2  3  1  2  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (11)  The GBF f can be rewritten as f = 3x3x2 + y2x1 + 2y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In this paper, we consider the array  size ̸= 2n × 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Hence, we define the truncated array f (L) corresponding to the 2D GBF f by  ignoring the last 2m − L columns of the corresponding array f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Example 2: Following the same notations given in Example 1, the truncated array f (6) is  given by  f (6) =  �  �  �  �  �  �  �  0  0  0  0  0  0  0  0  0  2  1  1  2  3  2  3  2  3  2  3  2  3  2  3  �  �  �  �  �  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (12)  For simplicity, we use f to stand for f (L) when L is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' System Model  Considering downlink transmission from a BS to UEs where each has one single antenna,  we suppose that the number of antennas at the BS is M = L1 × L2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=', the URA consists of  L1 rows and L2 columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 1 illustrates the diagram of data downlink transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' For an  January 4, 2023  DRAFT  8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Diagram of data transmission through STBC encoding and omnidirectional precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  L1 × L2 URA, the steering matrix A(ϕ, θ) at the direction (ϕ, θ) with the (g, i)-th entry can be  expressed as  (A(ϕ, θ))g,i =e−j 2π  λ gdy sin ϕ sin θ−j 2π  λ idx sin ϕ cos θ,  for g = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , L1 − 1, i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , L2 − 1,  θ ∈ [0, 2π], ϕ ∈ [0, π/2],  (13)  where dx and dy denote the vertical antenna and horizontal antenna inter-element spacings of  the URA, respectively, and λ denotes the carrier wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' To enhance the spatial diversity  and communication reliability, the STBC signal transmission scheme is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The N ×M STBC  is given by  S ≜  �  �  �  �  �  �  �  s0(0)  s0(1)  · ·  s0(M − 1)  s1(0)  s1(1)  · ·  s1(M − 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  sN−1(0)  sN−1(1)  · ·  sN−1(M − 1)  �  �  �  �  �  �  �  ∈ CN×M  (14)  January 4, 2023  DRAFT  So(t)  Xo(t)  7  Data  Omni-  Space-time  UE1  Si(t)  directional  xi(t)  block coding  y(t)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  precoding  V  SN-1(t)  XL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='L2  1(t)  y  d  d  (t)  xo(t)  x  Omnidirectional  precoding  UE19  where CN×M refers to the N-by-M complex space and sn(t) denotes the (n, t)-th element of  the STBC at time instant t for t = 0, 1, · · · , M − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We define the precoding matrix Wn of size  L1 × L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The encoded symbols is given by  x(t) = (x0(t), x1(t), · · · , xL1L2−1(t))T = vec  �N−1  �  n=0  Wn · sn(t)  �  , for t = 0, 1, · · · , M − 1,  (15)  which are transmitted by the L1L2 antennas of the URA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In the light-of-sight (LOS) channel  without multipaths, the received signal at the direction (ϕ, θ) can be written as  y(t) =  N−1  �  n=0  �  vec(A(ϕ, θ))Tvec(Wn)  �  sn(t) + η(t), t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=', M − 1,  (16)  where η(t) is the additive Gaussian white noise (AWGN) at time instant t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Omnidirectional Precoding Matrices Based on 2D Arrays  In this subsection, we list two necessary requirements for the design of OP matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then,  we will connect these two requirements with the conditions of 2D arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Requirement 1 (R1): Omnidirectional transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  We consider the MIMO system with URA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Following (16), the received power E at the angle  (ϕ, θ) is represented as  E =  N−1  �  n=0  ��[vec(A(ϕ, θ))Tvec(Wn)]  ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (17)  Therefore, to satisfy the omnidirectional transmission in the whole cell, (17) must be constant  for all ϕ and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Requirement 2 (R2): Equal average power on each antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  To enhance the efficiency of the power amplifier, the average transmission power on all L1×L2  antennas is required to be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We define  W = (vec(W0), vec(W1), · · · , vec(WN−1)) ,  (18)  where the array size of W is L1L2 × N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Hence, (15) can be rewritten as  X = (x(0), x(1), · · · , x(M − 1)) = W S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (19)  January 4, 2023  DRAFT  10  Let s(t) be the t-th column of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Throughout this paper, we assume E  �  s(t)s(t)H�  =IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The  transmitted signal on the (l1, l2)-th antenna is (W s)l2L1+l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The average power on the (l1, l2)-th  antenna can be expressed as  E  �  |(W s)l2L1+l1|2�  =  �  W E  �  s(t)s(t)H�  W H�  l2L1+l1,l2L1+l1  = (W W H)l2L1+l1,l2L1+l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (20)  Therefore, the condition to guarantee equal power on each antenna is equivalent to  diag(W W H) = N1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (21)  Next, we will derive two sufficient conditions on the precoding matrices to fulfill requirements  R1 and R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Lemma 1: [21] For an L1 × L2 URA, if the precoding matrices W0, W1, · · · , WN−1 of size  L1 × L2 form an (N, L1, L2)-GCAS, then the omnidirectional transmission is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Lemma 2: For an L1×L2 URA, if the precoding matrices W0, W1, · · · , WN−1 of size L1×L2  are unimodular, then the average power on each antenna is equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Proof: In order to meet the requirement for equal average power on each antenna, the  precoding matrix W must satisfy (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We let wi = vec(Wi), for i = 0, 1, · · · , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then,  diag  �  W W H�  =  �N−1  �  i=0  |(wi)0|2 ,  N−1  �  i=0  |(wi)1|2 , · · · ,  N−1  �  i=0  |(wi)L1L2−1|2  �T  = N1  (22)  since we have  |(wi)n|2 = 1,  for i = 0, 1, · · · , N − 1 and n = 0, 1, · · · , L1L2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (23)  According to (21), the requirement (R2) is fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  In the sequel, the design of OP matrices W0, W1, · · · , WN−1 are based on Lemma 1 and  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' That is, our goal is to construct unimodular GCASs with flexible sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' GCASS WITH FLEXIBLE ARRAY SIZE  In this section, two constructions of 2D GCASs with arbitrary array lengths based on 2D  GBFs will be proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' By recalling the function mapping in (9), we present our first theorem  in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  11  Theorem 1: For any integers q, m, n ≥ 2, and k < m, v is an integer satisfies 0 ≤ v ≤ m−k  and let π be a permutation of {1, 2, · · · m + n − k} satisfying {zπ(1), zπ(2), · · · , zπ(v+n)} =  {z1, z2, · · · , zv+n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The 2D generalized Boolean function can be written as  f = q  2  �m+n−k−1  �  l=1  zπ(l)zπ(l+1)  �  +  m+n  �  s=1  pszs + p0  (24)  where ps ∈ Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The array set  G =  �  f + q  2  k  �  α=1  λαzm+n−k+α + q  2λk+1zπ(1) : λα ∈ {0, 1}  �  (25)  is a q-ary (2k+1, 2n, 2m−1 + �k−1  α=1 dα2m−k+α−1 + d02v)-GCAS where dα ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Proof: Without loss of generality, we consider L1 = 2n and L2 = 2m−1+�k−1  α=1 2m−k+α−1+ 2v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  We need to show that  �  c∈G  L1−1−u1  �  g=0  L2−1−u2  �  i=0  �  ξcg+u1,i+u2−cg,i�  = 0  (26)  for 0 ≤ u1 < 2n, 0 ≤ u2 < 2m−1 + �k−1  α=1 2m−k+α−1 + 2v and (u1, u2) ̸= (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then we let  h = g + u1 and j = i + u2 for any integers g and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We also let (g1, g2, · · · , gn),(i1, i2, · · · , im),  (h1, h2, · · · , hn), and (j1, j2, · · · , jm) be the binary representations of g, i, h, and j, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  For the ease of presentation, we denote  al =  �  �  �  �  �  gl  for 1 ≤ l ≤ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  il−n for n < l ≤ n + m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  bl =  �  �  �  �  �  hl  for 1 ≤ l ≤ n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  jl−n for n < l ≤ n + m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (27)  In what follows, we consider four cases to show that the above formula holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Case 1: If aπ(1) ̸= bπ(1), we can find that c′ = c + (q/2)zπ(1) for any arrayc ∈ G satisfying  ch,j − cg,i − c′  h,j+c′  g,i = q  2(aπ(1) − bπ(1)) ≡ q  2  (mod q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (28)  Therefore, we have  ξch,j−cg,i + ξc′  h,j−c′  g,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (29)  Case 2: If am+n−k+α ̸= bm+n−k+α, we can find that c′ = c + (q/2)zm+n−k+α for any array  c ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Similar to Case 1, we have  ξch,j−cg,i + ξc′  h,j−c′  g,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (30)  January 4, 2023  DRAFT  12  Case 3: If aπ(1) = bπ(1) and am+n−k+α = bm+n−k+α for α = 1, 2, · · · , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Suppose that α′  is the largest integer satisfying am+n−k+α′ = bm+n−k+α′ = 0 for α′ ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then we assume β  is the smallest integer which satisfies aπ(β) ̸= bπ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Let a′ and b′ be integers distinct from a  and b, respectively, only in one position π(β − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' In other words, a′  π(β−1) = 1 − aπ(β−1) and  b′  π(β−1) = 1 − bπ(β−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' If 1 ≤ π(β − 1) ≤ n, by using the above definition, we have  cg′,i − cg,i  = q  2  �  aπ(β−2)g′  π(β−1) − aπ(β−2)gπ(β−1) + g′  π(β−1)aπ(β)  −gπ(β−1)aπ(β)  �  + pπ(β−1)g′  π2(β−1) − pπ(β−1)gπ(β−1)  ≡ q  2(aπ(β−2) + aπ(β)) + pπ(β−1)(1 − 2gπ(β−1))  (mod q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (31)  where a′  π(β−1) = g′  π(β−1) and aπ(β−1) = gπ(β−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Since aπ(β−2) = bπ(β−2) and aπ(β−1) = bπ(β−1),  we have  ch,j − cg,i − ch′,j + cg′,i  ≡ q  2(aπ(β−2) − bπ(β−2) + aπ(β) − bπ(β))  + pπ(β−1)(2hπ(β−1) − 2gπ(β−1))  ≡ q  2(aπ(β) − bπ(β)) ≡ q  2  (mod q)  (32)  implying ξch,j−cg,i/ξch′,j−cg′,i = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We can also obtain  ξch,j−cg,i + ξch′,j−cg′,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (33)  If n < π(β − 1) ≤ n + m, note that a′  π(β−1) = i′  π(β−1)−n and aπ(β−1) = iπ(β − 1) − n according  to (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Following the similar argument as given above, we can get ξch,j−cg,i + ξch,j′−cg,i′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Case 4: If aπ(1) = bπ(1) and am+n−k+α = bm+n−k+α = 1 for α = 1, 2, · · · , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We assume β is  the smallest integer such that aπ(β) ̸= bπ(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Since as = bs = 0 for s = v+n+1, v+n+2, · · · , m+  n−k, we can obtain π(β) ≤ v+n implying π(β −1) ≤ v+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' If 1 ≤ π(β−1) ≤ n, by following  the similar argument as given above, we have ξch,j−cg,i +ξch′,j−cg′,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' If n < π(β −1) ≤ v+n,  we have ξch,j−cg,i + ξch,j′−cg,i′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' From Cases 1 to 4, the theorem can be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Remark 1: The parameter 2m−1+�k−1  α=1 dα2m−k+α−1+d02v of the proposed GCASs in Theorem  1 can be any arbitrary length since m, k, v are flexible and dα ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Example 3: Taking q = 2, m = 6, n = 2, k = 1, and v = 0, we let π = (1, 2, 3, 4, 5, 6, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  The generalized Boolean function is f = z1z2 + z2z3 + z3z4 + z4z5 + z5z6 + z6z7 = x1x2 +  x2x3 + x3x4 + x4x5 + y1y2 + y2x1 by setting pk = 0 for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , m + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The array set  January 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 2023  DRAFT  13  TABLE II  THE CONSTRUCTED (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 33)-GCAS IN EXAMPLE 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='c0 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
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+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The summation of autocorrelations of constituent arrays in the GCAS in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  G = {f, f + x8, f + y1, f + x8 + y1} is a GCAS of size 4 and the array size is 4 × 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We let  G = {c0, c1, c2, c3} and list the constituent arrays in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 2 shows the AACF sum of  set G is zero at shift u1 ̸= 0 or u2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Thus, we can find that array set G is a (4, 4, 33)-GCAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  14  Next, we introduce a lemma which illustrates a construction of (4, 2n, 2m−1 +2v)-GCAS from  2D GBFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Lemma 3: [32, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 1] For nonnegative integers m, n, and v with 0 ≤ v < m − 1, let π1 be  a permutation of {1, 2, · · · , m − 1} and π2 be a permutation of {1, 2, · · · , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The 2D GBF is  given by  f =q  2  �m−2  �  k=1  xπ1(k)xπ1(k+1) +  n−1  �  k=1  yπ2(k)yπ2(k+1) + xπ1(m−1)xm + xmyπ2(1)  �  +  m  �  l=1  plxl +  n  �  s=1  κsys + p0  (34)  where pl, κs ∈ Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then the array set  G =  �  f, f + q  2xπ1(1), f + q  2yπ2(n), f + q  2xπ1(1) + q  2yπ2(n)  �  is a (4, 2n, 2m−1 + 2v)-GCAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Since the set size of the GCAS from Lemma 3 is limited to 4, we propose a general  construction of 2D GCASs with more flexible array sizes and set sizes which can include Lemma  3 as a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Theorem 2: For any integers q, m, n ≥ 2, and k < m, v is an integer satisfies 0 ≤ v ≤ m−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Assume that π1 is a permutation of {1, 2, · · · m} and π2 is a permutation of {1, 2, · · · n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The  2D generalized Boolean function can be written as  f =q  2  �m−k−1  �  l=1  xπ1(l)xπ1(l+1) +  n−1  �  s=1  yπ2(s)yπ2(s+1) + xπ1(m)yπ2(n)  �  +  m−k  �  l=1  µlxπ1(l)xπ1(m) +  m  �  l=1  plxk +  n  �  s=1  κsys + p0  (35)  where µl, pl, κs, ∈ Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The array set  G =  �  f + q  2  k−1  �  α=1  λαxπ1(m−k+α) + q  2λkyπ2(1) + q  2λk+1xπ1(1) : λα ∈ {0, 1}  �  (36)  is a q-ary (2k+1, 2n, 2m−1 + �k−1  α=1 dα2π1(m−k+α)−1 + d02v)-GCAS where dα ∈ {0, 1} if the  following three conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (C1) {π1(1), π1(2), · · · , π1(v)} = {1, 2, · · · , v} if v > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (C2) π1(m − k + α) < π1(m − k + α + 1) for 1 ≤ α ≤ k − 1 where π1(m) = m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (C3) For 1 ≤ α ≤ k − 1 and 2 ≤ β ≤ m − k, if π1(β) < π1(m − k + α), then π1(β − 1) <  π1(m − k + α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  15  Proof: Similarly, we consider L1 = 2n and L2 = 2m−1 + �k−1  α=1 2π1(m−k+α)−1 + 2v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then  we would like to prove that  �  C  ρ(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' u1, u2) =  �  c∈G  L1−1−u1  �  g=0  L2−1−u2  �  i=0  �  ξcg+u1,i+u2−cg,i�  = 0  (37)  for 0 ≤ u1 < 2n, 0 ≤ u2 < 2m−1 + �k−1  α=1 2π1(m−k+α)−1 + 2v and (u1, u2) ̸= (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' From (4)  we can find that  c = q  2  �m−k−1  �  l=1  xπ1(l)xπ1(l+1) +  n−1  �  s=1  yπ2(s)yπ2(s+1) + xπ1(m)yπ2(n)  �  +  m−k  �  l=1  µlxπ1(l)xπ1(m) +  m  �  l=1  plxl +  n  �  s=1  κsys + p0 · 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (38)  Then we let h = g + u1 and j = i + u2 for any integers g and i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Next, we discuss seven cases  to complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Case 1: Assuming u1 > 0, u2 ≥ 0, and gπ2(1) ̸= hπ2(1), we can find an array c′ = c +  (q/2)yπ2(1) ∈ G for any array c ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Therefore, we can obtain  ch,j − cg,i − c′  h,j+c′  g,i = q  2(gπ2(1) − hπ2(1)) ≡ q  2  (mod q)  (39)  Since gπ2(1) ̸= hπ2(1), we have  ξch,j−cg,i/ξc′  h,j−c′  g,i = ξ  q  2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (40)  Thus,  ξch,j−cg,i + ξc′  h,j−c′  g,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (41)  Case 2: If u1 > 0, u2 ≥ 0, and gπ2(1) = hπ2(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Let β be the smallest integer such that  gπ2(β) ̸= hπ2(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We define g′ and h′ are two integers which are distinct from g and h only in  one position π2(β − 1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then, similar to Case 2 of Theorem 1, we have  ξch,j−cg,i + ξch′,j−cg′,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (42)  Case 3: We suppose im ̸= jm, u1 = 0 and u2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We let g′ be an integer distinct from  i only in one position, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=', g′  π2(n) = 1 − gπ2(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Similar to Case 3 of Theorem 1, we have  ξcg,j−cg,i + ξcg′,j−cg′,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Case 4: If u1 = 0, u2 > 0, and iπ1(1) ̸= jπ1(1) or iπ1(m−k+α) ̸= jπ1(m−k+α), we can find an  array c′ = c + (q/2)xπ1(1) ∈ G or c′ = c + (q/2)xπ1(m−k+α) for any array c ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Similar to  Case 1, we can obtain ξcg,j−cg,i + ξc′  g,j−c′  g,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  16  Case 5: Suppose u1 = 0, u2 > 0, iπ1(1) = jπ1(1), and iπ1(m−k+α) = jπ1(m−k+α) for all  α = 1, 2, · · · , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Suppose that α′ is the largest non-negative integer satisfying iπ1(m−k+α′) =  jπ1(m−k+α′) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then we assume β is the smallest integer which satisfies iπ1(β) ̸= jπ1(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Here, we have is = js = 0 for s = π1(m − k + α′) + 1, π1(m − k + α′) + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , m − 1, and  s ̸= π1(m − k + α) for α = α′ + 1, α′ + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Hence, it implies π1(β) < π1(m − k + α′)  and π1(β − 1) < π1(m − k + α′) according to the condition (C-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Let i′ and j′ be integers that  differ from i and j, respectively, in the position π1(β − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Similar to Case 2, we have  ξcg,j−cg,i + ξcg,j′−cg,i′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (43)  Case 6: Suppose u1 = 0, u2 > 0, iπ1(1) = jπ1(1), and iπ1(m−k+α) = jπ1(m−k+α) = 1 for all  α = 1, 2, · · · , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Then we assume β is the smallest integer which satisfies iπ1(β) ̸= jπ1(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Since  is = js = 0 for s = v + 1, v + 2, · · · , m − k and s ̸= π1(m − k + α) for α = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , k − 1, we  can obtain π1(β) ≤ v implying π1(β − 1) ≤ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Similar to Case 2, we have  ξcg,j−cg,i + ξcg,j′−cg,i′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (44)  From Cases 1 to 6, the theorem can be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Remark 2: Taking σ2(l) = π2(n − l + 1) for l = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , n and π1(m − k + α) = m − k + α  for α = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' , k in Theorem 2, (34) can be represented as  f =q  2  �m−k−1  �  k=1  xπ1(k)xπ1(k+1) +  n−1  �  k=1  yσ2(k)yσ2(k+1) + xmyσ2(1)  �  +  m−k  �  l=1  µlxπ1(l)xm  +  m  �  l=1  plxl +  n  �  s=1  κsys + p0  (45)  where pl, κs ∈ Zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We can find that the result of Lemma 3 is a special case of Theorem 2 by  simply setting k = 1, µm−1 = q  2, and µl = 0 for l = 1, · · · , m − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Example 4: Taking q = 2, m = 5, n = 2, k = 2, and v = 0, we let π1 = (1, 2, 4, 3, 5) and  π2 = (1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The generalized Boolean function is f = x1x2 + x2x4 + y1y2 + x5y1 by setting  pl, κs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The array set G is a GCAS of size 8 when the truncated size L1 = 4 L2 = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We  let G = {c0, c1, · · · , c7} and list the constituent arrays in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Also, their AACF sum is  shown as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' SIMULATION RESULTS  In this section, we present the numerical results including the power radiation pattern and  BER performance by using our proposed 2D GCASs for massive MIMO systems with URA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 2023  DRAFT  17  TABLE III  THE CONSTRUCTED (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 21)-GCAS IN EXAMPLE 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='c0 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Power Radiation Pattern  According to (16), the power radiation pattern �N−1  n=0  ��[vec(A(ϕ, θ))Tvec(Wn)]  ��2 can be ob-  tained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We first consider the massive MIMO system equipped with a URA of size 4 × 33,  January 4, 2023  DRAFT  18  0  200  2  400  40  600  20  800  0  1000  0  20  2  40  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The summation of autocorrelations of constituent arrays in the GCAS in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=', L1 = 4 and L2 = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We take the GCS G = {c0, c1, c2, c3} listed in Table II to  generate the precoding matrices {W0, W1, W2, W3} = {(−1)c0, (−1)c1, (−1)c2, (−1)c3} with  the omnidirectional property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The power radiation pattern of the GCAS-based scheme with array  size 4 × 33 is perfectly omnidirectional as illustrated in Fig 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  For the purpose of comparison, we also show the power radiation patterns of the precoding  matrices based on Zadoff-Chu sequences and random-matrices whose elements are randomly  generated from “+1” and “−1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The ZC-based precoder consists of four 4 × 33 precoding  matrices, which are obtained based on a ZC sequence of length 4 and a ZC sequence of length  33 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 4(b) illustrates the power radiation pattern of the ZC-based precoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We can  find that its power radiation pattern is not omnidirectional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The random-matrix-based precoder  consists of four 4 × 33 precoding matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The elements in the random-matrix-based precoding  matrices are generated by selecting the elements from {1, −1} with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 4(c)  describes the power radiation pattern of the random matrix-based precoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We can observe that  the power radiation pattern is not omnidirectional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Next, we consider the massive MIMO system equipped with a URA of size 4 × 21, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=',  January 4, 2023  DRAFT  19  (a) GCAS-based precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (b) ZC-based precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (c) Random-matrix-based precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Power radiation pattern with 4 × 33 URA and 4 × 4 STBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  L1 = 4 and L2 = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We use the GCS G = {c0, c1, · · · , c7} listed in Table III for the precoding  matrix {W0, W1, · · · , W7} = {(−1)c0, (−1)c1, · · · , (−1)c7}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The power radiation pattern of the  GCAS-based scheme with array size 4×21 is described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The perfect omnidirectional  property can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We also see that the power radiation patterns of the ZC-based precoder  and the random-matrix precoder shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 5(b) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 5(c) are not omnidirectional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The  ZC-based precoding matrices are obtained by a ZC sequence of length 4 and ZC sequence of  21 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Bit Error Rate Performance  In this subsection, we present the BER performance of our proposed 2D GCAS-based schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  We first consider the massive MIMO system equipped with a URA of size 4×33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We let N = 4  January 4, 2023  DRAFT  X-axis-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='51Sy-ax1sX-axis550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='51Sy-ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  1S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='3X-axis00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='5Sy-ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  1S20  (a) GCAS-based precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (b) ZC-based precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  (c) Random-matrix-based precoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Power radiation pattern with 4 × 21 URA and 8 × 8 STBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  and then the 4 × 4 orthogonal real STBC be presented as  S =  �  �  �  �  �  �  �  s0  −s1  −s2  −s3  s1  s0  s3  −s2  s2  −s3  s0  s1  s3  s2  −s1  s0  �  �  �  �  �  �  �  ,  (46)  where s0, s1, s2, s3 are binary phase shift keying (BPSK) modulated symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We employ the  maximum likelihood (ML) decoding here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' For each realization, the elevation and the azimuth  angles are uniformly distributed at random between [0, π/2] and [0, 2π], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' For com-  parison, the ZC-based precoder and random-matrix-based precoder are the same as mentioned in  Section IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The BER performances of three different schemes are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We can  find that the 2D GCAS-based scheme outperform the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' At BER of 10−4, there are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='6 dB  and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='6 dB gains over the ZC-based scheme and the random-matrix-based scheme, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  X-axis-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='51Sy-ax1sX-axis-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='51Sy-ax1s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='8X-axis-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='2Sy-06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='ax1s21  2  0  2  4  6  8  10  12  14  SNR (dB)  10-8  10-6  10-4  10-2  100  BER  GCAS-based precoding (N=4)  ZC-based precoding (N=4)  Random-matrix-based precoding (N=4)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' BER performance of the different schemes for a 4 × 33 URA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Next, we consider the massive MIMO system equipped with a URA of size 4 × 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We  consider 8 × 8 STBC and the 8 × 8 orthogonal real STBC is given by  S =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  s0  s1  s2  s3  s4  s5  s6  s7  −s1  s0  s3  −s2  s5  −s4  −s7  s6  −s2  −s3  s0  s1  s6  s7  −s4  −s5  −s3  s2  −s1  s0  s7  −s6  s5  −s4  −s4  −s5  −s6  −s7  s0  s1  s2  s3  −s5  s4  −s7  s6  −s1  s0  −s3  s2  −s6  s7  s4  −s5  −s2  s3  s0  −s1  −s7  −s6  s5  −s4  −s3  s2  s1  s0  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  (47)  where s0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' s7 are BPSK modulated symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' We also take the ZC-based precoding and  random-matrix-based precoding for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The BER performance comparison for these  three different schemes is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' At BER of 10−4, there are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='2 dB and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='8 dB  gains over the ZC-based scheme and the random-matrix-based scheme, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' As a result,  the 2D GCASs are good candidates as precoding matrices for omnidirectional transmission in  January 4, 2023  DRAFT  22  2  0  2  4  6  8  10  12  14  SNR (dB)  10-8  10-6  10-4  10-2  100  BER  GCAS-based precoding (N=8)  ZC-based precoding (N=8)  Random-matrix-based precoding (N=8)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' BER performance of the different schemes for a 4 × 21 URA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  massive MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' CONCLUSION  In this paper, constructions of 2D GCASs with flexible array sizes have been proposed in  Theorems 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Our constructions can be obtained directly from 2D GBFs without the aid  of special sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Besides, our proposed GCASs have flexible array sizes which can fit  more antenna configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Furthermore, Theorem 2 can include the results in [32] as a special  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Simulation results showed that the omnidirectional transmission can be achieved when  the precoding matrices are based on the proposed GCASs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' The BER performance due to their  omnidirectional power radiation patterns, the ZC-based scheme and random-matix-based have  inferior performances because their power radiation patterns both are not ideally omnidirectional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  Although Theorems 1 and 2 can provide direct constructions of 2D GCASs, the first dimension  has size L1 limited to 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content=' Therefore, the future work includes the extension of constructions of  2D GCASs of which both dimensions have non-power-of-two sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
+page_content='  January 4, 2023  DRAFT  23  REFERENCES  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
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+page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfR_t3/content/2301.01225v1.pdf'}
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+arXiv:2301.05087v1  [math.DG]  12 Jan 2023
+SCALAR CURVATURE RIGIDITY OF CONVEX
+POLYTOPES
+SIMON BRENDLE
+Abstract. We prove a scalar curvature rigidity theorem for convex
+polytopes. The proof uses the Fredholm theory for Dirac operators on
+manifolds with boundary.
+A variant of a theorem of Fefferman and
+Phong plays a central role in our analysis.
+1. Introduction
+Let Ω be a compact polytope in Rn with non-empty interior. We may
+write Ω = �
+α∈A{uα ≤ 0}, where A is a finite set and the uα are linear
+functions in Rn. For each α ∈ A, we denote by Nα ∈ Sn−1 the outward-
+pointing unit normal vector to the halfspace {uα ≤ 0} with respect to the
+Euclidean metric.
+Let g be a Riemannian metric which is defined on an open set containing
+Ω. For each α ∈ A, we denote by να the outward-pointing unit normal
+vector to the halfspace {uα ≤ 0} with respect to the metric g. We will make
+the following assumption:
+Matching Angle Hypothesis. If x is point in ∂Ω and α1, α2 ∈ A satisfy
+uα1(x) = uα2(x) = 0, then ⟨να1, να2⟩ = ⟨Nα1, Nα2⟩ at the point x. Here, the
+inner product ⟨να1, να2⟩ is computed with respect to the metric g, and the
+inner product ⟨Nα1, Nα2⟩ is the standard inner product in Rn.
+Theorem 1.1. Suppose that n ≥ 3 is an odd integer, and Ω is a compact
+polytope in Rn with non-empty interior. Let g be a Riemannian metric which
+is defined on an open set containing Ω and has nonnegative scalar curvature
+at each point in Ω. For each α ∈ A, we assume that the mean curvature of
+the hypersurface {uα = 0} with respect to g is nonnegative at each point in
+Ω ∩ {uα = 0}. Moreover, we assume that the Matching Angle Hypothesis is
+satisfied. Then the Ricci tensor of g vanishes at each point in Ω.
+Theorem 1.1 also holds in the even-dimensional case. This can be seen
+by considering the Cartesian product Ω × [0, 1] ⊂ Rn+1.
+Scalar curvature comparison theorems for polytopes were first studied in
+seminal work of Gromov [6],[7],[8]. Li [9] has used minimal surface techniques
+to prove a scalar curvature comparison theorem for certain polytopes in
+The author was support by the National Science Foundation under grant DMS-2103573
+and by the Simons Foundation. The author acknowledges the hospitality of T¨ubingen
+University, where part of this work was carried out.
+1
+
+2
+SIMON BRENDLE
+dimension 3. Wang, Xie, and Yu [10] have proposed a different approach to
+this problem which is based on the study of Dirac operators on manifolds
+with corners.
+In this paper, we describe another approach to this problem. As in [10], we
+employ a spinor approach. In contrast to [10], we work with boundary value
+problems for Dirac operators on smooth domains, which are well understood
+thanks to the work of B¨ar and Ballmann [1],[2].
+In the following, we outline the main steps involved in the proof of The-
+orem 1.1. We approximate a given convex polytope Ω by a one-parameter
+family of smooth convex domains Ωλ. On each domain Ωλ, we solve the
+Dirac equation for an m-tuple of spinors s = (s1, . . . , sm) with a suitable
+local boundary condition. To prove the existence of a solution satisfying
+that particular boundary condition, we use the Fredholm theory developed
+by B¨ar and Ballmann [1],[2] together with the homotopy invariance of the
+Fredholm index. Having constructed an m-tuple of harmonic spinors on Ωλ
+satisfying this boundary condition, we apply a Weitzenb¨ock formula, and
+integrate over Ωλ. The resulting integral formula contains a term involving
+the scalar curvature, as well as a boundary term. Unfortunately, it is not
+clear if the boundary term has a favorable sign. We are able to control the
+boundary integral by adapting a theorem due to Fefferman and Phong [4].
+2. A boundary value problem for the Dirac operator on a
+smooth domain
+Let m = 2[ n
+2 ] denote the dimension of the space of spinors on Rn. Let
+{E1, . . . , En} denote the standard basis of Rn. Throughout this section, we
+fix an orthonormal basis {¯s1, . . . , ¯sm} of the space of spinors on flat Rn.
+We define ωaαβ = ⟨Ea · ¯sα, ¯sβ⟩ for a = 1, . . . , n and α, β = 1, . . . , m. The
+matrices ω1, . . . , ωn ∈ End(Cm) are skew-Hermitian, so that ωaαβ = −ωaβα.
+Moreover, ωaωb + ωbωa = −2δab id. In other words,
+m
+�
+β=1
+(ωaαβ ωbβγ + ωbαβ ωaβγ) = −2δab δαγ.
+We begin by stating a basic algebraic fact which will be needed later.
+Lemma 2.1. Assume that n is odd. Then there is no non-zero element of
+End(Cm) which anti-commutes with ωa ∈ End(Cm) for each a = 1, . . . , n.
+Proof. We recall the definition of the spin representation in odd dimen-
+sions. Let {E1, . . . , En} denote the standard basis of Rn. For k = 1, . . . , [n
+2 ],
+we define wk = E2k−1 − iE2k ∈ Cn. The spinor space is defined as the ex-
+terior algebra Λ∗W, where W = span{wk : k = 1, . . . , [n
+2 ]} ⊂ Cn. For each
+k ∈ {1, . . . , [n
+2 ]}, we define a linear map Pk ∈ End(Λ∗W) by
+Pk(wj1 ∧ . . . ∧ wjr) = wk ∧ wj1 ∧ . . . ∧ wjr.
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+3
+Moreover, for each k ∈ {1, . . . , [n
+2 ]}, we define a linear map Qk ∈ End(Λ∗W)
+by
+Qk(wj1 ∧ . . . ∧ wjr) = 0,
+Qk(wk ∧ wj1 ∧ . . . ∧ wjr) = wj1 ∧ . . . ∧ wjr
+for k /∈ {j1, . . . , jr}. Then PkPl + PlPk = QkQl + QlQk = 0 and PkQl +
+QlPk = δkl id for k, l ∈ {1, . . . , [n
+2 ]}. Finally, we define a linear map S ∈
+End(Λ∗W) so that
+S(wj1 ∧ . . . ∧ wjr) =
+�
+wj1 ∧ . . . ∧ wjr
+if r is even
+−wj1 ∧ . . . ∧ wjr
+if r is odd.
+Clearly, PkS + SPk = 0, QkS + SQk = 0, and S2 = id.
+Consequently,
+there is a natural algebra homomorphism from the Clifford algebra ClC(n)
+to End(Λ∗W) which maps wk to i
+√
+2 Pk, ¯wk to i
+√
+2 Qk, and En to iS. It is
+well known (see [5], Lemma 20.9) that
+span{Pk1 · · · PkrQl1 · · · Qls : r + s is even}
+= End(ΛevenW) ⊕ End(ΛoddW)
+and
+span{Pk1 · · · PkrQl1 · · · Qls : r + s is odd}
+= Hom(ΛevenW, ΛoddW) ⊕ Hom(ΛoddW, ΛevenW).
+We claim that there is no non-zero element of End(Λ∗W) which anti-commutes
+with Pk, Qk, S for each k ∈ {1, . . . , [n
+2 ]}.
+Suppose that L ∈ End(Λ∗W)
+is such an element.
+Since L anti-commutes with S, it follows that L ∈
+Hom(ΛevenW, ΛoddW)⊕Hom(ΛoddW, ΛevenW). Since L anti-commutes with
+Pk, Qk for each k ∈ {1, . . . , [n
+2 ]}, it follows that L anti-commutes with every
+element of Hom(ΛevenW, ΛoddW)⊕Hom(ΛoddW, ΛevenW). This implies that
+L = 0. This completes the proof of Lemma 2.1.
+Assume that Ω is a domain in Rn with smooth boundary ∂Ω = Σ. Let g
+be a Riemannian metric on Ω. We denote by ν the outward-pointing unit
+normal vector field with respect to the metric g. Let ∇ denote the spin
+connection. The Dirac operator is defined by
+Ds =
+n
+�
+i=1
+ei · ∇eis,
+where {e1, . . . , en} is a local orthonormal frame on Ω. The boundary Dirac
+operator DΣ is given by
+DΣs =
+n−1
+�
+i=1
+ν · ei · ∇eis + 1
+2 H s
+at each point on Σ, where {e1, . . . , en−1} is a local orthonormal frame on Σ.
+In the remainder of this section, we consider the Dirac operator act-
+ing on m-tuples of spinors with a suitable local boundary condition of
+
+4
+SIMON BRENDLE
+Lopatinsky-Shapiro type. To formulate the boundary condition, we assume
+that N : Σ → Sn−1 is a given smooth map.
+Definition 2.2. Consider an m-tuple of spinors s = (s1, . . . , sm). At each
+point on Σ, we define
+(χs)α = −
+n
+�
+a=1
+m
+�
+β=1
+⟨N, Ea⟩ ωaαβ ν · sβ
+and
+(Bs)α =
+n−1
+�
+i=1
+n
+�
+a=1
+m
+�
+β=1
+⟨dN(ei), Ea⟩ ωaαβ ei · sβ,
+where {e1, . . . , en−1} is a local orthonormal frame on Σ.
+Lemma 2.3. The map χ is self-adjoint. Moreover, χ2 is the identity.
+Proof. Suppose that s = (s1, . . . , sm) and t = (t1, . . . , tm) are two m-
+tuples of spinors. We compute
+(χ2s)α =
+n
+�
+a,b=1
+m
+�
+β,γ=1
+⟨N, Ea⟩ ⟨N, Eb⟩ ωaαβ ωbβγ ν · ν · sγ
+= −
+n
+�
+a,b=1
+m
+�
+β,γ=1
+⟨N, Ea⟩ ⟨N, Eb⟩ ωaαβ ωbβγ sγ
+= −1
+2
+n
+�
+a,b=1
+m
+�
+β,γ=1
+⟨N, Ea⟩ ⟨N, Eb⟩ (ωaαβ ωbβγ + ωbαβ ωaβγ) sγ
+=
+n
+�
+a,b=1
+m
+�
+γ=1
+⟨N, Ea⟩ ⟨N, Eb⟩ δab δαγ sγ
+= sα.
+Moreover,
+m
+�
+α=1
+⟨(χs)α, tα⟩ = −
+n
+�
+a=1
+m
+�
+α,β=1
+⟨N, Ea⟩ ωaαβ ⟨ν · sβ, tα⟩
+= −
+n
+�
+a=1
+m
+�
+α,β=1
+⟨N, Ea⟩ ωaβα ⟨sβ, ν · tα⟩
+=
+m
+�
+β=1
+⟨sβ, (χt)β⟩.
+This completes the proof of Lemma 2.3.
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+5
+Lemma 2.4. Assume that x ∈ Σ and ξ ∈ TxΣ. Then the map (s1, . . . , sm) �→
+(ν · ξ · s1, . . . , ν · ξ · sm) maps the eigenspace of χ with eigenvalue 1 to the
+eigenspace of χ with eigenvalue −1, and vice versa. In particular, the two
+eigenspaces have the same dimension.
+Proof. For each vector ξ ∈ TxΣ, the map
+(s1, . . . , sm) �→ (ν · ξ · s1, . . . , ν · ξ · sm)
+anti-commutes with χ. From this, the assertion follows.
+Lemma 2.5. The map B is self-adjoint. Moreover, χ and B commute.
+Proof. Let {e1, . . . , en−1} be a local orthonormal frame on Σ. Then
+m
+�
+α=1
+⟨(Bs)α, tα⟩ =
+n−1
+�
+i=1
+n
+�
+a=1
+m
+�
+α,β=1
+⟨dN(ei), Ea⟩ ωaαβ ⟨ei · sβ, tα⟩
+=
+n−1
+�
+i=1
+n
+�
+a=1
+m
+�
+α,β=1
+⟨dN(ei), Ea⟩ ωaβα ⟨sβ, ei · tα⟩
+=
+m
+�
+β=1
+⟨sβ, (Bt)β⟩.
+This shows that B is self-adjoint. Moreover,
+(χBs)α − (Bχs)α
+= −
+n−1
+�
+i=1
+n
+�
+a,b=1
+n
+�
+β,γ=1
+⟨N, Ea⟩ ⟨dN(ei), Eb⟩ ωaαβ ωbβγ ν · ei · sγ
++
+n−1
+�
+i=1
+n
+�
+a,b=1
+n
+�
+β,γ=1
+⟨dN(ei), Ea⟩ ⟨N, Eb⟩ ωaαβ ωbβγ ei · ν · sγ
+= −
+n−1
+�
+i=1
+n
+�
+a,b=1
+n
+�
+β,γ=1
+⟨N, Ea⟩ ⟨dN(ei), Eb⟩ (ωaαβ ωbβγ + ωbαβ ωaβγ) ν · ei · sγ
+= 2
+n−1
+�
+i=1
+n
+�
+a,b=1
+n
+�
+γ=1
+⟨N, Ea⟩ ⟨dN(ei), Eb⟩ δab δαγ ν · ei · sγ
+= 2
+n−1
+�
+i=1
+⟨N, dN(ei)⟩ ν · ei · sα
+= 0.
+Thus, χ and B commute. This completes the proof of Lemma 2.5.
+At this point, we recall a definition from linear algebra.
+
+6
+SIMON BRENDLE
+Definition 2.6. Let V and W be finite-dimensional vector spaces of the
+same dime, each of them equipped with an inner product. The trace norm
+of a linear map L : V → W is defined by ∥L∥tr = supQ tr(QL), where the
+supremum is taken over all linear isometries Q : W → V . Equivalently,
+∥L∥tr can be characterized as the sum of the singular values of L.
+It is easy to see from the definition that the trace norm satisfies the tri-
+angle inequality.
+Lemma 2.7. Suppose that s = (s1, . . . , sm) is an m-tuple of spinors. Then
+����
+m
+�
+α=1
+⟨(Bs)α, sα⟩
+���� ≤ ∥dN∥tr
+� m
+�
+α=1
+|sα|2
+�
+at each point x ∈ Σ. Here, ∥dN∥tr denotes the trace norm of the differential
+dN : TxΣ → TN(x)Sn−1. The tangent space TxΣ is equipped with the restric-
+tion of the inner product g, and the tangent space TN(x)Sn−1 is equipped
+with the restriction of the standard inner product on Rn.
+Proof. Fix a point x ∈ Σ. We can find an orthonormal basis {e1, . . . , en−1}
+of TxΣ so that dN(ei) = λi ˆEi, where { ˆE1, . . . , ˆEn−1} is an orthonormal ba-
+sis of TN(x)Sn−1 and λ1, . . . , λn−1 ≥ 0 denote the singular values of dN.
+Then
+m
+�
+α=1
+����
+n
+�
+a=1
+m
+�
+β=1
+⟨ ˆEi, Ea⟩ ωaαβ ei · sβ
+����
+2
+=
+n
+�
+a,b=1
+m
+�
+α,β,γ=1
+⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ ωaαβ ωbαγ ⟨ei · sβ, ei · sγ⟩
+= −
+n
+�
+a,b=1
+m
+�
+α,β,γ=1
+⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ ωaαβ ωbγα ⟨sβ, sγ⟩
+= −1
+2
+n
+�
+a,b=1
+m
+�
+α,β,γ=1
+⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ (ωaγα ωbαβ + ωbγα ωaαβ) ⟨sβ, sγ⟩
+=
+n
+�
+a,b=1
+m
+�
+β,γ=1
+⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ δab δγβ ⟨sβ, sγ⟩
+=
+m
+�
+α=1
+|sα|2
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+7
+for each i = 1, . . . , n − 1. Using the Cauchy-Schwarz inequality, we obtain
+����
+n
+�
+a=1
+m
+�
+α,β=1
+⟨ ˆEi, Ea⟩ ωaαβ ⟨ei · sβ, sα⟩
+����
+≤
+� m
+�
+α=1
+����
+n
+�
+a=1
+m
+�
+β=1
+⟨ ˆEi, Ea⟩ ωaαβ ei · sβ
+����
+2� 1
+2 � m
+�
+α=1
+|sα|2
+� 1
+2
+=
+m
+�
+α=1
+|sα|2
+for each i = 1, . . . , n − 1. Summation over i = 1, . . . , n − 1 gives
+����
+m
+�
+α=1
+⟨(Bs)α, sα⟩
+���� =
+����
+n−1
+�
+i=1
+n
+�
+a=1
+m
+�
+α,β=1
+⟨dN(ei), Ea⟩ ωaαβ ⟨ei · sβ, sα⟩
+����
+=
+����
+n−1
+�
+i=1
+λi
+�
+n
+�
+a=1
+m
+�
+α,β=1
+⟨ ˆEi, Ea⟩ ωaαβ ⟨ei · sβ, sα⟩
+�����
+≤
+� n−1
+�
+i=1
+λi
+� � m
+�
+α=1
+|sα|2
+�
+,
+as claimed.
+Proposition 2.8. Suppose that s = (s1, . . . , sm) and t = (t1, . . . , tm) are
+m-tuples of spinors. Then
+0 =
+�
+Σ
+m
+�
+α=1
+⟨DΣsα, (χt)α⟩ dσg +
+�
+Σ
+m
+�
+α=1
+⟨(χs)α, DΣtα⟩ dσg
++
+�
+Σ
+m
+�
+α=1
+⟨(Bs)α, tα⟩ dσg.
+Equivalently,
+0 =
+�
+Σ
+m
+�
+α=1
+⟨(As)α, (χt)α⟩ dσg +
+�
+Σ
+m
+�
+α=1
+⟨(χs)α, (At)α⟩ dσg,
+where A is defined by A = DΣ + 1
+2χB.
+Proof. Let {e1, . . . , en−1} be a local orthonormal frame on Σ. We define
+a tangential vector field Z on Σ by
+⟨Z, ei⟩ =
+n
+�
+a=1
+m
+�
+α,β=1
+⟨N, Ea⟩ ωaαβ ⟨ei · sβ, tα⟩
+
+8
+SIMON BRENDLE
+for i = 1, . . . , n − 1. Then
+divΣZ =
+n−1
+�
+i=1
+n
+�
+a=1
+m
+�
+α,β=1
+⟨N, Ea⟩ ωaαβ ⟨ei · ∇eisβ, tα⟩
++
+n−1
+�
+i=1
+n
+�
+a=1
+m
+�
+α,β=1
+⟨N, Ea⟩ ωaαβ ⟨ei · sβ, ∇eitα⟩
+−
+n
+�
+a=1
+m
+�
+α,β=1
+H ⟨N, Ea⟩ ωaαβ ⟨ν · sβ, tα⟩
++
+n−1
+�
+i=1
+n
+�
+a=1
+m
+�
+α,β=1
+⟨dN(ei), Ea⟩ ωaαβ ⟨ei · sβ, tα⟩
+= −
+n−1
+�
+i=1
+m
+�
+β=1
+⟨ei · ∇eisβ, ν · (χt)β⟩
++
+n−1
+�
+i=1
+m
+�
+α=1
+⟨ei · ν · (χs)α, ∇eitα⟩
++
+m
+�
+α=1
+H ⟨(χs)α, tα⟩ +
+m
+�
+α=1
+⟨(Bs)α, tα⟩
+=
+m
+�
+β=1
+⟨DΣsβ, (χt)β⟩ +
+m
+�
+α=1
+⟨(χs)α, DΣtα⟩ +
+m
+�
+α=1
+⟨(Bs)α, tα⟩.
+Integrating over Σ, we obtain
+0 =
+�
+Σ
+m
+�
+β=1
+⟨DΣsβ, (χt)β⟩ dσg +
+�
+Σ
+m
+�
+α=1
+⟨(χs)α, DΣtα⟩ dσg
++
+�
+Σ
+m
+�
+α=1
+⟨(Bs)α, tα⟩ dσg.
+This completes the proof of Proposition 2.8.
+Remark 2.9. It is well known that the boundary Dirac operator DΣ is
+formally self-adjoint. Moreover, it follows from Lemma 2.3 and Lemma 2.5
+that χB is self-adjoint. Consequently, the operator A = DΣ+ 1
+2χB is formally
+self-adjoint. Finally, Proposition 2.8 implies that A and χ anti-commute.
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+9
+Proposition 2.10. Suppose that s = (s1, . . . , sm) is an m-tuple of spinors.
+Then
+−
+�
+Ω
+m
+�
+α=1
+|Dsα|2 dvolg +
+�
+Ω
+n
+�
+α=1
+|∇sα|2 dvolg + 1
+4
+�
+Ω
+m
+�
+α=1
+R |sα|2 dvolg
+≤ 1
+2
+�
+Σ
+m
+�
+α=1
+⟨DΣsα, sα − (χs)α⟩ dσg + 1
+2
+�
+Σ
+m
+�
+α=1
+⟨sα − (χs)α, DΣsα⟩ dσg
+− 1
+2
+�
+Σ
+(H − ∥dN∥tr)
+� m
+�
+α=1
+|sα|2
+�
+dσg.
+Proof. By the Weitzenb¨ock formula, D2sα = −∆sα + 1
+4 R sα, where ∆
+denotes the connection Laplacian on the spinor bundle. Using the divergence
+theorem, we obtain
+−
+�
+Ω
+m
+�
+α=1
+|Dsα|2 dvolg +
+�
+Ω
+m
+�
+α=1
+|∇sα|2 dvolg + 1
+4
+�
+Ω
+m
+�
+α=1
+R |sα|2 dvolg
+=
+�
+Σ
+m
+�
+α=1
+⟨ν · Dsα, sα⟩ dσg +
+�
+Σ
+m
+�
+α=1
+⟨∇νsα, sα⟩ dσg
+=
+�
+Σ
+⟨DΣsα, sα⟩ dσg − 1
+2
+�
+Σ
+m
+�
+α=1
+H |sα|2 dσg.
+Applying Proposition 2.8 with s = t gives
+0 =
+�
+Σ
+m
+�
+α=1
+⟨DΣsα, (χs)α⟩ dσg +
+�
+Σ
+m
+�
+α=1
+⟨(χs)α, DΣsα⟩ dσg
++
+�
+Σ
+m
+�
+α=1
+⟨(Bs)α, sα⟩ dσg.
+This gives
+−
+�
+Ω
+m
+�
+α=1
+|Dsα|2 dvolg +
+�
+Ω
+n
+�
+α=1
+|∇sα|2 dvolg + 1
+4
+�
+Ω
+m
+�
+α=1
+R |sα|2 dvolg
+= 1
+2
+�
+Σ
+m
+�
+α=1
+⟨DΣsα, sα⟩ dσg + 1
+2
+�
+Σ
+m
+�
+α=1
+⟨sα, DΣsα⟩ dσg − 1
+2
+�
+Σ
+m
+�
+α=1
+H |sα|2 dσg
+= 1
+2
+�
+Σ
+m
+�
+α=1
+⟨DΣsα, sα − (χs)α⟩ dσg + 1
+2
+�
+Σ
+m
+�
+α=1
+⟨sα − (χs)α, DΣsα⟩ dσg
+− 1
+2
+�
+Σ
+m
+�
+α=1
+⟨(Bs)α, sα⟩ dσg − 1
+2
+�
+Σ
+m
+�
+α=1
+H |sα|2 dσg.
+Hence, the assertion follows from Lemma 2.7.
+
+10
+SIMON BRENDLE
+Corollary 2.11. Suppose that R ≥ 0 at each point in Ω and H ≥ ∥dN∥tr at
+each point on Σ. Then every m-tuple of harmonic spinors s = (s1, . . . , sm)
+with χs = s is parallel.
+Replacing N by −N, we can draw the following conclusion:
+Corollary 2.12. Suppose that R ≥ 0 at each point in Ω and H ≥ ∥dN∥tr at
+each point on Σ. Then every m-tuple of harmonic spinors s = (s1, . . . , sm)
+with χs = −s is parallel.
+Proposition 2.13. Suppose that Ω is a convex domain in Rn with smooth
+boundary ∂Ω = Σ.
+Let g be a Riemannian metric on Ω.
+Suppose that
+N : Σ → Sn−1 is a smooth map. Then the boundary condition χs = s is a
+D-elliptic boundary condition in the sense of B¨ar and Ballmann [2].
+Proof.
+We apply Corollary 3.18 in [2] with E′ = ker(id − χ) and
+E′′ = ker(id + χ).
+Lemma 2.4 implies that, for each point x ∈ Σ and
+each ξ ∈ TxΣ, the map (s1, . . . , sm) �→ (ν · ξ · s1, . . . , ν · ξ · sm) interchanges
+ker(id − χ) and ker(id + χ). Therefore, the boundary condition χs = s is a
+D-elliptic boundary condition in the sense of [2].
+Proposition 2.14. Assume that n ≥ 3 is an odd integer. Suppose that
+Ω is a convex domain in Rn with smooth boundary ∂Ω = Σ. Let g be a
+Riemannian metric on Ω. Suppose that N : Σ → Sn−1 is homotopic to
+the Gauss map of Σ with respect to the Euclidean metric. Then the Dirac
+operator with the boundary condition χs = s has Fredholm index at least 1.
+Proof. Since the Fredholm index is homotopy invariant, it suffices to
+prove the assertion in the special case when g is the Euclidean metric and
+N is the Gauss map of Σ with respect to the Euclidean metric.
+We first analyze the kernel of the Dirac operator with the boundary con-
+dition χs = s. Recall that ¯s1, . . . , ¯sm is a basis of spinors on flat Rn, and
+ωaαβ = ⟨Ea · ¯sα, ¯sβ⟩. Clearly, ¯s = (¯s1, . . . , ¯sm) is an m-tuple of harmonic
+spinors on Ω which satisfies the boundary condition χ¯s = ¯s. Therefore, the
+kernel has dimension at least 1.
+We next examine the cokernel.
+The cokernel can be identified with
+the space of all m-tuples of harmonic spinors s = (s1, . . . , sm) such that
+⟨ν · s, t⟩ = 0 for all points x ∈ Σ and all t ∈ ker(id − χ) (see [2], Example
+3.20). We claim that this space has dimension 0. To see this, suppose that
+s = (s1, . . . , sm) is an m-tuple of harmonic spinors such that ⟨ν · s, t⟩ = 0
+for all points x ∈ Σ and all t ∈ ker(id − χ). This implies s ∈ ker(id + χ) at
+each point on Σ. Since H = ∥dN∥tr at each point on Σ, Corollary 2.12 im-
+plies that s = (s1, . . . , sm) is parallel. In other words, s1, . . . , sm are constant
+spinors. Let us write sα = �m
+β=1 zαβ ¯sβ for some matrix z ∈ End(Cm). Since
+χs = −s at each point on Σ, it follows that the matrix z ∈ End(Cm) anti-
+commutes with the matrix �n
+a=1⟨N(x), Ea⟩ ωa ∈ End(Cm) for each point
+x ∈ Σ. It is easy to see that the Gauss map N : Σ → Sn−1 is surjective.
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+11
+Consequently, the matrix z ∈ End(Cm) anti-commutes with ωa ∈ End(Cm)
+for each a = 1, . . . , n. Since n is odd, Lemma 2.1 implies that z = 0, hence
+s = 0. This shows that the cokernel has dimension 0. This completes the
+proof of Proposition 2.14.
+3. Approximating a compact, convex polytope by smooth
+domains
+Let us consider a compact, convex polytope Ω ⊂ Rn with non-empty
+interior. We write Ω = �
+α∈A{uα ≤ 0}, where A is a finite set and the uα
+are linear functions in Rn. After eliminating redundant inequalities, we may
+assume that the following condition is satisfied.
+Assumption 3.1. For each α ∈ A, the set Ω ∩ {uα > 0} is non-empty.
+Let g be a Riemannian metric which is defined on an open set containing
+Ω. For each α ∈ A, ∇uα will denote the gradient of uα with respect to the
+metric g; |∇uα| will denote the norm of the gradient of uα with respect to
+the metric g; and να =
+∇uα
+|∇uα| will denote the outward-pointing unit normal
+vector to the halfspace {uα ≤ 0} with respect to the metric g. For each
+α ∈ A, we denote by Nα ∈ Sn−1 the outward-pointing unit normal vector
+to the halfspace {uα ≤ 0} with respect to the Euclidean metric.
+For each λ > 0, the function �
+α∈A eλuα is convex with respect to the
+Euclidean metric. Clearly, �
+α∈A eλuα > 1 on ∂Ω. Moreover, we can find
+large number λ0 such that infΩ
+�
+α∈A eλuα < 1 for each λ > λ0. For each
+λ > λ0, we define
+Ωλ =
+� �
+α∈A
+eλuα ≤ 1
+�
+.
+For each λ > λ0, Ωλ is a convex domain in Rn with smooth boundary
+Σλ = ∂Ωλ. The sets Ωλ form an increasing family of sets in the sense that
+Ωλ ⊂ Ωµ for λ0 < λ < µ. Moreover,
+�
+λ>λ0
+Ωλ =
+�
+α∈A
+{uα < 0}.
+Lemma 3.2. If λ is sufficiently large, then infΣλ
+�� �
+α∈A eλuα duα
+�� ≥ C−1
+for some large constant C which is independent of λ.
+Proof. We argue by contradiction. Suppose that the assertion is false.
+Then there exists a sequence of positive real numbers λl → ∞ and a se-
+quence of points xl ∈ Σλl such that
+�� �
+α∈A eλuα duα
+�� ≤ l−1 at the point
+xl. After passing to a subsequence, we may assume that the sequence xl
+converges to a point x0 ∈ Ω.
+Moreover, we may assume that, for each
+α ∈ A, the sequence eλluα(xl) converges to a nonnegative real number zα.
+Since �
+α∈A eλluα(xl) = 1 for each l, we know that �
+α∈A zα > 0.
+Let
+A0 := {α ∈ A : zα > 0}. Clearly, A0 is non-empty, and uα(x0) = 0 for all
+
+12
+SIMON BRENDLE
+α ∈ A0. Moreover, �
+α∈A0 zα duα = 0 at the point x0. On the other hand,
+since Ω is a convex set with non-empty interior, we can find a tangent vector
+ξ ∈ Tx0Ω such that duα(ξ) > 0 for all α ∈ A0. This is a contradiction. This
+completes the proof of Lemma 3.2.
+Lemma 3.3. If λ is sufficiently large, then infΣλ
+�� �
+α∈A eλuα |∇uα| Nα
+�� ≥
+C−1 for some large constant C which is independent of λ.
+Proof. We argue by contradiction. Suppose that the assertion is false.
+Then there exists a sequence of positive real numbers λl → ∞ and a se-
+quence of points xl ∈ Σλl such that
+�� �
+α∈A eλuα |∇uα| Nα
+�� ≤ l−1 at the
+point xl. After passing to a subsequence, we may assume that the sequence
+xl converges to a point x0 ∈ Ω. Moreover, we may assume that, for each
+α ∈ A, the sequence eλluα(xl) |∇uα(xl)| converges to a nonnegative real num-
+ber zα. Since �
+α∈A eλluα(xl) = 1 for each l, we know that �
+α∈A zα > 0.
+Let A0 := {α ∈ A : zα > 0}. Clearly, A0 is non-empty, and uα(x0) = 0
+for all α ∈ A0. Moreover, �
+α∈A0 zαNα = 0 at the point x0. On the other
+hand, since Ω is a convex set with non-empty interior, we can find a vector
+ξ ∈ Rn such that ⟨Nα, ξ⟩ > 0 for all α ∈ A0. This is a contradiction. This
+completes the proof of Lemma 3.3.
+The outward-pointing unit normal vector to the domain Ωλ with respect
+to the metric g is given by
+ν =
+�
+α∈A eλuα ∇uα
+�� �
+α∈A eλuα ∇uα
+�� =
+�
+α∈A eλuα |∇uα| να
+�� �
+α∈A eλuα |∇uα| να
+��.
+We define a map N : Σλ → Sn−1 by
+N =
+�
+α∈A eλuα |∇uα| Nα
+�� �
+α∈A eλuα |∇uα| Nα
+��.
+Lemma 3.4. The map N : Σλ → Sn−1 is homotopic to the Gauss map of
+Σλ with respect to the Euclidean metric.
+Proof. In the special case when g is the Euclidean metric, the map N
+coincides with the Gauss map of Σλ, and the assertion is trivially true. To
+prove the assertion in general, we deform the metric g to the Euclidean met-
+ric.
+Proposition 3.5. Let x ∈ Σλ. Let π : TxΩ → TxΩ denotes the orthogonal
+projection to the orthogonal complement of ν and P : Rn → Rn denotes
+the orthogonal projection to the orthogonal complement of N. Then H −
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+13
+∥dN∥tr ≥ Vλ, where the function Vλ : Σλ → R is defined by
+Vλ = λ
+�
+α∈A eλuα |∇uα|2 |π(να)|2
+�� �
+α∈A eλuα |∇uα| να
+��
+− λ
+�
+α∈A eλuα |∇uα|2 |π(να)| |P(Nα)|
+�� �
+α∈A eλuα |∇uα| Nα
+��
++
+�
+α∈A eλuα (∆uα − (D2uα)(ν, ν))
+�� �
+α∈A eλuα |∇uα| να
+��
+−
+�
+α∈A eλuα |∇(|∇uα|)| |P(Nα)|
+�� �
+α∈A eλuα |∇uα| Nα
+��
+.
+Proof. Let {e1, . . . , en−1} denote a local orthonormal frame on Σλ. The
+mean curvature of Σλ is given by
+H = λ
+�n−1
+i=1
+�
+α∈A eλuα ⟨∇uα, ei⟩2
+�� �
+α∈A eλuα ∇uα
+��
++
+�n−1
+i=1
+�
+α∈A eλuα (D2uα)(ei, ei)
+�� �
+α∈A eλuα ∇uα
+��
+= λ
+�
+α∈A eλuα |π(∇uα)|2
+�� �
+α∈A eλuα ∇uα
+��
++
+�
+α∈A eλuα (∆uα − (D2uα)(ν, ν))
+�� �
+α∈A eλuα ∇uα
+��
+= λ
+�
+α∈A eλuα |∇uα|2 |π(να)|2
+�� �
+α∈A eλuα |∇uα| να
+��
++
+�
+α∈A eλuα (∆uα − (D2uα)(ν, ν))
+�� �
+α∈A eλuα |∇uα| να
+��
+.
+If ξ is a tangent vector to Σλ, then
+dN(ξ)
+= λ
+�
+α∈A eλuα |∇uα| ⟨∇uα, ξ⟩ P(Nα)
+�� �
+α∈A eλuα |∇uα| Nα
+��
++
+�
+α∈A eλuα ⟨∇(|∇uα|), ξ⟩ P(Nα)
+�� �
+α∈A eλuα |∇uα| Nα
+��
+= λ
+�
+α∈A eλuα |∇uα|2 ⟨π(να), ξ⟩ P(Nα)
+�� �
+α∈A eλuα |∇uα| Nα
+��
++
+�
+α∈A eλuα ⟨∇(|∇uα|), ξ⟩ P(Nα)
+�� �
+α∈A eλuα |∇uα| Nα
+��
+.
+The trace norm of a linear transformation of the form ξ �→ ⟨X, ξ⟩ Y is given
+by |X| |Y |. Since the trace norm satisfies the triangle inequality, it follows
+that
+∥dN∥tr
+≤ λ
+�
+α∈A eλuα |∇uα|2 |π(να)| |P(Nα)|
+�� �
+α∈A eλuα |∇uα| Nα
+��
++
+�
+α∈A eλuα |∇(|∇uα|)| |P(Nα)|
+�� �
+α∈A eλuα |∇uα| Nα
+��
+.
+Putting these facts together, the assertion follows.
+In the following, we denote by Vλ,− = max{−Vλ, 0} the negative part of
+Vλ.
+Proposition 3.6. Suppose that the Matching Angle Hypothesis is satisfied.
+Then supΣλ Vλ,− ≤ o(λ) as λ → ∞.
+Proof. We argue by contradiction. Suppose that the assertion is false.
+Then there exists a sequence of positive real numbers λl → ∞ and a se-
+quence of points xl ∈ Σλl such that lim supl→∞ λ−1
+l
+Vλl(xl) < 0.
+After
+passing to a subsequence, we may assume that the sequence xl converges
+to a point x0 ∈ Ω.
+Moreover, we may assume that, for each α ∈ A,
+the sequence eλluα(xl) |∇uα(xl)| converges to a nonnegative real number zα.
+
+14
+SIMON BRENDLE
+Since �
+α∈A eλluα(xl) = 1 for each l, we know that �
+α∈A zα > 0.
+Let
+A0 := {α ∈ A : zα > 0}. Clearly, A0 is non-empty, and uα(x0) = 0 for
+all α ∈ A0. The Matching Angle Hypothesis implies that, at the point x0,
+⟨να1, να2⟩ = ⟨Nα1, Nα2⟩ for all α1, α2 ∈ A0. Let π : Tx0Ω → Tx0Ω denote
+the orthogonal projection to the orthogonal complement of �
+α∈A0 zανα,
+and let P : Rn → Rn denote the orthogonal projection to the orthogonal
+complement of �
+α∈A0 zαNα. For each β ∈ A0, we have
+|π(νβ)|2 = 1 −
+� �
+α∈A0 zανα, νβ
+�2
+�� �
+α∈A0 zανα
+��2
+= 1 −
+� �
+α∈A0 zαNα, Nβ
+�2
+�� �
+α∈A0 zαNα
+��2
+= |P(Nβ)|2
+at the point x0. Therefore, for each β ∈ A0, we obtain
+|π(νβ)|
+�� �
+α∈A0 zανα
+�� =
+|P(Nβ)|
+�� �
+α∈A0 zαNα
+��
+at the point x0. Using Proposition 3.5, we conclude that λ−1
+l
+Vλl(xl) → 0 as
+l → ∞. This is a contradiction.
+In the remainder of this section, we will estimate the Ls-norm Vλ,− on Σλ∩
+Br(p), where s ∈ [1, 3
+2) is a fixed exponent and Br(p) denotes a Euclidean
+ball of radius r. We begin by recalling a basic fact about the area of convex
+hypersurfaces in Rn.
+Lemma 3.7. Let Br(p) denote a Euclidean ball of radius r. Then the in-
+tersection Σλ ∩ Br(p) has area at most Crn−1.
+Proof. This follows from the fact that the hypersurface Σλ = ∂Ωλ is
+outward-minimizing with respect to the Euclidean metric.
+Definition 3.8. Consider three pairwise distinct elements α1, α2, α3 ∈ A.
+We denote by G(α1,α2,α3)
+λ
+the set of all points x ∈ Σλ with the property that
+uα1(x) ≥ uα2(x) ≥ uα3(x) and uα3(x) ≥ uα(x) for α ∈ A \ {α1, α2, α3}.
+Lemma 3.9. Assume that the mean curvature of the hypersurface {uα = 0}
+with respect to g is nonnegative at each point in Ω ∩ {uα = 0}. Let us fix an
+exponent s ∈ [1, 3
+2), and let Br(p) denote a Euclidean ball of radius r ≤ 1.
+If λr is sufficiently large, then
+�
+rs+1−n
+�
+G(α1,α2,α3)
+λ
+∩{uα2≤−λ− 7
+8 r
+1
+8 }∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ Cλr e−(λr)
+1
+8
+for all pairwise distinct elements α1, α2, α3 ∈ A.
+Proof. Let us consider an arbitrary point x ∈ G(α1,α2,α3)
+λ
+with uα2(x) ≤
+−λ− 7
+8 r
+1
+8 .
+By definition of G(α1,α2,α3)
+λ
+, it follows that uα(x) ≤ −λ− 7
+8r
+1
+8
+for all α ∈ A \ {α1}.
+Using the identity �
+α∈A eλuα(x) = 1, we obtain
+uα1(x) ≥ −Cλ−1 e−(λr)
+1
+8 . Moreover, |ν − να1| ≤ C e−(λr)
+1
+8 and |N − Nα1| ≤
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+15
+C e−(λr)
+1
+8 at the point x. From this, we deduce that |π(να1)| ≤ C e−(λr)
+1
+8
+and |P(Nα1)| ≤ C e−(λr)
+1
+8 at the point x. Therefore,
+Vλ ≥ ∆uα1 − (D2uα1)(να1, να1)
+|∇uα1|
+− Cλ e−(λr)
+1
+8
+at the point x. Since uα1(x) ≥ −Cλ−1 e−(λr)
+1
+8 and uα(x) ≤ −λ− 7
+8r
+1
+8 for
+all α ∈ A \ {α1}, we can find a point y ∈ Ω such that uα1(y) = 0 and
+d(x, y) ≤ Cλ−1 e−(λr)
+1
+8 . By assumption, the mean curvature of the hyper-
+surface {uα1 = 0} at the point y is nonnegative. This implies
+∆uα1 − (D2uα1)(να1, να1)
+|∇uα1|
+≥ 0
+at the point y. Consequently,
+∆uα1 − (D2uα1)(να1, να1)
+|∇uα1|
+≥ −C d(x, y)
+at the point x. Thus, we conclude that
+Vλ(x) ≥ −Cλ e−(λr)
+1
+8
+for each point x ∈ G(α1,α2,α3)
+λ
+∩ {uα2 ≤ −λ− 7
+8r
+1
+8}.
+On the other hand,
+Σλ ∩ Br(p) has area at most Crn−1. Consequently,
+�
+rs+1−n
+�
+G(α1,α2,α3)
+λ
+∩{uα2≤−λ− 7
+8 r
+1
+8 }∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ Cλr e−(λr)
+1
+8 .
+This completes the proof of Lemma 3.9.
+Lemma 3.10. Assume that the Matching Angle Hypothesis holds. Let us
+fix an exponent s ∈ [1, 3
+2), and let Br(p) denote a Euclidean ball of radius
+r ≤ 1. If λr is sufficiently large, then
+�
+rs+1−n
+�
+G(α1,α2,α3)
+λ
+∩{uα2≥−λ− 7
+8 r
+1
+8 }∩{uα3≤−λ− 3
+4 r
+1
+4 }∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ C (λr)
+1
+8 − 7
+8s
+for all pairwise distinct elements α1, α2, α3 ∈ A.
+Proof. We distinguish two cases:
+Case 1: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} = ∅. By continuity, we
+can find a real number δ such that Ω ∩ {uα1 ≥ −δ} ∩ {uα2 ≥ −δ} = ∅. If λr
+is sufficiently large, then λ− 7
+8r
+1
+8 ≤ δ. This implies
+G(α1,α2,α3)
+λ
+∩ {uα2 ≥ −λ− 7
+8 r
+1
+8 }
+⊂ Σλ ∩ {uα1 ≥ −δ} ∩ {uα2 ≥ −δ} = ∅.
+Hence, the assertion is trivially true in this case.
+
+16
+SIMON BRENDLE
+Case 2: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} ̸= ∅. It follows from
+Assumption 3.1 that the hypersurfaces {uα1 = 0} and {uα2 = 0} intersect
+transversally.
+Let us consider an arbitrary point x ∈ G(α1,α2,α3)
+λ
+with uα2(x) ≥ −λ− 7
+8r
+1
+8
+and uα3(x) ≤ −λ− 3
+4r
+1
+4. Clearly, uα1(x) ≥ −λ− 7
+8r
+1
+8 by definition of G(α1,α2,α3)
+λ
+.
+Moreover, uα(x) ≤ −λ− 3
+4r
+1
+4 for all α ∈ A \ {α1, α2}. Consequently, we can
+find a point y ∈ Ω such that uα1(y) = uα2(y) = 0 and d(x, y) ≤ Cλ− 7
+8r
+1
+8.
+The Matching Angle Hypothesis implies ⟨να1, να2⟩ = ⟨Nα1, Nα2⟩ at the point
+y. Consequently, |⟨να1, να2⟩ − ⟨Nα1, Nα2⟩| ≤ C d(x, y) at the point x. From
+this, we deduce that
+|π(να1)|
+�� �
+α∈A eλuα |∇uα| να
+�� −
+|P(Nα1)|
+�� �
+α∈A eλuα |∇uα| Nα
+�� ≥ −C d(x, y) − C e−(λr)
+1
+4
+and
+|π(να2)|
+�� �
+α∈A eλuα |∇uα| να
+�� −
+|P(Nα2)|
+�� �
+α∈A eλuα |∇uα| Nα
+�� ≥ −C d(x, y) − C e−(λr)
+1
+4
+at the point x. Thus, we conclude that
+Vλ(x) ≥ −Cλ
+1
+8 r− 7
+8
+for each point x ∈ G(α1,α2,α3)
+λ
+∩ {uα2 ≥ −λ− 7
+8 r
+1
+8 } ∩ {uα3 ≤ −λ− 3
+4r
+1
+4}. By
+transversality, the set {0 ≥ uα1 ≥ −λ− 7
+8r
+1
+8} ∩ {0 ≥ uα2 ≥ −λ− 7
+8 r
+1
+8} ∩ Br(p)
+can be covered by C (λr)
+7(n−2)
+8
+Euclidean balls of radius λ− 7
+8 r
+1
+8 .
+More-
+over, the intersection of Σλ with each ball of radius λ− 7
+8r
+1
+8 has area at
+most C (λr)− 7(n−1)
+8
+rn−1. This shows that Σλ ∩ {uα1 ≥ −λ− 7
+8 r
+1
+8 } ∩ {uα2 ≥
+−λ− 7
+8 r
+1
+8 } ∩ Br(p) has area at most C (λr)− 7
+8 rn−1. Since
+G(α1,α2,α3)
+λ
+∩ {uα2 ≥ −λ− 7
+8r
+1
+8} ∩ Br(p)
+⊂ Σλ ∩ {uα1 ≥ −λ− 7
+8 r
+1
+8 } ∩ {uα2 ≥ −λ− 7
+8r
+1
+8} ∩ Br(p),
+it follows that
+�
+rs+1−n
+�
+G(α1,α2,α3)
+λ
+∩{uα2≥−λ− 7
+8 r
+1
+8 }∩{uα3≤−λ− 3
+4 r
+1
+4 }∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ C (λr)
+1
+8 − 7
+8s .
+This completes the proof of Lemma 3.10.
+Lemma 3.11. Let us fix an exponent s ∈ [1, 3
+2), and let Br(p) denote a
+Euclidean ball of radius r ≤ 1. If λr is sufficiently large, then
+�
+rs+1−n
+�
+G(α1,α2,α3)
+λ
+∩{uα3≥−λ− 3
+4 r
+1
+4 }∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ C (λr)1− 3
+2s
+for all pairwise distinct elements α1, α2, α3 ∈ A.
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+17
+Proof. We distinguish two cases:
+Case 1: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} ∩ {uα3 = 0} = ∅. By
+continuity, we can find a real number δ such that Ω ∩ {uα1 ≥ −δ} ∩ {uα2 ≥
+−δ} ∩ {uα3 ≥ −δ} = ∅. If λr is sufficiently large, then λ− 3
+4r
+1
+4 ≤ δ. This
+implies
+G(α1,α2,α3)
+λ
+∩ {uα3 ≥ −λ− 3
+4 r
+1
+4 }
+⊂ Σλ ∩ {uα1 ≥ −δ} ∩ {uα2 ≥ −δ} ∩ {uα3 ≥ −δ} = ∅.
+Hence, the assertion is trivially true in this case.
+Case 2: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} ∩ {uα3 = 0} ̸= ∅. It
+follows from Assumption 3.1 that the hypersurfaces {uα1 = 0}, {uα2 = 0},
+and {uα3 = 0} intersect transversally.
+Let us consider an arbitrary point x ∈ G(α1,α2,α3)
+λ
+with uα3(x) ≥ −λ− 3
+4r
+1
+4.
+Clearly,
+Vλ(x) ≥ −Cλ
+for all points x ∈ G(α1,α2,α3)
+λ
+∩ {uα3 ≤ −λ− 3
+4 r
+1
+4 }. By transversality, the set
+{0 ≥ uα1 ≥ −λ− 3
+4r
+1
+4}∩{0 ≥ uα2 ≥ −λ− 3
+4 r
+1
+4}∩{0 ≥ uα3 ≥ −λ− 3
+4 r
+1
+4 }∩Br(p)
+can be covered by C (λr)
+3(n−3)
+4
+Euclidean balls of radius λ− 3
+4 r
+1
+4 .
+More-
+over, the intersection of Σλ with each ball of radius λ− 3
+4r
+1
+4 has area at
+most C (λr)− 3(n−1)
+4
+rn−1. This shows that Σλ ∩ {uα1 ≥ −λ− 3
+4 r
+1
+4 } ∩ {uα2 ≥
+−λ− 3
+4 r
+1
+4 }∩{uα3 ≥ −λ− 3
+4 r
+1
+4 }∩Br(p) has area at most C (λr)− 3
+2 rn−1. Since
+G(α1,α2,α3)
+λ
+∩ {uα3 ≥ −λ− 3
+4 r
+1
+4 } ∩ Br(p)
+⊂ Σλ ∩ {uα1 ≥ −λ− 3
+4 r
+1
+4 } ∩ {uα2 ≥ −λ− 3
+4 r
+1
+4 } ∩ {uα3 ≥ −λ− 3
+4r
+1
+4} ∩ Br(p),
+it follows that
+�
+rs+1−n
+�
+G(α1,α2,α3)
+λ
+∩{uα3≥−λ− 3
+4 r
+1
+4 }∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ C (λr)1− 3
+2s .
+This completes the proof of Lemma 3.11.
+Proposition 3.12. Assume that the mean curvature of the hypersurface
+{uα = 0} with respect to g is nonnegative at each point in Ω ∩ {uα = 0}
+and that the Matching Angle Hypothesis is satisfied. Let us fix an exponent
+s ∈ [1, 3
+2), and let Br(p) denote a Euclidean ball of radius r ≤ 1. If λr is
+sufficiently large, then
+�
+rs+1−n
+�
+Σλ∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ C (λr)−1 + C (λr)
+1
+8 − 7
+8s + C (λr)1− 3
+2s .
+
+18
+SIMON BRENDLE
+Proof. Combining Lemma 3.9, Lemma 3.10, and Lemma 3.11, we con-
+clude that
+�
+rs+1−n
+�
+G(α1,α2,α3)
+λ
+∩Br(p)
+V s
+λ,−
+� 1
+s
+≤ C (λr)−1 + C (λr)
+1
+8− 7
+8s + C (λr)1− 3
+2s
+for all pairwise distinct elements α1, α2, α3 ∈ A. On the other hand, Σλ =
+�
+α1,α2,α3 G(α1,α2,α3)
+λ
+, where the union is taken over all pairwise distinct el-
+ements α1, α2, α3 ∈ A.
+Hence, the assertion follows by summation over
+α1, α2, α3. This completes the proof of Proposition 3.12.
+Corollary 3.13. Assume that the mean curvature of the hypersurface {uα =
+0} with respect to g is nonnegative at each point in Ω ∩ {uα = 0} and that
+the Matching Angle Hypothesis is satisfied. Let us fix an exponent s ∈ [1, 3
+2).
+Then
+sup
+p∈Rn sup
+r≤1
+�
+rs+1−n
+�
+Σλ∩Br(p)
+V s
+λ,−
+� 1
+s
+→ 0
+as λ → ∞.
+Proof. Let us consider an arbitrary sequence λl → ∞. By Proposition
+3.6, we can find a sequence of positive real numbers δl → 0 such that
+sup
+p∈Rn
+sup
+r≤(δlλl)−1
+�
+rs+1−n
+�
+Σλl∩Br(p)
+V s
+λl,−
+� 1
+s
+→ 0
+as l → ∞. On the other hand, Proposition 3.12 implies that
+sup
+p∈Rn
+sup
+(δlλl)−1≤r≤1
+�
+rs+1−n
+�
+Σλl∩Br(p)
+V s
+λl,−
+� 1
+s
+→ 0
+as l → ∞. Putting these facts together, the assertion follows.
+4. Proof of the Theorem 1.1
+Throughout this section, we assume that n ≥ 3 is an odd integer, and Ω is
+a compact polytope in Rn with non-empty interior. Let g be a Riemannian
+metric which is defined on an open set containing Ω and has nonnegative
+scalar curvature at each point in Ω. We assume that the mean curvature of
+the hypersurface {uα = 0} with respect to g is nonnegative at each point in
+Ω ∩ {uα = 0} and that the Matching Angle Hypothesis is satisfied.
+Let U denote a Euclidean ball such that the closure of U is contained in
+the interior of Ω. Consider a sequence λl → ∞. Note that U ⊂ Ωλl if l is
+sufficiently large. By Proposition 2.14 we can find an m-tuple of harmonic
+spinors s(l) = (s(l)
+1 , . . . , s(l)
+m ) such that s(l) is defined on Ωλl; s(l) does not
+vanish identically; Ds(l) = 0 in Ωλl; and χs(l) = s(l) on Σλl.
+Standard
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+19
+unique continuation arguments imply that
+�
+U
+�m
+α=1 |s(l)
+α |2 dvolg > 0 if l is
+sufficiently large. By scaling, we can arrange that
+�
+U
+m
+�
+α=1
+|s(l)
+α |2 dvolg = 1
+for each l. Using Proposition 2.10, we obtain
+�
+Ωλl
+m
+�
+l=1
+|∇s(l)
+α |2 dvolg + 1
+4
+�
+Ωλl
+m
+�
+α=1
+R |s(l)
+α |2 dvolg
+≤ −1
+2
+�
+Σλl
+(H − ∥dN∥tr)
+� m
+�
+α=1
+|s(l)
+α |2
+�
+dσg.
+Proposition 3.5 implies that H − ∥dN∥tr ≥ Vλl at each point on Σλl. Con-
+sequently,
+�
+Ωλl
+m
+�
+l=1
+|∇s(l)
+α |2 dvolg + 1
+4
+�
+Ωλl
+m
+�
+α=1
+R |s(l)
+α |2 dvolg
+≤ 1
+2
+�
+Σλl
+Vλl,−
+� m
+�
+α=1
+|s(l)
+α |2
+�
+dσg.
+Note that the hypersurface Σλl = ∂Ωλl can be written as a radial graph
+with bounded slope. From this, it is easy to see that Ωλl is bi-Lipschitz
+equivalent to the Euclidean unit ball, with constants that are independent
+of λl. Using Theorem A.7 and Proposition 3.13, we obtain
+�
+Σλl
+Vλl,− F 2 dσg ≤ εl
+�
+Ωλl
+|∇F|2 dvolg + εl
+� �
+Σλl
+F dσg
+�2
+for every smooth function F on Ωλl, where εl → 0 as l → ∞. Moreover, the
+Sobolev trace theorem implies
+� �
+Σλl
+F dσg
+�2
+≤ C
+�
+Ωl
+|∇F|2 dvolg + C
+�
+U
+F 2 dvolg
+for every smooth function F on Ωλl, where C is a uniform constant inde-
+pendent of l. Putting these facts together, we conclude that
+�
+Σλl
+Vλl,− F 2 dσg ≤ Cεl
+�
+Ωλl
+|∇F|2 dvolg + Cεl
+�
+U
+F 2 dvolg
+
+20
+SIMON BRENDLE
+for every smooth function F on Ωλl. In the next step, we apply this inequal-
+ity with F =
+�
+δ2 + �m
+α=1 |s(l)
+α |2� 1
+2 , and send δ → 0. This gives
+�
+Ωλl
+m
+�
+l=1
+|∇s(l)
+α |2 dvolg + 1
+4
+�
+Ωλl
+m
+�
+α=1
+R |s(l)
+α |2 dvolg
+≤ 1
+2
+�
+Σλl
+Vλl,−
+� m
+�
+α=1
+|s(l)
+α |2
+�
+dσg
+≤ Cεl
+�
+Ωλl
+m
+�
+α=1
+|∇s(l)
+α |2 dvolg + Cεl
+�
+U
+m
+�
+α=1
+|s(l)
+α |2 dvolg
+for each l. Since the scalar curvature is nonnegative, it follows that
+�
+Ωλl
+m
+�
+α=1
+|∇s(l)
+α |2 dvolg ≤ Cεl
+�
+U
+m
+�
+α=1
+|s(l)
+α |2 dvolg
+if l is sufficiently large. Passing to the limit as l → ∞, we obtain a non-
+vanishing m-tuple of parallel spinors defined on the interior of Ω. Conse-
+quently, the Ricci tensor of g vanishes at each point in Ω. This completes
+the proof of Theorem 1.1.
+Appendix A. A variant of a theorem of Fefferman and Phong
+In this section, we describe a variant of an estimate due to Fefferman and
+Phong [4], which plays a central role in our argument. We denote by Q the
+collection of all (n − 1)-dimensional cubes of the form
+[2mj1, 2m(j1 + 1)] × . . . × [2mjn−1, 2m(jn−1 + 1)] × {0},
+where m ∈ Z and j1, . . . , jn−1 ∈ Z. For each Q ∈ Q, we denote by |Q| the
+(n − 1)-dimensional volume of Q.
+Theorem A.1. Fix an exponent s ∈ (1, n − 1). Let V be a nonnegative
+continuous function on the hyperplane Rn−1 × {0} ⊂ Rn with the property
+that
+diam(Q)s+1−n
+�
+Q
+V s ≤ 1
+for each (n − 1)-dimensional cube Q ∈ Q.
+Suppose that F is a smooth
+function on the half-space Rn
++ = {x ∈ Rn : xn ≥ 0}, and let f denote the
+restriction of F to the boundary ∂Rn
++ = Rn−1 × {0}. Then
+�
+Q
+V f 2 ≤ C
+�
+Q×[0,diam(Q)]
+|∇F|2 + C diam(Q)−1 |Q|−1
+� �
+Q
+|f|
+�2
+.
+for each (n − 1)-dimensional cube Q ∈ Q.
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+21
+The proof of Theorem A.1 involves a straightforward adaptation of the
+arguments of Fefferman and Phong [4]. Let us fix an exponent t such that
+s > t > 1. We define a nonnegative function W : Rn−1 × {0} → R by
+W(x) =
+sup
+Q∈Q,x∈Q
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+for each point x ∈ Rn−1 × {0}. In other words, W s is the maximal function
+associated with the function V s. Clearly, V ≤ W.
+We assume that F is a smooth function on the half-space Rn
++ = {x ∈
+Rn : xn ≥ 0}, and let f denote the restriction of F to the boundary ∂Rn
++ =
+Rn−1 × {0}.
+For each (n − 1)-dimensional cube Q ∈ Q, we denote by
+fQ = |Q|−1 �
+Q f the mean value of f over the cube Q.
+Lemma A.2. For each (n − 1)-dimensional cube Q0 ∈ Q, we have
+�
+|Q0|−1
+�
+Q0
+W t
+� 1
+t
+≤ C
+sup
+Q∈Q,Q0⊂Q
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+.
+Proof. For abbreviation, let
+Λ =
+sup
+Q∈Q,Q0⊂Q
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+.
+We define a nonnegative function W0 : Q0 → R by
+W0(x) =
+sup
+Q∈Q,x∈Q⊂Q0
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+for each point x ∈ Q0. Clearly,
+W(x) = max{Λ, W0(x)}
+for each point x ∈ Q0. The Hardy-Littlewood maximal inequality implies
+|Q0|−1 |{x ∈ Q0 : W0(x)s > α}| ≤ Cα−1 |Q0|−1
+�
+Q0
+V s ≤ Cα−1 Λs
+for all α > 0.
+We multiply both sides by α
+t
+s−1 and integrate over α ∈
+[Λs, ∞). This gives
+|Q0|−1
+�
+Q0
+W t
+0 ≤ C Λt,
+hence
+|Q0|−1
+�
+Q0
+W t ≤ C Λt.
+This completes the proof of Lemma A.2.
+
+22
+SIMON BRENDLE
+Lemma A.3. Given a real number ε > 0, we can find a real number δ > 0
+with the following property. If Q0 is an (n − 1)-dimensional cube in Q and
+A ⊂ Q0 is a Borel set with |A| ≤ δ |Q0|, then
+�
+A
+W ≤ ε
+�
+Q0
+W.
+Proof. Using Lemma A.2, we obtain
+�
+|Q0|−1
+�
+Q0
+W t
+� 1
+t
+≤ C
+sup
+Q∈Q,Q0⊂Q
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+.
+Moreover,
+sup
+Q∈Q,Q0⊂Q
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+≤ inf
+Q0 W ≤ |Q0|−1
+�
+Q0
+W.
+Therefore,
+�
+|Q0|−1
+�
+Q0
+W t
+� 1
+t
+≤ C |Q0|−1
+�
+Q0
+W.
+Hence, if A ⊂ Q0 is a Borel set with |A| ≤ δ Q0, then
+�
+A
+W ≤ |A|
+t−1
+t
+� �
+Q0
+W t
+� 1
+t
+≤ δ
+t−1
+t |Q0|
+t−1
+t
+� �
+Q0
+W t
+� 1
+t
+≤ Cδ
+t−1
+t
+�
+Q0
+W.
+This completes the proof of Lemma A.3.
+Lemma A.4. For each (n − 1)-dimensional cube Q0 ∈ Q, we have
+|Q0|−1
+�
+Q0
+W ≤ C diam(Q0)−1.
+Proof. Using Lemma A.2, we obtain
+�
+|Q0|−1
+�
+Q0
+W t
+� 1
+t
+≤ C
+sup
+Q∈Q,Q0⊂Q
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+.
+Moreover, our assumption implies that
+�
+|Q|−1
+�
+Q
+V s
+� 1
+s
+≤ C diam(Q)−1
+for each (n − 1)-dimensional cube Q ∈ Q. Putting these facts together, the
+assertion follows.
+Lemma A.5. For each (n − 1)-dimensional cube Q0 ∈ Q, we have
+�
+Q0
+V |f − fQ0|2 ≤ C
+�
+Q0
+Wg2,
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+23
+where the function g : Q0 → R is defined by
+g(x) =
+sup
+Q∈Q,x∈Q⊂Q0
+|Q|−1
+�
+Q
+|f − fQ|
+for x ∈ Q0.
+Proof. Fix an (n − 1)-dimensional cube Q0 ∈ Q. We define a function
+h : Q0 → R by
+h(x) =
+sup
+Q∈Q,x∈Q⊂Q0
+|Q|−1
+�
+Q
+|f − fQ0|
+for x ∈ Q0. Note that V ≤ W and |f −fQ0| ≤ h at each point in Q0. Hence,
+it suffices to prove that
+�
+Q0
+Wh2 ≤ C
+�
+Q0
+Wg2.
+To prove this inequality, let α0 = |Q0|−1 �
+Q0 |f − fQ0|. For each α > α0,
+we denote by Qα the set of all (n − 1)-dimensional cubes Q ∈ Q with the
+following properties:
+• Q ⊂ Q0.
+• |Q|−1 �
+Q |f − fQ0| > α.
+• If ˜Q is an (n − 1)-dimensional cube in Q with Q ⊊ ˜Q and ˜Q ⊂ Q0,
+then | ˜Q|−1 �
+˜Q |f − fQ0| ≤ α.
+It is easy to see that
+|Q|−1
+�
+Q
+|f − fQ0| ≤ 2n−1α
+for all Q ∈ Qα. In particular, |fQ − fQ0| ≤ 2n−1α for all Q ∈ Qα. Moreover,
+�
+Q∈Qα
+Q = {x ∈ Q0 : h(x) > α}.
+Finally, no point can be contained in the interior of more than one cube in
+Qα.
+Let K > 1 and δ ∈ (0, 1) be two real numbers that will be chosen later.
+For each (n−1)-dimensional cube Q ∈ Qα satisfying |Q|−1 �
+Q |f −fQ| ≤ δα,
+
+24
+SIMON BRENDLE
+we have
+(Kα − |fQ − fQ0|)
+�
+˜Q∈QKα, ˜Q⊂Q
+| ˜Q|
+≤
+�
+˜Q∈QKα, ˜Q⊂Q
+� �
+˜Q
+|f − fQ0| −
+�
+˜Q
+|fQ − fQ0|
+�
+≤
+�
+˜Q∈QKα, ˜Q⊂Q
+�
+˜Q
+|f − fQ|
+≤
+�
+Q
+|f − fQ|
+≤ δα |Q|.
+Recall that |fQ − fQ0| ≤ 2n−1α for all Q ∈ Qα. Hence, if we choose K > 2n,
+then we obtain
+�
+˜Q∈QKα, ˜Q⊂Q
+| ˜Q| ≤ 21−n δ |Q|
+for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 �
+Q |f − fQ| ≤ δα.
+We next apply Lemma A.3 with ε = 1
+2 K−2. Hence, we can choose δ ∈
+(0, 1) sufficiently small (depending on K) so that
+�
+˜Q∈QKα, ˜Q⊂Q
+�
+˜Q
+W ≤ 1
+2 K−2
+�
+Q
+W
+for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 �
+Q |f − fQ| ≤ δα.
+For each (n−1)-dimensional cube Q ∈ Qα, the set Q∩{h > Kα} is contained
+in the union �
+˜Q∈QKα, ˜Q⊂Q ˜Q. This implies
+�
+Q∩{h>Kα}
+W ≤ 1
+2 K−2
+�
+Q
+W
+for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 �
+Q |f − fQ| ≤ δα.
+On the other hand, if Q is an (n − 1)-dimensional cube in Qα satisfying
+|Q|−1 �
+Q |f − fQ| > δα, then g > δα at each point in Q. Therefore,
+�
+Q∩{h>Kα}
+W ≤
+�
+Q∩{g>δα}
+W
+for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 �
+Q |f − fQ| > δα.
+Putting these facts together, we conclude that
+�
+Q∩{h>Kα}
+W ≤ 1
+2 K−2
+�
+Q
+W +
+�
+Q∩{g>δα}
+W
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+25
+for each (n−1)-dimensional cube Q ∈ Qα. Summation over all cubes Q ∈ Qα
+gives
+�
+{h>Kα}
+W ≤ 1
+2 K−2
+�
+{h>α}
+W +
+�
+{g>δα}
+W.
+This inequality holds for each α > α0. Moreover, since g ≥ α0 at each point
+in Q0, the inequality is trivially true for α ≤ α0. Finally, we multiply the
+inequality by α
+2 and integrate over α ∈ (0, ∞). This gives
+K−2
+�
+Q0
+Wh2 ≤ 1
+2 K−2
+�
+Q0
+Wh2 + δ−2
+�
+Q0
+Wg2.
+This completes the proof of Lemma A.5.
+Lemma A.6. For each (n − 1)-dimensional cube Q0 ∈ Q, we have
+�
+Q0
+Wg2 ≤ C
+�
+Q0×[0,diam(Q)]
+|∇F|2,
+where the function g : Q0 → R is defined by
+g(x) =
+sup
+Q∈Q,x∈Q⊂Q0
+|Q|−1
+�
+Q
+|f − fQ|
+for x ∈ Q0.
+Proof. Fix an (n−1)-dimensional cube Q0 ∈ Q, and let α0 = |Q0|−1 �
+Q0 |f−
+fQ0|. For each α > α0, we denote by Qα the set of all (n − 1)-dimensional
+cubes Q ∈ Q with the following properties:
+• Q ⊂ Q0.
+• |Q|−1 �
+Q |f − fQ| > α.
+• If ˜Q is an (n − 1)-dimensional cube in Q with Q ⊊ ˜Q and ˜Q ⊂ Q0,
+then | ˜Q|−1 �
+˜Q |f − f ˜Q| ≤ α.
+It is easy to see that
+|Q|−1
+�
+Q
+|f − fQ| ≤ 2nα
+for all Q ∈ Qα. Moreover,
+�
+Q∈Qα
+Q = {x ∈ Q0 : g(x) > α}.
+Finally, no point can be contained in the interior of more than one cube in
+Qα.
+
+26
+SIMON BRENDLE
+Let K > 1 be a real number that will be chosen later. For each (n − 1)-
+dimensional cube Q ∈ Qα, we have
+Kα
+�
+˜Q∈QKα, ˜Q⊂Q
+| ˜Q| ≤
+�
+˜Q∈QKα, ˜Q⊂Q
+�
+˜Q
+|f − f ˜Q|
+≤ 2
+�
+˜Q∈QKα, ˜Q⊂Q
+�
+˜Q
+|f − fQ|
+≤ 2
+�
+Q
+|f − fQ|
+≤ 2n+1α |Q|.
+Hence, if we choose K > 2n+2, then
+�
+˜Q∈QKα, ˜Q⊂Q
+| ˜Q| ≤ 1
+2 |Q|
+for each cube Q ∈ Qα. For each (n − 1)-dimensional cube Q ∈ Qα, the set
+Q ∩ {g > Kα} is contained in the union �
+˜Q∈QKα, ˜Q⊂Q ˜Q. This implies
+|Q ∩ {g > Kα}| ≤
+�
+˜Q∈QKα, ˜Q⊂Q
+| ˜Q| ≤ 1
+2 |Q|,
+hence
+|Q ∩ {g ≤ Kα}| ≥ 1
+2 |Q|
+for each (n−1)-dimensional cube Q ∈ Qα. We define a nonnegative function
+ϕ : Rn−1 × {0} → R by
+ϕ(x1, . . . , xn−1, 0) =
+� � diam(Q0)
+0
+|∇F(x1, . . . , xn−1, xn)|2 dxn
+� 1
+2
+.
+Moreover, we define a nonnegative function ψ : Q0 → R by
+ψ(x) =
+sup
+Q∈Q,x∈Q⊂Q0
+|Q|−1
+�
+Q
+ϕ
+for each point x ∈ Q0. In other words, ψ is the maximal function associated
+with ϕ. Using the Sobolev trace theorem, we obtain
+α ≤ |Q|−1
+�
+Q
+|f − fQ|
+≤ 2 |Q|−1 inf
+a∈R
+�
+Q
+|f − a|
+≤ C |Q|−1 inf
+a∈R
+� �
+Q×[0,diam(Q)]
+|∇(F − a)|
++ diam(Q)−1
+�
+Q×[0,diam(Q)]
+|F − a|
+�
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+27
+for each (n − 1)-dimensional cube Q ∈ Qα. Using the Poincar´e inequality,
+we conclude that
+α ≤ C |Q|−1
+�
+Q×[0,diam(Q)]
+|∇F|
+≤ C diam(Q)
+1
+2 |Q|−1
+�
+Q
+ϕ
+≤ C diam(Q)
+1
+2 inf
+Q ψ
+for each (n − 1)-dimensional cube Q ∈ Qα. This implies
+α2 diam(Q)−1 |Q| ≤ C
+�
+Q∩{g≤Kα}
+ψ2
+for each (n − 1)-dimensional cube Q ∈ Qα. Combining this estimate with
+Lemma A.4, we obtain
+α2
+�
+Q
+W ≤ C
+�
+Q∩{g≤Kα}
+ψ2
+for each (n−1)-dimensional cube Q ∈ Qα. Summation over all cubes Q ∈ Qα
+gives
+α2
+�
+{g>α}
+W ≤
+�
+{α<g≤Kα}
+ψ2.
+This estimate holds for each α > α0. We now multiply both sides by α−1
+and integrate over α ∈ (2α0, ∞). This gives
+�
+{g>4α0}
+Wg2 ≤ C
+�
+Q0
+ψ2.
+In the next step, we bound the contribution from the set {g ≤ 4α0}. Using
+the Sobolev trace theorem, we obtain
+α0 = |Q0|−1
+�
+Q0
+|f − fQ0|
+≤ 2 |Q0|−1 inf
+a∈R
+�
+Q0
+|f − a|
+≤ C |Q0|−1 inf
+a∈R
+� �
+Q0×[0,diam(Q0)]
+|∇(F − a)|
++ diam(Q0)−1
+�
+Q0×[0,diam(Q)]
+|F − a|
+�
+.
+
+28
+SIMON BRENDLE
+Using the Poincar´e inequality, we conclude that
+α0 ≤ C |Q0|−1
+�
+Q0×[0,diam(Q0)]
+|∇F|
+≤ C diam(Q0)
+1
+2 |Q0|−1
+�
+Q0
+ϕ
+≤ C diam(Q0)
+1
+2 inf
+Q0 ψ.
+This implies
+α2
+0 diam(Q0)−1 |Q0| ≤ C
+�
+Q0
+ψ2.
+Combining this estimate with Lemma A.4, we obtain
+α2
+0
+�
+Q0
+W ≤ C
+�
+Q0
+ψ2,
+hence
+�
+{g≤4α0}
+Wg2 ≤ C
+�
+Q0
+ψ2.
+Putting these facts together, we conclude that
+�
+Q0
+Wg2 ≤ C
+�
+Q0
+ψ2.
+On the other hand, the Hardy-Littlewood maximal inequality implies
+�
+Q0
+ψ2 ≤ C
+�
+Q0
+ϕ2 = C
+�
+Q0×[0,diam(Q0)]
+|∇F|2.
+This completes the proof of Lemma A.6.
+After these preparations, we now complete the proof of Theorem A.1.
+Combining Lemma A.5 and Lemma A.6, we conclude that
+�
+Q0
+V |f − fQ0|2 ≤ C
+�
+Q0×[0,diam(Q0)]
+|∇F|2
+for each (n − 1)-dimensional cube Q0 ∈ Q. This implies
+�
+Q0
+V f 2 ≤ C
+�
+Q0×[0,diam(Q0)]
+|∇F|2 + C |Q0|−2
+� �
+Q0
+V
+� � �
+Q0
+|f|
+�2
+for each (n − 1)-dimensional cube Q0 ∈ Q. Using the estimate
+|Q0|−1
+�
+Q0
+V ≤
+�
+|Q0|−1
+�
+Q0
+V s
+� 1
+s
+≤ C diam(Q0)−1,
+we conclude that
+�
+Q0
+V f 2 ≤ C
+�
+Q0×[0,diam(Q0)]
+|∇F|2 + C diam(Q0)−1 |Q0|−1
+� �
+Q0
+|f|
+�2
+
+SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES
+29
+for each (n − 1)-dimensional cube Q0 ∈ Q.
+Corollary A.7. Fix an exponent s ∈ (1, n − 1). Let V be a nonnegative
+continuous function on the unit sphere Sn−1 ⊂ Rn with the property that
+rs+1−n
+�
+Sn−1∩Br(p)
+V s ≤ 1
+for all points p ∈ Rn and all r ≤ 1. Suppose that F is a smooth function on
+the unit ball Bn = {x ∈ Rn : |x| ≤ 1}, and let f denote the restriction of F
+to the boundary ∂Bn = Sn−1. Then
+�
+Sn−1 V f 2 ≤ C
+�
+Bn |∇F|2 + C
+� �
+Sn−1 |f|
+�2
+.
+References
+[1] C. B¨ar and W. Ballmann, Boundary value problems for elliptic differential opera-
+tors of first order, Surveys in Differential Geometry vol. 17, pp. 1–78, Intern. Press,
+Somerville, 2012
+[2] C. B¨ar and W. Ballmann, Guide to boundary value problems for Dirac-type operators,
+arXiv:1307.3021
+[3] C. B¨ar, B. Hanke, and T. Schick, Remarks on the paper ”On Gromov’s dihedral
+extremality and rigidity conjectures” by Jinmin Wang, Zhizhang Xie, and Guoliang
+Yu, arXiv:2202.05180
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+Math. France, Paris, 1982
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+arXiv:2207.13346
+[9] C. Li, A polyhedron comparison theorem for 3-manifolds with positive scalar curvature,
+Invent. Math. 219, 1–37 (2020)
+[10] J. Wang, Z. Xie, and G. Yu, On Gromov’s dihedral extremality and rigidity conjec-
+tures, arXiv:2112.01510
+Columbia University, 2990 Broadway, New York NY 10027, USA
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf,len=838
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='05087v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='DG]  12 Jan 2023  SCALAR CURVATURE RIGIDITY OF CONVEX  POLYTOPES  SIMON BRENDLE  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We prove a scalar curvature rigidity theorem for convex  polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The proof uses the Fredholm theory for Dirac operators on  manifolds with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  A variant of a theorem of Fefferman and  Phong plays a central role in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Introduction  Let Ω be a compact polytope in Rn with non-empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We may  write Ω = �  α∈A{uα ≤ 0}, where A is a finite set and the uα are linear  functions in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each α ∈ A, we denote by Nα ∈ Sn−1 the outward-  pointing unit normal vector to the halfspace {uα ≤ 0} with respect to the  Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Let g be a Riemannian metric which is defined on an open set containing  Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each α ∈ A, we denote by να the outward-pointing unit normal  vector to the halfspace {uα ≤ 0} with respect to the metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We will make  the following assumption:  Matching Angle Hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' If x is point in ∂Ω and α1, α2 ∈ A satisfy  uα1(x) = uα2(x) = 0, then ⟨να1, να2⟩ = ⟨Nα1, Nα2⟩ at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Here, the  inner product ⟨να1, να2⟩ is computed with respect to the metric g, and the  inner product ⟨Nα1, Nα2⟩ is the standard inner product in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Suppose that n ≥ 3 is an odd integer, and Ω is a compact  polytope in Rn with non-empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let g be a Riemannian metric which  is defined on an open set containing Ω and has nonnegative scalar curvature  at each point in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each α ∈ A, we assume that the mean curvature of  the hypersurface {uα = 0} with respect to g is nonnegative at each point in  Ω ∩ {uα = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Moreover, we assume that the Matching Angle Hypothesis is  satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then the Ricci tensor of g vanishes at each point in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='1 also holds in the even-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This can be seen  by considering the Cartesian product Ω × [0, 1] ⊂ Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Scalar curvature comparison theorems for polytopes were first studied in  seminal work of Gromov [6],[7],[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Li [9] has used minimal surface techniques  to prove a scalar curvature comparison theorem for certain polytopes in  The author was support by the National Science Foundation under grant DMS-2103573  and by the Simons Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The author acknowledges the hospitality of T¨ubingen  University, where part of this work was carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  1  2  SIMON BRENDLE  dimension 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Wang, Xie, and Yu [10] have proposed a different approach to  this problem which is based on the study of Dirac operators on manifolds  with corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  In this paper, we describe another approach to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' As in [10], we  employ a spinor approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' In contrast to [10], we work with boundary value  problems for Dirac operators on smooth domains, which are well understood  thanks to the work of B¨ar and Ballmann [1],[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  In the following, we outline the main steps involved in the proof of The-  orem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We approximate a given convex polytope Ω by a one-parameter  family of smooth convex domains Ωλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' On each domain Ωλ, we solve the  Dirac equation for an m-tuple of spinors s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , sm) with a suitable  local boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' To prove the existence of a solution satisfying  that particular boundary condition, we use the Fredholm theory developed  by B¨ar and Ballmann [1],[2] together with the homotopy invariance of the  Fredholm index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Having constructed an m-tuple of harmonic spinors on Ωλ  satisfying this boundary condition, we apply a Weitzenb¨ock formula, and  integrate over Ωλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The resulting integral formula contains a term involving  the scalar curvature, as well as a boundary term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Unfortunately, it is not  clear if the boundary term has a favorable sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We are able to control the  boundary integral by adapting a theorem due to Fefferman and Phong [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' A boundary value problem for the Dirac operator on a  smooth domain  Let m = 2[ n  2 ] denote the dimension of the space of spinors on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let  {E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , En} denote the standard basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Throughout this section, we  fix an orthonormal basis {¯s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , ¯sm} of the space of spinors on flat Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  We define ωaαβ = ⟨Ea · ¯sα, ¯sβ⟩ for a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , n and α, β = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The  matrices ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , ωn ∈ End(Cm) are skew-Hermitian, so that ωaαβ = −ωaβα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Moreover, ωaωb + ωbωa = −2δab id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' In other words,  m  �  β=1  (ωaαβ ωbβγ + ωbαβ ωaβγ) = −2δab δαγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  We begin by stating a basic algebraic fact which will be needed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Assume that n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then there is no non-zero element of  End(Cm) which anti-commutes with ωa ∈ End(Cm) for each a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We recall the definition of the spin representation in odd dimen-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let {E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , En} denote the standard basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , [n  2 ],  we define wk = E2k−1 − iE2k ∈ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The spinor space is defined as the ex-  terior algebra Λ∗W, where W = span{wk : k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , [n  2 ]} ⊂ Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each  k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , [n  2 ]}, we define a linear map Pk ∈ End(Λ∗W) by  Pk(wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' ∧ wjr) = wk ∧ wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' ∧ wjr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES  3  Moreover, for each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , [n  2 ]}, we define a linear map Qk ∈ End(Λ∗W)  by  Qk(wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' ∧ wjr) = 0,  Qk(wk ∧ wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' ∧ wjr) = wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' ∧ wjr  for k /∈ {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , jr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then PkPl + PlPk = QkQl + QlQk = 0 and PkQl +  QlPk = δkl id for k, l ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , [n  2 ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Finally, we define a linear map S ∈  End(Λ∗W) so that  S(wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' ∧ wjr) =  �  wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' ∧ wjr  if r is even  −wj1 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' ∧ wjr  if r is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Clearly, PkS + SPk = 0, QkS + SQk = 0, and S2 = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Consequently,  there is a natural algebra homomorphism from the Clifford algebra ClC(n)  to End(Λ∗W) which maps wk to i  √  2 Pk, ¯wk to i  √  2 Qk, and En to iS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' It is  well known (see [5], Lemma 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='9) that  span{Pk1 · · · PkrQl1 · · · Qls : r + s is even}  = End(ΛevenW) ⊕ End(ΛoddW)  and  span{Pk1 · · · PkrQl1 · · · Qls : r + s is odd}  = Hom(ΛevenW, ΛoddW) ⊕ Hom(ΛoddW, ΛevenW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  We claim that there is no non-zero element of End(Λ∗W) which anti-commutes  with Pk, Qk, S for each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , [n  2 ]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Suppose that L ∈ End(Λ∗W)  is such an element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Since L anti-commutes with S, it follows that L ∈  Hom(ΛevenW, ΛoddW)⊕Hom(ΛoddW, ΛevenW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Since L anti-commutes with  Pk, Qk for each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , [n  2 ]}, it follows that L anti-commutes with every  element of Hom(ΛevenW, ΛoddW)⊕Hom(ΛoddW, ΛevenW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This implies that  L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This completes the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Assume that Ω is a domain in Rn with smooth boundary ∂Ω = Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let g  be a Riemannian metric on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We denote by ν the outward-pointing unit  normal vector field with respect to the metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let ∇ denote the spin  connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The Dirac operator is defined by  Ds =  n  �  i=1  ei · ∇eis,  where {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , en} is a local orthonormal frame on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The boundary Dirac  operator DΣ is given by  DΣs =  n−1  �  i=1  ν · ei · ∇eis + 1  2 H s  at each point on Σ, where {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , en−1} is a local orthonormal frame on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  In the remainder of this section, we consider the Dirac operator act-  ing on m-tuples of spinors with a suitable local boundary condition of  4  SIMON BRENDLE  Lopatinsky-Shapiro type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' To formulate the boundary condition, we assume  that N : Σ → Sn−1 is a given smooth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Consider an m-tuple of spinors s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , sm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' At each  point on Σ, we define  (χs)α = −  n  �  a=1  m  �  β=1  ⟨N, Ea⟩ ωaαβ ν · sβ  and  (Bs)α =  n−1  �  i=1  n  �  a=1  m  �  β=1  ⟨dN(ei), Ea⟩ ωaαβ ei · sβ,  where {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , en−1} is a local orthonormal frame on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The map χ is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Moreover, χ2 is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Suppose that s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , sm) and t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , tm) are two m-  tuples of spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We compute  (χ2s)α =  n  �  a,b=1  m  �  β,γ=1  ⟨N, Ea⟩ ⟨N, Eb⟩ ωaαβ ωbβγ ν · ν · sγ  = −  n  �  a,b=1  m  �  β,γ=1  ⟨N, Ea⟩ ⟨N, Eb⟩ ωaαβ ωbβγ sγ  = −1  2  n  �  a,b=1  m  �  β,γ=1  ⟨N, Ea⟩ ⟨N, Eb⟩ (ωaαβ ωbβγ + ωbαβ ωaβγ) sγ  =  n  �  a,b=1  m  �  γ=1  ⟨N, Ea⟩ ⟨N, Eb⟩ δab δαγ sγ  = sα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Moreover,  m  �  α=1  ⟨(χs)α, tα⟩ = −  n  �  a=1  m  �  α,β=1  ⟨N, Ea⟩ ωaαβ ⟨ν · sβ, tα⟩  = −  n  �  a=1  m  �  α,β=1  ⟨N, Ea⟩ ωaβα ⟨sβ, ν · tα⟩  =  m  �  β=1  ⟨sβ, (χt)β⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  This completes the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES  5  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Assume that x ∈ Σ and ξ ∈ TxΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then the map (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , sm) �→  (ν · ξ · s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , ν · ξ · sm) maps the eigenspace of χ with eigenvalue 1 to the  eigenspace of χ with eigenvalue −1, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' In particular, the two  eigenspaces have the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each vector ξ ∈ TxΣ, the map  (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , sm) �→ (ν · ξ · s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , ν · ξ · sm)  anti-commutes with χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' From this, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The map B is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Moreover, χ and B commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , en−1} be a local orthonormal frame on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then  m  �  α=1  ⟨(Bs)α, tα⟩ =  n−1  �  i=1  n  �  a=1  m  �  α,β=1  ⟨dN(ei), Ea⟩ ωaαβ ⟨ei · sβ, tα⟩  =  n−1  �  i=1  n  �  a=1  m  �  α,β=1  ⟨dN(ei), Ea⟩ ωaβα ⟨sβ, ei · tα⟩  =  m  �  β=1  ⟨sβ, (Bt)β⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  This shows that B is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Moreover,  (χBs)α − (Bχs)α  = −  n−1  �  i=1  n  �  a,b=1  n  �  β,γ=1  ⟨N, Ea⟩ ⟨dN(ei), Eb⟩ ωaαβ ωbβγ ν · ei · sγ  +  n−1  �  i=1  n  �  a,b=1  n  �  β,γ=1  ⟨dN(ei), Ea⟩ ⟨N, Eb⟩ ωaαβ ωbβγ ei · ν · sγ  = −  n−1  �  i=1  n  �  a,b=1  n  �  β,γ=1  ⟨N, Ea⟩ ⟨dN(ei), Eb⟩ (ωaαβ ωbβγ + ωbαβ ωaβγ) ν · ei · sγ  = 2  n−1  �  i=1  n  �  a,b=1  n  �  γ=1  ⟨N, Ea⟩ ⟨dN(ei), Eb⟩ δab δαγ ν · ei · sγ  = 2  n−1  �  i=1  ⟨N, dN(ei)⟩ ν · ei · sα  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Thus, χ and B commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This completes the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  At this point, we recall a definition from linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  6  SIMON BRENDLE  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let V and W be finite-dimensional vector spaces of the  same dime, each of them equipped with an inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The trace norm  of a linear map L : V → W is defined by ∥L∥tr = supQ tr(QL), where the  supremum is taken over all linear isometries Q : W → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Equivalently,  ∥L∥tr can be characterized as the sum of the singular values of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  It is easy to see from the definition that the trace norm satisfies the tri-  angle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Suppose that s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , sm) is an m-tuple of spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then  ����  m  �  α=1  ⟨(Bs)α, sα⟩  ���� ≤ ∥dN∥tr  � m  �  α=1  |sα|2  �  at each point x ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Here, ∥dN∥tr denotes the trace norm of the differential  dN : TxΣ → TN(x)Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The tangent space TxΣ is equipped with the restric-  tion of the inner product g, and the tangent space TN(x)Sn−1 is equipped  with the restriction of the standard inner product on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Fix a point x ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We can find an orthonormal basis {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , en−1}  of TxΣ so that dN(ei) = λi ˆEi, where { ˆE1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , ˆEn−1} is an orthonormal ba-  sis of TN(x)Sn−1 and λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , λn−1 ≥ 0 denote the singular values of dN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Then  m  �  α=1  ����  n  �  a=1  m  �  β=1  ⟨ ˆEi, Ea⟩ ωaαβ ei · sβ  ����  2  =  n  �  a,b=1  m  �  α,β,γ=1  ⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ ωaαβ ωbαγ ⟨ei · sβ, ei · sγ⟩  = −  n  �  a,b=1  m  �  α,β,γ=1  ⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ ωaαβ ωbγα ⟨sβ, sγ⟩  = −1  2  n  �  a,b=1  m  �  α,β,γ=1  ⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ (ωaγα ωbαβ + ωbγα ωaαβ) ⟨sβ, sγ⟩  =  n  �  a,b=1  m  �  β,γ=1  ⟨ ˆEi, Ea⟩ ⟨ ˆEi, Eb⟩ δab δγβ ⟨sβ, sγ⟩  =  m  �  α=1  |sα|2  SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES  7  for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Using the Cauchy-Schwarz inequality, we obtain  ����  n  �  a=1  m  �  α,β=1  ⟨ ˆEi, Ea⟩ ωaαβ ⟨ei · sβ, sα⟩  ����  ≤  � m  �  α=1  ����  n  �  a=1  m  �  β=1  ⟨ ˆEi, Ea⟩ ωaαβ ei · sβ  ����  2� 1  2 � m  �  α=1  |sα|2  � 1  2  =  m  �  α=1  |sα|2  for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Summation over i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , n − 1 gives  ����  m  �  α=1  ⟨(Bs)α, sα⟩  ���� =  ����  n−1  �  i=1  n  �  a=1  m  �  α,β=1  ⟨dN(ei), Ea⟩ ωaαβ ⟨ei · sβ, sα⟩  ����  =  ����  n−1  �  i=1  λi  �  n  �  a=1  m  �  α,β=1  ⟨ ˆEi, Ea⟩ ωaαβ ⟨ei · sβ, sα⟩  �����  ≤  � n−1  �  i=1  λi  � � m  �  α=1  |sα|2  �  ,  as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Suppose that s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , sm) and t = (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , tm) are  m-tuples of spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then  0 =  �  Σ  m  �  α=1  ⟨DΣsα, (χt)α⟩ dσg +  �  Σ  m  �  α=1  ⟨(χs)α, DΣtα⟩ dσg  +  �  Σ  m  �  α=1  ⟨(Bs)α, tα⟩ dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Equivalently,  0 =  �  Σ  m  �  α=1  ⟨(As)α, (χt)α⟩ dσg +  �  Σ  m  �  α=1  ⟨(χs)α, (At)α⟩ dσg,  where A is defined by A = DΣ + 1  2χB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , en−1} be a local orthonormal frame on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We define  a tangential vector field Z on Σ by  ⟨Z, ei⟩ =  n  �  a=1  m  �  α,β=1  ⟨N, Ea⟩ ωaαβ ⟨ei · sβ, tα⟩  8  SIMON BRENDLE  for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then  divΣZ =  n−1  �  i=1  n  �  a=1  m  �  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='β=1  ⟨N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Ea⟩ ωaαβ ⟨ei · ∇eisβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' tα⟩  +  n−1  �  i=1  n  �  a=1  m  �  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='β=1  ⟨N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Ea⟩ ωaαβ ⟨ei · sβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' ∇eitα⟩  −  n  �  a=1  m  �  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='β=1  H ⟨N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Ea⟩ ωaαβ ⟨ν · sβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' tα⟩  +  n−1  �  i=1  n  �  a=1  m  �  α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='β=1  ⟨dN(ei),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Ea⟩ ωaαβ ⟨ei · sβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' tα⟩  = −  n−1  �  i=1  m  �  β=1  ⟨ei · ∇eisβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' ν · (χt)β⟩  +  n−1  �  i=1  m  �  α=1  ⟨ei · ν · (χs)α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' ∇eitα⟩  +  m  �  α=1  H ⟨(χs)α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' tα⟩ +  m  �  α=1  ⟨(Bs)α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' tα⟩  =  m  �  β=1  ⟨DΣsβ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' (χt)β⟩ +  m  �  α=1  ⟨(χs)α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' DΣtα⟩ +  m  �  α=1  ⟨(Bs)α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' tα⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Integrating over Σ, we obtain  0 =  �  Σ  m  �  β=1  ⟨DΣsβ, (χt)β⟩ dσg +  �  Σ  m  �  α=1  ⟨(χs)α, DΣtα⟩ dσg  +  �  Σ  m  �  α=1  ⟨(Bs)α, tα⟩ dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  This completes the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' It is well known that the boundary Dirac operator DΣ is  formally self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Moreover, it follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='3 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='5  that χB is self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Consequently, the operator A = DΣ+ 1  2χB is formally  self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Finally, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='8 implies that A and χ anti-commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES  9  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Suppose that s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , sm) is an m-tuple of spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Then  −  �  Ω  m  �  α=1  |Dsα|2 dvolg +  �  Ω  n  �  α=1  |∇sα|2 dvolg + 1  4  �  Ω  m  �  α=1  R |sα|2 dvolg  ≤ 1  2  �  Σ  m  �  α=1  ⟨DΣsα, sα − (χs)α⟩ dσg + 1  2  �  Σ  m  �  α=1  ⟨sα − (χs)α, DΣsα⟩ dσg  − 1  2  �  Σ  (H − ∥dN∥tr)  � m  �  α=1  |sα|2  �  dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' By the Weitzenb¨ock formula, D2sα = −∆sα + 1  4 R sα, where ∆  denotes the connection Laplacian on the spinor bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Using the divergence  theorem, we obtain  −  �  Ω  m  �  α=1  |Dsα|2 dvolg +  �  Ω  m  �  α=1  |∇sα|2 dvolg + 1  4  �  Ω  m  �  α=1  R |sα|2 dvolg  =  �  Σ  m  �  α=1  ⟨ν · Dsα, sα⟩ dσg +  �  Σ  m  �  α=1  ⟨∇νsα, sα⟩ dσg  =  �  Σ  ⟨DΣsα, sα⟩ dσg − 1  2  �  Σ  m  �  α=1  H |sα|2 dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Applying Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='8 with s = t gives  0 =  �  Σ  m  �  α=1  ⟨DΣsα, (χs)α⟩ dσg +  �  Σ  m  �  α=1  ⟨(χs)α, DΣsα⟩ dσg  +  �  Σ  m  �  α=1  ⟨(Bs)α, sα⟩ dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  This gives  −  �  Ω  m  �  α=1  |Dsα|2 dvolg +  �  Ω  n  �  α=1  |∇sα|2 dvolg + 1  4  �  Ω  m  �  α=1  R |sα|2 dvolg  = 1  2  �  Σ  m  �  α=1  ⟨DΣsα, sα⟩ dσg + 1  2  �  Σ  m  �  α=1  ⟨sα, DΣsα⟩ dσg − 1  2  �  Σ  m  �  α=1  H |sα|2 dσg  = 1  2  �  Σ  m  �  α=1  ⟨DΣsα, sα − (χs)α⟩ dσg + 1  2  �  Σ  m  �  α=1  ⟨sα − (χs)α, DΣsα⟩ dσg  − 1  2  �  Σ  m  �  α=1  ⟨(Bs)α, sα⟩ dσg − 1  2  �  Σ  m  �  α=1  H |sα|2 dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Hence, the assertion follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  10  SIMON BRENDLE  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Suppose that R ≥ 0 at each point in Ω and H ≥ ∥dN∥tr at  each point on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then every m-tuple of harmonic spinors s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , sm)  with χs = s is parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Replacing N by −N, we can draw the following conclusion:  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Suppose that R ≥ 0 at each point in Ω and H ≥ ∥dN∥tr at  each point on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then every m-tuple of harmonic spinors s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , sm)  with χs = −s is parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Suppose that Ω is a convex domain in Rn with smooth  boundary ∂Ω = Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Let g be a Riemannian metric on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Suppose that  N : Σ → Sn−1 is a smooth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then the boundary condition χs = s is a  D-elliptic boundary condition in the sense of B¨ar and Ballmann [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  We apply Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='18 in [2] with E′ = ker(id − χ) and  E′′ = ker(id + χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='4 implies that, for each point x ∈ Σ and  each ξ ∈ TxΣ, the map (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , sm) �→ (ν · ξ · s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , ν · ξ · sm) interchanges  ker(id − χ) and ker(id + χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Therefore, the boundary condition χs = s is a  D-elliptic boundary condition in the sense of [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Assume that n ≥ 3 is an odd integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Suppose that  Ω is a convex domain in Rn with smooth boundary ∂Ω = Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let g be a  Riemannian metric on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Suppose that N : Σ → Sn−1 is homotopic to  the Gauss map of Σ with respect to the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then the Dirac  operator with the boundary condition χs = s has Fredholm index at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Since the Fredholm index is homotopy invariant, it suffices to  prove the assertion in the special case when g is the Euclidean metric and  N is the Gauss map of Σ with respect to the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  We first analyze the kernel of the Dirac operator with the boundary con-  dition χs = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Recall that ¯s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , ¯sm is a basis of spinors on flat Rn, and  ωaαβ = ⟨Ea · ¯sα, ¯sβ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Clearly, ¯s = (¯s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , ¯sm) is an m-tuple of harmonic  spinors on Ω which satisfies the boundary condition χ¯s = ¯s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Therefore, the  kernel has dimension at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  We next examine the cokernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  The cokernel can be identified with  the space of all m-tuples of harmonic spinors s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , sm) such that  ⟨ν · s, t⟩ = 0 for all points x ∈ Σ and all t ∈ ker(id − χ) (see [2], Example  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We claim that this space has dimension 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' To see this, suppose that  s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , sm) is an m-tuple of harmonic spinors such that ⟨ν · s, t⟩ = 0  for all points x ∈ Σ and all t ∈ ker(id − χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This implies s ∈ ker(id + χ) at  each point on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Since H = ∥dN∥tr at each point on Σ, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='12 im-  plies that s = (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , sm) is parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' In other words, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , sm are constant  spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let us write sα = �m  β=1 zαβ ¯sβ for some matrix z ∈ End(Cm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Since  χs = −s at each point on Σ, it follows that the matrix z ∈ End(Cm) anti-  commutes with the matrix �n  a=1⟨N(x), Ea⟩ ωa ∈ End(Cm) for each point  x ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' It is easy to see that the Gauss map N : Σ → Sn−1 is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES  11  Consequently, the matrix z ∈ End(Cm) anti-commutes with ωa ∈ End(Cm)  for each a = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Since n is odd, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='1 implies that z = 0, hence  s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This shows that the cokernel has dimension 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This completes the  proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Approximating a compact, convex polytope by smooth  domains  Let us consider a compact, convex polytope Ω ⊂ Rn with non-empty  interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We write Ω = �  α∈A{uα ≤ 0}, where A is a finite set and the uα  are linear functions in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' After eliminating redundant inequalities, we may  assume that the following condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each α ∈ A, the set Ω ∩ {uα > 0} is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Let g be a Riemannian metric which is defined on an open set containing  Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each α ∈ A, ∇uα will denote the gradient of uα with respect to the  metric g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' |∇uα| will denote the norm of the gradient of uα with respect to  the metric g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' and να =  ∇uα  |∇uα| will denote the outward-pointing unit normal  vector to the halfspace {uα ≤ 0} with respect to the metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each  α ∈ A, we denote by Nα ∈ Sn−1 the outward-pointing unit normal vector  to the halfspace {uα ≤ 0} with respect to the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  For each λ > 0, the function �  α∈A eλuα is convex with respect to the  Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Clearly, �  α∈A eλuα > 1 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Moreover, we can find  large number λ0 such that infΩ  �  α∈A eλuα < 1 for each λ > λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each  λ > λ0, we define  Ωλ =  � �  α∈A  eλuα ≤ 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  For each λ > λ0, Ωλ is a convex domain in Rn with smooth boundary  Σλ = ∂Ωλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The sets Ωλ form an increasing family of sets in the sense that  Ωλ ⊂ Ωµ for λ0 < λ < µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Moreover,  �  λ>λ0  Ωλ =  �  α∈A  {uα < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' If λ is sufficiently large, then infΣλ  �� �  α∈A eλuα duα  �� ≥ C−1  for some large constant C which is independent of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Suppose that the assertion is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Then there exists a sequence of positive real numbers λl → ∞ and a se-  quence of points xl ∈ Σλl such that  �� �  α∈A eλuα duα  �� ≤ l−1 at the point  xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' After passing to a subsequence, we may assume that the sequence xl  converges to a point x0 ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Moreover, we may assume that, for each  α ∈ A, the sequence eλluα(xl) converges to a nonnegative real number zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Since �  α∈A eλluα(xl) = 1 for each l, we know that �  α∈A zα > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Let  A0 := {α ∈ A : zα > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Clearly, A0 is non-empty, and uα(x0) = 0 for all  12  SIMON BRENDLE  α ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Moreover, �  α∈A0 zα duα = 0 at the point x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' On the other hand,  since Ω is a convex set with non-empty interior, we can find a tangent vector  ξ ∈ Tx0Ω such that duα(ξ) > 0 for all α ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This  completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' If λ is sufficiently large, then infΣλ  �� �  α∈A eλuα |∇uα| Nα  �� ≥  C−1 for some large constant C which is independent of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Suppose that the assertion is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Then there exists a sequence of positive real numbers λl → ∞ and a se-  quence of points xl ∈ Σλl such that  �� �  α∈A eλuα |∇uα| Nα  �� ≤ l−1 at the  point xl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' After passing to a subsequence, we may assume that the sequence  xl converges to a point x0 ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Moreover, we may assume that, for each  α ∈ A, the sequence eλluα(xl) |∇uα(xl)| converges to a nonnegative real num-  ber zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Since �  α∈A eλluα(xl) = 1 for each l, we know that �  α∈A zα > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Let A0 := {α ∈ A : zα > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Clearly, A0 is non-empty, and uα(x0) = 0  for all α ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Moreover, �  α∈A0 zαNα = 0 at the point x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' On the other  hand, since Ω is a convex set with non-empty interior, we can find a vector  ξ ∈ Rn such that ⟨Nα, ξ⟩ > 0 for all α ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This  completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  The outward-pointing unit normal vector to the domain Ωλ with respect  to the metric g is given by  ν =  �  α∈A eλuα ∇uα  �� �  α∈A eλuα ∇uα  �� =  �  α∈A eλuα |∇uα| να  �� �  α∈A eλuα |∇uα| να  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  We define a map N : Σλ → Sn−1 by  N =  �  α∈A eλuα |∇uα| Nα  �� �  α∈A eλuα |∇uα| Nα  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The map N : Σλ → Sn−1 is homotopic to the Gauss map of  Σλ with respect to the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' In the special case when g is the Euclidean metric, the map N  coincides with the Gauss map of Σλ, and the assertion is trivially true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' To  prove the assertion in general, we deform the metric g to the Euclidean met-  ric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let x ∈ Σλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let π : TxΩ → TxΩ denotes the orthogonal  projection to the orthogonal complement of ν and P : Rn → Rn denotes  the orthogonal projection to the orthogonal complement of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then H −  SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES  13  ∥dN∥tr ≥ Vλ, where the function Vλ : Σλ → R is defined by  Vλ = λ  �  α∈A eλuα |∇uα|2 |π(να)|2  �� �  α∈A eλuα |∇uα| να  ��  − λ  �  α∈A eλuα |∇uα|2 |π(να)| |P(Nα)|  �� �  α∈A eλuα |∇uα| Nα  ��  +  �  α∈A eλuα (∆uα − (D2uα)(ν, ν))  �� �  α∈A eλuα |∇uα| να  ��  −  �  α∈A eλuα |∇(|∇uα|)| |P(Nα)|  �� �  α∈A eλuα |∇uα| Nα  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , en−1} denote a local orthonormal frame on Σλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The  mean curvature of Σλ is given by  H = λ  �n−1  i=1  �  α∈A eλuα ⟨∇uα, ei⟩2  �� �  α∈A eλuα ∇uα  ��  +  �n−1  i=1  �  α∈A eλuα (D2uα)(ei, ei)  �� �  α∈A eλuα ∇uα  ��  = λ  �  α∈A eλuα |π(∇uα)|2  �� �  α∈A eλuα ∇uα  ��  +  �  α∈A eλuα (∆uα − (D2uα)(ν, ν))  �� �  α∈A eλuα ∇uα  ��  = λ  �  α∈A eλuα |∇uα|2 |π(να)|2  �� �  α∈A eλuα |∇uα| να  ��  +  �  α∈A eλuα (∆uα − (D2uα)(ν, ν))  �� �  α∈A eλuα |∇uα| να  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  If ξ is a tangent vector to Σλ, then  dN(ξ)  = λ  �  α∈A eλuα |∇uα| ⟨∇uα, ξ⟩ P(Nα)  �� �  α∈A eλuα |∇uα| Nα  ��  +  �  α∈A eλuα ⟨∇(|∇uα|), ξ⟩ P(Nα)  �� �  α∈A eλuα |∇uα| Nα  ��  = λ  �  α∈A eλuα |∇uα|2 ⟨π(να), ξ⟩ P(Nα)  �� �  α∈A eλuα |∇uα| Nα  ��  +  �  α∈A eλuα ⟨∇(|∇uα|), ξ⟩ P(Nα)  �� �  α∈A eλuα |∇uα| Nα  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  The trace norm of a linear transformation of the form ξ �→ ⟨X, ξ⟩ Y is given  by |X| |Y |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Since the trace norm satisfies the triangle inequality, it follows  that  ∥dN∥tr  ≤ λ  �  α∈A eλuα |∇uα|2 |π(να)| |P(Nα)|  �� �  α∈A eλuα |∇uα| Nα  ��  +  �  α∈A eλuα |∇(|∇uα|)| |P(Nα)|  �� �  α∈A eλuα |∇uα| Nα  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Putting these facts together, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  In the following, we denote by Vλ,− = max{−Vλ, 0} the negative part of  Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Suppose that the Matching Angle Hypothesis is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Then supΣλ Vλ,− ≤ o(λ) as λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Suppose that the assertion is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Then there exists a sequence of positive real numbers λl → ∞ and a se-  quence of points xl ∈ Σλl such that lim supl→∞ λ−1  l  Vλl(xl) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  After  passing to a subsequence, we may assume that the sequence xl converges  to a point x0 ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Moreover, we may assume that, for each α ∈ A,  the sequence eλluα(xl) |∇uα(xl)| converges to a nonnegative real number zα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  14  SIMON BRENDLE  Since �  α∈A eλluα(xl) = 1 for each l, we know that �  α∈A zα > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Let  A0 := {α ∈ A : zα > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Clearly, A0 is non-empty, and uα(x0) = 0 for  all α ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The Matching Angle Hypothesis implies that, at the point x0,  ⟨να1, να2⟩ = ⟨Nα1, Nα2⟩ for all α1, α2 ∈ A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let π : Tx0Ω → Tx0Ω denote  the orthogonal projection to the orthogonal complement of �  α∈A0 zανα,  and let P : Rn → Rn denote the orthogonal projection to the orthogonal  complement of �  α∈A0 zαNα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each β ∈ A0, we have  |π(νβ)|2 = 1 −  � �  α∈A0 zανα, νβ  �2  �� �  α∈A0 zανα  ��2  = 1 −  � �  α∈A0 zαNα, Nβ  �2  �� �  α∈A0 zαNα  ��2  = |P(Nβ)|2  at the point x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Therefore, for each β ∈ A0, we obtain  |π(νβ)|  �� �  α∈A0 zανα  �� =  |P(Nβ)|  �� �  α∈A0 zαNα  ��  at the point x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='5, we conclude that λ−1  l  Vλl(xl) → 0 as  l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  In the remainder of this section, we will estimate the Ls-norm Vλ,− on Σλ∩  Br(p), where s ∈ [1, 3  2) is a fixed exponent and Br(p) denotes a Euclidean  ball of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We begin by recalling a basic fact about the area of convex  hypersurfaces in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let Br(p) denote a Euclidean ball of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then the in-  tersection Σλ ∩ Br(p) has area at most Crn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This follows from the fact that the hypersurface Σλ = ∂Ωλ is  outward-minimizing with respect to the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Consider three pairwise distinct elements α1, α2, α3 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  We denote by G(α1,α2,α3)  λ  the set of all points x ∈ Σλ with the property that  uα1(x) ≥ uα2(x) ≥ uα3(x) and uα3(x) ≥ uα(x) for α ∈ A \\ {α1, α2, α3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Assume that the mean curvature of the hypersurface {uα = 0}  with respect to g is nonnegative at each point in Ω ∩ {uα = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let us fix an  exponent s ∈ [1, 3  2), and let Br(p) denote a Euclidean ball of radius r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  If λr is sufficiently large, then  �  rs+1−n  �  G(α1,α2,α3)  λ  ∩{uα2≤−λ− 7  8 r  1  8 }∩Br(p)  V s  λ,−  � 1  s  ≤ Cλr e−(λr)  1  8  for all pairwise distinct elements α1, α2, α3 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let us consider an arbitrary point x ∈ G(α1,α2,α3)  λ  with uα2(x) ≤  −λ− 7  8 r  1  8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  By definition of G(α1,α2,α3)  λ  , it follows that uα(x) ≤ −λ− 7  8r  1  8  for all α ∈ A \\ {α1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Using the identity �  α∈A eλuα(x) = 1, we obtain  uα1(x) ≥ −Cλ−1 e−(λr)  1  8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Moreover, |ν − να1| ≤ C e−(λr)  1  8 and |N − Nα1| ≤  SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES  15  C e−(λr)  1  8 at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' From this, we deduce that |π(να1)| ≤ C e−(λr)  1  8  and |P(Nα1)| ≤ C e−(λr)  1  8 at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Therefore,  Vλ ≥ ∆uα1 − (D2uα1)(να1, να1)  |∇uα1|  − Cλ e−(λr)  1  8  at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Since uα1(x) ≥ −Cλ−1 e−(λr)  1  8 and uα(x) ≤ −λ− 7  8r  1  8 for  all α ∈ A \\ {α1}, we can find a point y ∈ Ω such that uα1(y) = 0 and  d(x, y) ≤ Cλ−1 e−(λr)  1  8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' By assumption, the mean curvature of the hyper-  surface {uα1 = 0} at the point y is nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This implies  ∆uα1 − (D2uα1)(να1, να1)  |∇uα1|  ≥ 0  at the point y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Consequently,  ∆uα1 − (D2uα1)(να1, να1)  |∇uα1|  ≥ −C d(x, y)  at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Thus, we conclude that  Vλ(x) ≥ −Cλ e−(λr)  1  8  for each point x ∈ G(α1,α2,α3)  λ  ∩ {uα2 ≤ −λ− 7  8r  1  8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  On the other hand,  Σλ ∩ Br(p) has area at most Crn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Consequently,  �  rs+1−n  �  G(α1,α2,α3)  λ  ∩{uα2≤−λ− 7  8 r  1  8 }∩Br(p)  V s  λ,−  � 1  s  ≤ Cλr e−(λr)  1  8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Assume that the Matching Angle Hypothesis holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let us  fix an exponent s ∈ [1, 3  2), and let Br(p) denote a Euclidean ball of radius  r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' If λr is sufficiently large, then  �  rs+1−n  �  G(α1,α2,α3)  λ  ∩{uα2≥−λ− 7  8 r  1  8 }∩{uα3≤−λ− 3  4 r  1  4 }∩Br(p)  V s  λ,−  � 1  s  ≤ C (λr)  1  8 − 7  8s  for all pairwise distinct elements α1, α2, α3 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We distinguish two cases:  Case 1: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' By continuity, we  can find a real number δ such that Ω ∩ {uα1 ≥ −δ} ∩ {uα2 ≥ −δ} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' If λr  is sufficiently large, then λ− 7  8r  1  8 ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This implies  G(α1,α2,α3)  λ  ∩ {uα2 ≥ −λ− 7  8 r  1  8 }  ⊂ Σλ ∩ {uα1 ≥ −δ} ∩ {uα2 ≥ −δ} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Hence, the assertion is trivially true in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  16  SIMON BRENDLE  Case 2: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' It follows from  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='1 that the hypersurfaces {uα1 = 0} and {uα2 = 0} intersect  transversally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Let us consider an arbitrary point x ∈ G(α1,α2,α3)  λ  with uα2(x) ≥ −λ− 7  8r  1  8  and uα3(x) ≤ −λ− 3  4r  1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Clearly, uα1(x) ≥ −λ− 7  8r  1  8 by definition of G(α1,α2,α3)  λ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Moreover, uα(x) ≤ −λ− 3  4r  1  4 for all α ∈ A \\ {α1, α2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Consequently, we can  find a point y ∈ Ω such that uα1(y) = uα2(y) = 0 and d(x, y) ≤ Cλ− 7  8r  1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  The Matching Angle Hypothesis implies ⟨να1, να2⟩ = ⟨Nα1, Nα2⟩ at the point  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Consequently, |⟨να1, να2⟩ − ⟨Nα1, Nα2⟩| ≤ C d(x, y) at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' From  this, we deduce that  |π(να1)|  �� �  α∈A eλuα |∇uα| να  �� −  |P(Nα1)|  �� �  α∈A eλuα |∇uα| Nα  �� ≥ −C d(x, y) − C e−(λr)  1  4  and  |π(να2)|  �� �  α∈A eλuα |∇uα| να  �� −  |P(Nα2)|  �� �  α∈A eλuα |∇uα| Nα  �� ≥ −C d(x, y) − C e−(λr)  1  4  at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Thus, we conclude that  Vλ(x) ≥ −Cλ  1  8 r− 7  8  for each point x ∈ G(α1,α2,α3)  λ  ∩ {uα2 ≥ −λ− 7  8 r  1  8 } ∩ {uα3 ≤ −λ− 3  4r  1  4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' By  transversality, the set {0 ≥ uα1 ≥ −λ− 7  8r  1  8} ∩ {0 ≥ uα2 ≥ −λ− 7  8 r  1  8} ∩ Br(p)  can be covered by C (λr)  7(n−2)  8  Euclidean balls of radius λ− 7  8 r  1  8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  More-  over, the intersection of Σλ with each ball of radius λ− 7  8r  1  8 has area at  most C (λr)− 7(n−1)  8  rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This shows that Σλ ∩ {uα1 ≥ −λ− 7  8 r  1  8 } ∩ {uα2 ≥  −λ− 7  8 r  1  8 } ∩ Br(p) has area at most C (λr)− 7  8 rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Since  G(α1,α2,α3)  λ  ∩ {uα2 ≥ −λ− 7  8r  1  8} ∩ Br(p)  ⊂ Σλ ∩ {uα1 ≥ −λ− 7  8 r  1  8 } ∩ {uα2 ≥ −λ− 7  8r  1  8} ∩ Br(p),  it follows that  �  rs+1−n  �  G(α1,α2,α3)  λ  ∩{uα2≥−λ− 7  8 r  1  8 }∩{uα3≤−λ− 3  4 r  1  4 }∩Br(p)  V s  λ,−  � 1  s  ≤ C (λr)  1  8 − 7  8s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let us fix an exponent s ∈ [1, 3  2), and let Br(p) denote a  Euclidean ball of radius r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' If λr is sufficiently large, then  �  rs+1−n  �  G(α1,α2,α3)  λ  ∩{uα3≥−λ− 3  4 r  1  4 }∩Br(p)  V s  λ,−  � 1  s  ≤ C (λr)1− 3  2s  for all pairwise distinct elements α1, α2, α3 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES  17  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We distinguish two cases:  Case 1: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} ∩ {uα3 = 0} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' By  continuity, we can find a real number δ such that Ω ∩ {uα1 ≥ −δ} ∩ {uα2 ≥  −δ} ∩ {uα3 ≥ −δ} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' If λr is sufficiently large, then λ− 3  4r  1  4 ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This  implies  G(α1,α2,α3)  λ  ∩ {uα3 ≥ −λ− 3  4 r  1  4 }  ⊂ Σλ ∩ {uα1 ≥ −δ} ∩ {uα2 ≥ −δ} ∩ {uα3 ≥ −δ} = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Hence, the assertion is trivially true in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Case 2: Suppose that Ω ∩ {uα1 = 0} ∩ {uα2 = 0} ∩ {uα3 = 0} ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' It  follows from Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='1 that the hypersurfaces {uα1 = 0}, {uα2 = 0},  and {uα3 = 0} intersect transversally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Let us consider an arbitrary point x ∈ G(α1,α2,α3)  λ  with uα3(x) ≥ −λ− 3  4r  1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Clearly,  Vλ(x) ≥ −Cλ  for all points x ∈ G(α1,α2,α3)  λ  ∩ {uα3 ≤ −λ− 3  4 r  1  4 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' By transversality, the set  {0 ≥ uα1 ≥ −λ− 3  4r  1  4}∩{0 ≥ uα2 ≥ −λ− 3  4 r  1  4}∩{0 ≥ uα3 ≥ −λ− 3  4 r  1  4 }∩Br(p)  can be covered by C (λr)  3(n−3)  4  Euclidean balls of radius λ− 3  4 r  1  4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  More-  over, the intersection of Σλ with each ball of radius λ− 3  4r  1  4 has area at  most C (λr)− 3(n−1)  4  rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This shows that Σλ ∩ {uα1 ≥ −λ− 3  4 r  1  4 } ∩ {uα2 ≥  −λ− 3  4 r  1  4 }∩{uα3 ≥ −λ− 3  4 r  1  4 }∩Br(p) has area at most C (λr)− 3  2 rn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Since  G(α1,α2,α3)  λ  ∩ {uα3 ≥ −λ− 3  4 r  1  4 } ∩ Br(p)  ⊂ Σλ ∩ {uα1 ≥ −λ− 3  4 r  1  4 } ∩ {uα2 ≥ −λ− 3  4 r  1  4 } ∩ {uα3 ≥ −λ− 3  4r  1  4} ∩ Br(p),  it follows that  �  rs+1−n  �  G(α1,α2,α3)  λ  ∩{uα3≥−λ− 3  4 r  1  4 }∩Br(p)  V s  λ,−  � 1  s  ≤ C (λr)1− 3  2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  This completes the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Assume that the mean curvature of the hypersurface  {uα = 0} with respect to g is nonnegative at each point in Ω ∩ {uα = 0}  and that the Matching Angle Hypothesis is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let us fix an exponent  s ∈ [1, 3  2), and let Br(p) denote a Euclidean ball of radius r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' If λr is  sufficiently large, then  �  rs+1−n  �  Σλ∩Br(p)  V s  λ,−  � 1  s  ≤ C (λr)−1 + C (λr)  1  8 − 7  8s + C (λr)1− 3  2s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  18  SIMON BRENDLE  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Combining Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='9, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='10, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='11, we con-  clude that  �  rs+1−n  �  G(α1,α2,α3)  λ  ∩Br(p)  V s  λ,−  � 1  s  ≤ C (λr)−1 + C (λr)  1  8− 7  8s + C (λr)1− 3  2s  for all pairwise distinct elements α1, α2, α3 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' On the other hand, Σλ =  �  α1,α2,α3 G(α1,α2,α3)  λ  , where the union is taken over all pairwise distinct el-  ements α1, α2, α3 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Hence, the assertion follows by summation over  α1, α2, α3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This completes the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Assume that the mean curvature of the hypersurface {uα =  0} with respect to g is nonnegative at each point in Ω ∩ {uα = 0} and that  the Matching Angle Hypothesis is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let us fix an exponent s ∈ [1, 3  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Then  sup  p∈Rn sup  r≤1  �  rs+1−n  �  Σλ∩Br(p)  V s  λ,−  � 1  s  → 0  as λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let us consider an arbitrary sequence λl → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' By Proposition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='6, we can find a sequence of positive real numbers δl → 0 such that  sup  p∈Rn  sup  r≤(δlλl)−1  �  rs+1−n  �  Σλl∩Br(p)  V s  λl,−  � 1  s  → 0  as l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' On the other hand, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='12 implies that  sup  p∈Rn  sup  (δlλl)−1≤r≤1  �  rs+1−n  �  Σλl∩Br(p)  V s  λl,−  � 1  s  → 0  as l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Putting these facts together, the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Proof of the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='1  Throughout this section, we assume that n ≥ 3 is an odd integer, and Ω is  a compact polytope in Rn with non-empty interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let g be a Riemannian  metric which is defined on an open set containing Ω and has nonnegative  scalar curvature at each point in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We assume that the mean curvature of  the hypersurface {uα = 0} with respect to g is nonnegative at each point in  Ω ∩ {uα = 0} and that the Matching Angle Hypothesis is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Let U denote a Euclidean ball such that the closure of U is contained in  the interior of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Consider a sequence λl → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Note that U ⊂ Ωλl if l is  sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='14 we can find an m-tuple of harmonic  spinors s(l) = (s(l)  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , s(l)  m ) such that s(l) is defined on Ωλl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' s(l) does not  vanish identically;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Ds(l) = 0 in Ωλl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' and χs(l) = s(l) on Σλl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Standard  SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES  19  unique continuation arguments imply that  �  U  �m  α=1 |s(l)  α |2 dvolg > 0 if l is  sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' By scaling, we can arrange that  �  U  m  �  α=1  |s(l)  α |2 dvolg = 1  for each l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Using Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='10, we obtain  �  Ωλl  m  �  l=1  |∇s(l)  α |2 dvolg + 1  4  �  Ωλl  m  �  α=1  R |s(l)  α |2 dvolg  ≤ −1  2  �  Σλl  (H − ∥dN∥tr)  � m  �  α=1  |s(l)  α |2  �  dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='5 implies that H − ∥dN∥tr ≥ Vλl at each point on Σλl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Con-  sequently,  �  Ωλl  m  �  l=1  |∇s(l)  α |2 dvolg + 1  4  �  Ωλl  m  �  α=1  R |s(l)  α |2 dvolg  ≤ 1  2  �  Σλl  Vλl,−  � m  �  α=1  |s(l)  α |2  �  dσg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Note that the hypersurface Σλl = ∂Ωλl can be written as a radial graph  with bounded slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' From this, it is easy to see that Ωλl is bi-Lipschitz  equivalent to the Euclidean unit ball, with constants that are independent  of λl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Using Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='7 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='13, we obtain  �  Σλl  Vλl,− F 2 dσg ≤ εl  �  Ωλl  |∇F|2 dvolg + εl  � �  Σλl  F dσg  �2  for every smooth function F on Ωλl, where εl → 0 as l → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Moreover, the  Sobolev trace theorem implies  � �  Σλl  F dσg  �2  ≤ C  �  Ωl  |∇F|2 dvolg + C  �  U  F 2 dvolg  for every smooth function F on Ωλl, where C is a uniform constant inde-  pendent of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Putting these facts together, we conclude that  �  Σλl  Vλl,− F 2 dσg ≤ Cεl  �  Ωλl  |∇F|2 dvolg + Cεl  �  U  F 2 dvolg  20  SIMON BRENDLE  for every smooth function F on Ωλl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' In the next step, we apply this inequal-  ity with F =  �  δ2 + �m  α=1 |s(l)  α |2� 1  2 , and send δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This gives  �  Ωλl  m  �  l=1  |∇s(l)  α |2 dvolg + 1  4  �  Ωλl  m  �  α=1  R |s(l)  α |2 dvolg  ≤ 1  2  �  Σλl  Vλl,−  � m  �  α=1  |s(l)  α |2  �  dσg  ≤ Cεl  �  Ωλl  m  �  α=1  |∇s(l)  α |2 dvolg + Cεl  �  U  m  �  α=1  |s(l)  α |2 dvolg  for each l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Since the scalar curvature is nonnegative, it follows that  �  Ωλl  m  �  α=1  |∇s(l)  α |2 dvolg ≤ Cεl  �  U  m  �  α=1  |s(l)  α |2 dvolg  if l is sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Passing to the limit as l → ∞, we obtain a non-  vanishing m-tuple of parallel spinors defined on the interior of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Conse-  quently, the Ricci tensor of g vanishes at each point in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This completes  the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' A variant of a theorem of Fefferman and Phong  In this section, we describe a variant of an estimate due to Fefferman and  Phong [4], which plays a central role in our argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We denote by Q the  collection of all (n − 1)-dimensional cubes of the form  [2mj1, 2m(j1 + 1)] × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' × [2mjn−1, 2m(jn−1 + 1)] × {0},  where m ∈ Z and j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , jn−1 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each Q ∈ Q, we denote by |Q| the  (n − 1)-dimensional volume of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Fix an exponent s ∈ (1, n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let V be a nonnegative  continuous function on the hyperplane Rn−1 × {0} ⊂ Rn with the property  that  diam(Q)s+1−n  �  Q  V s ≤ 1  for each (n − 1)-dimensional cube Q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Suppose that F is a smooth  function on the half-space Rn  + = {x ∈ Rn : xn ≥ 0}, and let f denote the  restriction of F to the boundary ∂Rn  + = Rn−1 × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then  �  Q  V f 2 ≤ C  �  Q×[0,diam(Q)]  |∇F|2 + C diam(Q)−1 |Q|−1  � �  Q  |f|  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  for each (n − 1)-dimensional cube Q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES  21  The proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='1 involves a straightforward adaptation of the  arguments of Fefferman and Phong [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let us fix an exponent t such that  s > t > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We define a nonnegative function W : Rn−1 × {0} → R by  W(x) =  sup  Q∈Q,x∈Q  �  |Q|−1  �  Q  V s  � 1  s  for each point x ∈ Rn−1 × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' In other words, W s is the maximal function  associated with the function V s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Clearly, V ≤ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  We assume that F is a smooth function on the half-space Rn  + = {x ∈  Rn : xn ≥ 0}, and let f denote the restriction of F to the boundary ∂Rn  + =  Rn−1 × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  For each (n − 1)-dimensional cube Q ∈ Q, we denote by  fQ = |Q|−1 �  Q f the mean value of f over the cube Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each (n − 1)-dimensional cube Q0 ∈ Q, we have  �  |Q0|−1  �  Q0  W t  � 1  t  ≤ C  sup  Q∈Q,Q0⊂Q  �  |Q|−1  �  Q  V s  � 1  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For abbreviation, let  Λ =  sup  Q∈Q,Q0⊂Q  �  |Q|−1  �  Q  V s  � 1  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  We define a nonnegative function W0 : Q0 → R by  W0(x) =  sup  Q∈Q,x∈Q⊂Q0  �  |Q|−1  �  Q  V s  � 1  s  for each point x ∈ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Clearly,  W(x) = max{Λ, W0(x)}  for each point x ∈ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' The Hardy-Littlewood maximal inequality implies  |Q0|−1 |{x ∈ Q0 : W0(x)s > α}| ≤ Cα−1 |Q0|−1  �  Q0  V s ≤ Cα−1 Λs  for all α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  We multiply both sides by α  t  s−1 and integrate over α ∈  [Λs, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This gives  |Q0|−1  �  Q0  W t  0 ≤ C Λt,  hence  |Q0|−1  �  Q0  W t ≤ C Λt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  This completes the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  22  SIMON BRENDLE  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Given a real number ε > 0, we can find a real number δ > 0  with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' If Q0 is an (n − 1)-dimensional cube in Q and  A ⊂ Q0 is a Borel set with |A| ≤ δ |Q0|, then  �  A  W ≤ ε  �  Q0  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='2, we obtain  �  |Q0|−1  �  Q0  W t  � 1  t  ≤ C  sup  Q∈Q,Q0⊂Q  �  |Q|−1  �  Q  V s  � 1  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Moreover,  sup  Q∈Q,Q0⊂Q  �  |Q|−1  �  Q  V s  � 1  s  ≤ inf  Q0 W ≤ |Q0|−1  �  Q0  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Therefore,  �  |Q0|−1  �  Q0  W t  � 1  t  ≤ C |Q0|−1  �  Q0  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Hence, if A ⊂ Q0 is a Borel set with |A| ≤ δ Q0, then  �  A  W ≤ |A|  t−1  t  � �  Q0  W t  � 1  t  ≤ δ  t−1  t |Q0|  t−1  t  � �  Q0  W t  � 1  t  ≤ Cδ  t−1  t  �  Q0  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  This completes the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each (n − 1)-dimensional cube Q0 ∈ Q, we have  |Q0|−1  �  Q0  W ≤ C diam(Q0)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='2, we obtain  �  |Q0|−1  �  Q0  W t  � 1  t  ≤ C  sup  Q∈Q,Q0⊂Q  �  |Q|−1  �  Q  V s  � 1  s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Moreover, our assumption implies that  �  |Q|−1  �  Q  V s  � 1  s  ≤ C diam(Q)−1  for each (n − 1)-dimensional cube Q ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Putting these facts together, the  assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each (n − 1)-dimensional cube Q0 ∈ Q, we have  �  Q0  V |f − fQ0|2 ≤ C  �  Q0  Wg2,  SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES  23  where the function g : Q0 → R is defined by  g(x) =  sup  Q∈Q,x∈Q⊂Q0  |Q|−1  �  Q  |f − fQ|  for x ∈ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Fix an (n − 1)-dimensional cube Q0 ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We define a function  h : Q0 → R by  h(x) =  sup  Q∈Q,x∈Q⊂Q0  |Q|−1  �  Q  |f − fQ0|  for x ∈ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Note that V ≤ W and |f −fQ0| ≤ h at each point in Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Hence,  it suffices to prove that  �  Q0  Wh2 ≤ C  �  Q0  Wg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  To prove this inequality, let α0 = |Q0|−1 �  Q0 |f − fQ0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each α > α0,  we denote by Qα the set of all (n − 1)-dimensional cubes Q ∈ Q with the  following properties:  Q ⊂ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  |Q|−1 �  Q |f − fQ0| > α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  If ˜Q is an (n − 1)-dimensional cube in Q with Q ⊊ ˜Q and ˜Q ⊂ Q0,  then | ˜Q|−1 �  ˜Q |f − fQ0| ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  It is easy to see that  |Q|−1  �  Q  |f − fQ0| ≤ 2n−1α  for all Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' In particular, |fQ − fQ0| ≤ 2n−1α for all Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Moreover,  �  Q∈Qα  Q = {x ∈ Q0 : h(x) > α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Finally, no point can be contained in the interior of more than one cube in  Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Let K > 1 and δ ∈ (0, 1) be two real numbers that will be chosen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  For each (n−1)-dimensional cube Q ∈ Qα satisfying |Q|−1 �  Q |f −fQ| ≤ δα,  24  SIMON BRENDLE  we have  (Kα − |fQ − fQ0|)  �  ˜Q∈QKα, ˜Q⊂Q  | ˜Q|  ≤  �  ˜Q∈QKα, ˜Q⊂Q  � �  ˜Q  |f − fQ0| −  �  ˜Q  |fQ − fQ0|  �  ≤  �  ˜Q∈QKα, ˜Q⊂Q  �  ˜Q  |f − fQ|  ≤  �  Q  |f − fQ|  ≤ δα |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Recall that |fQ − fQ0| ≤ 2n−1α for all Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Hence, if we choose K > 2n,  then we obtain  �  ˜Q∈QKα, ˜Q⊂Q  | ˜Q| ≤ 21−n δ |Q|  for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 �  Q |f − fQ| ≤ δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  We next apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='3 with ε = 1  2 K−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Hence, we can choose δ ∈  (0, 1) sufficiently small (depending on K) so that  �  ˜Q∈QKα, ˜Q⊂Q  �  ˜Q  W ≤ 1  2 K−2  �  Q  W  for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 �  Q |f − fQ| ≤ δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  For each (n−1)-dimensional cube Q ∈ Qα, the set Q∩{h > Kα} is contained  in the union �  ˜Q∈QKα, ˜Q⊂Q ˜Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This implies  �  Q∩{h>Kα}  W ≤ 1  2 K−2  �  Q  W  for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 �  Q |f − fQ| ≤ δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  On the other hand, if Q is an (n − 1)-dimensional cube in Qα satisfying  |Q|−1 �  Q |f − fQ| > δα, then g > δα at each point in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Therefore,  �  Q∩{h>Kα}  W ≤  �  Q∩{g>δα}  W  for each (n − 1)-dimensional cube Q ∈ Qα satisfying |Q|−1 �  Q |f − fQ| > δα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Putting these facts together, we conclude that  �  Q∩{h>Kα}  W ≤ 1  2 K−2  �  Q  W +  �  Q∩{g>δα}  W  SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES  25  for each (n−1)-dimensional cube Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Summation over all cubes Q ∈ Qα  gives  �  {h>Kα}  W ≤ 1  2 K−2  �  {h>α}  W +  �  {g>δα}  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  This inequality holds for each α > α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Moreover, since g ≥ α0 at each point  in Q0, the inequality is trivially true for α ≤ α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Finally, we multiply the  inequality by α  2 and integrate over α ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This gives  K−2  �  Q0  Wh2 ≤ 1  2 K−2  �  Q0  Wh2 + δ−2  �  Q0  Wg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  This completes the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each (n − 1)-dimensional cube Q0 ∈ Q, we have  �  Q0  Wg2 ≤ C  �  Q0×[0,diam(Q)]  |∇F|2,  where the function g : Q0 → R is defined by  g(x) =  sup  Q∈Q,x∈Q⊂Q0  |Q|−1  �  Q  |f − fQ|  for x ∈ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Fix an (n−1)-dimensional cube Q0 ∈ Q, and let α0 = |Q0|−1 �  Q0 |f−  fQ0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each α > α0, we denote by Qα the set of all (n − 1)-dimensional  cubes Q ∈ Q with the following properties:  Q ⊂ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  |Q|−1 �  Q |f − fQ| > α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  If ˜Q is an (n − 1)-dimensional cube in Q with Q ⊊ ˜Q and ˜Q ⊂ Q0,  then | ˜Q|−1 �  ˜Q |f − f ˜Q| ≤ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  It is easy to see that  |Q|−1  �  Q  |f − fQ| ≤ 2nα  for all Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Moreover,  �  Q∈Qα  Q = {x ∈ Q0 : g(x) > α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Finally, no point can be contained in the interior of more than one cube in  Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  26  SIMON BRENDLE  Let K > 1 be a real number that will be chosen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each (n − 1)-  dimensional cube Q ∈ Qα, we have  Kα  �  ˜Q∈QKα, ˜Q⊂Q  | ˜Q| ≤  �  ˜Q∈QKα, ˜Q⊂Q  �  ˜Q  |f − f ˜Q|  ≤ 2  �  ˜Q∈QKα, ˜Q⊂Q  �  ˜Q  |f − fQ|  ≤ 2  �  Q  |f − fQ|  ≤ 2n+1α |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Hence, if we choose K > 2n+2, then  �  ˜Q∈QKα, ˜Q⊂Q  | ˜Q| ≤ 1  2 |Q|  for each cube Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' For each (n − 1)-dimensional cube Q ∈ Qα, the set  Q ∩ {g > Kα} is contained in the union �  ˜Q∈QKα, ˜Q⊂Q ˜Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This implies  |Q ∩ {g > Kα}| ≤  �  ˜Q∈QKα, ˜Q⊂Q  | ˜Q| ≤ 1  2 |Q|,  hence  |Q ∩ {g ≤ Kα}| ≥ 1  2 |Q|  for each (n−1)-dimensional cube Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We define a nonnegative function  ϕ : Rn−1 × {0} → R by  ϕ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , xn−1, 0) =  � � diam(Q0)  0  |∇F(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' , xn−1, xn)|2 dxn  � 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Moreover, we define a nonnegative function ψ : Q0 → R by  ψ(x) =  sup  Q∈Q,x∈Q⊂Q0  |Q|−1  �  Q  ϕ  for each point x ∈ Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' In other words, ψ is the maximal function associated  with ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Using the Sobolev trace theorem, we obtain  α ≤ |Q|−1  �  Q  |f − fQ|  ≤ 2 |Q|−1 inf  a∈R  �  Q  |f − a|  ≤ C |Q|−1 inf  a∈R  � �  Q×[0,diam(Q)]  |∇(F − a)|  + diam(Q)−1  �  Q×[0,diam(Q)]  |F − a|  �  SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES  27  for each (n − 1)-dimensional cube Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Using the Poincar´e inequality,  we conclude that  α ≤ C |Q|−1  �  Q×[0,diam(Q)]  |∇F|  ≤ C diam(Q)  1  2 |Q|−1  �  Q  ϕ  ≤ C diam(Q)  1  2 inf  Q ψ  for each (n − 1)-dimensional cube Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This implies  α2 diam(Q)−1 |Q| ≤ C  �  Q∩{g≤Kα}  ψ2  for each (n − 1)-dimensional cube Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Combining this estimate with  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='4, we obtain  α2  �  Q  W ≤ C  �  Q∩{g≤Kα}  ψ2  for each (n−1)-dimensional cube Q ∈ Qα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Summation over all cubes Q ∈ Qα  gives  α2  �  {g>α}  W ≤  �  {α<g≤Kα}  ψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  This estimate holds for each α > α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' We now multiply both sides by α−1  and integrate over α ∈ (2α0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This gives  �  {g>4α0}  Wg2 ≤ C  �  Q0  ψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  In the next step, we bound the contribution from the set {g ≤ 4α0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Using  the Sobolev trace theorem, we obtain  α0 = |Q0|−1  �  Q0  |f − fQ0|  ≤ 2 |Q0|−1 inf  a∈R  �  Q0  |f − a|  ≤ C |Q0|−1 inf  a∈R  � �  Q0×[0,diam(Q0)]  |∇(F − a)|  + diam(Q0)−1  �  Q0×[0,diam(Q)]  |F − a|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  28  SIMON BRENDLE  Using the Poincar´e inequality, we conclude that  α0 ≤ C |Q0|−1  �  Q0×[0,diam(Q0)]  |∇F|  ≤ C diam(Q0)  1  2 |Q0|−1  �  Q0  ϕ  ≤ C diam(Q0)  1  2 inf  Q0 ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  This implies  α2  0 diam(Q0)−1 |Q0| ≤ C  �  Q0  ψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Combining this estimate with Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='4, we obtain  α2  0  �  Q0  W ≤ C  �  Q0  ψ2,  hence  �  {g≤4α0}  Wg2 ≤ C  �  Q0  ψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Putting these facts together, we conclude that  �  Q0  Wg2 ≤ C  �  Q0  ψ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  On the other hand, the Hardy-Littlewood maximal inequality implies  �  Q0  ψ2 ≤ C  �  Q0  ϕ2 = C  �  Q0×[0,diam(Q0)]  |∇F|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  This completes the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  After these preparations, we now complete the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Combining Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='5 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='6, we conclude that  �  Q0  V |f − fQ0|2 ≤ C  �  Q0×[0,diam(Q0)]  |∇F|2  for each (n − 1)-dimensional cube Q0 ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' This implies  �  Q0  V f 2 ≤ C  �  Q0×[0,diam(Q0)]  |∇F|2 + C |Q0|−2  � �  Q0  V  � � �  Q0  |f|  �2  for each (n − 1)-dimensional cube Q0 ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Using the estimate  |Q0|−1  �  Q0  V ≤  �  |Q0|−1  �  Q0  V s  � 1  s  ≤ C diam(Q0)−1,  we conclude that  �  Q0  V f 2 ≤ C  �  Q0×[0,diam(Q0)]  |∇F|2 + C diam(Q0)−1 |Q0|−1  � �  Q0  |f|  �2  SCALAR CURVATURE RIGIDITY OF CONVEX POLYTOPES  29  for each (n − 1)-dimensional cube Q0 ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Fix an exponent s ∈ (1, n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Let V be a nonnegative  continuous function on the unit sphere Sn−1 ⊂ Rn with the property that  rs+1−n  �  Sn−1∩Br(p)  V s ≤ 1  for all points p ∈ Rn and all r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Suppose that F is a smooth function on  the unit ball Bn = {x ∈ Rn : |x| ≤ 1}, and let f denote the restriction of F  to the boundary ∂Bn = Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Then  �  Sn−1 V f 2 ≤ C  �  Bn |∇F|2 + C  � �  Sn−1 |f|  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content='  References  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
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+page_content=' Li, A polyhedron comparison theorem for 3-manifolds with positive scalar curvature,  Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' 219, 1–37 (2020)  [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Wang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
+page_content=' Xie, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
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+page_content='01510  Columbia University, 2990 Broadway, New York NY 10027, USA' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/79E4T4oBgHgl3EQfdAwp/content/2301.05087v1.pdf'}
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+BARVINN: Arbitrary Precision DNN Accelerator Controlled by a
+RISC-V CPU
+Mohammadhossein Askarihemmat1, Sean Wagner2, Olexa Bilaniuk3,
+Yassine Hariri4, Yvon Savaria1, Jean-Pierre David1
+1Ecole Polytechnique Montreal, Canada, 2 IBM, Toronto, Canada, 3 Mila, Montreal, Canada,
+4 CMC Microsystems, Kingston, Canada
+{mohammad.hossein.askari-hemmat, yvon.savaria, jpdavid}@polymtl.ca, wagnerse@ca.ibm.com,
+olexa.bilaniuk@mila.quebec, hariri@cmc.ca
+ABSTRACT
+We present a DNN accelerator that allows inference at arbitrary
+precision with dedicated processing elements that are configurable
+at the bit level. Our DNN accelerator has 8 Processing Elements
+controlled by a RISC-V controller with a combined 8.2 TMACs of
+computational power when implemented with the recent Alveo
+U250 FPGA platform. We develop a code generator tool that ingests
+CNN models in ONNX format and generates an executable com-
+mand stream for the RISC-V controller. We demonstrate the scalable
+throughput of our accelerator by running different DNN kernels
+and models when different quantization levels are selected. Com-
+pared to other low precision accelerators, our accelerator provides
+run time programmability without hardware reconfiguration and
+can accelerate DNNs with multiple quantization levels, regardless
+of the target FPGA size. BARVINN is an open source project and it
+is available at https://github.com/hossein1387/BARVINN.
+KEYWORDS
+neural networks, hardware acceleration, FPGA, low-precision
+ACM Reference Format:
+Mohammadhossein Askarihemmat1, Sean Wagner2, Olexa Bilaniuk3,, Yas-
+sine Hariri4, Yvon Savaria1, Jean-Pierre David1. 2023. BARVINN: Arbi-
+trary Precision DNN Accelerator Controlled by a RISC-V CPU. In 28th
+Asia and South Pacific Design Automation Conference (ASPDAC ’23), Janu-
+ary 16–19, 2023, Tokyo, Japan. ACM, New York, NY, USA, 7 pages. https:
+//doi.org/10.1145/3566097.3567872
+1
+INTRODUCTION
+Deep neural networks (DNNs) traditionally rely on floating point
+computations. These operations are slow and costly in terms of
+power consumption and required silicon area compared to fixed-
+point/integer operations. One way to accelerate computation in
+a DNN is to use less precision for computation via quantization
+[12]. This also reduces memory consumption as well as energy
+consumption. For instance, in a 45 nm process, 8-bit integer multi-
+plication and addition take 0.2 pJ and 0.03 pJ, respectively, while the
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for components of this work owned by others than ACM
+must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
+to post on servers or to redistribute to lists, requires prior specific permission and/or a
+fee. Request permissions from permissions@acm.org.
+ASPDAC ’23, January 16–19, 2023, Tokyo, Japan
+© 2023 Association for Computing Machinery.
+ACM ISBN 978-1-4503-9783-4/23/01...$15.00
+https://doi.org/10.1145/3566097.3567872
+Table 1: Effects of Quantization on Accuracy and Model Size.
+Task
+Dataset
+Model
+Precision
+A/W
+Acc/
+MAP
+Size
+(MB)
+Classification
+CIFAR
+100
+ResNet18
+LSQ(2/2)
+76.81
+2.889
+LSQ(4/4)
+76.92
+5.559
+LSQ(8/8)
+78.45
+10.87
+FP32
+76.82
+42.8
+Object
+Detection
+VOC-
+2007
+SSD300-
+ResNet18
+LSQ(2/2)
+0.61
+10.34
+LSQ(4/4)
+0.60
+11.81
+LSQ(8/8)
+0.68
+14.77
+FP32
+0.59
+32.49
+same operations with 32-bit floating-point values requires 3.7 pJ
+for multiplication and 0.9 pJ for addition [11]. On an Intel Core i7
+4770 running at 3.4GHz, multiplication is more than 3 times faster
+for fixed-point compared to floating-point [15]. With recent quanti-
+zation techniques, these benefits are available with little to no loss
+in model performance and accuracy. In [9, 13], their quantization
+schemes showed accuracy losses of 1-3% at 2-bit precision on most
+classification and object detection models. Table 1 illustrates the
+result of applying Learned Scale Quantization (LSQ) [9] with differ-
+ent bit precisions on different models and tasks. Quantized models
+offer accuracy similar to full precision models, while having smaller
+size.
+Mixed-precision quantization [7, 16, 21, 23, 24] further provides
+finer control to reach an optimal solution by learning different
+precisions for each layer of a network. In [23], the authors illustrate
+that using their mixed-precision framework, they reduced model
+latency and energy consumption by a factor of almost 2× with little
+drop in accuracy compared with an 8-bit quantized model.
+Fully benefiting from low-precision in a DNN requires hardware
+that natively supports low-precision computations. Commodity
+hardware can perform arbitrary precision arithmetic by transform-
+ing data-layout and computing with bit-wise instructions [8]. How-
+ever, this approach is extremely costly for general processors, be-
+cause of the overhead for shifting, masking and packing bits to
+the correct format. At the time of writing this paper, there are no
+commercially available general processors (CPU or GPU) that can
+efficiently process data in arbitrary precision.
+In this paper, we propose an arbitrary low-precision DNN hard-
+ware accelerator called BARVINN. Our accelerator is software pro-
+grammable and can be integrated in the RISC-V standard devel-
+opment flow. It is designed as a highly optimized computational
+pipeline for DNNs that introduces low hardware overhead and of-
+fers low-power operation. The contributions of our paper are as
+follows:
+arXiv:2301.00290v1  [cs.AR]  31 Dec 2022
+
+ASPDAC ’23, January 16–19, 2023, Tokyo, Japan
+AskariHemmat et al
+• Implementation of a DNN hardware accelerator with arbi-
+trary fixed-point low-precision for matrix-vector multiply
+operations at high-throughput and low power.
+• Implementation of a custom embedded RISC-V CPU to con-
+trol an array of DNN accelerators by software.
+• Data structures for efficiently storing and processing weights
+and activations for high-throughput serial computation.
+• Development of a software code generator for transforming
+DNNs into RISC-V code that executes on our accelerator.
+In section 2, we review relevant DNN accelerators from the liter-
+ature. Section 3 presents the architecture of BARVINN. In section
+4, a detailed performance analysis of BARVINN is provided and
+compared with other DNN accelerators.
+2
+RELATED WORKS
+Several DNN hardware accelerators supporting quantization and
+low-precision have been presented in recent years for both FPGA
+and ASIC targets. Here, we discuss the accelerator architectures
+most relevant to our work.
+Recent FPGA-based accelerators include FINN [6, 22], DNNBuilder
+[25], and FILM-QNN [20]. In FINN and DNNBuilder, a software
+toolchain is used to map a trained DNN to generated logic modules
+that are integrated together. An overall processing pipeline is gen-
+erated and then synthesized for the target device. The advantage
+of this approach is that the logic efficiently implements a specific
+DNN with minimal overhead on a device that can be reconfigured
+to different DNNs at different times. However, this approach re-
+quires that all DNN layers be implemented in the logic all at once,
+which limits the size of the DNN to the amount of logic resources
+available on a given FPGA. While FINN supports low-precision
+down to binary and DNNBuilder down to 4-bit, neither supports
+arbitrary and mixed-precision at different DNN layers. In contrast,
+FILM-QNN does support DNN models of arbitrary sizes and quan-
+tized DNNs with mixed precisions. However, it is limited to only 4-
+or 8-bit weights and 5-bit activations due to a bit-packing scheme
+used with the DSP blocks in the FPGA.
+Several ASIC accelerator designs support arbitrary precision. Bit
+Fusion [19] uses a large array of 2-bit processing elements that
+can be fused together to perform up to 8-bit operations. Loom
+[18], and BitBlade [17] employ bit serial computation schemes for
+added flexibility. While bit-serial computation of any single math
+computation (e.g. multiplication) inherently requires additional
+clock cycles and latency over bit-parallel circuits, these designs
+exploit the large number of computations in a DNN that can be
+done in parallel. This is done by implementing a large number
+of bit-serial computational units operating simultaneously, which
+compensates high latency with high throughput. For example, the
+Loom engine consists of 128 × 16 = 2048 Serial Inner-Product units
+(SIPs), each of which performs 16 1 × 1-bit products per cycle.
+3
+ARCHITECTURE
+BARVINN is designed to provide high-throughput and software
+programmability, while supporting DNNs of arbitrary size and type.
+The high-level architecture of BARVINN is illustrated in Figure 1.
+It consists of the following main components: 1) an array of Matrix
+Vector Units (MVU) [5], and 2) a RISC-V CPU called Pito [4] as a
+controller for the MVU array. The MVUs accelerate common DNN
+computations such as GEMV, GEMM, and convolutions along with
+other operations such as batch normalization, ReLU activation, and
+quantization. Pito coordinates the computations in the MVU array
+while also handling data transfers to and from the host system.
+As it is not possible to foresee all possible neural networks that
+may crop up in the literature in the future, high-level sequencing of
+tensor operations for BARVINN is done in software. To control the
+array of processing elements, unlike the aforementioned accelera-
+tors, BARVINN uses the standard RISC-V RV32I ISA. This allows
+us to leverage the pre-existing software ecosystem. Furthermore,
+by using a CPU that supports a well known ISA, BARVINN is more
+flexible and it can be adapted to support new DNN architectures.
+3.1
+Matrix Vector Units
+The base configuration of BARVINN is implemented with an array
+of 8 MVUs. Figure 1 shows each MVU is a 64-element vector pipeline
+with several modules: a) a Matrix Vector Product unit (MVP), b)
+RAMs for activations/weights/scalers/biases, c) a scaler unit, d)
+a pooling/activation unit, and e) a quantizer. MVUs compute 64
+output vector elements per clock cycle using a 64 element input data
+vector from the activation RAM and a 64×64 element matrix from
+the weight RAM. Activation and weight RAMs store data in low-
+precision. MVP units operate in low-precision, while subsequent
+units in the pipeline operate in high-precision fixed-point.
+To justify our design choice of operating on 64 element vectors,
+we analysed over 50 models available at the ONNX Model Zoo
+[2] to check the input channel size of convolution layers. Figure 2
+illustrates a distribution of input channel size of all layers among
+those models. We found that 79% of these models use convolution
+with input channel sizes that are multiples of 64.
+3.1.1
+Matrix-Vector Product Units. Matrix operations are carried
+out by the MVP units. They compute on fixed-point arbitrary pre-
+cision operands from 1- to 16-bit. Each MVP has 64 vector-vector
+product (VVP) pipelines. Each VVP has 64 input lanes with 1-bit
+multipliers, followed by an addition tree with 8-bit output, as shown
+in Figure 4. On every cycle, 64 bits from the activation RAM are
+broadcasted to each of the 64 VVPs, while a 64×64 matrix tile from
+the weight RAM is read with each row of the tile sent to sepa-
+rate VVPs. The VVPs compute a 64-element dot product on 1-bit
+operands in each pipeline. With 64 VVPs per MVP, the overall
+output is a 64-element vector.
+MVPs compute arbitrary bit precision dot-products using the bit-
+serial scheme of [5]. Weights and activations can be unsigned or 2’s-
+complement signed fixed-point. Bit-depth is set independently for
+both, thus allowing for mixed precision. The bit-serial dot-product,
+shown in Algorithm 1, is a multi-cycle sequence starting with the
+most significant bits (MSB) from 64 elements of the activation and
+weight tensors. Bits are multiplied in each lane and results are
+added together across lanes in an addition tree producing an 8-bit
+dot product. This is added to an accumulator/shifter (see Figure 4).
+The MSB×MSB result represents the highest order-of-magnitude
+partial sum of the overall dot product. The MVP then computes the
+next lower order-of-magnitude partial sum by drawing the needed
+bit combinations of the operands. When a change in the order-of-
+magnitude is made, the accumulator is shifted left by 1-bit to align
+
+BARVINN: Arbitrary Precision DNN Accelerator Controlled by a RISC-V CPU
+ASPDAC ’23, January 16–19, 2023, Tokyo, Japan
+Figure 1: BARVINN hardware architecture with MVU array and Pito RISC-V controller. Right side is MVU detail.
+to the order-of-magnitude prior to adding the addition tree output.
+MVPs are fully pipelined, allowing them to work on different bit
+combinations at different stages without stalling. The operation
+completes when the dot products of the least significant bits (LSB)
+of the operands are computed and accumulated. For 𝑏𝑤-bit weights
+and 𝑏𝑎-bit activations, the overall operation takes 𝑏𝑤𝑏𝑎 cycles to
+compute one tile of the output vector. The precision of the operands
+is configured separately for each MVU, thus each MVU can process
+different layers with different bit precisions.
+Algorithm 1 Bit-serial dot-product
+1: 𝑏𝑎, 𝑏𝑤: activation and weight bit precisions
+2: 𝑥,𝑤: activation and weight vectors of size 𝑛
+3: 𝑗,𝑘: bit position for activations and weights
+4: 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 ← 0
+5: for 𝑖 ← 𝑏𝑤 + 𝑏𝑎 to 1 do
+6:
+for all (𝑗,𝑘) where 𝑗 + 𝑘 == 𝑖 do
+7:
+for 𝑙 ← 0 to 𝑛 − 1 do
+8:
+𝑜𝑛𝑒𝑏𝑖𝑡𝑝𝑟𝑜𝑑 = 𝑥𝑗 [𝑙] × 𝑤𝑘 [𝑙]
+9:
+𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 ← 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 + 𝑜𝑛𝑒𝑏𝑖𝑡𝑝𝑟𝑜𝑑
+10:
+end for
+11:
+end for
+11:
+shift 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 left 1-bit
+12: end for
+13: 𝑜𝑢𝑡𝑝𝑢𝑡 ← 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟
+Our bit-serial dot product scheme differs from other architec-
+tures. The computation scheme in BitFusion is based on computing
+the individual products of the overall dot-product, that are then
+summed. This requires a large number of shift-registers to align
+and sum partial products. BARVINN and BitBlade instead inter-
+change the ordering of the computation such that partial products
+of the same magnitude from all individual products are computed
+first and then summed. This reduces the number shifters needed.
+BARVINN additionally serializes the computation of partial prod-
+ucts of different magnitude, requiring only a single fixed shifter and
+a single adder tree, whereas BitBlade requires 16 variable shifters
+and 17 adder trees. BARVINN maintains throughput despite this
+serialized scheme by parallelizing across a wider number of input
+operands and producing a larger number of output products per
+clock cycle. BitFusion and BitBlade are further limited to operand
+sizes 2, 4, and 8-bit, whereas MVUs in BARVINN and SIPs in Loom
+support operands of any bit-depth down to 1-bit. However, Loom’s
+data loading scheme restricts the efficiency for general matrix mul-
+tiply operations when the weight bit depth is below 16, whereas
+BARVINNs is able to maintain full throughput down to 1-bit.
+3.1.2
+Memories and Data Layout. Activation and weight RAMs
+store data in a bit-transposed format shown in Figure 3 to exploit
+bit-serial computation. When precision is greater than 1 bit, tensor
+elements are organized in blocks where bits of the same order-of-
+magnitude are stored in the same memory word starting with the
+MSBs in the lowest address. A block of 𝑛 elements with precision 𝑏
+
+PITO RISC-V Core
+Matrix Vector Unit (MVU)
+Fetch
+Decode
+Execute
+Mem
+Commit
+Write Interconnect
+t  write Controller
+Read Controller Read Interco
+onnect
+Word
+Word
+Word
+Word
+imm :
+Instruction
+CSR WRITE
+ALU
+PRF1
+Memory
+D$
+8KB
+Decoder
+上
+CSR1
+Scaler
+PC+
+Activation Ram
+Data
+Ram
+select
+Memory
+Weight
+8KB
+Controller)
+Interface
+Ram
+pank3
+Bias
+APB][
+IRQ
+7
+Ram
+APB Bus
+1
+1
+Memory I
+64 bits
+64x64 bits
+MVU1
+MVU8
+MVP
+ Weight
+Input
+Bias
+Scaler
+Weight
+Bias
+Scaler
+Input
+32'bit 32'bit 32'bit
+32'bit
+Ram
+Ram
+Ram
+Ram
+Ram
+ Ram
+Ram
+Ram
+16bit
+4
+↑
+↑
+Scaler o
+ Scaler 1 
+Scaler 2
+Scaler 63
+32 bit
+32'bit 32'bit 32'bit
+32'bit
+Crossbar
+4
+Pol/ReLUPool/ReLu|Pool/ReLu
+Pool/ReLu
+Legend
+32'bit 32'bit 32'bit
+32'bit
+4
+[ Quantser
+PooIReLu
+> MVU to MVU Interface
+Quantser  Quantser
+<> Wire Interface
+1 bit
+1 bit
+1 bit
+1 bit
+> APB Bus CSR Config Interface
+AXlMemInterface
+L64 bit-ASPDAC ’23, January 16–19, 2023, Tokyo, Japan
+AskariHemmat et al
+Figure 2: Channel sizes in models from ONNX Model Zoo.
+requires 𝑏 memory words of width 𝑛. Activation vector elements
+are in blocks of 64 while weights matrix elements are in blocks of
+4096 bits in order to load a 64×64 matrix tile. A transposer module
+transforms input data from the host into the needed bit-transposed
+format. Transposition is only needed on the first layer of a DNN
+since MVUs write back to activation RAM in the bit-transposed
+format. Weights are pre-processed by a toolchain on the host and
+loaded into weights RAMs in the expected bit-transposed format.
+Figure 3: Bit-transposed data format for arbitrary precision.
+The layout of the tensors in the RAMs depends on the operation
+to be performed. For GEMV, activations are organized as vectors
+with blocks of 64 elements, while weight matrices are organized
+as a set of 64×64 tiles. For 2D convolutions, layout of activations
+is 𝑁𝐻𝑊𝐶, where the channel dimension 𝐶 is the innermost di-
+mension, followed by width 𝑊 , height 𝐻, and batch size 𝑁. The 𝐶
+dimension is the innermost since several common DNNs such as
+ResNet typically have hidden layer channel depths that are pow-
+ers of 2, and hence align to the 64 input lanes of the VVPs. When
+there are more than 64 channels, the first 64 channels are stored in
+the first block, the second 64 channels are stored into the second
+block and so on. As an example, an input tensor of [N=1, H=8, W=8,
+C=256] with 2-bit precision, will have 4 channel blocks, each block
+will have 64 rows of 2 by 64-bit elements.
+Our weight tensor memory layout for 2D convolutions is de-
+signed to support efficient execution by interleaving the input chan-
+nel dimension 𝐶𝑖 and output channel dimension 𝐶𝑜. Each weight
+memory word contains 64 subsets from the𝐶𝑜 dimension, with each
+subset containing 64 elements from the𝐶𝑖 dimension. A contiguous
+Figure 4: VVP unit with a shifter-accumulator. Bit 𝑗 from
+64 elements of the activation tensor 𝑥 and bit 𝑘 from 64 ele-
+ments of the weight tensor𝑤 are input in a bit-serial fashion.
+Note that some input bits and layers of the 5-deep adder tree
+are not shown.
+block of 𝑏𝑤 words that stores a complete set of bits for the needed
+weight precision is referred to as a channel block 𝐶𝑏. The layout
+for 2D convolution weights is 𝐶𝑜,𝑠𝐹𝐻 𝐹𝑊 𝐶𝑏, where 𝐶𝑜,𝑠 = 𝐶𝑜/64
+are output channel sets, and the kernel size is (𝐹𝑊 , 𝐹𝐻 ).
+3.1.3
+Job Configuration and Execution. MVUs are programmed to
+perform jobs such as GEMV or Conv2D operation. A controller sets
+configuration registers that orchestrate the sequence of calculations
+and memory reads to complete an operation in the MVUs. Once the
+job is finished, the MVU will generate an interrupt to the controller,
+indicating that the job is finished and results are ready to be sent
+back to the host or to trigger subsequent operations on the same
+MVU or other MVUs. While a MVU is busy, it can be programmed
+to prepare the next job to minimize idle time.
+Each MVU contains address generation units (AGU) that drive
+the memory access pattern across the activation and weight RAMs.
+The access pattern is managed by a set of up to five nested loops
+with parameters setting the number of iterations and the forward
+or backward address jumps to make on each iteration. The address
+jump scheme reduces the logic to a set of small accumulators to
+control the loops and small adders to compute addresses. Innermost
+loops are usually set to stride over the bit depth of the activations
+and weights. Outer loops are used to iterate over the bit combina-
+tions for the serial dot-product procedure and over the dimensions
+of the tensors. For GEMV, two nested loops are required for both
+activations and weights. Conv2D operations are programmed to
+compute one row of the output activation map per job, requiring
+four nested loops.
+3.1.4
+Pipeline Modules. Each MVU has modules downstream from
+the MVP to implement other DNN operations including a multi-
+plier/adder unit, a pooling/ReLU unit, and a quantizer/serializer
+unit. These modules operate at high-precision. Fixed-point mul-
+tiplier/adder units (Scaler in Figure 1), compute DNN operations
+such as batch normalization and quantization scaling as in LSQ [9].
+Scalers multiply the MVP output by a 16-bit operand sourced from
+the scaler RAM. In an FPGA, the multiplier is 27 × 16, which aligns
+with the port widths of on-chip fixed DSP units. An adder that
+follows adds 32-bit fixed-point bias terms from bias RAM. Scaler
+
+600
+Not Multiple of 64
+Multiple of 64
+500
+400
+ayers
+e-
+300
+#
+200
+100
+0
+128 256 512 1024
+1
+2
+4
+8
+16
+32
+64
+Input Channel Sizek elements
+address
+bit
+01
+k
+0
+n-1
+MSB
+1
+n-2
+block 0
+n-1
+0
+LSB
+MSB
+block 1
+LSBw,[0]
+x,[1] 
+32-bit shifter/accumulator
+w,[1]
+x,[2]
+W,[2]
+8-bit sum
+x[3]
+w,[3]
+x,[62]
+W,[62]
+x,[63]
+W.[63]BARVINN: Arbitrary Precision DNN Accelerator Controlled by a RISC-V CPU
+ASPDAC ’23, January 16–19, 2023, Tokyo, Japan
+and bias RAMs have independent AGUs. The module that follows
+combines max pooling and ReLU (Pool/ReLU in Figure 1), imple-
+mented as a comparator with an internal register. For ReLU, the
+incoming value is checked against the register initially set to 0. The
+combined MaxPool/ReLU is implemented by programming MVUs
+to produce data in the sequence needed for a MaxPool window.
+The pipeline ends at the quantization/serialization unit (QuantSer
+in Figure 1). It takes 32-bit fixed-point data from each of the 64 data
+paths and serializes them into 64 1-bit outputs. It is programmed to
+set the output bit-depth and the MSB position from the input word.
+Combined with scaler units, this is used to implement quantization
+schemes such as LSQ [9]. Serialized outputs of each datapath are
+grouped into a single 64-bit word that is sent either to the activation
+RAM of the same MVU, or to a different MVU via an interconnect.
+3.1.5
+Interconnect. MVUs can send data to each other via an inter-
+connect implemented as an 8-way crossbar switch with broadcast
+capability. A source MVU is programmed to send its output results
+in a serialized fashion to a given address in the activation memory
+of a destination MVU(s). At a destination MVU, a fixed-priority
+arbitration scheme to the write port of the target MVU activation
+RAM is used. The interconnect is given highest priority, followed
+by the controller, then lastly the MVU itself. When multiple MVUs
+attempt to write to the same destination MVU, a fixed priority
+scheme determines which MVU can write to its memory.
+3.1.6
+DNN Mapping. Each MVU can be assigned to different lay-
+ers of a DNN, such as convolutions and fully-connected layers.
+Alternatively, a single layer can be split between multiple MVUs
+with each MVUs processing a subset of the input activations and/or
+weights. Partial results are forwarded from one MVU to another
+via the interconnect to process subsequent layers of the network,
+thus creating an overall processing pipeline through the array. By
+sending partial results from one MVU to another, subsequent MVUs
+can begin processing as soon as sufficient data has been received
+from previous layers. For instance, a MVU processing a 3× 3 convo-
+lution requires only 3 rows of activations from the previous layer to
+produce one output row of the layer it is processing. This avoids the
+need to wait until all outputs from a layer are generated, which re-
+duces latency and idle time. Furthermore, the ability to immediately
+process partial layer outputs by subsequent MVUs keeps on-chip
+storage requirements low, since only the partial set of activations
+required to produce the next layer partial output needs to be stored.
+Depending on the performance goal, BARVINN can execute a
+DNN in either Pipelined mode or Distributed mode. In Pipelined
+mode (Figure 5.a), the MVU array can process up to 8 convolutions
+and fully-connected layers all at once. Each MVU can be configured
+to use different precisions. In cases where a DNN model contains
+more than 8 layers, the MVU array can be programmed to process
+the entire model by dividing it into subsets of up to 8 layers each.
+Each MVU can be loaded with weights from layers in each subset,
+either all from the start of processing if there is sufficient weight
+memory available in each MVU or on-the-fly from external memory
+if not. Output activations from the last MVU in the chain can also
+be stored temporarily in off-chip memory and fetched later in the
+case where the first MVU is still processing data from the current
+lap. In the Distributed mode, to minimize latency, the objective is
+to process single batch inputs as fast as possible. As can be seen
+in Figure 5.b, in this mode, the computation of a single layer is
+broken into 8 independent computation regions. All MVUs will be
+programmed to share the same set of weights. To make sure an MVU
+computation is independent of those performed on other MVUs, the
+user might need to copy the input regions that are shared between
+computation units. The programmability of BARVINN allows the
+user to mix and match these execution modes for different layers
+and models to achieve highest performance.
+a. Pipelined mode
+b. Distributed mode
+Figure 5: Execution flow of a DNN on the MVU array in
+Pipelined (a) and Distributed (b) modes. In Pipelined mode
+each MVU processes one layer at a time. In distributed mode,
+the computation of a single layer is distributed among mul-
+tiple MVUs.
+3.2
+Pito: RISC-V-based Controller
+To make use of MVUs for neural networks, a control unit is required.
+The controller is a barrel RISC-V processor designed to control
+the 8 MVUs using separate but communicating hardware threads
+(harts) that each manage their respective MVUs. DNN layers are ex-
+ecuted either in distributed or in a pipelined fashion, depending on
+whether the DNN is compiled to maximize throughput or minimize
+latency. This design allows MVUs to complete tensor operations
+independently of each other. The drawback is that it requires 8
+microprocessors to execute the 8 programs. We instead amortized
+the fixed costs of the processor by adopting barrel processing. With
+a 8-way threaded processor, we may assign one thread to control
+each of the MVUs. Because every thread comes up for execution
+only every 8 clock cycles, the five pipeline stages (fetch, decode,
+execute, data read & writes and commit) can be completely hidden.
+Branch prediction units are unnecessary. Since tensor operations
+can require hundreds of cycles to execute on a MVU, the barrel
+processor can fully turn over dozens of times in the interim, allow-
+ing each thread to issue the next command to its MVU in a few
+instructions.
+We adopted a Harvard architecture and divided the instruction
+and data RAM, 8KB each, and shared between all harts. This gives a
+
+MemoryInterface
+UART
+MvU Array
+MVU6
+MU5
+D$
+IS
+8KB
+8KB
+1
+CSR8 pe_irq
+pestart
+CSR2 pe irq
+Crossbar
+CSR1 pe_irq
+ant
+Pito RISC-V
+pe_start ld
+pe_command httus
+APB
+pe_quant
+pe_status
+Input
+Convo
+Conv4
+Conv1
+Conv5
+Conv2
+Conv6
+Input
+Weight
+Conv7
+Output
+Conv3
+[1x63x32x32]
+[64x64x3x3]
+[1x63x32x32]ASPDAC ’23, January 16–19, 2023, Tokyo, Japan
+AskariHemmat et al
+1K word space to store data and instructions to control each MVU.
+The processor executes instructions following compilation order
+and without any further scheduling. A hart scheduler provides
+access to the required resources for the hart at each stage. In the
+fetch stage, each hart loads instructions from the instruction RAM.
+The program counter (PC) and register file for each hart is different
+and the hart scheduler indicates which register should be accessed
+at a given time. The Decode stage decodes instruction and loads
+source registers or an immediate operand. Our RISC-V controller is
+compatible with RV32I RISC-V ISA with minimal support for privi-
+lege specification to make Control and Status Registers (CSRs) and
+Interrupts available to interface with the MVU array. In addition
+to the base CSRs, we have added 74 MVU-specific CSRs to allow
+software to control the processing element array. These CSRs con-
+trol different settings within an MVU such as weight and activation
+precision, AGU’s jump settings, input, weight and output memory
+address and pipeline module selection as described in 3.1.4.
+3.3
+Code Generator
+BARVINN performs GEMM/GEMV, Convolutions, Maxpooling and
+activation (ReLU). However, it is up to the user to sequence the
+operations within a DNN with software. To facilitate this, we de-
+veloped a code generator that takes a DNN described in ONNX
+[3] and configuration settings (weight/input/output precision), and
+generates RISC-V code for each operation. The code generator ex-
+ports weights to the bit-transposed format described in section 3.1.2.
+Since each MVU works on 64-bit words, the code generator tiles
+each weight tensor in blocks of 64×64. When this cannot be done
+(either tensor input channel or output channel is not a multiple of
+64), we pad the corresponding tile. Currently, our code generator
+does not apply graph optimization techniques. Also, for now, our
+code generator supports Pipelined mode execution. In the follow-
+ing section, we used our code generator to map PyTorch models to
+micro kernel codes which can then be directly used by BARVINN.
+4
+PERFORMANCE ANALYSIS AND RESULTS
+4.1
+Experimental Setup
+To illustrate the performance of BARVINN, we chose the ResNet9
+image classifier model for the CIFAR10 dataset. We trained and
+quantized a ResNet9 model on CIFAR10 using LSQ [9] and used the
+residual distillation [14] technique to remove shortcut connections
+(Plain CNN models). In many image classification DNN models such
+as ResNet [10], the input to the first layer typically consists of less
+than 64 channels. Furthermore, due to sensitivity of the first and
+last layer to information loss, most state-of-the-art compression
+and quantization methods do not apply optimization on input and
+output layers [9], hence keeping these layers untouched and in full
+precision. We have adopted the same technique to compute first
+and last layers on the host or on the RISC-V controller.
+Table 2 shows the performance of ResNet9 on CIFAR10 in the
+PyTorch framework. Once we were satisfied with the performance
+of our quantized model, we exported the trained model to ONNX
+and then used our code generator. Table 3 illustrates the per layer
+computation cost of running ResNet9 on BARVINN with 2-bit ac-
+tivations and weights. All convolutions use a padding of 1. As
+discussed before, we skipped running the first and last layer on
+Table 2: ResNet9 with different bit precision on CIFAR10
+ResNet9 Model
+Precision
+Accuracy
+Size (Bytes)
+Original
+Fp32
+90.8%
+19605141
+Plain-CNN
+Fp32
+91.1%
+18912487
+Quantized Plain-CNN
+Int2
+89.2%
+1181360
+Table 3: ResNet9 layers for CIFAR10 dataset and computa-
+tion cost. All layers are quantized to 2-bit for activation and
+weights, except for the first and last layers.
+Layer
+Input
+Kernel
+Output
+Cycles
+conv0
+[3, 32, 32]
+[64, 3, 3, 3]
+[64, 32, 32]
+N/A
+conv1
+[64, 32, 32]
+[64, 64, 3, 3]
+[64, 32, 32]
+34560
+conv2
+[64, 32, 32]
+[64, 64, 3, 3]
+[64, 32, 32]
+34560
+conv3
+[64, 32, 32]
+[128, 64, 3, 3]
+[128, 16, 16]
+17280
+conv4
+[128, 16, 16]
+[128, 128, 3, 3]
+[128, 8, 8]
+32256
+conv5
+[128, 8, 8]
+[256, 128, 3, 3]
+[128, 8, 8]
+16128
+conv6
+[128, 8, 8]
+[256, 256, 3, 3]
+[256, 4, 4]
+27648
+conv7
+[256, 4, 4]
+[512, 256, 3, 3]
+[256, 4, 4]
+13824
+conv8
+[256, 4, 4]
+[512, 512, 3, 3]
+[512, 4, 4]
+18432
+fc
+[512, 4, 4]
+[10, 512]
+[10]
+N/A
+Total:
+194688
+BARVINN and we kept them in their original format. The overall
+computation takes 194,688 cycles to complete.
+Our design was written in Verilog and synthesized using Xilinx
+Vivado 2021.1 for the Xilinx Alveo U250 accelerator card. Synthesis
+results for the RISC-V controller, the processing array, and the accel-
+erator are presented in Table 4. Power consumption was estimated
+using the software tools in Vivado.
+4.2
+Discussion
+We compared BARVINN with FINN [22], which is a templated Vi-
+vado HLS C++ library of common DNN layers. Like BARVINN,
+FINN can generate hardware for arbitrary precision, but is not
+software programmable. Hence, once the FINN hardware is gen-
+erated, the user cannot change the computation data stream. We
+attempted to compare the performance of BARVINN with FINN
+using the ResNet9 model we used earlier. However, at the time of
+writing, FINN supports simple linear topologies and we were not
+able to get performance metrics for our model. Instead, we used the
+available CIFAR10-CNV model from the FINN repository that was
+tuned for the FINN dataflow for our comparison. Table 5 shows the
+performance of BARVINN and FINN. For this experiment, we used
+different precisions for weights and activation. For both tools, we
+used the performance estimation numbers for frames per second
+(FPS). For FINN, we used the default folding configurations publicly
+available in FINN-example repository [1]. As illustrated in Table
+5, we provide 7-15 times better throughput albeit with higher LUT
+usage. On the other hand, for higher bit precisions, FINN provides
+a better FPS/LUT, suggesting a scalable solution for bigger models.
+We also compared the performance on a ResNet-50 model. Table
+6 shows our estimated FPS for BARVINN executing in Pipelined
+mode along with reported performance for FINN [1] synthesized for
+the Xilinx U250 and for FILM-QNN [20] synthesized for the Xilinx
+ZCU102 FPGA. While FINN has the highest FPS, BARVINN shows
+
+BARVINN: Arbitrary Precision DNN Accelerator Controlled by a RISC-V CPU
+ASPDAC ’23, January 16–19, 2023, Tokyo, Japan
+Table 4: Post-synthesis resource utilization of BARVINN.
+Resource
+Pito RISC-V
+MVU Array
+Overall
+LUT
+10454
+190625
+201079
+BRAM
+15
+1312
+1327
+DSP
+0
+512
+512
+Dynamic Power
+0.410 W
+21.066 W
+21.504 W
+Frequency
+250 MHz
+250 MHz
+250 MHz
+Table 5: Estimated performance of running CNV model on
+CIFAR10 on Alveo U250 when different bit precision is used.
+Bits
+(W/A)
+kLUT
+BRAM
+DSP
+FPS
+FPS/
+kLUT
+Ours
+1/1
+201.1 (15.0%)
+1327
+512
+61035
+303.5
+1/2
+201.1 (15.0%)
+1327
+512
+30517
+151.7
+2/2
+201.1 (15.0%)
+1327
+512
+15258
+75.8
+FINN
+1/1
+28.2 (2.1%)
+150
+0
+7716
+273.6
+1/2
+19.8(1.47%)
+103
+0
+2170
+109.6
+2/2
+24.3(1.81%)
+202
+0
+2170
+89.3
+Table 6: Performance for ResNet-50 model on ImageNet.
+Bits (W/A)
+Clock Freq.
+FPS
+FPS/Watt
+Ours
+1/2
+250 MHz
+2296
+106.8
+FINN-R [1][6]
+1/2
+178 MHz
+2873
+41.0
+FILM-QNN [20]
+4(8)/5
+150 MHz
+109
+8.4
+the best performance per Watt. According to the FINN-example
+repository [1], a fine-tuned ResNet50 model, requires more than
+87% of Alveo U250 accelerator’s resources. This shows the limits
+of FINN dealing with bigger models. BARVINN requires the same
+LUT usage regardless of the model size and bit-width.
+5
+CONCLUSION
+In this paper, we presented an FPGA-based DNN accelerator that
+supports arbitrary bit precision computations. We tested the perfor-
+mance of BARVINN over different DNN kernels and models with
+different bit precision. For model deployment, we developed a code
+generator tool that takes in a model in ONNX format and generates
+RISC-V assembly code for the controller. Compared to other low
+precision accelerators, we provide a programmable solution which
+makes BARVINN more flexible. With the programmable MVUs, the
+user can run different models regardless of their size. BARVINN
+allows trading off throughput and latency by running DNN lay-
+ers either in distributed or in pipeline modes. Unlike other low
+precision accelerators, our proposed solution offers implementing
+various trade-offs through software and the end user can control
+them for each individual layer without FPGA reconfiguration at
+run time. Compared to programmable accelerators, BARVINN was
+shown to provide a better throughput per Watt performance.
+ACKNOWLEDGMENTS
+The authors acknowledge support for this project from the IBM
+AI Horizons Network, CMC Microsystems, Fonds de Recherche du
+Quebec–Nature et Technologies (FRQNT), MITACS and from the
+NSERC COHESA Strategic Research Network.
+REFERENCES
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+page_content='BARVINN: Arbitrary Precision DNN Accelerator Controlled by a  RISC-V CPU  Mohammadhossein Askarihemmat1, Sean Wagner2, Olexa Bilaniuk3,  Yassine Hariri4, Yvon Savaria1, Jean-Pierre David1  1Ecole Polytechnique Montreal, Canada, 2 IBM, Toronto, Canada, 3 Mila, Montreal, Canada,  4 CMC Microsystems, Kingston, Canada  {mohammad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content='bilaniuk@mila.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content='ca  ABSTRACT  We present a DNN accelerator that allows inference at arbitrary  precision with dedicated processing elements that are configurable  at the bit level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Our DNN accelerator has 8 Processing Elements  controlled by a RISC-V controller with a combined 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='2 TMACs of  computational power when implemented with the recent Alveo  U250 FPGA platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' We develop a code generator tool that ingests  CNN models in ONNX format and generates an executable com-  mand stream for the RISC-V controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' We demonstrate the scalable  throughput of our accelerator by running different DNN kernels  and models when different quantization levels are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Com-  pared to other low precision accelerators, our accelerator provides  run time programmability without hardware reconfiguration and  can accelerate DNNs with multiple quantization levels, regardless  of the target FPGA size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' BARVINN is an open source project and it  is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='com/hossein1387/BARVINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  KEYWORDS  neural networks, hardware acceleration, FPGA, low-precision  ACM Reference Format:  Mohammadhossein Askarihemmat1, Sean Wagner2, Olexa Bilaniuk3,, Yas-  sine Hariri4, Yvon Savaria1, Jean-Pierre David1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' BARVINN: Arbi-  trary Precision DNN Accelerator Controlled by a RISC-V CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In 28th  Asia and South Pacific Design Automation Conference (ASPDAC ’23), Janu-  ary 16–19, 2023, Tokyo, Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' ACM, New York, NY, USA, 7 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' https:  //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content='3567872  1  INTRODUCTION  Deep neural networks (DNNs) traditionally rely on floating point  computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' These operations are slow and costly in terms of  power consumption and required silicon area compared to fixed-  point/integer operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' One way to accelerate computation in  a DNN is to use less precision for computation via quantization  [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' This also reduces memory consumption as well as energy  consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' For instance, in a 45 nm process, 8-bit integer multi-  plication and addition take 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='2 pJ and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='03 pJ, respectively, while the  Permission to make digital or hard copies of all or part of this work for personal or  classroom use is granted without fee provided that copies are not made or distributed  for profit or commercial advantage and that copies bear this notice and the full citation  on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Copyrights for components of this work owned by others than ACM  must be honored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Abstracting with credit is permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' To copy otherwise, or republish,  to post on servers or to redistribute to lists, requires prior specific permission and/or a  fee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Request permissions from permissions@acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  ASPDAC ’23, January 16–19, 2023, Tokyo, Japan  © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  ACM ISBN 978-1-4503-9783-4/23/01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content='3567872  Table 1: Effects of Quantization on Accuracy and Model Size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Task  Dataset  Model  Precision  A/W  Acc/  MAP  Size  (MB)  Classification  CIFAR  100  ResNet18  LSQ(2/2)  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='81  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='889  LSQ(4/4)  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='92  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='559  LSQ(8/8)  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='45  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='87  FP32  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='82  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='8  Object  Detection  VOC-  2007  SSD300-  ResNet18  LSQ(2/2)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='61  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='34  LSQ(4/4)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='60  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='81  LSQ(8/8)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='68  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='77  FP32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='59  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='49  same operations with 32-bit floating-point values requires 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='7 pJ  for multiplication and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='9 pJ for addition [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' On an Intel Core i7  4770 running at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='4GHz, multiplication is more than 3 times faster  for fixed-point compared to floating-point [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' With recent quanti-  zation techniques, these benefits are available with little to no loss  in model performance and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In [9, 13], their quantization  schemes showed accuracy losses of 1-3% at 2-bit precision on most  classification and object detection models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Table 1 illustrates the  result of applying Learned Scale Quantization (LSQ) [9] with differ-  ent bit precisions on different models and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Quantized models  offer accuracy similar to full precision models, while having smaller  size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Mixed-precision quantization [7, 16, 21, 23, 24] further provides  finer control to reach an optimal solution by learning different  precisions for each layer of a network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In [23], the authors illustrate  that using their mixed-precision framework, they reduced model  latency and energy consumption by a factor of almost 2× with little  drop in accuracy compared with an 8-bit quantized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Fully benefiting from low-precision in a DNN requires hardware  that natively supports low-precision computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Commodity  hardware can perform arbitrary precision arithmetic by transform-  ing data-layout and computing with bit-wise instructions [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' How-  ever, this approach is extremely costly for general processors, be-  cause of the overhead for shifting, masking and packing bits to  the correct format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' At the time of writing this paper, there are no  commercially available general processors (CPU or GPU) that can  efficiently process data in arbitrary precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  In this paper, we propose an arbitrary low-precision DNN hard-  ware accelerator called BARVINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Our accelerator is software pro-  grammable and can be integrated in the RISC-V standard devel-  opment flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' It is designed as a highly optimized computational  pipeline for DNNs that introduces low hardware overhead and of-  fers low-power operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The contributions of our paper are as  follows:  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='00290v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='AR]  31 Dec 2022  ASPDAC ’23, January 16–19, 2023, Tokyo, Japan  AskariHemmat et al  Implementation of a DNN hardware accelerator with arbi-  trary fixed-point low-precision for matrix-vector multiply  operations at high-throughput and low power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Implementation of a custom embedded RISC-V CPU to con-  trol an array of DNN accelerators by software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Data structures for efficiently storing and processing weights  and activations for high-throughput serial computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Development of a software code generator for transforming  DNNs into RISC-V code that executes on our accelerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  In section 2, we review relevant DNN accelerators from the liter-  ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Section 3 presents the architecture of BARVINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In section  4, a detailed performance analysis of BARVINN is provided and  compared with other DNN accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  2  RELATED WORKS  Several DNN hardware accelerators supporting quantization and  low-precision have been presented in recent years for both FPGA  and ASIC targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Here, we discuss the accelerator architectures  most relevant to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Recent FPGA-based accelerators include FINN [6, 22], DNNBuilder  [25], and FILM-QNN [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In FINN and DNNBuilder, a software  toolchain is used to map a trained DNN to generated logic modules  that are integrated together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' An overall processing pipeline is gen-  erated and then synthesized for the target device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The advantage  of this approach is that the logic efficiently implements a specific  DNN with minimal overhead on a device that can be reconfigured  to different DNNs at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' However, this approach re-  quires that all DNN layers be implemented in the logic all at once,  which limits the size of the DNN to the amount of logic resources  available on a given FPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' While FINN supports low-precision  down to binary and DNNBuilder down to 4-bit, neither supports  arbitrary and mixed-precision at different DNN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In contrast,  FILM-QNN does support DNN models of arbitrary sizes and quan-  tized DNNs with mixed precisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' However, it is limited to only 4-  or 8-bit weights and 5-bit activations due to a bit-packing scheme  used with the DSP blocks in the FPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Several ASIC accelerator designs support arbitrary precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Bit  Fusion [19] uses a large array of 2-bit processing elements that  can be fused together to perform up to 8-bit operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Loom  [18], and BitBlade [17] employ bit serial computation schemes for  added flexibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' While bit-serial computation of any single math  computation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' multiplication) inherently requires additional  clock cycles and latency over bit-parallel circuits, these designs  exploit the large number of computations in a DNN that can be  done in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' This is done by implementing a large number  of bit-serial computational units operating simultaneously, which  compensates high latency with high throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' For example, the  Loom engine consists of 128 × 16 = 2048 Serial Inner-Product units  (SIPs), each of which performs 16 1 × 1-bit products per cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  3  ARCHITECTURE  BARVINN is designed to provide high-throughput and software  programmability, while supporting DNNs of arbitrary size and type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  The high-level architecture of BARVINN is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  It consists of the following main components: 1) an array of Matrix  Vector Units (MVU) [5], and 2) a RISC-V CPU called Pito [4] as a  controller for the MVU array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The MVUs accelerate common DNN  computations such as GEMV, GEMM, and convolutions along with  other operations such as batch normalization, ReLU activation, and  quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Pito coordinates the computations in the MVU array  while also handling data transfers to and from the host system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  As it is not possible to foresee all possible neural networks that  may crop up in the literature in the future, high-level sequencing of  tensor operations for BARVINN is done in software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' To control the  array of processing elements, unlike the aforementioned accelera-  tors, BARVINN uses the standard RISC-V RV32I ISA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' This allows  us to leverage the pre-existing software ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Furthermore,  by using a CPU that supports a well known ISA, BARVINN is more  flexible and it can be adapted to support new DNN architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1  Matrix Vector Units  The base configuration of BARVINN is implemented with an array  of 8 MVUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Figure 1 shows each MVU is a 64-element vector pipeline  with several modules: a) a Matrix Vector Product unit (MVP), b)  RAMs for activations/weights/scalers/biases, c) a scaler unit, d)  a pooling/activation unit, and e) a quantizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' MVUs compute 64  output vector elements per clock cycle using a 64 element input data  vector from the activation RAM and a 64×64 element matrix from  the weight RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Activation and weight RAMs store data in low-  precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' MVP units operate in low-precision, while subsequent  units in the pipeline operate in high-precision fixed-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  To justify our design choice of operating on 64 element vectors,  we analysed over 50 models available at the ONNX Model Zoo  [2] to check the input channel size of convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Figure 2  illustrates a distribution of input channel size of all layers among  those models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' We found that 79% of these models use convolution  with input channel sizes that are multiples of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1  Matrix-Vector Product Units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Matrix operations are carried  out by the MVP units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' They compute on fixed-point arbitrary pre-  cision operands from 1- to 16-bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Each MVP has 64 vector-vector  product (VVP) pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Each VVP has 64 input lanes with 1-bit  multipliers, followed by an addition tree with 8-bit output, as shown  in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' On every cycle, 64 bits from the activation RAM are  broadcasted to each of the 64 VVPs, while a 64×64 matrix tile from  the weight RAM is read with each row of the tile sent to sepa-  rate VVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The VVPs compute a 64-element dot product on 1-bit  operands in each pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' With 64 VVPs per MVP, the overall  output is a 64-element vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  MVPs compute arbitrary bit precision dot-products using the bit-  serial scheme of [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Weights and activations can be unsigned or 2’s-  complement signed fixed-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Bit-depth is set independently for  both, thus allowing for mixed precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The bit-serial dot-product,  shown in Algorithm 1, is a multi-cycle sequence starting with the  most significant bits (MSB) from 64 elements of the activation and  weight tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Bits are multiplied in each lane and results are  added together across lanes in an addition tree producing an 8-bit  dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' This is added to an accumulator/shifter (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  The MSB×MSB result represents the highest order-of-magnitude  partial sum of the overall dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The MVP then computes the  next lower order-of-magnitude partial sum by drawing the needed  bit combinations of the operands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' When a change in the order-of-  magnitude is made, the accumulator is shifted left by 1-bit to align  BARVINN: Arbitrary Precision DNN Accelerator Controlled by a RISC-V CPU  ASPDAC ’23, January 16–19, 2023, Tokyo, Japan  Figure 1: BARVINN hardware architecture with MVU array and Pito RISC-V controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Right side is MVU detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  to the order-of-magnitude prior to adding the addition tree output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  MVPs are fully pipelined, allowing them to work on different bit  combinations at different stages without stalling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The operation  completes when the dot products of the least significant bits (LSB)  of the operands are computed and accumulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' For 𝑏𝑤-bit weights  and 𝑏𝑎-bit activations, the overall operation takes 𝑏𝑤𝑏𝑎 cycles to  compute one tile of the output vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The precision of the operands  is configured separately for each MVU, thus each MVU can process  different layers with different bit precisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Algorithm 1 Bit-serial dot-product  1: 𝑏𝑎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 𝑏𝑤: activation and weight bit precisions  2: 𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='𝑤: activation and weight vectors of size 𝑛  3: 𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='𝑘: bit position for activations and weights  4: 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 ← 0  5: for 𝑖 ← 𝑏𝑤 + 𝑏𝑎 to 1 do  6:  for all (𝑗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='𝑘) where 𝑗 + 𝑘 == 𝑖 do  7:  for 𝑙 ← 0 to 𝑛 − 1 do  8:  𝑜𝑛𝑒𝑏𝑖𝑡𝑝𝑟𝑜𝑑 = 𝑥𝑗 [𝑙] × 𝑤𝑘 [𝑙]  9:  𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 ← 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 + 𝑜𝑛𝑒𝑏𝑖𝑡𝑝𝑟𝑜𝑑  10:  end for  11:  end for  11:  shift 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟 left 1-bit  12: end for  13: 𝑜𝑢𝑡𝑝𝑢𝑡 ← 𝑎𝑐𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑜𝑟  Our bit-serial dot product scheme differs from other architec-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The computation scheme in BitFusion is based on computing  the individual products of the overall dot-product, that are then  summed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' This requires a large number of shift-registers to align  and sum partial products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' BARVINN and BitBlade instead inter-  change the ordering of the computation such that partial products  of the same magnitude from all individual products are computed  first and then summed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' This reduces the number shifters needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  BARVINN additionally serializes the computation of partial prod-  ucts of different magnitude, requiring only a single fixed shifter and  a single adder tree, whereas BitBlade requires 16 variable shifters  and 17 adder trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' BARVINN maintains throughput despite this  serialized scheme by parallelizing across a wider number of input  operands and producing a larger number of output products per  clock cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' BitFusion and BitBlade are further limited to operand  sizes 2, 4, and 8-bit, whereas MVUs in BARVINN and SIPs in Loom  support operands of any bit-depth down to 1-bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' However, Loom’s  data loading scheme restricts the efficiency for general matrix mul-  tiply operations when the weight bit depth is below 16, whereas  BARVINNs is able to maintain full throughput down to 1-bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='2  Memories and Data Layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Activation and weight RAMs  store data in a bit-transposed format shown in Figure 3 to exploit  bit-serial computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' When precision is greater than 1 bit, tensor  elements are organized in blocks where bits of the same order-of-  magnitude are stored in the same memory word starting with the  MSBs in the lowest address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' A block of 𝑛 elements with precision 𝑏  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='PITO RISC-V Core  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Matrix Vector Unit (MVU)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Fetch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Decode  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Execute  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Mem  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Commit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Write Interconnect  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='t  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content='ALU  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Memory I  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='64 bits  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='64x64 bits  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='MVU1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='MVU8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='MVP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Weight  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Input  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Bias  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Scaler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Weight  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Bias  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Scaler  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Input  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content="32'bit 32'bit 32'bit  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content="32'bit  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Ram  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Ram  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Ram  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Ram  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Ram  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Ram  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Ram  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Ram  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='16bit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='↑  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='↑  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Scaler o  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Scaler 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Scaler 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Scaler 63  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='32 bit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content="32'bit 32'bit 32'bit  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content="32'bit  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Crossbar  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Pol/ReLUPool/ReLu|Pool/ReLu  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Pool/ReLu  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Legend  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content="32'bit 32'bit 32'bit  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content="32'bit  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='[ Quantser  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='PooIReLu  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='> MVU to MVU Interface  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Quantser  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='Quantser  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='<> Wire Interface  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1 bit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1 bit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1 bit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1 bit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='> APB Bus CSR Config Interface  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='AXlMemInterface  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='L64 bit-ASPDAC ’23,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' January 16–19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 2023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Japan  AskariHemmat et al  Figure 2: Channel sizes in models from ONNX Model Zoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  requires 𝑏 memory words of width 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Activation vector elements  are in blocks of 64 while weights matrix elements are in blocks of  4096 bits in order to load a 64×64 matrix tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' A transposer module  transforms input data from the host into the needed bit-transposed  format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Transposition is only needed on the first layer of a DNN  since MVUs write back to activation RAM in the bit-transposed  format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Weights are pre-processed by a toolchain on the host and  loaded into weights RAMs in the expected bit-transposed format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Figure 3: Bit-transposed data format for arbitrary precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  The layout of the tensors in the RAMs depends on the operation  to be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' For GEMV, activations are organized as vectors  with blocks of 64 elements, while weight matrices are organized  as a set of 64×64 tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' For 2D convolutions, layout of activations  is 𝑁𝐻𝑊𝐶, where the channel dimension 𝐶 is the innermost di-  mension, followed by width 𝑊 , height 𝐻, and batch size 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The 𝐶  dimension is the innermost since several common DNNs such as  ResNet typically have hidden layer channel depths that are pow-  ers of 2, and hence align to the 64 input lanes of the VVPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' When  there are more than 64 channels, the first 64 channels are stored in  the first block, the second 64 channels are stored into the second  block and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' As an example, an input tensor of [N=1, H=8, W=8,  C=256] with 2-bit precision, will have 4 channel blocks, each block  will have 64 rows of 2 by 64-bit elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Our weight tensor memory layout for 2D convolutions is de-  signed to support efficient execution by interleaving the input chan-  nel dimension 𝐶𝑖 and output channel dimension 𝐶𝑜.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Each weight  memory word contains 64 subsets from the𝐶𝑜 dimension, with each  subset containing 64 elements from the𝐶𝑖 dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' A contiguous  Figure 4: VVP unit with a shifter-accumulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Bit 𝑗 from  64 elements of the activation tensor 𝑥 and bit 𝑘 from 64 ele-  ments of the weight tensor𝑤 are input in a bit-serial fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Note that some input bits and layers of the 5-deep adder tree  are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  block of 𝑏𝑤 words that stores a complete set of bits for the needed  weight precision is referred to as a channel block 𝐶𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The layout  for 2D convolution weights is 𝐶𝑜,𝑠𝐹𝐻 𝐹𝑊 𝐶𝑏, where 𝐶𝑜,𝑠 = 𝐶𝑜/64  are output channel sets, and the kernel size is (𝐹𝑊 , 𝐹𝐻 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='3  Job Configuration and Execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' MVUs are programmed to  perform jobs such as GEMV or Conv2D operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' A controller sets  configuration registers that orchestrate the sequence of calculations  and memory reads to complete an operation in the MVUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Once the  job is finished, the MVU will generate an interrupt to the controller,  indicating that the job is finished and results are ready to be sent  back to the host or to trigger subsequent operations on the same  MVU or other MVUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' While a MVU is busy, it can be programmed  to prepare the next job to minimize idle time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Each MVU contains address generation units (AGU) that drive  the memory access pattern across the activation and weight RAMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  The access pattern is managed by a set of up to five nested loops  with parameters setting the number of iterations and the forward  or backward address jumps to make on each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The address  jump scheme reduces the logic to a set of small accumulators to  control the loops and small adders to compute addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Innermost  loops are usually set to stride over the bit depth of the activations  and weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Outer loops are used to iterate over the bit combina-  tions for the serial dot-product procedure and over the dimensions  of the tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' For GEMV, two nested loops are required for both  activations and weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Conv2D operations are programmed to  compute one row of the output activation map per job, requiring  four nested loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='4  Pipeline Modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Each MVU has modules downstream from  the MVP to implement other DNN operations including a multi-  plier/adder unit, a pooling/ReLU unit, and a quantizer/serializer  unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' These modules operate at high-precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Fixed-point mul-  tiplier/adder units (Scaler in Figure 1), compute DNN operations  such as batch normalization and quantization scaling as in LSQ [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Scalers multiply the MVP output by a 16-bit operand sourced from  the scaler RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In an FPGA, the multiplier is 27 × 16, which aligns  with the port widths of on-chip fixed DSP units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' An adder that  follows adds 32-bit fixed-point bias terms from bias RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Scaler  600  Not Multiple of 64  Multiple of 64  500  400  ayers  e-  300  #  200  100  0  128 256 512 1024  1  2  4  8  16  32  64  Input Channel Sizek elements  address  bit  01  k  0  n-1  MSB  1  n-2  block 0  n-1  0  LSB  MSB  block 1  LSBw,[0]  x,[1]  32-bit shifter/accumulator  w,[1]  x,[2]  W,[2]  8-bit sum  x[3]  w,[3]  x,[62]  W,[62]  x,[63]  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='[63]BARVINN: Arbitrary Precision DNN Accelerator Controlled by a RISC-V CPU  ASPDAC ’23, January 16–19, 2023, Tokyo, Japan  and bias RAMs have independent AGUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The module that follows  combines max pooling and ReLU (Pool/ReLU in Figure 1), imple-  mented as a comparator with an internal register.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' For ReLU, the  incoming value is checked against the register initially set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The  combined MaxPool/ReLU is implemented by programming MVUs  to produce data in the sequence needed for a MaxPool window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  The pipeline ends at the quantization/serialization unit (QuantSer  in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' It takes 32-bit fixed-point data from each of the 64 data  paths and serializes them into 64 1-bit outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' It is programmed to  set the output bit-depth and the MSB position from the input word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Combined with scaler units, this is used to implement quantization  schemes such as LSQ [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Serialized outputs of each datapath are  grouped into a single 64-bit word that is sent either to the activation  RAM of the same MVU, or to a different MVU via an interconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='5  Interconnect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' MVUs can send data to each other via an inter-  connect implemented as an 8-way crossbar switch with broadcast  capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' A source MVU is programmed to send its output results  in a serialized fashion to a given address in the activation memory  of a destination MVU(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' At a destination MVU, a fixed-priority  arbitration scheme to the write port of the target MVU activation  RAM is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The interconnect is given highest priority, followed  by the controller, then lastly the MVU itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' When multiple MVUs  attempt to write to the same destination MVU, a fixed priority  scheme determines which MVU can write to its memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='6  DNN Mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Each MVU can be assigned to different lay-  ers of a DNN, such as convolutions and fully-connected layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Alternatively, a single layer can be split between multiple MVUs  with each MVUs processing a subset of the input activations and/or  weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Partial results are forwarded from one MVU to another  via the interconnect to process subsequent layers of the network,  thus creating an overall processing pipeline through the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' By  sending partial results from one MVU to another, subsequent MVUs  can begin processing as soon as sufficient data has been received  from previous layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' For instance, a MVU processing a 3× 3 convo-  lution requires only 3 rows of activations from the previous layer to  produce one output row of the layer it is processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' This avoids the  need to wait until all outputs from a layer are generated, which re-  duces latency and idle time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Furthermore, the ability to immediately  process partial layer outputs by subsequent MVUs keeps on-chip  storage requirements low, since only the partial set of activations  required to produce the next layer partial output needs to be stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Depending on the performance goal, BARVINN can execute a  DNN in either Pipelined mode or Distributed mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In Pipelined  mode (Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='a), the MVU array can process up to 8 convolutions  and fully-connected layers all at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Each MVU can be configured  to use different precisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In cases where a DNN model contains  more than 8 layers, the MVU array can be programmed to process  the entire model by dividing it into subsets of up to 8 layers each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Each MVU can be loaded with weights from layers in each subset,  either all from the start of processing if there is sufficient weight  memory available in each MVU or on-the-fly from external memory  if not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Output activations from the last MVU in the chain can also  be stored temporarily in off-chip memory and fetched later in the  case where the first MVU is still processing data from the current  lap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In the Distributed mode, to minimize latency, the objective is  to process single batch inputs as fast as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' As can be seen  in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='b, in this mode, the computation of a single layer is  broken into 8 independent computation regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' All MVUs will be  programmed to share the same set of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' To make sure an MVU  computation is independent of those performed on other MVUs, the  user might need to copy the input regions that are shared between  computation units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The programmability of BARVINN allows the  user to mix and match these execution modes for different layers  and models to achieve highest performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Pipelined mode  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Distributed mode  Figure 5: Execution flow of a DNN on the MVU array in  Pipelined (a) and Distributed (b) modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In Pipelined mode  each MVU processes one layer at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In distributed mode,  the computation of a single layer is distributed among mul-  tiple MVUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='2  Pito: RISC-V-based Controller  To make use of MVUs for neural networks, a control unit is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  The controller is a barrel RISC-V processor designed to control  the 8 MVUs using separate but communicating hardware threads  (harts) that each manage their respective MVUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' DNN layers are ex-  ecuted either in distributed or in a pipelined fashion, depending on  whether the DNN is compiled to maximize throughput or minimize  latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' This design allows MVUs to complete tensor operations  independently of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The drawback is that it requires 8  microprocessors to execute the 8 programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' We instead amortized  the fixed costs of the processor by adopting barrel processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' With  a 8-way threaded processor, we may assign one thread to control  each of the MVUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Because every thread comes up for execution  only every 8 clock cycles, the five pipeline stages (fetch, decode,  execute, data read & writes and commit) can be completely hidden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Branch prediction units are unnecessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Since tensor operations  can require hundreds of cycles to execute on a MVU, the barrel  processor can fully turn over dozens of times in the interim, allow-  ing each thread to issue the next command to its MVU in a few  instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  We adopted a Harvard architecture and divided the instruction  and data RAM, 8KB each, and shared between all harts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' This gives a  MemoryInterface  UART  MvU Array  MVU6  MU5  D$  IS  8KB  8KB  1  CSR8 pe_irq  pestart  CSR2 pe irq  Crossbar  CSR1 pe_irq  ant  Pito RISC-V  pe_start ld  pe_command httus  APB  pe_quant  pe_status  Input  Convo  Conv4  Conv1  Conv5  Conv2  Conv6  Input  Weight  Conv7  Output  Conv3  [1x63x32x32]  [64x64x3x3]  [1x63x32x32]ASPDAC ’23, January 16–19, 2023, Tokyo, Japan  AskariHemmat et al  1K word space to store data and instructions to control each MVU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  The processor executes instructions following compilation order  and without any further scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' A hart scheduler provides  access to the required resources for the hart at each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In the  fetch stage, each hart loads instructions from the instruction RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  The program counter (PC) and register file for each hart is different  and the hart scheduler indicates which register should be accessed  at a given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The Decode stage decodes instruction and loads  source registers or an immediate operand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Our RISC-V controller is  compatible with RV32I RISC-V ISA with minimal support for privi-  lege specification to make Control and Status Registers (CSRs) and  Interrupts available to interface with the MVU array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In addition  to the base CSRs, we have added 74 MVU-specific CSRs to allow  software to control the processing element array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' These CSRs con-  trol different settings within an MVU such as weight and activation  precision, AGU’s jump settings, input, weight and output memory  address and pipeline module selection as described in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='3  Code Generator  BARVINN performs GEMM/GEMV, Convolutions, Maxpooling and  activation (ReLU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' However, it is up to the user to sequence the  operations within a DNN with software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' To facilitate this, we de-  veloped a code generator that takes a DNN described in ONNX  [3] and configuration settings (weight/input/output precision), and  generates RISC-V code for each operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The code generator ex-  ports weights to the bit-transposed format described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Since each MVU works on 64-bit words, the code generator tiles  each weight tensor in blocks of 64×64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' When this cannot be done  (either tensor input channel or output channel is not a multiple of  64), we pad the corresponding tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Currently, our code generator  does not apply graph optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Also, for now, our  code generator supports Pipelined mode execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In the follow-  ing section, we used our code generator to map PyTorch models to  micro kernel codes which can then be directly used by BARVINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  4  PERFORMANCE ANALYSIS AND RESULTS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1  Experimental Setup  To illustrate the performance of BARVINN, we chose the ResNet9  image classifier model for the CIFAR10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' We trained and  quantized a ResNet9 model on CIFAR10 using LSQ [9] and used the  residual distillation [14] technique to remove shortcut connections  (Plain CNN models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In many image classification DNN models such  as ResNet [10], the input to the first layer typically consists of less  than 64 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Furthermore, due to sensitivity of the first and  last layer to information loss, most state-of-the-art compression  and quantization methods do not apply optimization on input and  output layers [9], hence keeping these layers untouched and in full  precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' We have adopted the same technique to compute first  and last layers on the host or on the RISC-V controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Table 2 shows the performance of ResNet9 on CIFAR10 in the  PyTorch framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Once we were satisfied with the performance  of our quantized model, we exported the trained model to ONNX  and then used our code generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Table 3 illustrates the per layer  computation cost of running ResNet9 on BARVINN with 2-bit ac-  tivations and weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' All convolutions use a padding of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' As  discussed before, we skipped running the first and last layer on  Table 2: ResNet9 with different bit precision on CIFAR10  ResNet9 Model  Precision  Accuracy  Size (Bytes)  Original  Fp32  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='8%  19605141  Plain-CNN  Fp32  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1%  18912487  Quantized Plain-CNN  Int2  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='2%  1181360  Table 3: ResNet9 layers for CIFAR10 dataset and computa-  tion cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' All layers are quantized to 2-bit for activation and  weights, except for the first and last layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Layer  Input  Kernel  Output  Cycles  conv0  [3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 32]  [64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 3]  [64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 32]  N/A  conv1  [64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content=' 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 32]  34560  conv2  [64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 32]  [64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content=' 32]  34560  conv3  [64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 32]  [128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 3]  [128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 16]  17280  conv4  [128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 16]  [128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content=' 3]  [128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 8]  32256  conv5  [128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 8]  [256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 8]  16128  conv6  [128,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content=' 3]  [256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 4]  13824  conv8  [256,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 4]  [512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 3]  [512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 4]  18432  fc  [512,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 4]  [10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 512]  [10]  N/A  Total:  194688  BARVINN and we kept them in their original format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' The overall  computation takes 194,688 cycles to complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Our design was written in Verilog and synthesized using Xilinx  Vivado 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1 for the Xilinx Alveo U250 accelerator card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Synthesis  results for the RISC-V controller, the processing array, and the accel-  erator are presented in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Power consumption was estimated  using the software tools in Vivado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='2  Discussion  We compared BARVINN with FINN [22], which is a templated Vi-  vado HLS C++ library of common DNN layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Like BARVINN,  FINN can generate hardware for arbitrary precision, but is not  software programmable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Hence, once the FINN hardware is gen-  erated, the user cannot change the computation data stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' We  attempted to compare the performance of BARVINN with FINN  using the ResNet9 model we used earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' However, at the time of  writing, FINN supports simple linear topologies and we were not  able to get performance metrics for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Instead, we used the  available CIFAR10-CNV model from the FINN repository that was  tuned for the FINN dataflow for our comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Table 5 shows the  performance of BARVINN and FINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' For this experiment, we used  different precisions for weights and activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' For both tools, we  used the performance estimation numbers for frames per second  (FPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' For FINN, we used the default folding configurations publicly  available in FINN-example repository [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' As illustrated in Table  5, we provide 7-15 times better throughput albeit with higher LUT  usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' On the other hand, for higher bit precisions, FINN provides  a better FPS/LUT, suggesting a scalable solution for bigger models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  We also compared the performance on a ResNet-50 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Table  6 shows our estimated FPS for BARVINN executing in Pipelined  mode along with reported performance for FINN [1] synthesized for  the Xilinx U250 and for FILM-QNN [20] synthesized for the Xilinx  ZCU102 FPGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' While FINN has the highest FPS, BARVINN shows  BARVINN: Arbitrary Precision DNN Accelerator Controlled by a RISC-V CPU  ASPDAC ’23, January 16–19, 2023, Tokyo, Japan  Table 4: Post-synthesis resource utilization of BARVINN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Resource  Pito RISC-V  MVU Array  Overall  LUT  10454  190625  201079  BRAM  15  1312  1327  DSP  0  512  512  Dynamic Power  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='410 W  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='066 W  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='504 W  Frequency  250 MHz  250 MHz  250 MHz  Table 5: Estimated performance of running CNV model on  CIFAR10 on Alveo U250 when different bit precision is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Bits  (W/A)  kLUT  BRAM  DSP  FPS  FPS/  kLUT  Ours  1/1  201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='0%)  1327  512  61035  303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='5  1/2  201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='0%)  1327  512  30517  151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='7  2/2  201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='0%)  1327  512  15258  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='8  FINN  1/1  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='1%)  150  0  7716  273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='6  1/2  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='8(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='47%)  103  0  2170  109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='6  2/2  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='3(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='81%)  202  0  2170  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='3  Table 6: Performance for ResNet-50 model on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  Bits (W/A)  Clock Freq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  FPS  FPS/Watt  Ours  1/2  250 MHz  2296  106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='8  FINN-R [1][6]  1/2  178 MHz  2873  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='0  FILM-QNN [20]  4(8)/5  150 MHz  109  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='4  the best performance per Watt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' According to the FINN-example  repository [1], a fine-tuned ResNet50 model, requires more than  87% of Alveo U250 accelerator’s resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' This shows the limits  of FINN dealing with bigger models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' BARVINN requires the same  LUT usage regardless of the model size and bit-width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  5  CONCLUSION  In this paper, we presented an FPGA-based DNN accelerator that  supports arbitrary bit precision computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' We tested the perfor-  mance of BARVINN over different DNN kernels and models with  different bit precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' For model deployment, we developed a code  generator tool that takes in a model in ONNX format and generates  RISC-V assembly code for the controller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Compared to other low  precision accelerators, we provide a programmable solution which  makes BARVINN more flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' With the programmable MVUs, the  user can run different models regardless of their size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' BARVINN  allows trading off throughput and latency by running DNN lay-  ers either in distributed or in pipeline modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Unlike other low  precision accelerators, our proposed solution offers implementing  various trade-offs through software and the end user can control  them for each individual layer without FPGA reconfiguration at  run time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Compared to programmable accelerators, BARVINN was  shown to provide a better throughput per Watt performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The authors acknowledge support for this project from the IBM  AI Horizons Network, CMC Microsystems, Fonds de Recherche du  Quebec–Nature et Technologies (FRQNT), MITACS and from the  NSERC COHESA Strategic Research Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  REFERENCES  [1] 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' Bit-  Slicing FPGA Accelerator for Quantized Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' In 2019 IEEE Interna-  tional Symposium on Circuits and Systems (ISCAS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content=' 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
+page_content='  [6] Michaela Blott, Thomas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content=' Fraser, Giulio Gambardella, Ken-  neth O’brien, Yaman Umuroglu, Miriam Leeser, and Kees Vissers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content=' Bit-Mixer: Mixed-Precision  Networks With Runtime Bit-Width Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content=' Automatic Generation of High-Performance Quantized Machine Learning  Kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content='org/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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+page_content=' DNNBuilder: an Automated Tool for Building High-  Performance DNN Hardware Accelerators for FPGAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NAyT4oBgHgl3EQfcvfE/content/2301.00290v1.pdf'}
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diff --git a/7NE1T4oBgHgl3EQf7QUc/content/tmp_files/2301.03531v1.pdf.txt b/7NE1T4oBgHgl3EQf7QUc/content/tmp_files/2301.03531v1.pdf.txt
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+ 
+ 
+1 
+Abstract— Objectives: Identifying suicidality including suicidal 
+ideation, attempts, and risk factors in electronic health record data 
+in clinical notes is difficult. A major difficulty is the lack of training 
+samples given the small number of true positive instances among 
+the increasingly large number of patients being screened.  This 
+paper describes a novel methodology that identifies suicidality in 
+clinical notes by addressing this data sparsity issue through zero-
+shot learning. Materials and Methods: U.S. Veterans Affairs 
+clinical notes served as data. The training dataset label was 
+determined using diagnostic codes of suicide attempt and self-
+harm. A base string associated with the target label of suicidality 
+was used to provide auxiliary information by narrowing the 
+positive training cases to those containing the base string. A deep 
+neural network was trained by mapping the training documents’ 
+contents to a semantic space.  For comparison, we trained another 
+deep neural network using the identical training dataset labels and 
+bag-of-words features. Results: The zero shot learning model 
+outperformed the baseline model in terms of AUC, sensitivity, 
+specificity, and positive predictive value at multiple probability 
+thresholds. In applying a 0.90 probability threshold, the 
+methodology identified notes not associated with a relevant ICD-
+10-CM code that documented suicidality, with 94% accuracy. 
+Conclusion: This new method can effectively identify suicidality 
+without requiring manual annotation.   
+ 
+Keywords— Suicide, Clinical Notes, NLP, Zero-Shot Learning 
+I. INTRODUCTION 
+uicide is a significant problem in the United States, 
+increasing 35.2% from 1999 to 2018, and from 10.5 to 14.2 
+suicides per every 100,000 individuals in that same time period 
+[1] In 2020, 45,979 people died from suicide, and 
+approximately 1.2 million attempted suicide in the United 
+States [2] Its estimated cost is over $70 billion annually in lost 
+productivity and medical care [3]; this calculation does not 
+include residual costs from the estimated 4-17 people closely 
+tied to the suicide decedent who are left bereaved [4]. Suicide, 
+however, is a complicated problem that includes a dynamic web 
+of individual-level risk factors (e.g., depression, substance use 
+behaviors, personality traits), interpersonal risk factors (e.g., 
+violence, victimization), and community-level factors (e.g., 
+unemployment, stigmatization of mental illness) [5, 6].   
+ 
+1Biomedical Informatics Center; The George Washington University; 
+Washington DC, USA; 
+2VA Medical Center, Washington, DC, USA; 
+3Department of Emergency Medicine, Yale School of Medicine, New Haven, 
+CT, USA; 4PRIME Center, VA Connecticut Healthcare System, West Haven, 
+CT, USA; 5Research, VA Connecticut Healthcare System, West Haven, CT, 
+Veterans are especially affected by suicide, with an age- and 
+sex-adjusted rate that is 1.5 times higher than nonveterans [7]. 
+The Department of Veterans Affairs (VA) operates the single 
+largest integrated health care system in the U.S., and has 
+devoted resources to suicide prevention, including the Suicide 
+Prevention Applications Network (SPAN), embedding suicide 
+prevention coordinators and special reporting measures in 
+facilities [8], increased mental health staffing, partnerships with 
+community care organizations, and enhanced surveillance and 
+monitoring through its electronic health record (EHR) system 
+[9, 10]. Additionally, the VA has continual efforts to develop 
+predictive analytics to identify patients at the highest risk of 
+suicide [8, 11] The data elements for these predictive analytic 
+algorithms rely on structured data (e.g., International 
+Classification of Disease [ICD] diagnosis codes, prescription 
+data, socio-demographic data, care utilization metrics) [12] 
+which often provide an incomplete record [13, 14]. Less is 
+known about how unstructured data, such as contained in 
+clinical notes, can contribute to suicidality (i.e., suicidal 
+ideation or attempt) identification and prevention. Given that a 
+suicide attempt is one of the greatest risk factors for subsequent 
+suicide death, a more thorough means of detecting such events 
+is warranted [15].  
+A. Background and Significance 
+Natural language processing (NLP) combined with machine 
+learning may add value to suicide documentation research.  
+Supervised machine learning methods use “supervised”, or pre-
+classified data.  However, naïve attempts at note retrieval using 
+keyword search alone quickly demonstrate the difficulty of this 
+problem, as words such as “suicide” occur in standard 
+questionnaires which are included in many notes, with few 
+actually documenting suicidality. For instance, in a prior 
+experiment we carried out, we randomly collected 1,000 VA 
+notes containing the term “suicidal” or “suicide” from 1,000 
+individual patients and performed manual chart review for 
+affirmed suicidality. Only 1.57% of these notes documented 
+actual suicidality. Patient reluctance to disclose suicidal 
+ideation provides a further complicating factor [16, 17]. As a 
+result, a patient’s negative response to a suicide ideation inquiry 
+may not reflect their real feelings or intentions.  Additionally, 
+USA; 6VA Connecticut Healthcare System, West Haven, CT, USA; 7Suzanne 
+Dworak-Peck School of Social Work, University of Southern California, Los 
+Angeles, CA, USA; 8Department of Internal Medicine, Yale School of 
+Medicine, West Haven, CT; 
+ 
+Leveraging Contextual Relatedness to Identify Suicide 
+Documentation in Clinical Notes through Zero Shot 
+Learning 
+T. Elizabeth Workman, Ph.D.1,2, Joseph L. Goulet, Ph.D.3,6, Cynthia A. Brandt, M.D.3,6, Allison R. 
+Warren, Ph.D.4, Jacob Eleazer, Ph.D.4, Melissa Skanderson, M.S.W.5, Luke Lindemann, Ph.D.6, John 
+R. Blosnich, Ph.D.7, John O’Leary, M.Ed.6,8, Qing Zeng-Treitler, Ph.D.1,2  
+S 
+
+ 
+ 
+2 
+relying on structured data alone will result in incomplete 
+identification of patients who have or are experiencing 
+suicidality, because relevant coding is prone to underuse [8].  
+However, not all clinical notes associated with relevant 
+structured data document suicidality.  For example, a note 
+documenting a secondary service such as group therapy, or a 
+note documenting fluid intake may not directly document 
+suicidality.  
+Prior attempts to apply NLP and machine learning are often 
+limited to mental health-oriented notes and may suffer if using 
+imbalanced data. Levis et al.[18] applied sentiment analysis and 
+various machine learning algorithms to classify suicide, using 
+VA psychotherapy notes, yielding area under the curve (AUC) 
+ratings comparable to chance. Fernandes et al.[19] obtained 
+excellent NLP performance in their study of clinical notes from 
+the Clinical Record Initiative Search (CRIS), but performance 
+was computed after removing neutral (non-suicide) results from 
+their machine learning output. Carson et al. enriched notes 
+associated with suicide attempt that were then used to train a 
+random forest model achieving 83% sensitivity, but only 22% 
+specificity [20].  Cook et al. [21] applied a bag-of-words 
+approach with machine learning to identify suicide ideation and 
+psychiatric symptoms using notes for patients identified as 
+having performed self-harm, achieving 61% PPV (positive 
+predictive value), 59% sensitivity, and 60% specificity, with 
+results varying  depending on the task. Zhang et al. sought to 
+identify psychological stressors using a pre-annotated dataset of 
+psychiatric evaluation records from the CEGS N-GRID 2016 
+challenge [22] as a gold standard, for a conditional random 
+fields machine learning model, [23] yielding final F scores of 
+73.91% and 89.01%, respectively, on exact and inexact stressor 
+matching, and 97.73% and 100% respectively, for exact and 
+inexact suicide recognition on instances of the positive 
+keywords with the stressors; however, their evaluation methods 
+for this are not detailed.   
+Zhong et al. applied structured data and NLP to identify 
+suicidal behavior in pregnant women, achieving PPV of 76% 
+and 30%, for women identified through relevant diagnostic 
+codes and through NLP for women not receiving a relevant 
+diagnostic code, respectively [24].  Obeid et al.[25] trained a 
+convolutional neural network that achieved an AUC of 0.882 
+and an F1 score of 0.769 in predicting relevant suicide ICD 
+codes in subsequent years.  Using notes from psychiatric 
+encounters, Cusick et al. [26] developed a rule-based NLP tool 
+to identify positive instances of suicide-oriented keywords that 
+leveraged NegEx. [27] They also developed different weakly-
+supervised machine learning models. A convolutional neural 
+network receiving Word2Vec [28] word embeddings as input 
+achieved precision, recall, F1 score, and AUC values of 0.81, 
+0.83, 0.82, and 0.946.  In a subsequent evaluation the 
+convolutional neural network correctly classified 87% of the 23 
+notes (of 5000 clinical notes) receiving a positive classification, 
+from notes for patients diagnosed with depression or prescribed 
+an antidepressant.  In a related task Tsui et al. [29] used prior 
+structured and unstructured data (clinical notes from history, 
+physical examination, progress notes and discharge summaries) 
+of inpatient and emergency room patients with a coded suicide 
+attempt, to identify first-time suicide attempts in a case-control 
+study.  An ensemble of extreme gradient boosting (EXGB) 
+yielded best performance, with an AUC ranging from 91.9% to 
+93.2%, according to time window between prior data and 
+suicide attempt diagnosis.  Recently, Rozova et al. obtained 
+promising results (87% AUC) using a gradient boosting model, 
+although the study was limited to emergency room triage notes 
+[30]. 
+Seeking suicidality in all types of clinical notes, among all 
+types of patients, or when hampered by imbalanced data, is 
+indeed a complex task. Some of the methods in the papers cited 
+above tend to suffer from low precision, specificity, and 
+possibly also low sensitivity (recall).  Identifying probability 
+thresholds addresses these problems, providing flexibility for a 
+given task. For example, a high probability threshold (e.g., the 
+top ten percent) can serve as a means for identifying 
+documentation indicating suicidality and its risk with high 
+precision. When the prevalence is very low, which is often the 
+case of true positive suicidality documentation, the optimal 
+threshold needs to balance metrics such as the true positive rate 
+(sensitivity, also known as recall), specificity, and the positive 
+predictive value (precision).  A strategic implementation of a 
+technique like Zero-Shot Learning may also provide accurate 
+identification of suicidality in clinical notes. 
+B.   Zero-Shot Learning 
+Zero-Shot Learning (ZSL) enables predictions on unseen 
+data using a model trained on data that has labels that are 
+different than those of the unseen data [31, 32].  It largely 
+operates by mapping select properties of the data (i.e., the 
+“feature space”) to a semantic representation (i.e., the “semantic 
+space”) that enables prediction of unseen classes [33]. In other 
+words, auxiliary information must be provided on the labels of 
+the unseen classes to make it possible for a trained model to 
+recognize them in the testing data.   
+ ZSL has been applied in several computer vision tasks [34, 
+35], as well as NLP tasks [36].  Accordingly, a feature space 
+can consist of data derived from images [37] or text [36].  The 
+semantic representation can be based on several different 
+approaches, including data attributes, semantic word vectors as 
+those provided by skip-gram or continuous-bag-of-word  
+architectures [33] or BERT output [38], or knowledge graphs 
+[33].  Examples in NLP applications include semantic utterance 
+classification [39] multilingual translation [40] and emotion 
+detection [41]. However, other than Sivarajkumar and Wang’s 
+work [38] there is little ZSL research in unstructured clinical 
+text data.  
+Naturally, different semantic representations affect the 
+accuracy of ZSL [42]. In this study, we leveraged word 
+embedding and usage context.    
+C. Objectives 
+We investigated a ZSL methodology applied to a binary 
+suicidality classification task. The training dataset was 
+constructed using diagnostic codes (ICD-10-CM codes) related 
+to suicide. Our target label is the broader concept of suicidality. 
+To enable ZSL, a base string representing suicidality was 
+
+ 
+ 
+3 
+selected. We then built the semantic space by identifying key 
+features associated with suicidality in the training dataset. A 
+DNN model was developed using the training data and tested 
+on two different sets of unseen data with the unseen label of 
+suicidality. Specifically, we sought to answer: 
+ Will ZSL effectively identify suicidality documentation 
+from among all types of clinical notes, using review by 
+clinicians as the reference standard? 
+ Will ZSL effectively identify suicidality or suicide risk 
+documentation from among clinical notes not associated with 
+a relevant ICD-10-CM code, by probability threshold, in terms 
+of precision, using the same reference standard? 
+We are unaware of previous descriptions of this methodology 
+and to our knowledge it has not been used prior to this study.  
+II. METHODS 
+A. Training Data 
+A training dataset was created using two corpora. The first 
+corpus consisted of 50,000 randomly selected VA clinical notes 
+from outpatient encounters recorded between 2016 and 2019 
+which contained the base string “suicid” (e.g. “suicide”, 
+“suicidal” ) and were associated with at least one ICD-CM-10 
+code identified by the National Health Statistics Report from 
+the Centers for Disease Control and Prevention (CDC) 
+indicating suicide attempt or intentional self-harm.[43]  This 
+corpus is referred to as stringAndDx (9170 unique patients). 
+The second corpus consisted of 50,000 randomly selected VA 
+clinical notes from outpatient encounters recorded between 
+2016 and 2019 that were associated with other ICD-CM-10 
+codes that were irrelevant to suicidality or self-harm. These 
+notes were extracted from patients matching the stringAndDx 
+patients in age (at the time of document retrieval), race, and 
+ethnicity.  This second corpus is referred to as noDx (8638 
+unique patients).  Each corpus was preprocessed by 
+transforming all letters to lower case, removing basic 
+formatting markup and punctuation, separating character 
+strings into tokens (words), separating relevant concatenated 
+tokens (e.g., “suicidalhomicidal” to “suicidal” ”homicidal”), 
+and removing all tokens that did not entirely consist of letters.   
+B.  Semantic Space Feature Extraction and Mapping 
+The task to build the semantic space was carried out in three 
+steps: First, we identified a list of features that are potentially 
+relevant for the positive training label. Second, we created word 
+embeddings using a skip-gram architecture. Third, we 
+identified context words of the selected features using the word 
+embeddings.  In a fourth step, a contextual weight is assigned 
+to each feature for each document in mapping the semantic 
+space to the feature space.  
+In the first step, inverse document frequency (TF-IDF) 
+analysis was used to identify the n most important terms in each 
+corpus.  For this investigation, n = 1000.  TF-IDF evaluates 
+term frequency using the count of documents containing a given 
+term.  In each document, the relative frequency of each term is 
+weighted by the log of the number of documents in the corpus 
+divided by the number of documents containing the term, as 
+shown in (1) 
+𝑡𝑖,𝑗 = 𝑡𝑓𝑖,𝑗 ∗ 𝑙𝑜𝑔( 𝑛
+𝑑𝑓𝑖
+) 
+ 
+where ti,j is term i in document j, tfi,j is the relative frequency of 
+term i in document j, n is the total number of documents, and 
+dfi is the number of documents containing term i.  Because TF-
+IDF is a document-based measurement, we used the mean TF-
+IDF value for each term in its respective corpus.   The words 
+with the top TF-IDF scores that are unique to the stringAndDx 
+corpus were treated as features.  Figure 1 illustrates this process.  
+Each circle represents terms from one of the corpora. Sets a and 
+b are the words with the top n TFIDF scores for stringAndDx 
+and noDx, respectively. Set c is the overlap between a and b. 
+The feature set F contains words that are in set a, but not in the 
+overlap set c or in set b (f  a and f  c and f  b).  
+ 
+ 
+Figure 1. Feature identification. Words that are deemed as features are in set a, 
+excluding words in c and b. 
+ 
+In the second step, we created a Word2Vec model using the 
+stringAndDx corpus. In this study, the model was a shallow 
+neural network with the hidden layer containing 300 nodes, 
+applying the skip-gram architecture, with an analytic window 
+size of 5, trained through 10 iterations. 
+In the third step, we identified the top m context words for 
+each feature word using the word embeddings from the 
+Word2Vec model. The m words most similar to each feature 
+word, according to cosine similarity values, served as its 
+context words. In this investigation, m = 50. 
+In the fourth step, we map the feature space, i.e. a document’s 
+preprocessed content, to the semantic space. A weight v is 
+assigned to each feature word for each document, based on its 
+occurrence with its context words in a window in the 
+document’s text.  This weight is the summed total of the cosine 
+similarity between the feature and a co-occurring context word 
+multiplied by the mean TF-IDF value of the feature word. The 
+formula is shown in (2) 
+ 
+𝑣 =
+∑
+𝑐𝑜𝑠𝑆𝑖𝑚(𝑥, 𝑦) ∗ 𝑡𝑓𝑖𝑑𝑓(𝑥)
+𝑥∈𝐹,𝑦∈𝐷
+ 
+ 
+where x is a feature in F, the set of features in the semantic 
+space, and y is a context word of set D, the context words for x 
+in the semantic space, which occurs in a five-word window 
+around x in the document’s text. This process is illustrated in 
+Figure 2, where “pattern” (highlighted in light gray) is a feature 
+word, and “internalizing” and “fitful” (highlighted in dark gray) 
+are among its set of context words and appear in a five-word 
+window. 
+(2) 
+(1) 
+
+stringAndDx
+noDx
+a
+c
+6 
+ 
+4 
+ 
+ 
+Figure 2.  Example of deriving a feature weight using (2) 
+ 
+If a feature word is not in the text, its value is zero for the 
+given document. 
+C. Model Development  
+20,000 documents were randomly selected from each corpus 
+(stringAndDx and noDx). We trained a DNN model (here 
+referred to as the ZSL DNN) consisting of five fully-connected 
+hidden layers of alternating sizes of 30 or 70 nodes, with each 
+layer implementing a dropout rate of 0.5.  We implemented the 
+Adam optimizer [44], with a learning rate of 0.0012, beta 1 
+value of 0.92, beta 2 value of 0.9992, and an epsilon value of 
+1e-08, with binary cross entropy as the loss function, and the 
+sigmoid function in the output layer, since it was a binary 
+classification task.  The architecture and hyperparameters were 
+chosen on empirical grounds, after experimentation.  Each 
+document from the stringAndDx corpus was classified as “1” (a 
+generic positive instance), and each document from the noDx 
+corpus was classified as “0” (a generic negative instance).  
+These labels do not indicate whether or not the given document 
+directly pertains, or not pertains, to suicidality or its risks, but 
+an association with a structured data element, and for those 
+labeled “1”, also containing a base string.  Balancing the 
+positive and negative approximated training datasets in this 
+manner (i.e., providing balanced training examples) addressed 
+the problematic issue of otherwise training a model with few 
+positive and many negative instances. We implemented a 60% 
+training, 20% validation, and 20% testing split in developing 
+the ZSL DNN.  Figure 3 illustrates the method. 
+ 
+   
+Figure 3. Method. The corpora stringAndDx (2016-19), noDx (2016-19), 
+testSet1 (2020), and testSet2 (2020) are unique and extracted from all clinical 
+notes based on associated ICD-10-CM codes, and in the case of 
+stringAndDx, where a base string is also present; corpora content is 
+preprocessed.  The stringAndDx and noDx corpora are used in the TF-IDF 
+analysis to identify feature words that are unique to stringAndDx (step 1). 
+stringAndDx is applied to a skip-gram model to produce word embeddings 
+(step 2). Feature words and their significant context words (determined 
+through the word embeddings) form the semantic space (step 3). The 
+contents of stringAndDx and noDx are mapped to the semantic space, using 
+a function to determine feature word weights (step 4). The mapped contents 
+of stringAndDx and noDx documents are used to train the ZSL DNN, using 
+generic labels 1 and 0, respectively.    The mapped contents of unseen 
+testSet1 and testSet2 notes were classified by the trained ZSL DNN, for the 
+classes (a) containing suicidality documentation, or (b) not containing 
+suicidality documentation.  Human annotation independently classified 
+random documents from testSet1 and testSet2 for the same classes (a) 
+containing suicidality documentation, or (b) not containing suicidality 
+documentation; human annotation also assessed documents from testSet2 
+containing the base string that received a probability of 0.90 or greater, for 
+these classes and suicidality risk factors. 
+ 
+D. Evaluation 
+The authors randomly retrieved 5,000 different clinical notes 
+recorded in 2020 that were associated with at least one of the 
+relevant IDC-10-CM codes. This corpus is subsequently 
+labeled as testSet1. The authors also randomly retrieved 5,000 
+different clinical notes recorded in 2020 that were associated 
+with other IDC-10-CM codes irrelevant to suicidality or self-
+harm.  This corpus is subsequently labeled testSet2.   
+The contents of each of the notes in testSet1 and testSet2 were 
+mapped to the semantic space, i.e., deriving a weight for each 
+feature word as described earlier in the fourth step.  Then, the 
+trained ZSL DNN was used to classify the notes in testSet1 and 
+testSet2 as (a) containing suicidality documentation, or (b) not 
+containing suicidality documentation.   
+In joint sessions, two clinical psychologists familiar with VA 
+clinical note documentation together identified suicidality (i.e., 
+current or past suicide ideation or attempt) in 200 notes 
+randomly selected from testSet1 and testSet2 (100 from each 
+test set), after being instructed to look for documentation for 
+these specific events. They addressed differences of opinion 
+through discussion and mutual consensus during the joint 
+sessions.  In a second evaluation, to explore how the 
+application’s output may serve to identify patients who had 
+experienced or were at risk for suicidality, but never formally 
+diagnosed as such, the clinicians examined the testSet2 notes 
+containing the base string “suicid" that received a probability 
+value of 0.90 or greater from the trained ZSL DNN, for 
+documentation of suicidality and/or its risk factors, according 
+to NIH guidelines.[45] This threshold was chosen in order to 
+explore how high-probability documents (i.e. the top 10% in 
+terms of probability) would be representative in identifying 
+documented suicidality or its risk factors with high precision, 
+thus addressing our second question.   
+1) Baseline Comparison 
+For comparative purposes, the 163 most frequent bigrams 
+unique to the stringAndDx corpus were identified and used in a 
+bag-of-words baseline model.  We trained a DNN (here referred 
+to as the Baseline DNN) using these 163 bigrams as features for 
+the 20,000 stringAndDx documents and the 20,000 noDx 
+documents. This baseline DNN was also used to classify the 
+Document Text: “The patient has a pattern of internalizing 
+criticism from his family.  This pattern sometimes results in fitful 
+outbursts.” 
+ 
+TF-IDF value of feature word “pattern”: 0.0062 
+Cosine similarity of “pattern” and “internalizing”: 0.4673 
+Cosine similarity of “pattern” and “fitful”: 0.3824 
+Feature weight for “pattern”:  
+(0.0062 * 0.4673) + (0.0062 * 0.3824) = 0.0053 
+
+All Clinical
+Notes
+ICDcodes
+ICDcodes
+Base String
+Human
+testSetl
+testSet2
+stringAndDx
+noDx
+Annotation
+(suicidality)
+TFIDF
+Analysis
+Word
+Embeddings
+Semantic
+Map content
+Space
+Mapcontent
+Testing
+DNN
+Training
+Output is a classification of (a) containing suicidality
+documentation or (b) not containing suicidality documentation 
+ 
+5 
+notes in testSet1 and testSet2, for (a) containing suicidality 
+documentation, or (b) not containing suicidality documentation, 
+using the 163 most frequent bigrams as features. 
+III. RESULTS 
+The first step of the new method (described in Methods) 
+identified 163 feature words associated with suicidality 
+diagnosis.  The top thirty feature words are listed in Table I. No 
+form of the base string “suicid” was found among the 163 final 
+feature words. Both “suicide” and “suicidal” were prominent 
+terms in both the noDx and stringAndDx corpora, along with 
+terms like “psychiatrist” and “psychosocial”; this is likely due 
+to the proliferation of objects like questionnaires, and mental 
+health care documentation in notes that are unrelated to 
+suicidality. 
+TABLE I 
+TOP 30 FEATURE WORDS 
+flag 
+overdose 
+coordinator 
+took 
+spc 
+observation 
+called 
+warning 
+pills 
+prf 
+unknown 
+interrupted 
+gun 
+placement 
+lcsw 
+lethal 
+outcome 
+reportedly 
+notified 
+sdv 
+occurred 
+police 
+protocol 
+od 
+supports 
+seeking 
+category 
+preparatory 
+cut 
+determined 
+ 
+A. ZSL DNN and Baseline DNN Performance 
+The classifications by the clinicians and the probabilities 
+assigned by the ZSL DNN and the Baseline DNN were first 
+assessed by AUC score. The results are in Table II and Figure 
+4. 
+ 
+TABLE II 
+AUC PERFORMANCE 
+ZSL DNN 
+Baseline DNN 
+0.946 
+0.47 
+ 
+ 
+Figure 4. ZSL DNN AUC results (left), Baseline DNN AUC results (right) 
+ 
+In terms of AUC, the ZSL DNN trained through mapping the 
+semantic space to the feature space outperformed the Baseline 
+DNN trained with the bigram bag-of-words features.  
+The sensitivity, specificity, and PPV results at 0.15, 0.5, and 
+0.85 probability thresholds for each DNN are in Tables III-V.  
+Probability refers to the probability the DNN assigned to each 
+note for positive suicidality documentation. We applied the 
+median probability (0.1499, rounded) assigned by the ZSL 
+DNN to the testSet2 documents (the test set containing random 
+notes associated with irrelevant ICD-10-CM codes) in forming 
+minimum and maximum thresholds; 0.5 is a standard midpoint 
+probability threshold. The combined scores in these tables were 
+computed with all true positives, true negatives, false positives, 
+and false negatives for both test sets, for the indicated metrics.  
+Values of NaN (not a number) occurred where there were no 
+true positives or false positives. 
+ 
+TABLE III 
+EVALUATION RESULTS AT 0.15 PROBABILITY THRESHOLD 
+ZSL DNN 
+Sensitivity/Recall  
+Specificity 
+Precision/PPV 
+testSet1 
+97% 
+100% 
+91% 
+testSet2 
+100% 
+64% 
+05% 
+Combined 
+97% 
+59% 
+67% 
+Baseline DNN 
+ 
+ 
+ 
+testSet1 
+99% 
+0% 
+90% 
+testSet2 
+50% 
+09% 
+01% 
+Combined 
+98% 
+08% 
+48% 
+ 
+TABLE IV 
+EVALUATION RESULTS AT 0.5 PROBABILITY THRESHOLD 
+ZSL DNN 
+Sensitivity/Recall  
+Specificity 
+Precision/PPV 
+testSet1 
+92% 
+40% 
+93% 
+testSet2 
+50% 
+97% 
+25% 
+Combined 
+91% 
+92% 
+90% 
+Baseline DNN 
+ 
+ 
+ 
+testSet1 
+92% 
+0% 
+89% 
+testSet2 
+50% 
+10% 
+1% 
+Combined 
+91% 
+9% 
+46% 
+ 
+TABLE V 
+EVALUATION RESULTS AT 0.85 PROBABILITY THRESHOLD 
+ZSL DNN 
+Sensitivity/Recall  
+Specificity 
+Precision/PPV 
+testSet1 
+77% 
+70% 
+96% 
+testSet2 
+50% 
+100% 
+100% 
+Combined 
+76% 
+97% 
+96% 
+Baseline DNN 
+ 
+ 
+ 
+testSet1 
+0% 
+100% 
+NaN/div by 0 
+testSet2 
+0% 
+100% 
+NaN/div by 0 
+Combined 
+0% 
+100% 
+NaN/div by 0 
+ 
+The ZSL DNN outperformed the Baseline DNN in most 
+metrics at all probability thresholds.   
+B. Second Evaluation 
+To explore how this new methodology can identify clinical 
+notes documenting suicidality that are not associated with a 
+relevant ICD-10-CM code with high precision, the clinicians 
+also reviewed the 16 notes from testSet2 containing the base 
+string “suicid’ that received a probability at or above 0.90 from 
+the trained ZSL DNN. The clinicians noted suicide ideation or 
+attempt, and the presence of the following suicide risk factors, 
+based on National Institute of Mental Health guidelines [45]: 
+ Depression and other mental health disorders 
+ Substance abuse disorder 
+ Family history of a mental health or substance abuse 
+disorder 
+ Family history of suicide 
+ Family violence, including physical or sexual abuse 
+ Having guns or other firearms in the home 
+ Being in prison or jail 
+ Being exposed to others’ suicidal behavior 
+Of these 16 clinical notes (associated with 16 different 
+patients), 7 documented current or past suicide ideation or 
+attempt.  Eight of the remaining notes included one or more 
+risk factors for suicide (nearly all included multiple risk 
+factors). In all, 15 of the 16 notes contained documentation of 
+current or past suicide ideation or attempt, and/or suicide risk 
+ 
+ 
+
+1.0
+model results
+0.0
+0.2
+0.4
+0.8
+1.Dno distinction
+model results
+0.8-
+2 0.4
+0.2 
+0.0 
+0.0
+Q2
+0.4
+False Positive Rate
+0.8
+10  
+ 
+6 
+factors, for patients who had never received a suicidality ICD-
+10-CM code diagnosis during the study period, achieving a 
+PPV of 93.8%. 
+IV. DISCUSSION 
+Regarding the study’s original questions, our ZSL approach 
+effectively identified suicidality in all types of clinical notes, 
+surpassing the performance of the bag-of-words baseline in 
+conjunction with deep learning. It also effectively identified 
+suicidality or suicide risk documentation from among clinical 
+notes not associated with a relevant ICD-10-CM code with high 
+precision, on probability threshold. 
+A. Semantic Space 
+In this work, the semantic space development is framed as  
+feature extraction where mapping is enhanced by attaching 
+weights to features found in the data, an approach also used in 
+computer vision ZSL [46]. The semantic space captures natural 
+data properties by identifying salient terms and relevant 
+contextual 
+terms 
+in 
+collective 
+clinical 
+suicidality 
+documentation (i.e., a corpus of notes associated with relevant 
+ICD codes). Table 1 lists 30 prominent feature words associated 
+with collective suicidality documentation after removing terms 
+associated with other kinds of documents. There is an intuitive 
+sense to these words; “flag” is found in the phrase “high risk for 
+suicide flag”; “overdose” and “cut” refer to suicide methods; 
+“pills” and “gun” refer to suicide instruments. Identifying terms 
+contextually similar to these provides patterns in relevant 
+documentation.  Again, this has an intuitive logic. The most 
+contextually similar terms to “flag” include “reactivate” and 
+“deactivate” (for a high suicide risk flag) and “high” (the level 
+of risk).  The most contextually similar terms to “pills” include 
+“handful”, “fistfuls”, and “bunch”, implying large quantities, 
+along with “overdosing” and “took”, the associated actions.  
+The feature word  “spc” indicates VA’s suicide prevention 
+coordinators, which is a structural change that VA implemented 
+for suicide prevention [10]. Concordantly, “police” and “lcsw” 
+(i.e., licensed clinical social worker) refer to other professions 
+highly associated with individuals at risk for suicide. For 
+example, police may be activated for a rescue, and a licensed 
+clinical social worker may be involved in treatment planning or 
+referral connections for suicidal individuals. The feature words 
+“prf” and “sdv” refer to “patient record flag” and “self-directed 
+violence”, respectively.  The semantic space provided an 
+efficient representation for effective mapping to the feature 
+space.   
+B. Data Retrieval and Model Training 
+Using associated structured data elements like ICD-10-CM 
+codes, and a base string provides a means to locate equally sized 
+corpora for training that could be generically labeled “0” or “1”.  
+These labels were primarily based on a structured data 
+association, since their individual unstructured content was 
+mostly unknown. This approach solves the issue of imbalanced 
+training data. The predominant clinical note types (Appendix) 
+also illustrate this.  Most of the frequent note types associated 
+with one of the relevant CDC ICD-10-CM codes and containing 
+the base string are relevant to suicidality.  Addendum is a 
+common note type [47] associated with many domains [48]. 
+The most frequent note types not associated with a relevant 
+code resemble frequencies of all note types in the VA [47].  
+C. Identifying Suicidality Documentation 
+To our knowledge, this method has not been applied in other 
+studies.  Unlike VA surveillance methods using structured data, 
+it also leverages information found in EHR notes. Also, unlike 
+other NLP methods [18, 20, 21, 23, 24, 26, 29, 30] it can be 
+applied to all patients and note types.  In other studies, a bag-
+of-words approach has been applied to suicidality identification 
+and other machine learning tasks [21, 49, 50].  However, the 
+results of this current study suggest that the complexity of 
+suicidality documentation demands a more targeted approach.   
+This method could complement existing measures like 
+SPAN, alerting suicide prevention coordinators of additional 
+patients at risk.  The results of the two clinical psychologists’ 
+evaluations demonstrate the method’s efficiency in identifying 
+suicidality documentation for documents where there is no 
+relevant ICD-10-CM code. The performance on both test sets 
+demonstrates the methodology’s effectiveness in classifying 
+notes that are mixed in terms of ICD-10-CM coding. 
+Tables III - V suggest that the probability threshold can be 
+adjusted to suit a specific task like finding suicidality and its 
+risk factors with high precision among notes not associated with 
+a relevant ICD-10-CM code. This is especially true considering 
+the small prevalence of suicidality documentation in clinical 
+notes.  The second evaluation (which yielded 93.8% PPV) 
+demonstrates this.  By applying a high probability threshold of 
+0.90 to all 5000 testSet2 documents and focusing on clinical 
+notes containing the base string, of the 16 documents (for 16 
+different patients), 94% contained suicidality and/or suicidality 
+risk factor documentation, based on clinician review. These 
+results exceed those of Cusick et al.’s [26] similar task, where 
+87% of notes were correctly classified, among notes for patients 
+diagnosed with depression or prescribed an antidepressant.  In 
+this current study’s second evaluation, none of the 16 patients 
+identified had ever received a suicide ICD-10-CM code during 
+the study’s time period.  It is impossible to know if the patients 
+in the 8 notes simply containing documented risk factors were 
+suicidal or not based solely on electronic health records. 
+Suicidal patients sometimes deny suicide ideation or attempt 
+[16, 51].  For example, in one note from the chart review 
+associated with a relevant ICD-10-CM code, the patient 
+reportedly denied suicide ideation, even after checking into the 
+hospital hours earlier for a self-reported suicide attempt. 
+D. Future Work 
+ This work is part of a larger study of patients at risk for 
+suicide.[52] The next step is to combine these findings with 
+prior work. We also plan an analysis of patients from first 
+suicide ideation or attempt documented in the VA system, to 
+understand their evolution of care. 
+E. Limitations 
+VHA data largely cover a population of older men. However, 
+the amount of women and younger patients is increasing, thus 
+
+ 
+ 
+7 
+also increasing the generalizability of these findings. The 
+corpora retrieval method we used to train the ZSL DNN is 
+dependent on clinicians’ use of the relevant ICD-10-CM codes 
+in documenting care, which may be prone to underuse [8]. 
+However, the results of this study indicate the method’s utility. 
+Due to environmental computational limitations, we randomly 
+selected 20,000 notes from the stringAndDx corpus, and 20,000 
+notes from the noDx corpus for training the ZSL DNN.  
+V. CONCLUSION 
+We developed a new methodology to identify suicidality in 
+clinical notes using zero-shot learning (ZSL). A trained ZSL 
+deep neural network (DNN) outperformed a DNN trained using 
+a baseline bag-of-words method in AUC scores and other 
+metrics assessed at various probability thresholds on unseen 
+data, according to expert review. This novel methodology 
+identifies suicidality and its risk factors with high precision, 
+when applying a 0.90 probability threshold, in VA clinical notes 
+not associated with a relevant ICD-10-CM code.  This 
+methodology 
+could 
+complement 
+existing 
+suicidality 
+identification measures. These findings hold promise for future 
+research. 
+APPENDIX 
+ 
+Most Frequent Note Types in Training Data by Corpus 
+stringAndDx 
+noDx 
+Note Type 
+Count 
+Note Type 
+Count 
+Addendum 
+2844 
+Addendum 
+5683 
+Suicide Behavior 
+and Report 
+843 
+Primary Care Secure 
+Messaging 
+291 
+Suicide Prevention 
+Telephone Note 
+811 
+Nursing Note 
+228 
+Suicide Behavior 
+and Overdose 
+Report 
+613 
+Administrative Note 
+207 
+Suicide Prevention 
+Note 
+452 
+State Prescription 
+Drug Monitoring 
+Program 
+110 
+Suicide Prevention 
+Safety Plan 
+448 
+Care Flow Sheet 
+88 
+Mental Health 
+Nursing 
+Assessment Note 
+374 
+Telephone Contact 
+75 
+Veterans Crisis 
+Line Note 
+222 
+Mental Health 
+Diagnostic Study 
+Note 
+71 
+Social Work Note 
+213 
+Non VA Care 
+Consult Result Note 
+69 
+Suicide Prevention 
+Contact 
+212 
+Operation Report 
+64 
+ 
+ACKNOWLEDGMENT 
+The views expressed are those of the authors and do not 
+necessarily reflect those of the Department of Veterans Affairs, 
+the United States Government, or the academic affiliate 
+institutions. This work was funded by Veterans Affairs Health 
+Services Research and Development Services grant IIR 18-035 
+Understanding Suicide Risks among LGBT Veterans in VA 
+Care, and NIH National Center for Advancing Translational 
+Sciences grant UL1TR001876. 
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+
diff --git a/7NE1T4oBgHgl3EQf7QUc/content/tmp_files/load_file.txt b/7NE1T4oBgHgl3EQf7QUc/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf,len=909
+page_content='1  \uf020Abstract— Objectives: Identifying suicidality including suicidal  ideation, attempts, and risk factors in electronic health record data  in clinical notes is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' A major difficulty is the lack of training  samples given the small number of true positive instances among  the increasingly large number of patients being screened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' This  paper describes a novel methodology that identifies suicidality in  clinical notes by addressing this data sparsity issue through zero-  shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Materials and Methods: U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Veterans Affairs  clinical notes served as data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The training dataset label was  determined using diagnostic codes of suicide attempt and self-  harm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' A base string associated with the target label of suicidality  was used to provide auxiliary information by narrowing the  positive training cases to those containing the base string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' A deep  neural network was trained by mapping the training documents’  contents to a semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' For comparison, we trained another  deep neural network using the identical training dataset labels and  bag-of-words features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Results: The zero shot learning model  outperformed the baseline model in terms of AUC, sensitivity,  specificity, and positive predictive value at multiple probability  thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' In applying a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='90 probability threshold, the  methodology identified notes not associated with a relevant ICD-  10-CM code that documented suicidality, with 94% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Conclusion: This new method can effectively identify suicidality  without requiring manual annotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Keywords— Suicide, Clinical Notes, NLP, Zero-Shot Learning  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' INTRODUCTION  uicide is a significant problem in the United States,  increasing 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='2% from 1999 to 2018, and from 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='5 to 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='2  suicides per every 100,000 individuals in that same time period  [1] In 2020, 45,979 people died from suicide, and  approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='2 million attempted suicide in the United  States [2] Its estimated cost is over $70 billion annually in lost  productivity and medical care [3];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' this calculation does not  include residual costs from the estimated 4-17 people closely  tied to the suicide decedent who are left bereaved [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Suicide,  however, is a complicated problem that includes a dynamic web  of individual-level risk factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=', depression, substance use  behaviors, personality traits), interpersonal risk factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=',  violence, victimization), and community-level factors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=',  unemployment, stigmatization of mental illness) [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  1Biomedical Informatics Center;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The George Washington University;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Washington DC, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  2VA Medical Center, Washington, DC, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  3Department of Emergency Medicine, Yale School of Medicine, New Haven,  CT, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' 4PRIME Center, VA Connecticut Healthcare System, West Haven,  CT, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' 5Research, VA Connecticut Healthcare System, West Haven, CT,  Veterans are especially affected by suicide, with an age- and  sex-adjusted rate that is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='5 times higher than nonveterans [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  The Department of Veterans Affairs (VA) operates the single  largest integrated health care system in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=', and has  devoted resources to suicide prevention, including the Suicide  Prevention Applications Network (SPAN), embedding suicide  prevention coordinators and special reporting measures in  facilities [8], increased mental health staffing, partnerships with  community care organizations, and enhanced surveillance and  monitoring through its electronic health record (EHR) system  [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Additionally, the VA has continual efforts to develop  predictive analytics to identify patients at the highest risk of  suicide [8, 11] The data elements for these predictive analytic  algorithms rely on structured data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=', International  Classification of Disease [ICD] diagnosis codes, prescription  data, socio-demographic data, care utilization metrics) [12]  which often provide an incomplete record [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Less is  known about how unstructured data, such as contained in  clinical notes, can contribute to suicidality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=', suicidal  ideation or attempt) identification and prevention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Given that a  suicide attempt is one of the greatest risk factors for subsequent  suicide death, a more thorough means of detecting such events  is warranted [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Background and Significance  Natural language processing (NLP) combined with machine  learning may add value to suicide documentation research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Supervised machine learning methods use “supervised”, or pre-  classified data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' However, naïve attempts at note retrieval using  keyword search alone quickly demonstrate the difficulty of this  problem, as words such as “suicide” occur in standard  questionnaires which are included in many notes, with few  actually documenting suicidality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' For instance, in a prior  experiment we carried out, we randomly collected 1,000 VA  notes containing the term “suicidal” or “suicide” from 1,000  individual patients and performed manual chart review for  affirmed suicidality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='57% of these notes documented  actual suicidality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Patient reluctance to disclose suicidal  ideation provides a further complicating factor [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' As a  result, a patient’s negative response to a suicide ideation inquiry  may not reflect their real feelings or intentions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Additionally,  USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' 6VA Connecticut Healthcare System, West Haven, CT, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' 7Suzanne  Dworak-Peck School of Social Work, University of Southern California, Los  Angeles, CA, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' 8Department of Internal Medicine, Yale School of  Medicine, West Haven, CT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Leveraging Contextual Relatedness to Identify Suicide  Documentation in Clinical Notes through Zero Shot  Learning  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Elizabeth Workman, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='1,2, Joseph L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Goulet, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='3,6, Cynthia A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Brandt, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='3,6, Allison R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Warren, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='4, Jacob Eleazer, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='4, Melissa Skanderson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='5, Luke Lindemann, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='6, John  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Blosnich, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='7, John O’Leary, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='6,8, Qing Zeng-Treitler, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='1,2  S  2  relying on structured data alone will result in incomplete  identification of patients who have or are experiencing  suicidality, because relevant coding is prone to underuse [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  However, not all clinical notes associated with relevant  structured data document suicidality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' For example, a note  documenting a secondary service such as group therapy, or a  note documenting fluid intake may not directly document  suicidality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Prior attempts to apply NLP and machine learning are often  limited to mental health-oriented notes and may suffer if using  imbalanced data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Levis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' [18] applied sentiment analysis and  various machine learning algorithms to classify suicide, using  VA psychotherapy notes, yielding area under the curve (AUC)  ratings comparable to chance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Fernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' [19] obtained  excellent NLP performance in their study of clinical notes from  the Clinical Record Initiative Search (CRIS), but performance  was computed after removing neutral (non-suicide) results from  their machine learning output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Carson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' enriched notes  associated with suicide attempt that were then used to train a  random forest model achieving 83% sensitivity, but only 22%  specificity [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' [21] applied a bag-of-words  approach with machine learning to identify suicide ideation and  psychiatric symptoms using notes for patients identified as  having performed self-harm, achieving 61% PPV (positive  predictive value), 59% sensitivity, and 60% specificity, with  results varying  depending on the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' sought to  identify psychological stressors using a pre-annotated dataset of  psychiatric evaluation records from the CEGS N-GRID 2016  challenge [22] as a gold standard, for a conditional random  fields machine learning model, [23] yielding final F scores of  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='91% and 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='01%, respectively, on exact and inexact stressor  matching, and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='73% and 100% respectively, for exact and  inexact suicide recognition on instances of the positive  keywords with the stressors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' however, their evaluation methods  for this are not detailed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Zhong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' applied structured data and NLP to identify  suicidal behavior in pregnant women, achieving PPV of 76%  and 30%, for women identified through relevant diagnostic  codes and through NLP for women not receiving a relevant  diagnostic code, respectively [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Obeid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' [25] trained a  convolutional neural network that achieved an AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='882  and an F1 score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='769 in predicting relevant suicide ICD  codes in subsequent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Using notes from psychiatric  encounters, Cusick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' [26] developed a rule-based NLP tool  to identify positive instances of suicide-oriented keywords that  leveraged NegEx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' [27] They also developed different weakly-  supervised machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' A convolutional neural  network receiving Word2Vec [28] word embeddings as input  achieved precision, recall, F1 score, and AUC values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='81,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='83, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='82, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' In a subsequent evaluation the  convolutional neural network correctly classified 87% of the 23  notes (of 5000 clinical notes) receiving a positive classification,  from notes for patients diagnosed with depression or prescribed  an antidepressant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' In a related task Tsui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' [29] used prior  structured and unstructured data (clinical notes from history,  physical examination, progress notes and discharge summaries)  of inpatient and emergency room patients with a coded suicide  attempt, to identify first-time suicide attempts in a case-control  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' An ensemble of extreme gradient boosting (EXGB)  yielded best performance, with an AUC ranging from 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='9% to  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='2%, according to time window between prior data and  suicide attempt diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Recently, Rozova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' obtained  promising results (87% AUC) using a gradient boosting model,  although the study was limited to emergency room triage notes  [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Seeking suicidality in all types of clinical notes, among all  types of patients, or when hampered by imbalanced data, is  indeed a complex task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Some of the methods in the papers cited  above tend to suffer from low precision, specificity, and  possibly also low sensitivity (recall).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Identifying probability  thresholds addresses these problems, providing flexibility for a  given task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' For example, a high probability threshold (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=', the  top ten percent) can serve as a means for identifying  documentation indicating suicidality and its risk with high  precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' When the prevalence is very low, which is often the  case of true positive suicidality documentation, the optimal  threshold needs to balance metrics such as the true positive rate  (sensitivity, also known as recall), specificity, and the positive  predictive value (precision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' A strategic implementation of a  technique like Zero-Shot Learning may also provide accurate  identification of suicidality in clinical notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='   Zero-Shot Learning  Zero-Shot Learning (ZSL) enables predictions on unseen  data using a model trained on data that has labels that are  different than those of the unseen data [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' It largely  operates by mapping select properties of the data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=', the  “feature space”) to a semantic representation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=', the “semantic  space”) that enables prediction of unseen classes [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' In other  words, auxiliary information must be provided on the labels of  the unseen classes to make it possible for a trained model to  recognize them in the testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  ZSL has been applied in several computer vision tasks [34,  35], as well as NLP tasks [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Accordingly, a feature space  can consist of data derived from images [37] or text [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The  semantic representation can be based on several different  approaches, including data attributes, semantic word vectors as  those provided by skip-gram or continuous-bag-of-word  architectures [33] or BERT output [38], or knowledge graphs  [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Examples in NLP applications include semantic utterance  classification [39] multilingual translation [40] and emotion  detection [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' However, other than Sivarajkumar and Wang’s  work [38] there is little ZSL research in unstructured clinical  text data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Naturally, different semantic representations affect the  accuracy of ZSL [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' In this study, we leveraged word  embedding and usage context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Objectives  We investigated a ZSL methodology applied to a binary  suicidality classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The training dataset was  constructed using diagnostic codes (ICD-10-CM codes) related  to suicide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Our target label is the broader concept of suicidality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  To enable ZSL, a base string representing suicidality was  3  selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' We then built the semantic space by identifying key  features associated with suicidality in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' A  DNN model was developed using the training data and tested  on two different sets of unseen data with the unseen label of  suicidality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Specifically, we sought to answer:  Will ZSL effectively identify suicidality documentation  from among all types of clinical notes, using review by  clinicians as the reference standard?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Will ZSL effectively identify suicidality or suicide risk  documentation from among clinical notes not associated with  a relevant ICD-10-CM code, by probability threshold, in terms  of precision, using the same reference standard?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  We are unaware of previous descriptions of this methodology  and to our knowledge it has not been used prior to this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' METHODS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Training Data  A training dataset was created using two corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The first  corpus consisted of 50,000 randomly selected VA clinical notes  from outpatient encounters recorded between 2016 and 2019  which contained the base string “suicid” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' “suicide”,  “suicidal” ) and were associated with at least one ICD-CM-10  code identified by the National Health Statistics Report from  the Centers for Disease Control and Prevention (CDC)  indicating suicide attempt or intentional self-harm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' [43]  This  corpus is referred to as stringAndDx (9170 unique patients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  The second corpus consisted of 50,000 randomly selected VA  clinical notes from outpatient encounters recorded between  2016 and 2019 that were associated with other ICD-CM-10  codes that were irrelevant to suicidality or self-harm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' These  notes were extracted from patients matching the stringAndDx  patients in age (at the time of document retrieval), race, and  ethnicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' This second corpus is referred to as noDx (8638  unique patients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Each corpus was preprocessed by  transforming all letters to lower case, removing basic  formatting markup and punctuation, separating character  strings into tokens (words), separating relevant concatenated  tokens (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=', “suicidalhomicidal” to “suicidal” ”homicidal”),  and removing all tokens that did not entirely consist of letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Semantic Space Feature Extraction and Mapping  The task to build the semantic space was carried out in three  steps: First, we identified a list of features that are potentially  relevant for the positive training label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Second, we created word  embeddings using a skip-gram architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Third, we  identified context words of the selected features using the word  embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' In a fourth step, a contextual weight is assigned  to each feature for each document in mapping the semantic  space to the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  In the first step, inverse document frequency (TF-IDF)  analysis was used to identify the n most important terms in each  corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' For this investigation, n = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' TF-IDF evaluates  term frequency using the count of documents containing a given  term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' In each document, the relative frequency of each term is  weighted by the log of the number of documents in the corpus  divided by the number of documents containing the term, as  shown in (1)  𝑡𝑖,𝑗 = 𝑡𝑓𝑖,𝑗 ∗ 𝑙𝑜𝑔( 𝑛  𝑑𝑓𝑖  )  where ti,j is term i in document j, tfi,j is the relative frequency of  term i in document j, n is the total number of documents, and  dfi is the number of documents containing term i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Because TF-  IDF is a document-based measurement, we used the mean TF-  IDF value for each term in its respective corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The words  with the top TF-IDF scores that are unique to the stringAndDx  corpus were treated as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Figure 1 illustrates this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Each circle represents terms from one of the corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Sets a and  b are the words with the top n TFIDF scores for stringAndDx  and noDx, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Set c is the overlap between a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  The feature set F contains words that are in set a, but not in the  overlap set c or in set b (f \uf0ce a and f \uf0cf c and f \uf0cf b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Feature identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Words that are deemed as features are in set a,  excluding words in c and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  In the second step, we created a Word2Vec model using the  stringAndDx corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' In this study, the model was a shallow  neural network with the hidden layer containing 300 nodes,  applying the skip-gram architecture, with an analytic window  size of 5, trained through 10 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  In the third step, we identified the top m context words for  each feature word using the word embeddings from the  Word2Vec model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The m words most similar to each feature  word, according to cosine similarity values, served as its  context words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' In this investigation, m = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  In the fourth step, we map the feature space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' a document’s  preprocessed content, to the semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' A weight v is  assigned to each feature word for each document, based on its  occurrence with its context words in a window in the  document’s text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' This weight is the summed total of the cosine  similarity between the feature and a co-occurring context word  multiplied by the mean TF-IDF value of the feature word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The  formula is shown in (2)  𝑣 =  ∑  𝑐𝑜𝑠𝑆𝑖𝑚(𝑥, 𝑦) ∗ 𝑡𝑓𝑖𝑑𝑓(𝑥)  𝑥∈𝐹,𝑦∈𝐷  where x is a feature in F, the set of features in the semantic  space, and y is a context word of set D, the context words for x  in the semantic space, which occurs in a five-word window  around x in the document’s text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' This process is illustrated in  Figure 2, where “pattern” (highlighted in light gray) is a feature  word, and “internalizing” and “fitful” (highlighted in dark gray)  are among its set of context words and appear in a five-word  window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  (2)  (1)  stringAndDx  noDx  a  c  6  4  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Example of deriving a feature weight using (2)  If a feature word is not in the text, its value is zero for the  given document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Model Development  20,000 documents were randomly selected from each corpus  (stringAndDx and noDx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' We trained a DNN model (here  referred to as the ZSL DNN) consisting of five fully-connected  hidden layers of alternating sizes of 30 or 70 nodes, with each  layer implementing a dropout rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' We implemented the  Adam optimizer [44], with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='0012, beta 1  value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='92, beta 2 value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='9992, and an epsilon value of  1e-08, with binary cross entropy as the loss function, and the  sigmoid function in the output layer, since it was a binary  classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The architecture and hyperparameters were  chosen on empirical grounds, after experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Each  document from the stringAndDx corpus was classified as “1” (a  generic positive instance), and each document from the noDx  corpus was classified as “0” (a generic negative instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  These labels do not indicate whether or not the given document  directly pertains, or not pertains, to suicidality or its risks, but  an association with a structured data element, and for those  labeled “1”, also containing a base string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Balancing the  positive and negative approximated training datasets in this  manner (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=', providing balanced training examples) addressed  the problematic issue of otherwise training a model with few  positive and many negative instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' We implemented a 60%  training, 20% validation, and 20% testing split in developing  the ZSL DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Figure 3 illustrates the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The corpora stringAndDx (2016-19), noDx (2016-19),  testSet1 (2020), and testSet2 (2020) are unique and extracted from all clinical  notes based on associated ICD-10-CM codes, and in the case of  stringAndDx, where a base string is also present;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' corpora content is  preprocessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The stringAndDx and noDx corpora are used in the TF-IDF  analysis to identify feature words that are unique to stringAndDx (step 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  stringAndDx is applied to a skip-gram model to produce word embeddings  (step 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Feature words and their significant context words (determined  through the word embeddings) form the semantic space (step 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The  contents of stringAndDx and noDx are mapped to the semantic space, using  a function to determine feature word weights (step 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The mapped contents  of stringAndDx and noDx documents are used to train the ZSL DNN, using  generic labels 1 and 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The mapped contents of unseen  testSet1 and testSet2 notes were classified by the trained ZSL DNN, for the  classes (a) containing suicidality documentation, or (b) not containing  suicidality documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Human annotation independently classified  random documents from testSet1 and testSet2 for the same classes (a)  containing suicidality documentation, or (b) not containing suicidality  documentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' human annotation also assessed documents from testSet2  containing the base string that received a probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='90 or greater, for  these classes and suicidality risk factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Evaluation  The authors randomly retrieved 5,000 different clinical notes  recorded in 2020 that were associated with at least one of the  relevant IDC-10-CM codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' This corpus is subsequently  labeled as testSet1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The authors also randomly retrieved 5,000  different clinical notes recorded in 2020 that were associated  with other IDC-10-CM codes irrelevant to suicidality or self-  harm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' This corpus is subsequently labeled testSet2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  The contents of each of the notes in testSet1 and testSet2 were  mapped to the semantic space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=', deriving a weight for each  feature word as described earlier in the fourth step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Then, the  trained ZSL DNN was used to classify the notes in testSet1 and  testSet2 as (a) containing suicidality documentation, or (b) not  containing suicidality documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  In joint sessions, two clinical psychologists familiar with VA  clinical note documentation together identified suicidality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=',  current or past suicide ideation or attempt) in 200 notes  randomly selected from testSet1 and testSet2 (100 from each  test set), after being instructed to look for documentation for  these specific events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' They addressed differences of opinion  through discussion and mutual consensus during the joint  sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' In a second evaluation, to explore how the  application’s output may serve to identify patients who had  experienced or were at risk for suicidality, but never formally  diagnosed as such, the clinicians examined the testSet2 notes  containing the base string “suicid" that received a probability  value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='90 or greater from the trained ZSL DNN, for  documentation of suicidality and/or its risk factors, according  to NIH guidelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' [45] This threshold was chosen in order to  explore how high-probability documents (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' the top 10% in  terms of probability) would be representative in identifying  documented suicidality or its risk factors with high precision,  thus addressing our second question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  1) Baseline Comparison  For comparative purposes, the 163 most frequent bigrams  unique to the stringAndDx corpus were identified and used in a  bag-of-words baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' We trained a DNN (here referred  to as the Baseline DNN) using these 163 bigrams as features for  the 20,000 stringAndDx documents and the 20,000 noDx  documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' This baseline DNN was also used to classify the  Document Text: “The patient has a pattern of internalizing  criticism from his family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' This pattern sometimes results in fitful  outbursts.”  TF-IDF value of feature word “pattern”: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='0062  Cosine similarity of “pattern” and “internalizing”: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='4673  Cosine similarity of “pattern” and “fitful”: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='3824  Feature weight for “pattern”:  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='0062 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='4673) + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='0062 * 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='3824) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='0053  All Clinical  Notes  ICDcodes  ICDcodes  Base String  Human  testSetl  testSet2  stringAndDx  noDx  Annotation  (suicidality)  TFIDF  Analysis  Word  Embeddings  Semantic  Map content  Space  Mapcontent  Testing  DNN  Training  Output is a classification of (a) containing suicidality  documentation or (b) not containing suicidality documentation  5  notes in testSet1 and testSet2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' for (a) containing suicidality  documentation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' or (b) not containing suicidality documentation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  using the 163 most frequent bigrams as features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' RESULTS  The first step of the new method (described in Methods)  identified 163 feature words associated with suicidality  diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The top thirty feature words are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' No  form of the base string “suicid” was found among the 163 final  feature words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Both “suicide” and “suicidal” were prominent  terms in both the noDx and stringAndDx corpora, along with  terms like “psychiatrist” and “psychosocial”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' this is likely due  to the proliferation of objects like questionnaires, and mental  health care documentation in notes that are unrelated to  suicidality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  TABLE I  TOP 30 FEATURE WORDS  flag  overdose  coordinator  took  spc  observation  called  warning  pills  prf  unknown  interrupted  gun  placement  lcsw  lethal  outcome  reportedly  notified  sdv  occurred  police  protocol  od  supports  seeking  category  preparatory  cut  determined  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' ZSL DNN and Baseline DNN Performance  The classifications by the clinicians and the probabilities  assigned by the ZSL DNN and the Baseline DNN were first  assessed by AUC score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The results are in Table II and Figure  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  TABLE II  AUC PERFORMANCE  ZSL DNN  Baseline DNN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='946  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='47  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' ZSL DNN AUC results (left), Baseline DNN AUC results (right)  In terms of AUC, the ZSL DNN trained through mapping the  semantic space to the feature space outperformed the Baseline  DNN trained with the bigram bag-of-words features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  The sensitivity, specificity, and PPV results at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='5, and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='85 probability thresholds for each DNN are in Tables III-V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Probability refers to the probability the DNN assigned to each  note for positive suicidality documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' We applied the  median probability (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='1499, rounded) assigned by the ZSL  DNN to the testSet2 documents (the test set containing random  notes associated with irrelevant ICD-10-CM codes) in forming  minimum and maximum thresholds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='5 is a standard midpoint  probability threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The combined scores in these tables were  computed with all true positives, true negatives, false positives,  and false negatives for both test sets, for the indicated metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Values of NaN (not a number) occurred where there were no  true positives or false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  TABLE III  EVALUATION RESULTS AT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='15 PROBABILITY THRESHOLD  ZSL DNN  Sensitivity/Recall  Specificity  Precision/PPV  testSet1  97%  100%  91%  testSet2  100%  64%  05%  Combined  97%  59%  67%  Baseline DNN  testSet1  99%  0%  90%  testSet2  50%  09%  01%  Combined  98%  08%  48%  TABLE IV  EVALUATION RESULTS AT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='5 PROBABILITY THRESHOLD  ZSL DNN  Sensitivity/Recall  Specificity  Precision/PPV  testSet1  92%  40%  93%  testSet2  50%  97%  25%  Combined  91%  92%  90%  Baseline DNN  testSet1  92%  0%  89%  testSet2  50%  10%  1%  Combined  91%  9%  46%  TABLE V  EVALUATION RESULTS AT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='85 PROBABILITY THRESHOLD  ZSL DNN  Sensitivity/Recall  Specificity  Precision/PPV  testSet1  77%  70%  96%  testSet2  50%  100%  100%  Combined  76%  97%  96%  Baseline DNN  testSet1  0%  100%  NaN/div by 0  testSet2  0%  100%  NaN/div by 0  Combined  0%  100%  NaN/div by 0  The ZSL DNN outperformed the Baseline DNN in most  metrics at all probability thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Second Evaluation  To explore how this new methodology can identify clinical  notes documenting suicidality that are not associated with a  relevant ICD-10-CM code with high precision, the clinicians  also reviewed the 16 notes from testSet2 containing the base  string “suicid’ that received a probability at or above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='90 from  the trained ZSL DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The clinicians noted suicide ideation or  attempt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' and the presence of the following suicide risk factors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  based on National Institute of Mental Health guidelines [45]:  Depression and other mental health disorders  Substance abuse disorder  Family history of a mental health or substance abuse  disorder  Family history of suicide  Family violence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' including physical or sexual abuse  Having guns or other firearms in the home  Being in prison or jail  Being exposed to others’ suicidal behavior  Of these 16 clinical notes (associated with 16 different  patients),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' 7 documented current or past suicide ideation or  attempt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Eight of the remaining notes included one or more  risk factors for suicide (nearly all included multiple risk  factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' In all, 15 of the 16 notes contained documentation of  current or past suicide ideation or attempt, and/or suicide risk  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='0  model results  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Dno distinction  model results  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='8-  2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='0  Q2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='4  False Positive Rate  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='8  10  6  factors, for patients who had never received a suicidality ICD-  10-CM code diagnosis during the study period, achieving a  PPV of 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' DISCUSSION  Regarding the study’s original questions, our ZSL approach  effectively identified suicidality in all types of clinical notes,  surpassing the performance of the bag-of-words baseline in  conjunction with deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' It also effectively identified  suicidality or suicide risk documentation from among clinical  notes not associated with a relevant ICD-10-CM code with high  precision, on probability threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Semantic Space  In this work, the semantic space development is framed as  feature extraction where mapping is enhanced by attaching  weights to features found in the data, an approach also used in  computer vision ZSL [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The semantic space captures natural  data properties by identifying salient terms and relevant  contextual  terms  in  collective  clinical  suicidality  documentation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=', a corpus of notes associated with relevant  ICD codes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Table 1 lists 30 prominent feature words associated  with collective suicidality documentation after removing terms  associated with other kinds of documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' There is an intuitive  sense to these words;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' “flag” is found in the phrase “high risk for  suicide flag”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' “overdose” and “cut” refer to suicide methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  “pills” and “gun” refer to suicide instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Identifying terms  contextually similar to these provides patterns in relevant  documentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Again, this has an intuitive logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The most  contextually similar terms to “flag” include “reactivate” and  “deactivate” (for a high suicide risk flag) and “high” (the level  of risk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The most contextually similar terms to “pills” include  “handful”, “fistfuls”, and “bunch”, implying large quantities,  along with “overdosing” and “took”, the associated actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  The feature word  “spc” indicates VA’s suicide prevention  coordinators, which is a structural change that VA implemented  for suicide prevention [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Concordantly, “police” and “lcsw”  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=', licensed clinical social worker) refer to other professions  highly associated with individuals at risk for suicide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' For  example, police may be activated for a rescue, and a licensed  clinical social worker may be involved in treatment planning or  referral connections for suicidal individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The feature words  “prf” and “sdv” refer to “patient record flag” and “self-directed  violence”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The semantic space provided an  efficient representation for effective mapping to the feature  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Data Retrieval and Model Training  Using associated structured data elements like ICD-10-CM  codes, and a base string provides a means to locate equally sized  corpora for training that could be generically labeled “0” or “1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  These labels were primarily based on a structured data  association, since their individual unstructured content was  mostly unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' This approach solves the issue of imbalanced  training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The predominant clinical note types (Appendix)  also illustrate this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Most of the frequent note types associated  with one of the relevant CDC ICD-10-CM codes and containing  the base string are relevant to suicidality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Addendum is a  common note type [47] associated with many domains [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  The most frequent note types not associated with a relevant  code resemble frequencies of all note types in the VA [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Identifying Suicidality Documentation  To our knowledge, this method has not been applied in other  studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Unlike VA surveillance methods using structured data,  it also leverages information found in EHR notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Also, unlike  other NLP methods [18, 20, 21, 23, 24, 26, 29, 30] it can be  applied to all patients and note types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' In other studies, a bag-  of-words approach has been applied to suicidality identification  and other machine learning tasks [21, 49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' However, the  results of this current study suggest that the complexity of  suicidality documentation demands a more targeted approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  This method could complement existing measures like  SPAN, alerting suicide prevention coordinators of additional  patients at risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The results of the two clinical psychologists’  evaluations demonstrate the method’s efficiency in identifying  suicidality documentation for documents where there is no  relevant ICD-10-CM code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The performance on both test sets  demonstrates the methodology’s effectiveness in classifying  notes that are mixed in terms of ICD-10-CM coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Tables III - V suggest that the probability threshold can be  adjusted to suit a specific task like finding suicidality and its  risk factors with high precision among notes not associated with  a relevant ICD-10-CM code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' This is especially true considering  the small prevalence of suicidality documentation in clinical  notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The second evaluation (which yielded 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='8% PPV)  demonstrates this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' By applying a high probability threshold of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='90 to all 5000 testSet2 documents and focusing on clinical  notes containing the base string, of the 16 documents (for 16  different patients), 94% contained suicidality and/or suicidality  risk factor documentation, based on clinician review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' These  results exceed those of Cusick et al.’s [26] similar task, where  87% of notes were correctly classified, among notes for patients  diagnosed with depression or prescribed an antidepressant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' In  this current study’s second evaluation, none of the 16 patients  identified had ever received a suicide ICD-10-CM code during  the study’s time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' It is impossible to know if the patients  in the 8 notes simply containing documented risk factors were  suicidal or not based solely on electronic health records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Suicidal patients sometimes deny suicide ideation or attempt  [16, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' For example, in one note from the chart review  associated with a relevant ICD-10-CM code, the patient  reportedly denied suicide ideation, even after checking into the  hospital hours earlier for a self-reported suicide attempt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Future Work  This work is part of a larger study of patients at risk for  suicide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' [52] The next step is to combine these findings with  prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' We also plan an analysis of patients from first  suicide ideation or attempt documented in the VA system, to  understand their evolution of care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' Limitations  VHA data largely cover a population of older men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' However,  the amount of women and younger patients is increasing, thus  7  also increasing the generalizability of these findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' The  corpora retrieval method we used to train the ZSL DNN is  dependent on clinicians’ use of the relevant ICD-10-CM codes  in documenting care, which may be prone to underuse [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  However, the results of this study indicate the method’s utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  Due to environmental computational limitations, we randomly  selected 20,000 notes from the stringAndDx corpus, and 20,000  notes from the noDx corpus for training the ZSL DNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' CONCLUSION  We developed a new methodology to identify suicidality in  clinical notes using zero-shot learning (ZSL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' A trained ZSL  deep neural network (DNN) outperformed a DNN trained using  a baseline bag-of-words method in AUC scores and other  metrics assessed at various probability thresholds on unseen  data, according to expert review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' This novel methodology  identifies suicidality and its risk factors with high precision,  when applying a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='90 probability threshold, in VA clinical notes  not associated with a relevant ICD-10-CM code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' This  methodology  could  complement  existing  suicidality  identification measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content=' These findings hold promise for future  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='APPENDIX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Most Frequent Note Types in Training Data by Corpus  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='stringAndDx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='noDx  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Note Type  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Count  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Note Type  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Count  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Addendum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='2844  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Addendum  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='5683  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Suicide Behavior  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='and Report  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='843  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Primary Care Secure  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Messaging  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='291  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Suicide Prevention  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Telephone Note  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='811  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Nursing Note  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='228  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Suicide Behavior  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='and Overdose  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Report  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='613  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Administrative Note  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='207  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Suicide Prevention  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Note  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='452  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='State Prescription  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Drug Monitoring  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Program  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='110  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Suicide Prevention  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Safety Plan  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='448  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Care Flow Sheet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='88  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Mental Health  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Nursing  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Assessment Note  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='374  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Telephone Contact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='75  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Veterans Crisis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Line Note  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='222  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Mental Health  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Diagnostic Study  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Note  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='71  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Social Work Note  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='213  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Non VA Care  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Consult Result Note  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='69  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Suicide Prevention  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Contact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='212  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='Operation Report  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='64  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='ACKNOWLEDGMENT  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='The views expressed are those of the authors and do not  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='necessarily reflect those of the Department of Veterans Affairs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
+page_content='  the United States Government,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
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+page_content=' This work was funded by Veterans Affairs Health  Services Research and Development Services grant IIR 18-035  Understanding Suicide Risks among LGBT Veterans in VA  Care, and NIH National Center for Advancing Translational  Sciences grant UL1TR001876.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE1T4oBgHgl3EQf7QUc/content/2301.03531v1.pdf'}
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+A Bayesian latent position approach for community detection in
+single- and multi-layer networks with continuous attributes
+Zhumengmeng Jina, Juan Sosab and Brenda Betancourtc
+a University of Florida
+b Universidad Nacional de Colombia
+c NORC at the University of Chicago
+December 2022
+Abstract
+The increasing prevalence of multiplex networks has spurred a critical need to take into account po-
+tential dependencies across different layers, especially when the goal is community detection, which is a
+fundamental learning task in network analysis. We propose a full Bayesian mixture model for community
+detection in both single-layer and multi-layer networks. A key feature of our model is the joint modeling
+of the nodal attributes that often come with the network data as a spatial process over the latent space.
+In addition, our model for multi-layer networks allows layers to have different strengths of dependency
+in the unique latent position structure and assumes that the probability of a relation between two actors
+(in a layer) depends on the distances between their latent positions (multiplied by a layer-specific factor)
+and the difference between their nodal attributes. Under our prior specifications, the actors’ positions
+in the latent space arise from a finite mixture of Gaussian distributions, each corresponding to a cluster.
+Simulated examples show that our model performs favorably compared to the existing ones. The model
+is also applied to a real three-layer network of employees in a law firm.
+1
+Introduction
+Network data conveniently describes the relationships between actors in complex systems and is ubiquitous
+in many statistical applications, including finance, social science, criminology, biology, epidemiology, and
+computer science, among others. Understanding the relationships between actors can aid domain experts.
+Key words and phrases. multiplex network, community detection, latent position model, mixture model, spatial process, visu-
+alization
+1
+arXiv:2301.00055v1  [stat.AP]  30 Dec 2022
+
+For instance, in epidemiology, people in a certain area can be portrayed in a contact network that can be
+studied to detect infectious disease outbreaks. In criminology, communications between terrorists form a
+terrorist network, helping intelligence agencies to better counter terrorism.
+Many models have been developed for the inference of networks over the past decades (e.g., Erdös and
+Rényi, 1959, Frank and Strauss, 1986), among which the broad class of latent space models is one of the
+most widely used (see, e.g., Sosa, 2021 for an exhaustive review). Suppose the network under study has
+N actors, then under latent space models, there are N independent and identically distributed (i.i.d.) latent
+variables z1, . . . , zN, one for each actor. Under a mild exchangeability assumption in Hoff [2007], results
+in Aldous [1985] and Hoover [1982] show that edge variables yi,j depend on latent variables through a
+symmetric function γ(zi, zj) that is meant to capture any pattern in the network beyond any known covariate
+information.
+Many well-known models fall into the category of latent space models, which can be distinguished between
+two cases depending on whether latent variables are discrete or continuous [Matias and Robin, 2014]. For in-
+stance, stochastic block models [Nowicki and Snijders, 2001, Wang and Wong, 1987] – hereafter SBM – are
+special cases of latent space models with discrete latent variables zi ∈ {1, 2, . . . , K}. When latent variables
+are assumed to be continuous, another approach using latent variables is the class of latent position models
+(LPM) proposed by Hoff et al. [2002] which our model in the paper is built upon. In its basic formulation,
+LPMs model the edge variables yi,j as conditionally independent given the distance between latent variables
+γ(zi, zj) = −∥zi − zj∥, which naturally accounts for transitivity effects through the latent space (typically
+a Euclidean K-dimensional space for a predetermined K) where zi lives. Later on, Handcock et al. [2007]
+proposed an extension on Hoff et al.’s LPM, namely the latent position cluster model (LPCM), by imposing
+a Gaussian mixture prior on the latent positions to perform clustering tasks. Krivitsky et al. [2009] further
+extended Handcock et al.’s model by adding the random sender and receiver effects proposed by Hoff [2005].
+Other formulations of γ(·, ·) can be found in Schweinberger and Snijders [2003], Hoff [2005, 2009], Athreya
+et al. [2017], Minhas et al. [2019], among others.
+Besides edge information of a network, extra information like node and edge attributes and different types
+of edges are often available, and should ideally be leveraged for inference. Typical ways to incorporate
+attributes in a network model include: (1) modeling the network as a function of the attributes (see, e.g.,
+Hoff et al., 2002, Hoff, 2005); (2) modeling the attributes as a function of the network [Guha and Rodriguez,
+2021]; (3) jointly modeling the network and attributes (Linkletter, 2007, Kim and Leskovec, 2012, Fosdick
+and Hoff, 2015, Ciminelli et al., 2019). The first approach is arguably the most common approach to incor-
+porate covariates into the model, but we consider an approach of joint modeling proposed by Ciminelli et al.
+2
+
+[2019], namely the social network spatial model (SNSM), where the authors modeled edges yi,j as condi-
+tionally independent given ∥zi − zj∥ and the distance of the continuous node attributes ∥xi − xj∥, and node
+attributes are further modeled as a spatial process over the latent space. Note that joint modeling does not
+require the network or the attributes to be fully observed as the first two approaches, hence one could predict
+missing network and attribute data (if there is any). In addition, it improves model fitting by capturing the
+dependence structure between latent variables and the attributes (when such dependency exists), as we will
+see in Section 3.
+We propose a full hierarchical Bayesian model that builds on Ciminelli et al.’s SNSM. Instead of using a
+Gaussian distribution as the prior for latent positions as in Ciminelli et al. [2019], we impose a Gaussian
+mixture prior as in Handcock et al. [2007], so that our model could also capture the group structure in the
+network. Detecting communities or clusters among actors in the network is an important task in network
+analysis and has spurred the development of many models and algorithms, among which the SBM has
+motivated an active line of research that deals with community detection (see, e.g., Lee and Wilkinson
+[2019] for a review). However, SBM may not fit well when many actors fall between clusters [Hoff et al.,
+2002]. We will compare our model with an SBM that incorporates covariates as fixed effects (i.e., model
+the edge variables as a function of latent classes and covariates [Leger, 2016]), and we call this model a
+covariate-assisted stochastic block model (CSBM). We will show that our model presents improved model
+fitting while producing similar clustering results as CSBM.
+We also propose an extension of our model to multi-layer network settings. Multi-layer networks can gen-
+erally be categorized into two cases: cross-sectional networks that have different types of connections (e.g.,
+social networks of friendship, coworker-ship, etc.) and time-varying networks where the same type of con-
+nections are measured over time (e.g., a trade network that changes over time). We consider a type of
+cross-sectional multi-layer network where each layer has a common set of actors. Substantial work has been
+done on latent space models for cross-sectional multi-layer networks that take a Bayesian approach (see, e.g.,
+Gollini and Murphy, 2016, Salter-Townshend and McCormick, 2017, D’Angelo et al., 2019, Sosa and Betan-
+court, 2022, Durante and Dunson, 2018, Wang et al., 2019, MacDonald et al., 2020). In extending our model
+to the multiple networks setting, we adopt the approach in Sosa and Betancourt [2022] in a parsimonious
+way, where latent positions are assumed to be the same for all layers, but the strength of borrowing such
+latent structure information is allowed to be different across different layers. Note that, the original model
+in Sosa and Betancourt [2022] assumed different latent positions for different layers and had an additional
+hierarchy on the hyperparameters. The specification of our model is given in the next section.
+The remainder of the paper is organized as follows. Section 2 contains general background on the spatial
+3
+
+process and introduces the proposed model (for single- and multi-layer network settings) which we call
+the latent position joint mixture model (LPJMM) in the rest of the paper. In addition, prior specification,
+identifiable problem, and inference will also be discussed in this section. Several simulation studies are
+conducted in section 3, where LPJMM is compared with Handcock et al.’s LPCM, Ciminelli et al.’s SNSM
+and CSBM in single-layer settings and the model is also evaluated in multi-layer settings. In section 4, we
+apply LPJMM to a real-world multi-layer network data set. Finally, we conclude with some discussion in
+section 5.
+2
+Models
+We first review the LPM introduced in Hoff et al. [2002], and then build upon it with a spatial process to allow
+for joint modeling of the network and the nodal attributes, and with a finite Gaussian mixture distribution for
+latent positions to allow for clustering.
+Consider a binary single-layer network with N actors. Denote its adjacency matrix as Y = (yi,j) ∈
+{0, 1}N×N, where yi,j = 1 if actors i and j are connected, and yi,j = 0 if they are not connected. Suppose
+the network data comes with a one-dimensional nodal attribute xi for each actor, and denote the covariate as
+x = (xi) ∈ RN. The LPM assumes that each actor i has an observed latent position zi in a K-dimensional
+Euclidean latent space, the so-called latent space, for some K ∈ N. Let z = (zi) ∈ RN×K, then LPM
+models edge yi,j as conditionally independent given distances between nodal attributes as well as distances
+between latent positions via logistic regression. But instead of the logistic link, we use the probit link in our
+model. The analysis of probit regression models can often be facilitated by a Gibbs sampler constructed using
+the data augmentation approach that introduces latent variables with truncated normal distributions [Albert
+and Chib, 1993]. (See also Sosa and Betancourt (2022) for a discussion on the choice of link functions.)
+Specifically, for i, j ∈ {1, . . . , N} and i ̸= j,
+yi,j | z, x, a, b, θ ind
+∼ Ber
+�
+Φ(a + b|xi − xj| − θ∥zi − zj∥)
+�
+,
+(1)
+where a, b ∈ R and θ ∈ R+, Ber(p) is a Bernoulli distribution that takes value 1 with some probability p,
+∥ · ∥ is the Euclidean norm on RK and Φ(·) is the cumulative distribution function of the standard normal
+distribution. Note that we impose a factor θ for the distance between latent positions, which is different from
+Hoff et al. [2002] and Krivitsky et al. [2009]. Although θ is unidentifiable in single-layer networks, it plays
+a non-trivial role in multi-layer network settings (introduced in Section 2.1). We defer a detailed discussion
+of θ to Section 2.4.
+4
+
+To allow for joint modeling of the network and nodal attributes, we model the nodal attributes as a spatial
+process over the latent space RK. Hence, nodal attributes are treated as random variables indexed by their
+latent positions, and the distance between these random variables is found by the distance between their
+corresponding positions. As in Ciminelli et al. [2019], we specify the spatial process as a Gaussian process
+that is stationary with mean β and isotropic (see Banerjee et al., 2015 for definitions). In this case, the
+process is completely defined by its covariance function Cov(d), where d is the distance between two random
+variables in the Gaussian process. In particular, we specify Cov(d) with an exponential kernel, that is,
+Cov(d) =
+�
+�
+�
+�
+�
+τ 2 + σ2,
+if d = 0;
+σ2 exp(−φd),
+if d > 0,
+where τ ≥ 0, σ > 0 and φ > 0. It is well-known that such a covariance structure is valid, i.e., the covariance
+matrix for any finite collection of random variables in the process is positive definite [Banerjee et al., 2015].
+Let Mz = (mij) ∈ RN×N where mij = exp(−φ∥zi − zj∥) and denote IN as the N-dimensional identity
+matrix, then the Gaussian process of the nodal attributes is constructed as follows,
+x | z, β, σ, τ, φ ∼ NN(β111N, σ2M(z, φ) + τ 2IN),
+(2)
+where Nd is a d-dimensional multivariate normal distribution for some dimension d ∈ {2, 3, . . . }, and 111N is
+an N-dimensional vector with all 1s.
+As in Krivitsky et al. [2009], we impose a Gaussian mixture distribution on latent positions, which allows us
+to cluster actors into different groups. Suppose there are H < ∞ predetermined number of components in
+the Gaussian mixture distribution, then
+zi | ωωω,µµµ,κκκ ind
+∼
+H
+�
+h=1
+ωhNK(µh, κ2
+hIK) ,
+(3)
+where ωωω = {ω1, . . . , ωH}, µµµ = {µ1, . . . , µH}, κκκ = {κ1, . . . , κH}. Note that µh is a K-dimensional mean
+vector where h ∈ {1, . . . , H}, and ωh is the probability that an actor belongs to the h-th group such that
+ωh ∈ (0, 1) and �H
+h=1 ωh = 1.
+In single-layer network settings, the model is given by Eqs. (1) to (3). Under our model, nodal attributes
+of two actors whose latent positions are close are more likely to be similar according to the exponential
+covariance structure. If b < 0 (b > 0), actors with similar attributes are more (less) likely to be connected.
+When b = 0, nodal attributes do not affect the distribution of the network directly (but it still has an indirect
+5
+
+Figure 1: DAG representation of the LPJMM in multi-layer settings.
+impact on the network through latent positions by Eq. (2)).
+2.1
+An extension to multi-layer networks.
+Our model can also be extended to multi-layer network settings in the following way. Suppose we have L
+layers Y1, . . . , YL in the network, where all layers are defined over the same set of actors. We assume the
+same latent positions z for all layers but allow the strength of borrowing such latent structure information to
+be different by imposing layer-specific factors θℓ for ℓ ∈ {1, . . . , L}. Our model in multi-layer settings is
+then presented as follows
+yi,j,ℓ | z, x, aℓ, bℓ, θℓ
+ind
+∼ Ber
+�
+Φ(aℓ + bℓ|xi − xj| − θℓ∥zi − zj∥)
+�
+,
+(4)
+x | z, β, σ, τ, φ ∼ NN(β111N, σ2M(z, φ) + τ 2IN) ,
+(5)
+zi | ωωω,µµµ,κκκ i.i.d.
+∼
+H
+�
+h=1
+ωhNK(µh, κ2
+hIK) ,
+(6)
+where yi,j,ℓ is the edge variable between actors i and j in layer ℓ ∈ {1, . . . , L}, aℓ, bℓ and θℓ are layer-specific
+parameters. Note that Eqs. (5) and (6) are the same as Eqs. (2) and (3). Fig. 1 shows a directed acyclic graph
+(DAG) representation of the model given by Eqs. (4) to (6).
+6
+
+2.2
+Prior specification
+We take a Bayesian approach to estimate the model parameters. Without loss of generality, a Bayesian ver-
+sion of the model given by Eqs. (4) to (6) is formed by placing prior distributions on the unknown parameters
+aℓ, bℓ, θℓ, β, σ, τ, φ, ωωω, µµµh, κh, for ℓ = {1, . . . , L} and h = {1, . . . , H}. In the model we consider, these
+parameters are assumed a priori independent. For parameters in the probit regression tier as specified by
+Eq. (4), their priors are specified as follows:
+aℓ
+i.i.d.
+∼ N(ma, ν2
+a) ,
+bℓ
+i.i.d.
+∼ N(mb, ν2
+b ) ,
+θℓ
+i.i.d.
+∼ Gamma(λ1, λ2) .
+The priors for the parameters in the spatial process tier as given in Eq. (5) are given as follows:
+β ∼ N(0, ν2
+β) ,
+σ2 ∼ InvG(η1, η2) ,
+τ 2 ∼ InvG(ξ1, ξ2) ,
+φ ∼ U(u1, u2) .
+Finally, we put the following priors on the rest of the parameters:
+ωωω ∼ Dir(α) ,
+µh
+i.i.d.
+∼ NK(mµ, ν2
+µIK) ,
+κ2
+h
+i.i.d.
+∼ InvG(γ1, γ2) .
+Note that, ma, νa, mb, νb, λ1, λ2, νβ, η1, η2, ξ1, ξ2, u1, u2, α, mµ, νµ, γ1 and γ2 are user-specified
+hyperparameters, and Gamma(·, ·), InvG(·, ·), U(·, ·), Dir(·) represents Gamma, Inverse-Gamma, uniform,
+and Dirichlet distributions respectively.
+2.3
+Posterior distribution and model estimation
+As is standard in Bayesian estimation of mixture models (see, e.g., Diebolt and Robert [1994]), we define a
+new variable gi that serves as the missing data of group membership of actor i whose distribution depends
+on ωωω. In particular, gi = h if actor i belongs to the h-th group. The joint density of (zi, gi) given ωωω, µµµ and κκκ
+is then given by
+H
+�
+h=1
+�
+ωh
+1
+�
+2πκ2
+h
+exp
+�
+−
+1
+2κ2
+h
+∥zi − µh∥2��I{gi=h}
+,
+where the indicator function I{gi=h} = 1 if gi = h, and I{gi=h} = 0 otherwise. Let g = (gi)N
+i=1 be the group
+membership for all actors and L(·) be the law of a random variable. Then the posterior distribution of z, g
+7
+
+and the parameters upon which priors are specified in Section 2.2 is given by
+Π(z, g, a1, . . . , aL, b1, . . . , bL, θ1, . . . , θL, β, τ 2, σ2, φ,ωωω,µµµ,κκκ | Y1, . . . , YL, x)
+∝
+� L
+�
+ℓ=1
+L(Yℓ | z, x, aℓ, bℓ, θℓ)
+�
+L(x | z, σ, τ, φ)L(z, g | ωωω,µµµ,κκκ)
+� L
+�
+ℓ=1
+L(aℓ)L(bℓ)L(θℓ)
+�
+× L(β)L(σ2)L(τ 2)L(φ)L(ωωω)L(µµµ)L(κκκ) .
+Note that the dimension of the posterior distribution has dimension NK + N + 3L + 3H + 4 and the
+corresponding posterior density is presented as follows,
+π(z, g, a1, . . . , aL, b1, . . . , bL, θ1, . . . , θL, β, τ 2, σ2, φ,ωωω,µµµ,κκκ | Y1, . . . , YL, x)
+∝
+N
+�
+i,j=1
+i̸=j
+L
+�
+ℓ=1
+�
+Φ(aℓ + bℓ|xi − xj| − θℓ∥zi − zj∥)
+�yi,j,ℓ�
+1 − Φ(aℓ + bℓ|xi − xj| − θℓ∥zi − zj∥)
+�1−yi,j,ℓ
+× |σ2M(z, φ) + τ 2IN|− 1
+2 exp
+�
+− 1
+2(x − β1)⊺�
+σ2M(z, φ) + τ 2IN
+�−1(x − β1)
+�
+×
+N
+�
+i=1
+H
+�
+h=1
+� ωh
+�
+κ2
+h
+exp
+�
+−
+1
+2κ2
+h
+∥zi − µh∥2��I{gi=h}
+× exp
+� 1
+2ν2a
+L
+�
+ℓ=1
+(aℓ − ma)2 +
+1
+2ν2
+b
+L
+�
+ℓ=1
+(bℓ − mb)2� L
+�
+ℓ=1
+θλ1−1
+ℓ
+exp(−λ2θℓ)
+× exp
+� β2
+2ν2
+β
+�
+(σ2)−η1−1(τ 2)−ξ1−1 exp
+�
+− η2
+σ2 − ξ2
+τ 2
+�
+I{φ∈[u1,u2]}
+×
+H
+�
+h=1
+�
+ωαh−1
+h
+I{�H
+h=1 ωh=1} exp
+�
+−
+1
+2ν2µ
+∥µh − mµ∥2�
+(κ2
+h)−γ1−1 exp
+�
+− γ2
+κ2
+h
+��
+.
+2.4
+Inference and identifiability of parameters
+Note that the posterior distribution is highly intractable, hence we must resort to Markov chain Monte Carlo
+(MCMC) methods for inferences on model parameters. A Markov chain of the parameters is generated via
+the program “Just Another Gibbs Sampler” (JAGS) which is implemented in R [R Core Team, 2021] using
+the rjags package [Plummer, 2022].
+Several parameters are not identifiable in our model. Firstly, due to factors θℓ and φ, and the fact that latent
+positions are incorporated in the posterior only through their distances, the posterior is, therefore, invariant
+to θℓs and φ, and is invariant to scaling, reflection, rotation, and translation of the latent positions z. (Note
+that, Hoff et al., 2002 and Krivitsky et al., 2009 did not have θℓs, hence their posterior is not invariant to the
+8
+
+scaling of latent positions.) Although θℓs are not identifiable and do not affect the model fitting, in multi-
+layer settings, their ratios θℓ1/θℓ2 still provide valid information on layer’s relative strength of borrowing
+information from the latent space.
+Despite being unidentifiable, one can still make inferences on the latent positions and find a reasonable
+estimate for z through a post-process which we now describe. Similar to the definition in [Hoff et al., 2002],
+we define the equivalence class of z ∈ RN×K, denoted as [z], to be the set of positions that are equivalent
+to z under scaling, reflection, rotation, and translation. Given a fixed reference position zref, a position
+z∗ is found in [z] such that z∗ = arg minz′∈[z] tr(zref − z′)⊺(zref − z′), which is the so-called Procrustes
+transformation. In simulation studies, zref is naturally chosen to be the true latent position, while in practical
+applications, we could use the last iteration of the Markov chain of latent positions as the reference. The
+Procrustes transformation is performed for each iteration of the Markov chain of the latent positions {zn},
+and an estimate for z is taken as the mean of the Procrustes transformations of {zn}.
+As occurs in Bayesian mixture models, the label-switching problem for the group membership g is an-
+other source of non-identifiability. That is, the posterior is invariant under permutations of clustering labels.
+Many algorithms have been proposed to obtain a single clustering estimate based on the MCMC sample of
+the group membership {gn}, including an optimization method (which we call “MaxPEAR” hereafter) in
+Fritsch and Ickstadt [2009] that finds a clustering that maximizes posterior expected adjusted rand index, an
+optimization method (“MinBinder”) in Lau and Green [2007] that minimizes Binder’s loss function, and a
+greedy algorithm (“GreedyEPL”) in Rastelli and Friel [2018] that aims to minimize the variation of informa-
+tion, among others. These approaches may generate different clustering estimates, and to get a better under-
+standing of the model performance, all aforementioned algorithms (MaxPEAR, MinBinder and GreedyEPL)
+are used to assess the model. Estimates based on these approaches are found using the packages GreedyEPL
+[Rastelli, 2021] and mcclust [Fritsch, 2022].
+3
+Simulation
+Two simulation studies are carried out in this section to evaluate our model. A single-layer network is
+considered in the first simulation where we compare LPJMM with three other models designed only for
+single-layer networks, namely LPCM in Handcock et al. [2007], SNSM in Ciminelli et al. [2019], and CSBM
+in Leger [2016], where SNSM is also implemented using the rjags package, and LPCM and CSBM are
+implemented using the latentnet [Krivitsky and Handcock, 2022] and sbm [Chiquet et al., 2022] packages
+respectively. The model specifications for these models can be found in Appendix A. Models assessments
+include how well a model could recover the group membership and the latent position configuration, and a
+9
+
+ 
+1
+Figure 2: Left: A visualization of the network based on the true latent position and color indicates group
+membership g. Right: Heatmap of the adjacency matrix (where actors are reordered according to g).
+goodness-of-fit test using summaries of networks including density, transitivity, and assortative coefficient
+with respect to the group membership g (see Kolaczyk and Csárdi, 2020 for definitions). We also evaluate
+our model by how accurately it could estimate certain parameters in the model. The second simulation is
+conducted in two-layer network settings, where the performance of our model could be further evaluated by
+how well the ratio θ1/θ2 can be recovered that reflects differences in each layer’s dependency on the latent
+position structure.
+3.1
+Simulation 1: a single-layer network
+Consider a single-layer network (i.e., L = 1) with N = 100 actors generated as follows. Firstly, generate
+latent positions z from a mixture of H = 5 multivariate normal distributions, and then generate attributes x
+jointly from a multivariate normal distribution with mean β1N = 0 and covariance matrix given by Cov(d)
+in Section 2 where φ = 0.5, τ 2 = 0.3, σ2 = 1. Finally, the network data is generated according to Eq. (1)
+with a = 5, b = −2, and θ = 2.72. See Fig. 2 for a visualization of the simulated network. The network is
+fairly sparse with a density equal to 0.1531, and shows fairly strong transitivity and assortative mixing with
+coefficients 0.5049 and 0.5512 respectively.
+As for the prior specifications, we set ma = mb = 0, and ν2
+a = ν2
+b = 9 to allow a wide range of values for a
+and b. Let θ ∼ Gamma(1, 1) so that θ has mean 1. An almost flat prior is imposed on β by setting νβ = 104.
+The same uniform prior U(0, 1) as in Ciminelli et al. [2019] is specified for φ. We suggest the sum of the prior
+means of τ 2 and σ2 to be on the same scale as the sample variance of x, and here we use σ2 ∼ InvG(2, 1)
+and τ 2 ∼ InvG(2, 1). Let α = 1 so that the prior on ωωω is a flat Dirichlet distribution. Following the
+heuristics in Sosa and Betancourt [2022], we specify µh
+i.i.d.
+∼ NK(0, 2/3IK) and κ2
+h
+i.i.d.
+∼ InvG(3, 2/3) so
+that var(zij|gi) = 1.
+10
+
+our model name
+LPCM
+CSBM
+MaxPEAR
+0.737 (5)
+0.707 (4)
+–
+MinBinder
+0.712 (11)
+0.748 (10)
+–
+GreedyEPL
+0.664 (4)
+0.688 (4)
+–
+Variational-EM
+–
+–
+0.707 (6)
+Table 1: Adjusted Rand indices corresponding to different estimation methods for group membership. Num-
+bers in the parentheses represent numbers of estimated groups.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+39
+40
+41
+42
+43
+44
+45
+46
+47
+48
+49
+50
+51
+52
+53
+54
+55
+56
+57
+58
+59
+60
+61
+62
+63
+64
+65
+66
+67
+68
+69
+70
+71
+72
+73
+74
+75
+76
+77
+78
+79
+80
+81
+82
+83
+84
+85
+86
+87
+88
+89
+90
+91
+92
+93
+94
+95
+96
+97
+98
+99
+100
+(a) Truth
+1
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+(b) LPJMM
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+(c) LPCM
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+(d) CSBM
+Figure 3: (A): Color indicates the true group membership g. (B)-(D): Color indicates the estimated group
+memberships ˆg of LPJMM, LPCM and CSBM respectively. Positions of the points in all plots are true latent
+positions z.
+Note that the latent space dimension K and the number of clusters H in the model need to be prespecified
+along with the priors. We take K to be the true dimensions of the latent space (i.e., K = 2) since this
+facilitates model assessment by allowing visualizations of the estimated latent positions. One could also
+use the Watanabe-Akaike Information Criterion (WAIC) to select a K with the smallest WAIC as in Sosa
+and Betancourt [2022]. However, WAIC and other information criteria like Deviance Information Criterion
+(DIC) are not helpful in choosing the number of clusters H. We noticed that the model assessment is
+significantly worse when H is chosen to be smaller than the truth. However, model assessments are similar
+among models whose H is at least as large as the truth. A comparison of the model assessment for different
+specified H is given in Appendix B. From the comparison, we could also see that when H is specified to be
+larger, the number of clusters in the estimated group membership ˆg also tends to be larger. Therefore, we
+suggest choosing H to be the largest number of groups that one is willing to accept, and in this example, we
+choose H to be 5.
+We then fit LPJMM using MCMC sampling with 20 000 burn-in iterations and a further 10 000 iterations
+which are kept for posterior analysis. The Markov chain mixes reasonably well and shows no signs of lack
+of convergence (see Appendix C for the traceplot of the log-likelihood chain).
+11
+
+LPCM
+LPJMM
+(simulation 1)
+LPJMM
+(simulation 2)
+Sum of Euclidean distances
+23.08
+28.06
+12.20
+Table 2: Sum of distances between the estimated and true latent positions.
+To evaluate a model’s ability to recover the group membership, we first find estimates of clustering using
+the optimization algorithms (i.e., MaxPEAR, MinBinder and GreedyEPL) mentioned in Section 2.4. The
+adjusted Rand index is then calculated for each clustering estimate. Note that SNSM does not define clusters,
+therefore we only compare the adjusted Rand index between LPJMM, LPCM, and CSBM. Since the sbm
+package takes a non-Bayesian approach that uses a Variational-EM algorithm to find a point estimator for
+the group membership g, optimization methods like MaxPEAR are not necessary to analyze results from
+CSBM. The results shown in Table 1 suggest that these three models have a similar ability in recovering
+group membership, with rand indices of LPJMM using the MaxPEAR and MinBinder algorithms being
+higher than the rand index (0.707) under the CSBM model. A visualization of the estimated clusters based
+on the true latent positions is given in Fig. 3. Also, notice that the MinBinder algorithm tends to overestimate
+the number of clusters in the network.
+To further compare the ability to recover latent position configuration between LPJMM and LPCM, we find
+an estimate of the latent positions as follows. Firstly, we perform the Procrustes transformation on zn for each
+iteration n, and then take the estimate ˆz of z to be the average of zn. We then calculate the Euclidean distance
+between the estimated latent position ˆzi (which is the i-th row in ˆz) and the true latent position zi (i.e., the
+i-th row in z) for each actor i and use the sum of distances of all actors to quantify the similarity between
+the estimated and the true latent position configurations. The results are shown in Table 2 which suggests
+that these two models have similar recovery of the latent positions. Plots of the estimated latent positions of
+LPJMM and LPCM can be found in Appendix D, which also suggest similar estimated configurations of z
+as Table 2.
+Following Sosa and Betancourt [2022], we assess if models have a good fit in the sense of good reproduction
+of a variety of summary statistics, which are calculated based on a collection of simulated networks generated
+as follows. For LPJMM and SNSM, a network is simulated for every 10-th iteration using the parameters in
+that iteration. For LPCM and CSBM, 1000 networks are simulated using their respective packages. Then
+for each model, we calculate the density, transitivity, and assortative coefficient (if applicable) with respect
+to the true group membership for each simulated network. Boxplots of these summary statistics are given in
+Fig. 4 and the averages of these summary statistics for each model are given in Table 3. Note that our model
+12
+
+density
+transitivity
+assortativity
+0.14
+0.15
+0.16
+0.35 0.40 0.45 0.50 0.55
+0.45
+0.50
+0.55
+0.60
+LPJMM
+LPCM
+SNSM
+CSBM
+Figure 4: Boxplots of summary statistics for each model. Red dotted lines indicate the true values for
+network characteristics respectively.
+true value
+LPJMM
+LPCM
+SNSM
+CSBM
+density
+0.1531
+0.1539
+0.1530
+0.1504
+0.1499
+transitivity
+0.5049
+0.5144
+0.5467
+0.4027
+0.3776
+assortativity
+0.5512
+0.5468
+0.5475
+0.4811
+0.4954
+Table 3: Means of the summary statistics of the simulated networks for each model in simulation 1.
+appropriately captures these structural features of the network data, while LPCM tends to overestimate tran-
+sitivity in the network, and both SNSM and CSBM tend to underestimate both transitivity and assortativity
+in the network.
+3.2
+Simulation 2: a two-layer network
+Continue using the parameter setup in simulation 1 and its generated network as the first layer (i.e., a1 = 5,
+b1 = −2, θ1 = 2.72), we generate a second layer of the network with a2 = 3, b2 = 1, θ2 = 4. As in
+simulation 1, we fit LPJMM with K = 2 and H = 5 and evaluate the model’s ability to recover the group
+membership using the adjusted Rand indices based on four clustering summaries. The results are given in
+Table 4, which shows similar clustering estimates as in simulation 1 where only one layer is considered.
+However, the sum of Euclidean distances between the estimated and true latent positions of all actors (see
+Table 2) in simulation 2 is 12.20, which is a significant improvement compared to 28.06 in simulation 1.
+The plot of the estimated latent position configurations is given in Fig. 5 (B), which visualizes the model’s
+recovery of latent positions and group membership.
+We also carry out the goodness-of-fit test as in simulation 1 and the result is given in Table 5, which shows
+that LPJMM captures these structural features accurately, and the result for layer 1 is similar to the result in
+simulation 1.
+13
+
+MaxPEAR
+MinBinder
+GreedyEPL
+adjusted Rand index
+0.748 (6)
+0.753 (12)
+0.662 (4)
+Table 4: Adjusted Rand indices corresponding to different estimation methods for group membership in
+simulation 2. Numbers in the parentheses represent numbers of estimated groups.
+1
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+(a) Truth
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+(b) Estimated z and g
+Figure 5: (A): Points are plotted based on true latent position z and true group membership g. (B): Points
+are plotted using the estimated latent positions in simulation 2, and color represents the estimated group
+membership using the MaxPEAR method.
+Recall that θ1 and θ2 are of no direct interest since they are not identifiable. However, we are still interested
+in the ratio θ1/θ2 since it reflects the relative strength of borrowing information from the latent space of each
+layer. Although aℓ and bℓ are of no direct interest, we pay attention to their signs, especially that of bℓ because
+different signs of bℓ have different interpretations of the effect of attributes as discussed in Section 2. We also
+assess the model’s ability to estimate parameters β, τ 2, and σ2 using posterior means and 95% credible inter-
+vals. The results are given in Table 6. Overall, the performance of LPJMM in recovering the true values of
+these model parameters is pretty well, except for τ 2 and σ2. Both LPJMM and SNSM tend to underestimate
+σ2 and overestimate τ 2. That is, the covariance of the attributes tends to be underestimated, and although τ 2
+is slightly overestimated, the variance of the attributes (τ 2 + σ2) still tends to be underestimated.
+4
+Real data analysis
+In this section, we consider a three-layer network data set collected by [Lazega, 2001] from a corporate
+law firm from 1988-1991 in New England. This network describes three types of relationships (namely,
+networks of advice, friendship, and coworker contacts) between 71 lawyers in the law firm. Several actor
+attributes are also collected: age, gender, seniority (years with the firm), office (located in Boston, Hartford,
+or Providence), practice (litigation or corporate law), law school the lawyers attended (Harvard or Yale,
+University of Connecticut, or other universities) and status (partner or associate). A principal component
+14
+
+true value
+mean
+density
+layer 1
+0.1531
+0.1535
+layer 2
+0.1024
+0.1023
+transitivity
+layer 1
+0.5049
+0.5088
+layer 2
+0.5477
+0.5546
+assortativity
+layer 1
+0.5512
+0.5466
+layer 2
+0.6923
+0.6890
+Table 5: Means of the summary statistics in simulation 2.
+true value
+posterior mean
+95% credible interval
+θ1/θ2
+0.680
+0.653
+(0.579, 0.721)
+a1
+5
+5.01
+(4.719, 5.262)
+a2
+3
+3.25
+(2.976, 3.572)
+b1
+-2
+-1.919
+(-2.053, -1.766)
+b2
+1
+1.058
+(0.901, 1.252)
+β
+0
+-0.047
+(-1.027, 1.01 )
+τ 2
+0.3
+0.409
+(0.261, 0.592)
+σ2
+1
+0.642
+(0.230, 1.684)
+Table 6: Posterior means and 95% credible intervals.
+analysis (PCA) is performed on age and seniority attributes, and the first principal component explains
+89% of the variance which is of no surprise since age and seniority are highly correlated with a correlation
+coefficient being 0.78. We chose the first principal component to be the attribute x and let H = 8 since it
+is the largest number of clusters we expect in the network. Then the model is fitted to the network using the
+same prior and Markov chain setup as in Section 3.
+The study of the Lazega network in this paper is meant to find out how the three types of relations can be
+explained by the findings deduced from the model fitting. We first visualize the estimated latent positions
+z colored by different categorical attributes (gender, office, practice, law school, and status) in Fig. 6. As
+we can see from these plots, the estimated positions z are well separated by the office (especially offices
+in Boston and Hartford) and practice. Compare these plots with z colored by MaxPEAR and GreedyEPL
+estimated clustering g in Fig. 7, we can see that both estimated g roughly clusters lawyers into three groups:
+lawyers in Hartford office, litigation lawyers in Boston or Providence offices, and corporate lawyers in
+Boston or Providence offices.
+Plots of adjacency matrices of the three layers (where lawyers are grouped by the MaxPEAR estimate of g)
+and their corresponding networks are given in Fig. 8, where we could see that the coworker network shows
+15
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+male
+female
+gender
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+Boston
+Hartford
+Providence
+office
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+corporate
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+Harvard/Yale
+Ucon
+other
+law school
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+partner
+associate
+status
+Figure 6: Points in all plots are drawn based on the estimated latent positions z, and are colored based on
+their categories in gender, office, practice, law school, and status.
+the most estimated clustering pattern, while the advice network presents the least of such pattern, which
+could also be seen from the relative ratios of θℓs in Table 7. This means that lawyers from the same office
+and doing the same practice are more likely to become coworkers and friends, but who they seek advice
+from does not depend much on office and practice. Furthermore, we can deduce from the posteriors of bℓ in
+Table 7 that these lawyers tend to seek advice from people of similar age (or seniority) since the posterior
+estimate of b1 is negative, while lawyers of different ages (or seniority) are more likely to become friends and
+coworkers. This conclusion is in line with the assortativity coefficients with respect to the nodal attributes
+(lawyer’s age) given in Table 8.
+5
+Discussion
+This paper presents a latent position model that extends LPCM of Handcock et al. [2007] and SNSM of
+Ciminelli et al. [2019] to jointly model network data and the nodal attributes and perform model-based
+clustering. By jointly modeling the network and the attributes, we are able to describe how the attributes
+16
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+71
+(a) MaxPEAR
+1
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+65
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+68
+69
+70
+71
+(b) GreedyEPL
+Figure 7: Points are plotted using the estimated latent positions and color indicates the estimated group
+membership using MaxPEAR and GreedyEPL methods respectively.
+posterior mean
+95% credible interval
+θ1/θ2
+0.3229
+(0.2352, 0.4152)
+θ1/θ3
+0.2035
+(0.1479, 0.2606)
+θ2/θ3
+0.6319
+(0.5536, 0.7198)
+b1
+-0.0986
+(-0.1401, -0.0579)
+b2
+0.0708
+(0.0263, 0.1137)
+b3
+0.133
+(0.0854, 0.186)
+Table 7: Posterior means and 95% credible intervals.
+change over the network and explain how relations could be influenced by attributes. LPJMM also provides
+an extension to multi-layer network settings on the assumption that all layers share the same latent position
+structure but with different strengths of borrowing such latent structure information. We applied our method
+to two simulated networks, one with a single layer and another with two layers, and found our model to
+give satisfactory fits to these two data sets and is competitive in terms of goodness-of-fit and group detection
+compared with SNSM, LPCM, and CSBM. The model is also applied to a three-layer real network data set
+and we are able to draw reasonable conclusions from the modeling results.
+We have suggested choosing the number of groups H to be the largest number of groups that one is willing to
+accept in the network because we have found that varying the number of groups has almost no impact on the
+model fit and prediction outcome as long as it is in a reasonable range. One could also fit the CSBM to the
+network first, and choose H based on its estimated number of groups. One problem we have not addressed
+in the paper is of choosing the dimension of the latent space. This can be done by using Bayesian model
+selection like WAIC as in Sosa and Betancourt [2022].
+Our model could be extended in several ways. Firstly, other extensions of our model to multi-layer settings
+could be considered. For example, Sosa and Betancourt [2022] assumed conditionally independent layer-
+17
+
+advice
+friendship
+coworker
+ 
+ 
+ 
+Figure 8: Upper: Heatmaps of the adjacency matrices (where lawyers are reordered according to the Max-
+PEAR estimate of g). Lower: A visualization of the three layers based on the estimated z and color indicates
+the MaxPEAR estimate of g.
+advice
+friendship
+coworker
+assortativity
+0.2536
+-0.1107
+-0.1224
+Table 8: Assortativity coefficients with respect to lawyer’s age.
+specific latent positions, whereas MacDonald et al. [2022] assumed that the latent position of an actor in all
+layers is (d0 + d1)-dimensional, where the first d0 components of the latent position are the same across all
+layers, and only the last d1 components are layer-specific. Secondly, instead of assigning a user-specified
+number of groups H to the model, we could learn the number of groups by using a Bayesian nonpara-
+metric approach with a Dirichlet Process prior to model community memberships (see, e.g., Amini et al.,
+2019).
+LPJMM could also be extended to leverage multivariate covariates. So far, we have limited ourselves to mod-
+eling univariate nodal attributes that are approximately Gaussian. For continuous nodal attributes with more
+than one dimension, we have used the first principal component from the principal component analysis. To
+take full advantage of high-dimensional nodal attributes, one could use multivariate spatial process modeling
+to replace Eq. (2). Other extensions of more sophisticated spatial modeling include spatiotemporal modeling
+of attributes for time-varying networks, which would help to describe changes in actors over time.
+18
+
+Appendix
+A
+Model Specifications for SNSM, LPCM and CSBM
+Note that the original SNSM in Ciminelli et al. [2019] uses the logit link. In order to make a fair comparison,
+we also use the probit link in SNSM as in LPJMM. The model specification for SNSM used in this paper is
+given as follows:
+yi,j | z, x, aℓ, bℓ, θℓ
+ind
+∼ Ber
+�
+Φ(a + b|xi − xj| − ∥zi − zj∥)
+�
+,
+x | z, β, σ, τ, φ ∼ NN(β111N, σ2M(z, φ) + τ 2IN) ,
+and the priors are set to be the same as the priors in LPJMM (if possible). To be specific,
+zi
+i.i.d.
+∼ N2(000, I2) ,
+β ∼ N(0, 104) ,
+σ2 ∼ InvG(2, 1) ,
+τ 2 ∼ InvG(2, 1) ,
+φ ∼ U(0, 1) ,
+and the priors on the parameters in the probit regression tier are given by:
+a i.i.d.
+∼ N(0, 9) ,
+b i.i.d.
+∼ N(0, 9) .
+SNSM in this paper is implemented using JAGS.
+The model specification for LPCM (see Handcock et al., 2007) is given as the follows,
+yi,j | z, x, β0, β1
+ind
+∼ Ber
+�
+logit(β⊺
+0xi,j − β1∥zi − zj∥)
+�
+,
+zi | ωωω,µµµ,κκκ i.i.d.
+∼
+5
+�
+h=1
+ωhN5(µh, κ2
+hIK) ,
+and we use the default priors given in the latentnet package for prior specifications.
+We first introduce several notations before presenting CSBM in Leger [2016]. Suppose there are Q groups
+in the network. Denote the N × Q group membership matrix as ZZZ = {Ziq}, and Ziq = 1 if actor i belongs
+to group q, Ziq = 0 if otherwise. It is assumed that an actor can only belong to one group. The model
+specification for CSBM is given as follows,
+yi,j | Zi, Zj, x, β ind
+∼ Ber
+�
+logit(mqi,qj + β⊺xi,j)
+�
+,
+where Zi is the i-th row of ZZZ, qi is the group membership for actor i and the group effect mqi,qj ∈ R.
+19
+
+B
+Comparing model performances for different number of groups
+We conduct a comparison of LPJMM with different H ∈ {3, 4, . . . , 9} using the data set in simulation 1.
+Table 9 presents the adjusted rand indices, and the results are similar for models that assume H to be equal to
+or larger than the true number of groups (which is 5 in this example). However, the adjusted rand indices for
+all three estimates are significantly smaller when the model assumes H to be smaller than 5. Also, notice that
+the estimated number of groups increases with H. Visualizations of how adjusted rand indices and estimated
+number of groups changes over H are given in Fig. 9.
+H
+MaxPEAR
+MinBinder
+GreedyEPL
+3
+0.4067 (3)
+0.4008 (5)
+0.4321 (3)
+4
+0.4882 (3)
+0.4977 (6 )
+0.6521 (4)
+5
+0.7374 (5)
+0.7115 (11)
+0.6635 (4)
+6
+0.7237 (6)
+0.7442 (20)
+0.7134 (4)
+7
+0.7449 (7)
+0.6624 (25)
+0.7313 (4)
+8
+0.7422 (8)
+0.6674 (25)
+0.7293 (8)
+9
+0.7056 (12)
+0.7041 (25)
+0.7043 (11)
+Table 9: Adjusted Rand indices of different estimates under LPJMM with different H. Numbers in the
+parentheses denote the numbers of estimated groups.
+3
+4
+5
+6
+7
+8
+9
+H
+Adjusted rand index
+0.4
+0.6
+0.8
+3
+4
+5
+6
+7
+8
+9
+H
+number of groups
+5
+10
+15
+20
+25
+MaxPEAR
+MinBinder
+GreedyEPL
+Figure 9: Left: Adjusted rand indices of the clustering estimates found by using the MaxPear, MinBinder,
+and GreedyEPL methods. Right: Estimated number of groups using the three methods.
+The goodness-of-fit test outlined in Section 3 is also carried out here to compare the means of several sum-
+mary statistics, which are plotted in Fig. 10. As we can see from the plots, the model’s fit is not affected by
+the choice of H even for H smaller than the actual number of clusters in the network.
+20
+
+3
+4
+5
+6
+7
+8
+9
+H
+0.1536
+0.1540
+0.1544
+(a) density
+3
+4
+5
+6
+7
+8
+9
+H
+0.513
+0.514
+0.515
+(b) transitivity
+3
+4
+5
+6
+7
+8
+9
+H
+0.538
+0.546
+0.554
+(c) assortativity
+Figure 10: The means of summary statistics for different H.
+C
+Traceplots of log-likelihood
+The traceplots of the log-likelihood (after thinning the Markov chain every 10 iterations) in simulation stud-
+ies and real applications in Sections 3 and 4 are given in Fig. 11.
+0
+2000
+6000
+10000
+−1100
+−1080
+−1060
+−1040
+(a) simulation 1
+0
+2000
+6000
+10000
+−1920
+−1900
+−1880
+−1860
+(b) simulation 2
+0
+2000
+6000
+10000
+−5230
+−5214
+−5198
+−5182
+(c) Lazega network
+Figure 11: Traceplots of the log-likelihood.
+D
+Visualizations of results from LPJMM and LPCM
+Visualizations of the estimated latent positions and estimated group membership using the MaxPEAR, Min-
+Binder, and GreedyEPL methods under LPJMM and LPCM are shown in Figs. 12 and 13 respectively.
+21
+
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+39
+40
+41
+42
+43
+44
+45
+46
+47
+48
+49
+50
+51
+52
+53
+54
+55
+56
+57
+58
+59
+60
+61
+62
+63
+64
+65
+66
+67
+68
+69
+70
+71
+72
+73
+74
+75
+76
+77
+78
+79
+80
+81
+82
+83
+84
+85
+86
+87
+88
+89
+90
+91
+92
+93
+94
+95
+96
+97
+98
+99
+100
+(a) MaxPEAR
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+39
+40
+41
+42
+43
+44
+45
+46
+47
+48
+49
+50
+51
+52
+53
+54
+55
+56
+57
+58
+59
+60
+61
+62
+63
+64
+65
+66
+67
+68
+69
+70
+71
+72
+73
+74
+75
+76
+77
+78
+79
+80
+81
+82
+83
+84
+85
+86
+87
+88
+89
+90
+91
+92
+93
+94
+95
+96
+97
+98
+99
+100
+(b) MinBinder
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+39
+40
+41
+42
+43
+44
+45
+46
+47
+48
+49
+50
+51
+52
+53
+54
+55
+56
+57
+58
+59
+60
+61
+62
+63
+64
+65
+66
+67
+68
+69
+70
+71
+72
+73
+74
+75
+76
+77
+78
+79
+80
+81
+82
+83
+84
+85
+86
+87
+88
+89
+90
+91
+92
+93
+94
+95
+96
+97
+98
+99
+100
+(c) GreedyEPL
+Figure 12: Points are plotted based on the estimated latent position z and three estimated group memberships
+ˆg of LPJMM.
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+39
+40
+41
+42
+43
+44
+45
+46
+47
+48
+49
+50
+51
+52
+53
+54
+55
+56
+57
+58
+59
+60
+61
+62
+63
+64
+65
+66
+67
+68
+69
+70
+71
+72
+73
+74
+75
+76
+77
+78
+79
+80
+81
+82
+83
+84
+85
+86
+87
+88
+89
+90
+91
+92
+93
+94
+95
+96
+97
+98
+99
+100
+(a) MaxPEAR (LPCM)
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+39
+40
+41
+42
+43
+44
+45
+46
+47
+48
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+50
+51
+52
+53
+54
+55
+56
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+59
+60
+61
+62
+63
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+65
+66
+67
+68
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+70
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+72
+73
+74
+75
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+77
+78
+79
+80
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+82
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+84
+85
+86
+87
+88
+89
+90
+91
+92
+93
+94
+95
+96
+97
+98
+99
+100
+(b) MinBinder (LPCM)
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+23
+24
+25
+26
+27
+28
+29
+30
+31
+32
+33
+34
+35
+36
+37
+38
+39
+40
+41
+42
+43
+44
+45
+46
+47
+48
+49
+50
+51
+52
+53
+54
+55
+56
+57
+58
+59
+60
+61
+62
+63
+64
+65
+66
+67
+68
+69
+70
+71
+72
+73
+74
+75
+76
+77
+78
+79
+80
+81
+82
+83
+84
+85
+86
+87
+88
+89
+90
+91
+92
+93
+94
+95
+96
+97
+98
+99
+100
+(c) GreedyEPL (LPCM)
+Figure 13: Points are plotted based on estimated z and three estimated ˆg of LPCM.
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+
diff --git a/8tAyT4oBgHgl3EQfQ_YL/content/tmp_files/load_file.txt b/8tAyT4oBgHgl3EQfQ_YL/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,2653 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf,len=2652
+page_content='A Bayesian latent position approach for community detection in  single- and multi-layer networks with continuous attributes  Zhumengmeng Jina,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Juan Sosab and Brenda Betancourtc  a University of Florida  b Universidad Nacional de Colombia  c NORC at the University of Chicago  December 2022  Abstract  The increasing prevalence of multiplex networks has spurred a critical need to take into account po-  tential dependencies across different layers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' especially when the goal is community detection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' which is a  fundamental learning task in network analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' We propose a full Bayesian mixture model for community  detection in both single-layer and multi-layer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' A key feature of our model is the joint modeling  of the nodal attributes that often come with the network data as a spatial process over the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  In addition, our model for multi-layer networks allows layers to have different strengths of dependency  in the unique latent position structure and assumes that the probability of a relation between two actors  (in a layer) depends on the distances between their latent positions (multiplied by a layer-specific factor)  and the difference between their nodal attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Under our prior specifications, the actors’ positions  in the latent space arise from a finite mixture of Gaussian distributions, each corresponding to a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Simulated examples show that our model performs favorably compared to the existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The model  is also applied to a real three-layer network of employees in a law firm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  1  Introduction  Network data conveniently describes the relationships between actors in complex systems and is ubiquitous  in many statistical applications, including finance, social science, criminology, biology, epidemiology, and  computer science, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Understanding the relationships between actors can aid domain experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' multiplex network, community detection, latent position model, mixture model, spatial process, visu-  alization  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='00055v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='AP]  30 Dec 2022  For instance, in epidemiology, people in a certain area can be portrayed in a contact network that can be  studied to detect infectious disease outbreaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' In criminology, communications between terrorists form a  terrorist network, helping intelligence agencies to better counter terrorism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Many models have been developed for the inference of networks over the past decades (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', Erdös and  Rényi, 1959, Frank and Strauss, 1986), among which the broad class of latent space models is one of the  most widely used (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', Sosa, 2021 for an exhaustive review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Suppose the network under study has  N actors, then under latent space models, there are N independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=') latent  variables z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , zN, one for each actor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Under a mild exchangeability assumption in Hoff [2007], results  in Aldous [1985] and Hoover [1982] show that edge variables yi,j depend on latent variables through a  symmetric function γ(zi, zj) that is meant to capture any pattern in the network beyond any known covariate  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Many well-known models fall into the category of latent space models, which can be distinguished between  two cases depending on whether latent variables are discrete or continuous [Matias and Robin, 2014].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' For in-  stance, stochastic block models [Nowicki and Snijders, 2001, Wang and Wong, 1987] – hereafter SBM – are  special cases of latent space models with discrete latent variables zi ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' When latent variables  are assumed to be continuous, another approach using latent variables is the class of latent position models  (LPM) proposed by Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2002] which our model in the paper is built upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' In its basic formulation,  LPMs model the edge variables yi,j as conditionally independent given the distance between latent variables  γ(zi, zj) = −∥zi − zj∥, which naturally accounts for transitivity effects through the latent space (typically  a Euclidean K-dimensional space for a predetermined K) where zi lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Later on, Handcock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2007]  proposed an extension on Hoff et al.’s LPM, namely the latent position cluster model (LPCM), by imposing  a Gaussian mixture prior on the latent positions to perform clustering tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Krivitsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2009] further  extended Handcock et al.’s model by adding the random sender and receiver effects proposed by Hoff [2005].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Other formulations of γ(·, ·) can be found in Schweinberger and Snijders [2003], Hoff [2005, 2009], Athreya  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2017], Minhas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2019], among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Besides edge information of a network, extra information like node and edge attributes and different types  of edges are often available, and should ideally be leveraged for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Typical ways to incorporate  attributes in a network model include: (1) modeling the network as a function of the attributes (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=',  Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', 2002, Hoff, 2005);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (2) modeling the attributes as a function of the network [Guha and Rodriguez,  2021];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (3) jointly modeling the network and attributes (Linkletter, 2007, Kim and Leskovec, 2012, Fosdick  and Hoff, 2015, Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The first approach is arguably the most common approach to incor-  porate covariates into the model, but we consider an approach of joint modeling proposed by Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  2  [2019], namely the social network spatial model (SNSM), where the authors modeled edges yi,j as condi-  tionally independent given ∥zi − zj∥ and the distance of the continuous node attributes ∥xi − xj∥, and node  attributes are further modeled as a spatial process over the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Note that joint modeling does not  require the network or the attributes to be fully observed as the first two approaches, hence one could predict  missing network and attribute data (if there is any).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' In addition, it improves model fitting by capturing the  dependence structure between latent variables and the attributes (when such dependency exists), as we will  see in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  We propose a full hierarchical Bayesian model that builds on Ciminelli et al.’s SNSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Instead of using a  Gaussian distribution as the prior for latent positions as in Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2019], we impose a Gaussian  mixture prior as in Handcock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2007], so that our model could also capture the group structure in the  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Detecting communities or clusters among actors in the network is an important task in network  analysis and has spurred the development of many models and algorithms, among which the SBM has  motivated an active line of research that deals with community detection (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', Lee and Wilkinson  [2019] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' However, SBM may not fit well when many actors fall between clusters [Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=',  2002].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' We will compare our model with an SBM that incorporates covariates as fixed effects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', model  the edge variables as a function of latent classes and covariates [Leger, 2016]), and we call this model a  covariate-assisted stochastic block model (CSBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' We will show that our model presents improved model  fitting while producing similar clustering results as CSBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  We also propose an extension of our model to multi-layer network settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Multi-layer networks can gen-  erally be categorized into two cases: cross-sectional networks that have different types of connections (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=',  social networks of friendship, coworker-ship, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=') and time-varying networks where the same type of con-  nections are measured over time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', a trade network that changes over time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' We consider a type of  cross-sectional multi-layer network where each layer has a common set of actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Substantial work has been  done on latent space models for cross-sectional multi-layer networks that take a Bayesian approach (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=',  Gollini and Murphy, 2016, Salter-Townshend and McCormick, 2017, D’Angelo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', 2019, Sosa and Betan-  court, 2022, Durante and Dunson, 2018, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', 2019, MacDonald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' In extending our model  to the multiple networks setting, we adopt the approach in Sosa and Betancourt [2022] in a parsimonious  way, where latent positions are assumed to be the same for all layers, but the strength of borrowing such  latent structure information is allowed to be different across different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Note that, the original model  in Sosa and Betancourt [2022] assumed different latent positions for different layers and had an additional  hierarchy on the hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The specification of our model is given in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Section 2 contains general background on the spatial  3  process and introduces the proposed model (for single- and multi-layer network settings) which we call  the latent position joint mixture model (LPJMM) in the rest of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' In addition, prior specification,  identifiable problem, and inference will also be discussed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Several simulation studies are  conducted in section 3, where LPJMM is compared with Handcock et al.’s LPCM, Ciminelli et al.’s SNSM  and CSBM in single-layer settings and the model is also evaluated in multi-layer settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' In section 4, we  apply LPJMM to a real-world multi-layer network data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Finally, we conclude with some discussion in  section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  2  Models  We first review the LPM introduced in Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2002], and then build upon it with a spatial process to allow  for joint modeling of the network and the nodal attributes, and with a finite Gaussian mixture distribution for  latent positions to allow for clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Consider a binary single-layer network with N actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Denote its adjacency matrix as Y = (yi,j) ∈  {0, 1}N×N, where yi,j = 1 if actors i and j are connected, and yi,j = 0 if they are not connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Suppose  the network data comes with a one-dimensional nodal attribute xi for each actor, and denote the covariate as  x = (xi) ∈ RN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The LPM assumes that each actor i has an observed latent position zi in a K-dimensional  Euclidean latent space, the so-called latent space, for some K ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Let z = (zi) ∈ RN×K, then LPM  models edge yi,j as conditionally independent given distances between nodal attributes as well as distances  between latent positions via logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' But instead of the logistic link, we use the probit link in our  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The analysis of probit regression models can often be facilitated by a Gibbs sampler constructed using  the data augmentation approach that introduces latent variables with truncated normal distributions [Albert  and Chib, 1993].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (See also Sosa and Betancourt (2022) for a discussion on the choice of link functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=')  Specifically, for i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , N} and i ̸= j,  yi,j | z, x, a, b, θ ind  ∼ Ber  �  Φ(a + b|xi − xj| − θ∥zi − zj∥)  �  ,  (1)  where a, b ∈ R and θ ∈ R+, Ber(p) is a Bernoulli distribution that takes value 1 with some probability p,  ∥ · ∥ is the Euclidean norm on RK and Φ(·) is the cumulative distribution function of the standard normal  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Note that we impose a factor θ for the distance between latent positions, which is different from  Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2002] and Krivitsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Although θ is unidentifiable in single-layer networks, it plays  a non-trivial role in multi-layer network settings (introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' We defer a detailed discussion  of θ to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  4  To allow for joint modeling of the network and nodal attributes, we model the nodal attributes as a spatial  process over the latent space RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Hence, nodal attributes are treated as random variables indexed by their  latent positions, and the distance between these random variables is found by the distance between their  corresponding positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' As in Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2019], we specify the spatial process as a Gaussian process  that is stationary with mean β and isotropic (see Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', 2015 for definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' In this case, the  process is completely defined by its covariance function Cov(d), where d is the distance between two random  variables in the Gaussian process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' In particular, we specify Cov(d) with an exponential kernel, that is,  Cov(d) =  �  �  �  �  �  τ 2 + σ2,  if d = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  σ2 exp(−φd),  if d > 0,  where τ ≥ 0, σ > 0 and φ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' It is well-known that such a covariance structure is valid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', the covariance  matrix for any finite collection of random variables in the process is positive definite [Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Let Mz = (mij) ∈ RN×N where mij = exp(−φ∥zi − zj∥) and denote IN as the N-dimensional identity  matrix, then the Gaussian process of the nodal attributes is constructed as follows,  x | z, β, σ, τ, φ ∼ NN(β111N, σ2M(z, φ) + τ 2IN),  (2)  where Nd is a d-dimensional multivariate normal distribution for some dimension d ∈ {2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' }, and 111N is  an N-dimensional vector with all 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  As in Krivitsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2009], we impose a Gaussian mixture distribution on latent positions, which allows us  to cluster actors into different groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Suppose there are H < ∞ predetermined number of components in  the Gaussian mixture distribution, then  zi | ωωω,µµµ,κκκ ind  ∼  H  �  h=1  ωhNK(µh, κ2  hIK) ,  (3)  where ωωω = {ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , ωH}, µµµ = {µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , µH}, κκκ = {κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , κH}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Note that µh is a K-dimensional mean  vector where h ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , H}, and ωh is the probability that an actor belongs to the h-th group such that  ωh ∈ (0, 1) and �H  h=1 ωh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  In single-layer network settings, the model is given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (1) to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Under our model, nodal attributes  of two actors whose latent positions are close are more likely to be similar according to the exponential  covariance structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' If b < 0 (b > 0), actors with similar attributes are more (less) likely to be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  When b = 0, nodal attributes do not affect the distribution of the network directly (but it still has an indirect  5  Figure 1: DAG representation of the LPJMM in multi-layer settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  impact on the network through latent positions by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='1  An extension to multi-layer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Our model can also be extended to multi-layer network settings in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Suppose we have L  layers Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , YL in the network, where all layers are defined over the same set of actors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' We assume the  same latent positions z for all layers but allow the strength of borrowing such latent structure information to  be different by imposing layer-specific factors θℓ for ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Our model in multi-layer settings is  then presented as follows  yi,j,ℓ | z, x, aℓ, bℓ, θℓ  ind  ∼ Ber  �  Φ(aℓ + bℓ|xi − xj| − θℓ∥zi − zj∥)  �  ,  (4)  x | z, β, σ, τ, φ ∼ NN(β111N, σ2M(z, φ) + τ 2IN) ,  (5)  zi | ωωω,µµµ,κκκ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  ∼  H  �  h=1  ωhNK(µh, κ2  hIK) ,  (6)  where yi,j,ℓ is the edge variable between actors i and j in layer ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , L}, aℓ, bℓ and θℓ are layer-specific  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Note that Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (5) and (6) are the same as Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' 1 shows a directed acyclic graph  (DAG) representation of the model given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (4) to (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='2  Prior specification  We take a Bayesian approach to estimate the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Without loss of generality, a Bayesian ver-  sion of the model given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (4) to (6) is formed by placing prior distributions on the unknown parameters  aℓ, bℓ, θℓ, β, σ, τ, φ, ωωω, µµµh, κh, for ℓ = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , L} and h = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' In the model we consider, these  parameters are assumed a priori independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' For parameters in the probit regression tier as specified by  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (4), their priors are specified as follows:  aℓ  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  ∼ N(ma, ν2  a) ,  bℓ  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  ∼ N(mb, ν2  b ) ,  θℓ  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  ∼ Gamma(λ1, λ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  The priors for the parameters in the spatial process tier as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (5) are given as follows:  β ∼ N(0, ν2  β) ,  σ2 ∼ InvG(η1, η2) ,  τ 2 ∼ InvG(ξ1, ξ2) ,  φ ∼ U(u1, u2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Finally, we put the following priors on the rest of the parameters:  ωωω ∼ Dir(α) ,  µh  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  ∼ NK(mµ, ν2  µIK) ,  κ2  h  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  ∼ InvG(γ1, γ2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Note that, ma, νa, mb, νb, λ1, λ2, νβ, η1, η2, ξ1, ξ2, u1, u2, α, mµ, νµ, γ1 and γ2 are user-specified  hyperparameters, and Gamma(·, ·), InvG(·, ·), U(·, ·), Dir(·) represents Gamma, Inverse-Gamma, uniform,  and Dirichlet distributions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='3  Posterior distribution and model estimation  As is standard in Bayesian estimation of mixture models (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', Diebolt and Robert [1994]), we define a  new variable gi that serves as the missing data of group membership of actor i whose distribution depends  on ωωω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' In particular, gi = h if actor i belongs to the h-th group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The joint density of (zi, gi) given ωωω, µµµ and κκκ  is then given by  H  �  h=1  �  ωh  1  �  2πκ2  h  exp  �  −  1  2κ2  h  ∥zi − µh∥2��I{gi=h}  ,  where the indicator function I{gi=h} = 1 if gi = h, and I{gi=h} = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Let g = (gi)N  i=1 be the group  membership for all actors and L(·) be the law of a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Then the posterior distribution of z, g  7  and the parameters upon which priors are specified in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='2 is given by  Π(z, g, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , aL, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , bL, θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , θL, β, τ 2, σ2, φ,ωωω,µµµ,κκκ | Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , YL, x)  ∝  � L  �  ℓ=1  L(Yℓ | z, x, aℓ, bℓ, θℓ)  �  L(x | z, σ, τ, φ)L(z, g | ωωω,µµµ,κκκ)  � L  �  ℓ=1  L(aℓ)L(bℓ)L(θℓ)  �  × L(β)L(σ2)L(τ 2)L(φ)L(ωωω)L(µµµ)L(κκκ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Note that the dimension of the posterior distribution has dimension NK + N + 3L + 3H + 4 and the  corresponding posterior density is presented as follows,  π(z, g, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , aL, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , bL, θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , θL, β, τ 2, σ2, φ,ωωω,µµµ,κκκ | Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' YL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' x)  ∝  N  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='j=1  i̸=j  L  �  ℓ=1  �  Φ(aℓ + bℓ|xi − xj| − θℓ∥zi − zj∥)  �yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='ℓ�  1 − Φ(aℓ + bℓ|xi − xj| − θℓ∥zi − zj∥)  �1−yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='ℓ  × |σ2M(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' φ) + τ 2IN|− 1  2 exp  �  − 1  2(x − β1)⊺�  σ2M(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' φ) + τ 2IN  �−1(x − β1)  �  ×  N  �  i=1  H  �  h=1  � ωh  �  κ2  h  exp  �  −  1  2κ2  h  ∥zi − µh∥2��I{gi=h}  × exp  � 1  2ν2a  L  �  ℓ=1  (aℓ − ma)2 +  1  2ν2  b  L  �  ℓ=1  (bℓ − mb)2� L  �  ℓ=1  θλ1−1  ℓ  exp(−λ2θℓ)  × exp  � β2  2ν2  β  �  (σ2)−η1−1(τ 2)−ξ1−1 exp  �  − η2  σ2 − ξ2  τ 2  �  I{φ∈[u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='u2]}  ×  H  �  h=1  �  ωαh−1  h  I{�H  h=1 ωh=1} exp  �  −  1  2ν2µ  ∥µh − mµ∥2�  (κ2  h)−γ1−1 exp  �  − γ2  κ2  h  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='4  Inference and identifiability of parameters  Note that the posterior distribution is highly intractable, hence we must resort to Markov chain Monte Carlo  (MCMC) methods for inferences on model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' A Markov chain of the parameters is generated via  the program “Just Another Gibbs Sampler” (JAGS) which is implemented in R [R Core Team, 2021] using  the rjags package [Plummer, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Several parameters are not identifiable in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Firstly, due to factors θℓ and φ, and the fact that latent  positions are incorporated in the posterior only through their distances, the posterior is, therefore, invariant  to θℓs and φ, and is invariant to scaling, reflection, rotation, and translation of the latent positions z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (Note  that, Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', 2002 and Krivitsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', 2009 did not have θℓs, hence their posterior is not invariant to the  8  scaling of latent positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=') Although θℓs are not identifiable and do not affect the model fitting, in multi-  layer settings, their ratios θℓ1/θℓ2 still provide valid information on layer’s relative strength of borrowing  information from the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Despite being unidentifiable, one can still make inferences on the latent positions and find a reasonable  estimate for z through a post-process which we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Similar to the definition in [Hoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', 2002],  we define the equivalence class of z ∈ RN×K, denoted as [z], to be the set of positions that are equivalent  to z under scaling, reflection, rotation, and translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Given a fixed reference position zref, a position  z∗ is found in [z] such that z∗ = arg minz′∈[z] tr(zref − z′)⊺(zref − z′), which is the so-called Procrustes  transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' In simulation studies, zref is naturally chosen to be the true latent position, while in practical  applications, we could use the last iteration of the Markov chain of latent positions as the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The  Procrustes transformation is performed for each iteration of the Markov chain of the latent positions {zn},  and an estimate for z is taken as the mean of the Procrustes transformations of {zn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  As occurs in Bayesian mixture models, the label-switching problem for the group membership g is an-  other source of non-identifiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' That is, the posterior is invariant under permutations of clustering labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Many algorithms have been proposed to obtain a single clustering estimate based on the MCMC sample of  the group membership {gn},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' including an optimization method (which we call “MaxPEAR” hereafter) in  Fritsch and Ickstadt [2009] that finds a clustering that maximizes posterior expected adjusted rand index,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' an  optimization method (“MinBinder”) in Lau and Green [2007] that minimizes Binder’s loss function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' and a  greedy algorithm (“GreedyEPL”) in Rastelli and Friel [2018] that aims to minimize the variation of informa-  tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' These approaches may generate different clustering estimates, and to get a better under-  standing of the model performance, all aforementioned algorithms (MaxPEAR, MinBinder and GreedyEPL)  are used to assess the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Estimates based on these approaches are found using the packages GreedyEPL  [Rastelli, 2021] and mcclust [Fritsch, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  3  Simulation  Two simulation studies are carried out in this section to evaluate our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' A single-layer network is  considered in the first simulation where we compare LPJMM with three other models designed only for  single-layer networks, namely LPCM in Handcock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2007], SNSM in Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2019], and CSBM  in Leger [2016], where SNSM is also implemented using the rjags package, and LPCM and CSBM are  implemented using the latentnet [Krivitsky and Handcock, 2022] and sbm [Chiquet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', 2022] packages  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The model specifications for these models can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Models assessments  include how well a model could recover the group membership and the latent position configuration, and a  9  1  Figure 2: Left: A visualization of the network based on the true latent position and color indicates group  membership g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Right: Heatmap of the adjacency matrix (where actors are reordered according to g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  goodness-of-fit test using summaries of networks including density, transitivity, and assortative coefficient  with respect to the group membership g (see Kolaczyk and Csárdi, 2020 for definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' We also evaluate  our model by how accurately it could estimate certain parameters in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The second simulation is  conducted in two-layer network settings, where the performance of our model could be further evaluated by  how well the ratio θ1/θ2 can be recovered that reflects differences in each layer’s dependency on the latent  position structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='1  Simulation 1: a single-layer network  Consider a single-layer network (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', L = 1) with N = 100 actors generated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Firstly, generate  latent positions z from a mixture of H = 5 multivariate normal distributions, and then generate attributes x  jointly from a multivariate normal distribution with mean β1N = 0 and covariance matrix given by Cov(d)  in Section 2 where φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='5, τ 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='3, σ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Finally, the network data is generated according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (1)  with a = 5, b = −2, and θ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' 2 for a visualization of the simulated network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The network is  fairly sparse with a density equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='1531, and shows fairly strong transitivity and assortative mixing with  coefficients 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='5049 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='5512 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  As for the prior specifications, we set ma = mb = 0, and ν2  a = ν2  b = 9 to allow a wide range of values for a  and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Let θ ∼ Gamma(1, 1) so that θ has mean 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' An almost flat prior is imposed on β by setting νβ = 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  The same uniform prior U(0, 1) as in Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2019] is specified for φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' We suggest the sum of the prior  means of τ 2 and σ2 to be on the same scale as the sample variance of x, and here we use σ2 ∼ InvG(2, 1)  and τ 2 ∼ InvG(2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Let α = 1 so that the prior on ωωω is a flat Dirichlet distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Following the  heuristics in Sosa and Betancourt [2022], we specify µh  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  ∼ NK(0, 2/3IK) and κ2  h  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  ∼ InvG(3, 2/3) so  that var(zij|gi) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  10  our model name  LPCM  CSBM  MaxPEAR  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='737 (5)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='707 (4)  –  MinBinder  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='712 (11)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='748 (10)  –  GreedyEPL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='664 (4)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='688 (4)  –  Variational-EM  –  –  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='707 (6)  Table 1: Adjusted Rand indices corresponding to different estimation methods for group membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Num-  bers in the parentheses represent numbers of estimated groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content='96  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='97  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='98  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='99  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='(d) CSBM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='Figure 3: (A): Color indicates the true group membership g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (B)-(D): Color indicates the estimated group  memberships ˆg of LPJMM, LPCM and CSBM respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Positions of the points in all plots are true latent  positions z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Note that the latent space dimension K and the number of clusters H in the model need to be prespecified  along with the priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' We take K to be the true dimensions of the latent space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', K = 2) since this  facilitates model assessment by allowing visualizations of the estimated latent positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' One could also  use the Watanabe-Akaike Information Criterion (WAIC) to select a K with the smallest WAIC as in Sosa  and Betancourt [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' However, WAIC and other information criteria like Deviance Information Criterion  (DIC) are not helpful in choosing the number of clusters H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' We noticed that the model assessment is  significantly worse when H is chosen to be smaller than the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' However, model assessments are similar  among models whose H is at least as large as the truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' A comparison of the model assessment for different  specified H is given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' From the comparison, we could also see that when H is specified to be  larger, the number of clusters in the estimated group membership ˆg also tends to be larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Therefore, we  suggest choosing H to be the largest number of groups that one is willing to accept, and in this example, we  choose H to be 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  We then fit LPJMM using MCMC sampling with 20 000 burn-in iterations and a further 10 000 iterations  which are kept for posterior analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The Markov chain mixes reasonably well and shows no signs of lack  of convergence (see Appendix C for the traceplot of the log-likelihood chain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  11  LPCM  LPJMM  (simulation 1)  LPJMM  (simulation 2)  Sum of Euclidean distances  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='08  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='06  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='20  Table 2: Sum of distances between the estimated and true latent positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  To evaluate a model’s ability to recover the group membership, we first find estimates of clustering using  the optimization algorithms (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', MaxPEAR, MinBinder and GreedyEPL) mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The  adjusted Rand index is then calculated for each clustering estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Note that SNSM does not define clusters,  therefore we only compare the adjusted Rand index between LPJMM, LPCM, and CSBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Since the sbm  package takes a non-Bayesian approach that uses a Variational-EM algorithm to find a point estimator for  the group membership g, optimization methods like MaxPEAR are not necessary to analyze results from  CSBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The results shown in Table 1 suggest that these three models have a similar ability in recovering  group membership, with rand indices of LPJMM using the MaxPEAR and MinBinder algorithms being  higher than the rand index (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='707) under the CSBM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' A visualization of the estimated clusters based  on the true latent positions is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Also, notice that the MinBinder algorithm tends to overestimate  the number of clusters in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  To further compare the ability to recover latent position configuration between LPJMM and LPCM, we find  an estimate of the latent positions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Firstly, we perform the Procrustes transformation on zn for each  iteration n, and then take the estimate ˆz of z to be the average of zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' We then calculate the Euclidean distance  between the estimated latent position ˆzi (which is the i-th row in ˆz) and the true latent position zi (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', the  i-th row in z) for each actor i and use the sum of distances of all actors to quantify the similarity between  the estimated and the true latent position configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The results are shown in Table 2 which suggests  that these two models have similar recovery of the latent positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Plots of the estimated latent positions of  LPJMM and LPCM can be found in Appendix D, which also suggest similar estimated configurations of z  as Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Following Sosa and Betancourt [2022], we assess if models have a good fit in the sense of good reproduction  of a variety of summary statistics, which are calculated based on a collection of simulated networks generated  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' For LPJMM and SNSM, a network is simulated for every 10-th iteration using the parameters in  that iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' For LPCM and CSBM, 1000 networks are simulated using their respective packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Then  for each model, we calculate the density, transitivity, and assortative coefficient (if applicable) with respect  to the true group membership for each simulated network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Boxplots of these summary statistics are given in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' 4 and the averages of these summary statistics for each model are given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Note that our model  12  density  transitivity  assortativity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='60  LPJMM  LPCM  SNSM  CSBM  Figure 4: Boxplots of summary statistics for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Red dotted lines indicate the true values for  network characteristics respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  true value  LPJMM  LPCM  SNSM  CSBM  density  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='1531  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='1539  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content='1499  transitivity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='5049  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='5144  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='5467  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='4027  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='3776  assortativity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='5512  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='5468  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='5475  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='4811  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='4954  Table 3: Means of the summary statistics of the simulated networks for each model in simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  appropriately captures these structural features of the network data, while LPCM tends to overestimate tran-  sitivity in the network, and both SNSM and CSBM tend to underestimate both transitivity and assortativity  in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='2  Simulation 2: a two-layer network  Continue using the parameter setup in simulation 1 and its generated network as the first layer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', a1 = 5,  b1 = −2, θ1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='72), we generate a second layer of the network with a2 = 3, b2 = 1, θ2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' As in  simulation 1, we fit LPJMM with K = 2 and H = 5 and evaluate the model’s ability to recover the group  membership using the adjusted Rand indices based on four clustering summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The results are given in  Table 4, which shows similar clustering estimates as in simulation 1 where only one layer is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  However, the sum of Euclidean distances between the estimated and true latent positions of all actors (see  Table 2) in simulation 2 is 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='20, which is a significant improvement compared to 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='06 in simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  The plot of the estimated latent position configurations is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' 5 (B), which visualizes the model’s  recovery of latent positions and group membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  We also carry out the goodness-of-fit test as in simulation 1 and the result is given in Table 5, which shows  that LPJMM captures these structural features accurately, and the result for layer 1 is similar to the result in  simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  13  MaxPEAR  MinBinder  GreedyEPL  adjusted Rand index  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='748 (6)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='753 (12)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='662 (4)  Table 4: Adjusted Rand indices corresponding to different estimation methods for group membership in  simulation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Numbers in the parentheses represent numbers of estimated groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content='(b) Estimated z and g  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='Figure 5: (A): Points are plotted based on true latent position z and true group membership g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (B): Points  are plotted using the estimated latent positions in simulation 2, and color represents the estimated group  membership using the MaxPEAR method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Recall that θ1 and θ2 are of no direct interest since they are not identifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' However, we are still interested  in the ratio θ1/θ2 since it reflects the relative strength of borrowing information from the latent space of each  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Although aℓ and bℓ are of no direct interest, we pay attention to their signs, especially that of bℓ because  different signs of bℓ have different interpretations of the effect of attributes as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' We also  assess the model’s ability to estimate parameters β, τ 2, and σ2 using posterior means and 95% credible inter-  vals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The results are given in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Overall, the performance of LPJMM in recovering the true values of  these model parameters is pretty well, except for τ 2 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Both LPJMM and SNSM tend to underestimate  σ2 and overestimate τ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' That is, the covariance of the attributes tends to be underestimated, and although τ 2  is slightly overestimated, the variance of the attributes (τ 2 + σ2) still tends to be underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  4  Real data analysis  In this section, we consider a three-layer network data set collected by [Lazega, 2001] from a corporate  law firm from 1988-1991 in New England.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' This network describes three types of relationships (namely,  networks of advice, friendship, and coworker contacts) between 71 lawyers in the law firm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Several actor  attributes are also collected: age, gender, seniority (years with the firm), office (located in Boston, Hartford,  or Providence), practice (litigation or corporate law), law school the lawyers attended (Harvard or Yale,  University of Connecticut, or other universities) and status (partner or associate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' A principal component  14  true value  mean  density  layer 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='1531  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='1535  layer 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='1024  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='1023  transitivity  layer 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='5049  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='5088  layer 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='5477  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='5546  assortativity  layer 1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='5512  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='5466  layer 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='6923  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='6890  Table 5: Means of the summary statistics in simulation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  true value  posterior mean  95% credible interval  θ1/θ2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='680  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='653  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='579, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='721)  a1  5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='01  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='719, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='262)  a2  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='25  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='976, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='572)  b1  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='919  (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='053, -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='766)  b2  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='058  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='901, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='252)  β  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='047  (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='027, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='01 )  τ 2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='409  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='261, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='592)  σ2  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='642  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='230, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='684)  Table 6: Posterior means and 95% credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  analysis (PCA) is performed on age and seniority attributes, and the first principal component explains  89% of the variance which is of no surprise since age and seniority are highly correlated with a correlation  coefficient being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' We chose the first principal component to be the attribute x and let H = 8 since it  is the largest number of clusters we expect in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Then the model is fitted to the network using the  same prior and Markov chain setup as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  The study of the Lazega network in this paper is meant to find out how the three types of relations can be  explained by the findings deduced from the model fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' We first visualize the estimated latent positions  z colored by different categorical attributes (gender, office, practice, law school, and status) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' As  we can see from these plots, the estimated positions z are well separated by the office (especially offices  in Boston and Hartford) and practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Compare these plots with z colored by MaxPEAR and GreedyEPL  estimated clustering g in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' 7, we can see that both estimated g roughly clusters lawyers into three groups:  lawyers in Hartford office, litigation lawyers in Boston or Providence offices, and corporate lawyers in  Boston or Providence offices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Plots of adjacency matrices of the three layers (where lawyers are grouped by the MaxPEAR estimate of g)  and their corresponding networks are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content=' where we could see that the coworker network shows  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content='Figure 6: Points in all plots are drawn based on the estimated latent positions z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' and are colored based on  their categories in gender,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' office,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content='  the most estimated clustering pattern, while the advice network presents the least of such pattern, which  could also be seen from the relative ratios of θℓs in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' This means that lawyers from the same office  and doing the same practice are more likely to become coworkers and friends, but who they seek advice  from does not depend much on office and practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Furthermore, we can deduce from the posteriors of bℓ in  Table 7 that these lawyers tend to seek advice from people of similar age (or seniority) since the posterior  estimate of b1 is negative, while lawyers of different ages (or seniority) are more likely to become friends and  coworkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' This conclusion is in line with the assortativity coefficients with respect to the nodal attributes  (lawyer’s age) given in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  5  Discussion  This paper presents a latent position model that extends LPCM of Handcock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2007] and SNSM of  Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2019] to jointly model network data and the nodal attributes and perform model-based  clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' By jointly modeling the network and the attributes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content='(b) GreedyEPL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='Figure 7: Points are plotted using the estimated latent positions and color indicates the estimated group  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='membership using MaxPEAR and GreedyEPL methods respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  posterior mean  95% credible interval  θ1/θ2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='3229  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content='4152)  θ1/θ3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content='2606)  θ2/θ3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content='7198)  b1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content='0579)  b2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content='1137)  b3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='133  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='0854, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='186)  Table 7: Posterior means and 95% credible intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  change over the network and explain how relations could be influenced by attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' LPJMM also provides  an extension to multi-layer network settings on the assumption that all layers share the same latent position  structure but with different strengths of borrowing such latent structure information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' We applied our method  to two simulated networks, one with a single layer and another with two layers, and found our model to  give satisfactory fits to these two data sets and is competitive in terms of goodness-of-fit and group detection  compared with SNSM, LPCM, and CSBM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The model is also applied to a three-layer real network data set  and we are able to draw reasonable conclusions from the modeling results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  We have suggested choosing the number of groups H to be the largest number of groups that one is willing to  accept in the network because we have found that varying the number of groups has almost no impact on the  model fit and prediction outcome as long as it is in a reasonable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' One could also fit the CSBM to the  network first, and choose H based on its estimated number of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' One problem we have not addressed  in the paper is of choosing the dimension of the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' This can be done by using Bayesian model  selection like WAIC as in Sosa and Betancourt [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Our model could be extended in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Firstly, other extensions of our model to multi-layer settings  could be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' For example, Sosa and Betancourt [2022] assumed conditionally independent layer-  17  advice  friendship  coworker  Figure 8: Upper: Heatmaps of the adjacency matrices (where lawyers are reordered according to the Max-  PEAR estimate of g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Lower: A visualization of the three layers based on the estimated z and color indicates  the MaxPEAR estimate of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  advice  friendship  coworker  assortativity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='2536  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='1107  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='1224  Table 8: Assortativity coefficients with respect to lawyer’s age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  specific latent positions, whereas MacDonald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2022] assumed that the latent position of an actor in all  layers is (d0 + d1)-dimensional, where the first d0 components of the latent position are the same across all  layers, and only the last d1 components are layer-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Secondly, instead of assigning a user-specified  number of groups H to the model, we could learn the number of groups by using a Bayesian nonpara-  metric approach with a Dirichlet Process prior to model community memberships (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', Amini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=',  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  LPJMM could also be extended to leverage multivariate covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' So far, we have limited ourselves to mod-  eling univariate nodal attributes that are approximately Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' For continuous nodal attributes with more  than one dimension, we have used the first principal component from the principal component analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' To  take full advantage of high-dimensional nodal attributes, one could use multivariate spatial process modeling  to replace Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Other extensions of more sophisticated spatial modeling include spatiotemporal modeling  of attributes for time-varying networks, which would help to describe changes in actors over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  18  Appendix  A  Model Specifications for SNSM, LPCM and CSBM  Note that the original SNSM in Ciminelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' [2019] uses the logit link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' In order to make a fair comparison,  we also use the probit link in SNSM as in LPJMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The model specification for SNSM used in this paper is  given as follows:  yi,j | z, x, aℓ, bℓ, θℓ  ind  ∼ Ber  �  Φ(a + b|xi − xj| − ∥zi − zj∥)  �  ,  x | z, β, σ, τ, φ ∼ NN(β111N, σ2M(z, φ) + τ 2IN) ,  and the priors are set to be the same as the priors in LPJMM (if possible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' To be specific,  zi  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  ∼ N2(000, I2) ,  β ∼ N(0, 104) ,  σ2 ∼ InvG(2, 1) ,  τ 2 ∼ InvG(2, 1) ,  φ ∼ U(0, 1) ,  and the priors on the parameters in the probit regression tier are given by:  a i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  ∼ N(0, 9) ,  b i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  ∼ N(0, 9) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  SNSM in this paper is implemented using JAGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  The model specification for LPCM (see Handcock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=', 2007) is given as the follows,  yi,j | z, x, β0, β1  ind  ∼ Ber  �  logit(β⊺  0xi,j − β1∥zi − zj∥)  �  ,  zi | ωωω,µµµ,κκκ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  ∼  5  �  h=1  ωhN5(µh, κ2  hIK) ,  and we use the default priors given in the latentnet package for prior specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  We first introduce several notations before presenting CSBM in Leger [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Suppose there are Q groups  in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Denote the N × Q group membership matrix as ZZZ = {Ziq}, and Ziq = 1 if actor i belongs  to group q, Ziq = 0 if otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' It is assumed that an actor can only belong to one group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' The model  specification for CSBM is given as follows,  yi,j | Zi, Zj, x, β ind  ∼ Ber  �  logit(mqi,qj + β⊺xi,j)  �  ,  where Zi is the i-th row of ZZZ, qi is the group membership for actor i and the group effect mqi,qj ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  19  B  Comparing model performances for different number of groups  We conduct a comparison of LPJMM with different H ∈ {3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' , 9} using the data set in simulation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Table 9 presents the adjusted rand indices, and the results are similar for models that assume H to be equal to  or larger than the true number of groups (which is 5 in this example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' However, the adjusted rand indices for  all three estimates are significantly smaller when the model assumes H to be smaller than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Also, notice that  the estimated number of groups increases with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Visualizations of how adjusted rand indices and estimated  number of groups changes over H are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  H  MaxPEAR  MinBinder  GreedyEPL  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='4067 (3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='4008 (5)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='4321 (3)  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='4882 (3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='4977 (6 )  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='6521 (4)  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='7374 (5)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='7115 (11)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='6635 (4)  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='7237 (6)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='7442 (20)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='7134 (4)  7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='7449 (7)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='6624 (25)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='7313 (4)  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='7422 (8)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='6674 (25)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='7293 (8)  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='7056 (12)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='7041 (25)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='7043 (11)  Table 9: Adjusted Rand indices of different estimates under LPJMM with different H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Numbers in the  parentheses denote the numbers of estimated groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  3  4  5  6  7  8  9  H  Adjusted rand index  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='8  3  4  5  6  7  8  9  H  number of groups  5  10  15  20  25  MaxPEAR  MinBinder  GreedyEPL  Figure 9: Left: Adjusted rand indices of the clustering estimates found by using the MaxPear, MinBinder,  and GreedyEPL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Right: Estimated number of groups using the three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  The goodness-of-fit test outlined in Section 3 is also carried out here to compare the means of several sum-  mary statistics, which are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' As we can see from the plots, the model’s fit is not affected by  the choice of H even for H smaller than the actual number of clusters in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  20  3  4  5  6  7  8  9  H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='1536  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='1540  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='1544  (a) density  3  4  5  6  7  8  9  H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='513  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='514  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='515  (b) transitivity  3  4  5  6  7  8  9  H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='538  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='546  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='554  (c) assortativity  Figure 10: The means of summary statistics for different H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  C  Traceplots of log-likelihood  The traceplots of the log-likelihood (after thinning the Markov chain every 10 iterations) in simulation stud-  ies and real applications in Sections 3 and 4 are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  0  2000  6000  10000  −1100  −1080  −1060  −1040  (a) simulation 1  0  2000  6000  10000  −1920  −1900  −1880  −1860  (b) simulation 2  0  2000  6000  10000  −5230  −5214  −5198  −5182  (c) Lazega network  Figure 11: Traceplots of the log-likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  D  Visualizations of results from LPJMM and LPCM  Visualizations of the estimated latent positions and estimated group membership using the MaxPEAR, Min-  Binder, and GreedyEPL methods under LPJMM and LPCM are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' 12 and 13 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content='(c) GreedyEPL (LPCM)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='Figure 13: Points are plotted based on estimated z and three estimated ˆg of LPCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  References  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content=' CRC Press, 2nd  edition, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content=' Settings in social networks: a measurement model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Sociological  Methodology, 33:307–341, 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content=' A review on latent space models for social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Revista Colombiana de Estadistica, 44:  171–200, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Sosa and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Betancourt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' A latent space model for multilayer network data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+page_content=' Zhang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Dunson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Common and individual structure of brain networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Annals of Applied  Statistics, 13:85–112, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content='  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Wang and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Wong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Stochastic blockmodels for directed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
+page_content=' Journal of the American Statistical  Association, 82:8–19, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tAyT4oBgHgl3EQfQ_YL/content/2301.00055v1.pdf'}
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+Perspective: How to overcome dynamical density functional theory
+Daniel de las Heras,1 Toni Zimmermann,1 Florian Samm¨uller,1 Sophie Hermann,1 and Matthias Schmidt1
+1Theoretische Physik II, Physikalisches Institut, Universit¨at Bayreuth, D-95447 Bayreuth, Germany
+(Dated: 28 January 2023)
+We argue in favour of developing a comprehensive dynamical theory for rationalizing, predicting,
+and machine learning nonequilibrium phenomena that occur in soft matter. To give guidance for
+navigating the theoretical and practical challenges that lie ahead, we discuss and exemplify the
+limitations of dynamical density functional theory. Instead of the implied adiabatic sequence of
+equilibrium states that this approach provides as a makeshift for the true time evolution, we posit
+that the pending theoretical tasks lie in developing a systematic understanding of the dynamical
+functional relationships that govern the genuine nonequilibrium physics. While static density func-
+tional theory gives a comprehensive account of the equilibrium properties of many-body systems,
+we argue that power functional theory is the only present contender to shed similar insights into
+nonequilibrium dynamics, including the recognition and implementation of exact sum rules that
+result from the Noether theorem. As a demonstration of the power functional point of view, we
+consider an idealized steady sedimentation flow of the three-dimensional Lennard-Jones fluid and
+machine-learn the kinematic map from the mean motion to the internal force field. This proof of con-
+cept demonstrates the significant potential of machine learning the inherent functional relationships
+that govern nonequilibrium many-body physics.
+I.
+INTRODUCTION
+The coupled dynamics of the microscopic degrees of
+freedom in typical soft matter systems generates a wide
+array of relevant and also often unsolved nonequilibrium
+phenomena [1, 2].
+One central quantity for the char-
+acterization of self-assembly and structure formation in
+complex systems is the microscopically resolved one-body
+density distribution ρ(r, t), where r indicates position
+and t denotes time. The “density profile” ρ(r, t) acts as a
+central order parameter both due to its intuitive physical
+interpretation and clearcut mathematical definition [3].
+According to the dynamical density functional theory
+(DDFT), as originally proposed by Evans in 1979 [4],
+the time evolution of the microscopic density profile is
+assumed to be determined by the following partial differ-
+ential equation:
+∂ρ(r, t)
+∂t
+= γ−1∇ · ρ(r, t)∇
+� δF[ρ]
+δρ(r, t) + Vext(r, t)
+�
+.
+(1)
+Here γ is a friction constant, F[ρ] is an intrinsic free
+energy functional that depends functionally on the den-
+sity profile, and the external potential Vext(r, t) repre-
+sents interactions of the system with the environment.
+The system is set into motion by a temporal variation
+of Vext(r, t), such as e.g. step-like switching at an initial
+time.
+The time evolution according to Eq. (1) conserves the
+particle number locally and hence it constitutes dynam-
+ics of model B type [5].
+In standard applications one
+starts with an equilibrium state of the system and then
+the dynamics are monitored on the basis of numerical
+time integration of Eq. (1). In order to provide reference
+data and to allow for the generation of benchmark results
+to assess the quality of the theory, resorting to many-
+body computer simulations is common, with overdamped
+Brownian dynamics (BD) being a popular choice (Ref. [6]
+describes a modern and stable algorithm). Comparison
+of DDFT data with experimental results are more scarce,
+but notable exceptions include non-equilibrium sedimen-
+tation of colloids [7], the self-diffusion of particles in com-
+plex fluids [8], and the bulk dynamics of Brownian hard
+disks [9].
+The DDFT time evolution reaches a stationary state
+if the gradient on the right hand side of Eq. (1) vanishes,
+i.e. provided that the expression inside of the parentheses
+is constant:
+δF[ρ]
+δρ(r) + Vext(r) = µ.
+(2)
+Here we have dropped the dependence on time in the
+notation, as the situation is now static. The constant µ
+can be identified with the chemical potential, which in a
+grand canonical statistical mechanical setting is the con-
+jugate control parameter of the mean particle number.
+Equation (2) is exact in equilibrium, as was shown by
+Evans [4]. He proved the equilibrium intrinsic free en-
+ergy functional F[ρ] to exist, to be unique, and to form
+the starting point for a modern equilibrium theory of
+spatially inhomogeneous liquids and crystals [10, 11].
+In practice one needs to rely on approximations
+for F[ρ], given a microscopic fluid model under consid-
+eration.
+Once one has solved Eq. (2) for given values
+of µ and temperature T (the dependence of F[ρ] on T
+is suppressed in the notation), then in principle com-
+plete knowledge of the thermal system is available. The
+value of the density functional F[ρ] is the true intrinsic
+free energy, and higher-order correlation functions are
+determined via higher-order derivatives of the free en-
+ergy functional or via test-particle procedures. In par-
+ticular two-body correlations functions, such as the bulk
+pair correlation function g(r) as well as its generalization
+to inhomogeneous systems are accessible. These exhibit
+defining characteristics of liquids and more general soft
+arXiv:2301.12156v1  [cond-mat.soft]  28 Jan 2023
+
+2
+matter systems and they are formally fully contained in
+the static density functional theory framework.
+Together with a number of available reliable approxi-
+mate free energy functionals, density functional theory is
+a powerful theoretical framework that has been used to
+elucidate much intricate and complex behaviour in soft
+matter.
+Recent representative highlights include trac-
+ing hydrophobicity to critical drying at substrates [12–
+14], resolving three-dimensional structures of electrolyte
+aqueous solutions near surfaces [15, 16], and addressing
+the magnitude of the decay lengths in electrolytes [17].
+Rosenfeld’s celebrated hard sphere fundamental measure
+free energy functional [18–21] is at the core of much of
+this research activity.
+In the following we wish to address whether or not
+the DDFT has the prowess to play a similar role in
+nonequilibrium, as is often at least implicitly assumed.
+We demonstrate on the basis of an explicit and generic
+example, i.e., that of uniaxial compressional flow of the
+three-dimensional Lennard-Jones fluid, that the DDFT is
+fundamentally flawed and that in reality, as represented
+by many-body simulations, recognizing the flow field as
+a further relevant degree of freedom is required to rep-
+resent true nonequilibrium. These conclusions are based
+on analytical power functional approximations, adaptive
+BD simulation data, and explicit machine learning of the
+power functional map from motion to the interparticle
+one-body force field.
+This Perspective is organized as follows. We first make
+some key aspects of DDFT explicit in Sec. II and describe
+several prominent shortcomings of this theory. We then
+give an account of how to go towards the formally exact
+one-body dynamics in Sec. III and provide in Sec. IV a
+description of key aspects of the power functional frame-
+work, which as we wish to argue overcomes the funda-
+mental defects of DDFT. We describe the exemplary sta-
+tionary compressional flow situation in Sec. V and lay
+put the application of Noether’s theorem in this statis-
+tical mechanical setting in Sec. VI. We present machine
+learning results for the kinematic functional relationships
+of the streaming Lennard-Jones fluid in Sec. VII. We give
+conclusion and an outlook in Sec. VIII.
+II.
+LIMITS AND LIMITATIONS OF
+ADIABATIC DYNAMICS
+We go into some detail and describe why the DDFT
+represents adiabatic dynamics in the sense of a temporal
+sequence of spatially inhomogeneous equilibrium states.
+The equilibrium intrinsic free energy functional splits into
+ideal and excess (over ideal gas) contributions according
+to F[ρ] = Fid[ρ] + Fexc[ρ]. Here the excess free energy
+functional Fexc[ρ] accounts for the effects of the inter-
+particle interactions on the equilibrium properties of the
+system and it is in general unknown and requires approx-
+imations to be made. The ideal gas free energy functional
+however is exactly given by
+Fid[ρ] = kBT
+�
+drρ(r) ln(ρ(r)Λ3) − 1],
+(3)
+where kB denotes the Boltzmann constant, Λ is the
+thermal de Broglie wavelength, and we consider three-
+dimensional systems. The functional derivative, as it is
+relevant for Eq. (1), is δFid[ρ]/δρ(r) = kBT ln(ρ(r)Λ3).
+When disregarding the excess contribution and in-
+serting this result alone into the DDFT equation
+of motion (1), its right hand side becomes γ−1∇ ·
+ρ(r, t)∇[kBT ln(ρ(r, t)Λ3) + Vext(r, t)]. This can be re-
+written further such that for the case of the ideal gas,
+where Fexc[ρ] = 0 and F[ρ] = Fid[ρ], the equation of
+motion (1) attains the following form:
+∂ρ(r, t)
+∂t
+= D0∇2ρ(r, t) − ∇ · ρ(r, t)fext(r, t)/γ.
+(4)
+Here D0 = kBT/γ is the diffusion constant, ∇2 is the
+Laplace operator and the external force field is given
+(here) as fext(r, t) = −∇Vext(r, t). Equation (4) is the
+exact drift-diffusion equation for overdamped motion of
+a mutually noninteracting system, i.e., the ideal gas.
+Besides Evans’ original proposal [4] based on the con-
+tinuity equation and undoubtedly his physical intuition,
+derivations of the DDFT (1) were founded much more
+recently on Dean’s equation of motion for the density op-
+erator [22], the Smoluchowski equation [23], a stationary
+action principle for the density [24], the projection op-
+erator formalism [25], a phase-space approach [26], the
+mean-field approximation [27], a local equilibrium as-
+sumption [28], and a non-equilibrium free energy [29].
+The question of the well-posedness of the DDFT was ad-
+dressed [30] and several extensions beyond overdamped
+Brownian dynamics were formulated, such as e.g. for dy-
+namics including inertia [31–34] and for particles that ex-
+perience hydrodynamic interactions [34, 35] or undergo
+chemical reactions [36, 37].
+The DDFT was also used beyond the description of flu-
+ids, such as e.g. for opinion dynamics [38] and epidemic
+spreading [39].
+Recent reviews of DDFT are given in
+Refs. [40, 41]. The theory is put into a wider perspective,
+together with much background pedagogical material in
+Ref. [42]. A modern and well-accessible account of the
+general strategy of dynamical coarse-graining in statisti-
+cal physics, of which the DDFT can be viewed as being a
+representative, has recently been given by Schilling [43].
+The fact that both the static limit for the fully in-
+teracting system (2) as well as the full dynamics of the
+noninteracting system (4) are exact, taken together with
+the heft of the DDFT literature, appears to give much
+credibility to the equation of motion (1). However, de-
+spite the range of theoretical techniques employed [22–29]
+neither of these approaches has provided us with a con-
+crete way of going beyond Eq. (1). Apart from several
+case-by-case and rather ad hoc modifications, no system-
+atic or even only practical identification of what is miss-
+ing has been formulated. (We turn to power functional
+
+3
+theory in Sec. IV.) This is a problematic situation as
+two defects of Eq. (1) are immediately obvious upon in-
+spection: i) the description is local in time and there is
+no natural mechanism for the inclusion of memory while
+time-locality is not sufficient for general nonequilibrium
+situations; ii) only flow that leads to direct changes in
+the density profile is captured and hence effects of rota-
+tional flow, such as shearing, as well as of nonequilibrium
+effects in compression and expansion are lost (see below).
+Here we argue that these defects are indicative of a
+broader failure of Eq. (1) to describe nonequilibrium
+physics. We show that the DDFT is only fit to describe
+situations in which the dynamics follow an adiabatic path
+through a sequence of equilibrium states. The description
+of genuine nonequilibrium dynamics in a functional set-
+ting on the one-body level rather requires recognition of
+the local velocity field as a further relevant physical vari-
+able besides the density profile, and this is provided by
+power functional theory [42]. Before laying out key prin-
+ciples of this approach in Sec. IV, we first describe the mi-
+croscopically sharp coarse-graining on the one-body level
+of correlation functions.
+III.
+TOWARDS EXACT ONE-BODY
+DYNAMICS
+Evans based his original derivation [4] of Eq. (1) on the
+continuity equation,
+∂ρ(r, t)
+∂t
+= −∇ · J(r, t),
+(5)
+where J(r, t) is the microscopically resolved one-body
+current distribution. Equation (5) is exact in a variety of
+contexts, including overdamped Brownian dynamics, as
+described either on the Fokker-Planck level by the Smolu-
+chowski equation or by the corresponding overdamped
+Langevin equation that governs the trajectories, as they
+are realized in simulation work [6]. For BD the one-body
+current distribution is given exactly by [42]:
+γJ(r, t) = −kBT∇ρ(r, t) + Fint(r, t) + ρ(r, t)fext(r, t).
+(6)
+This identity expresses the force density balance of the
+negative friction force density (left hand side) with the
+force densities due to ideal thermal diffusion, interparti-
+cle interactions, and external influence (three contribu-
+tions on the right hand side). Here the interparticle force
+density distribution is given by the statistical average
+Fint(r, t) = −
+� �
+i
+δ(r − ri)∇iu(rN)
+����
+t,
+(7)
+where the angular brackets indicate an average at fixed
+time t over the nonequilibrium many-body distribu-
+tion, u(rN) is the interparticle interaction potential
+that depends on all particle position coordinates rN ≡
+r1, . . . , rN and ∇i indicates the derivative with respect to
+the position ri of particle i. The formulation of Eq. (7) is
+based on the concept of static operators and a dynami-
+cally evolving probability distribution. This is analogous
+to the Schr¨odinger picture of quantum mechanics. The
+Heisenberg picture is more closely related to simulation
+work. Here the probability distribution is that of the ini-
+tial microstates and the operators move forward in time,
+i.e., the position ri(t) of particle i changes over the course
+of time. Then the Dirac distribution in Eq. (7) becomes
+δ(r − ri(t)), with the generic position variable r however
+remaining static. The forces are those that act in the
+given microstate rN(t) at time t, i.e., the interparticle
+force on particle i at time t is −∇iu(rN(t)).
+In practice, using BD simulations, carrying out the
+average in Eq. (7) requires to build the mean over suf-
+ficiently many separate realizations of the microscopic
+evolution of the many-body system that differ in the ini-
+tial state and in the realization of the thermal noise. As
+Eq. (7) measures both the probability to find particle i at
+position r (via the delta function) and the interparticle
+force that acts via the negative gradient −∇iu(rN), we
+refer to Fint(r, t) as a force density. The corresponding
+force field fint(r, t) is obtained by simple normalization
+with the density profile, i.e. fint(r, t) = Fint(r, t)/ρ(r, t).
+Building this ratio scales out the probability effect and
+the force field then carries physical units of force, i.e.
+energy per length.
+In equilibrium the definition (7) remains intact. Com-
+plementing the statistical average, static density func-
+tional theory allows to express the equilibrium force den-
+sity as being functionally dependent on the density pro-
+file via the functional derivative of the excess free energy
+functional according to:
+Fint(r)
+��
+eq = −ρ(r)∇δFexc[ρ]
+δρ(r) .
+(8)
+Crucially, and in contrast to Eq. (7), here the internal
+force density is directly expressed as a density functional.
+This dependence has superseded the original dependence
+on the external potential, as is manifest in the probability
+distribution for building the average (7) in equilibrium.
+As a self-consistency check we insert the force density
+functional (8) into the equilibrium limit of the force den-
+sity balance (6). The current vanishes in the equilibrium
+case, J(r, t) ≡ 0, and we obtain
+−kBT∇ρ(r) + Fint(r)|eq + ρ(r)fext(r) = 0.
+(9)
+This result is independent of time and it consti-
+tutes the gradient of the static Euler-Lagrange equa-
+tion (2) when divided by the density profile.
+(Insert
+Eq. (8), identify the ideal gas contribution −kBT∇ρ(r) =
+−ρ(r)δFid[ρ]/δρ(r), and divide by ρ(r).)
+The classical
+force density balance result (9) by Yvon, Born and Green
+[3] has recently been derived from systematically address-
+ing thermal Noether invariance [44, 45] against locally
+resolved spatial deformations of the statistical ensemble
+[46–48], as also valid quantum mechanically [48] and at
+
+4
+second order in the displacement field [49, 50]; we give a
+brief account of this theory in Sec. VI below.
+A naive transfer of Eq. (8) to nonequilibrium lets
+one simply evaluate the equilibrium excess free energy
+functional at the instantaneous nonequilibrium density
+ρ(r, t). In order to separate this contribution from true
+static equilibrium, we refer to this force density as being
+adiabatic (subscript “ad”) and to be defined as
+Fad(r, t) = −ρ(r, t)∇δFexc[ρ]
+δρ(r, t) .
+(10)
+We recall that the right hand side offers a concrete com-
+putational structure that is of practical usefulness in ac-
+tual applications, as considerable knowledge about ap-
+proximative forms of the excess free energy density func-
+tional Fexc[ρ] is available. Using the adiabatic force den-
+sity as a proxy for the true nonequilibrium intrinsic force
+density distribution (7), i.e. setting Fint(r, t) = Fad(r, t)
+in the force density balance (6) together with the conti-
+nuity equation (5) leads to the DDFT equation of mo-
+tion (1). The adiabatic force density approximation is
+uncontrolled though and the theory inherently yields the
+dynamics as an adiabatic sequence of equilibrium states.
+Surely, more than 40 years after the conception of the
+DDFT [4], we have to be able to do better!
+IV.
+POWER FUNCTIONAL TECHNIQUES
+Power functional theory [42] offers a concrete math-
+ematical structure to go forward.
+We describe the es-
+sential steps that enable one to go beyond the DDFT
+and to hence address a significantly expanded realm of
+nonequilibrium physics which Eq. (1) is oblivious of.
+The interparticle force density profile (7) is identified
+to consist of two contributions according to:
+Fint(r, t) = Fad(r, t) + Fsup(r, t).
+(11)
+Here Fad(r, t) is the adiabatic force density profile, as
+given formally via the explicit equilibrium free energy
+derivative (10) and directly accessible in simulations via
+the custom flow method [51, 52]. The custom flow al-
+gorithm allows to systematically construct a hypotheti-
+cal adiabatic (equilibrium) system that shares its density
+profile with the nonequilibrium system at the given time.
+Then sampling the internal force density in the adiabatic
+system yields results for Fad(r, t).
+The second, superadiabatic contribution in Eq. (11),
+Fsup(r, t), contains all effects that are not expressible
+as an instantaneous density functional.
+This includes
+forces that lead to viscous and to nonequilibrium struc-
+ture forming phenomena, as we exemplify below in a con-
+crete model compressional flow situation. Formally, the
+superadiabatic force density is generated from the su-
+peradiabatic excess free power functional P exc
+t
+[ρ, J] upon
+functional differentiation with respect to the one-body
+current via [42, 53]:
+Fsup(r, t) = −ρ(r, t)δP exc
+t
+[ρ, J]
+δJ(r, t)
+.
+(12)
+The functional dependence of P exc
+t
+[ρ, J] on the density
+and current is causal, i.e. on the values of these fields
+at prior times to t; density and current need to satisfy
+the continuity equation. Upon using Eqs. (11) the force
+density balance (6) attains the following form:
+γJ(r, t) = −kBT∇ρ(r, t) + Fad(r, t)
++ Fsup(r, t) + ρ(r, t)fext(r, t).
+(13)
+This
+relationship
+holds
+beyond
+gradient
+forms
+of
+fext(r, t), i.e. for external force fields that contain non-
+conservative contributions.
+Crucially Fsup(r, t) will in
+general also acquire nonconservative contributions, such
+as e.g. damping effects that represent viscous behaviour.
+Moreover, nonequilibrium structure-forming effects will
+also arise in general. These affect directly the shape of
+the density profile, whether this evolves in time or per-
+sists in a nonequilibrium steady state.
+If one wishes to eliminate the explicit occurrence of the
+current from the dynamics, then inputting the force den-
+sity balance (13) into the continuity equation (5) leads
+to the following formally exact form of the equation of
+motion for the density profile:
+∂ρ(r, t)
+∂t
+= D0∇2ρ(r, t) + ∇ · ρ(r, t)
+γ
+∇δFexc[ρ]
+δρ(r, t)
+− ∇ · ρ(r, t)
+γ
+[fsup(r, t) + fext(r, t)].
+(14)
+Here it is apparent that the superadiabatic force field
+fsup(r, t) = Fsup(r, t)/ρ(r, t) has a direct effect on the
+system dynamics. The effect is similar to that of the ex-
+ternal force field. Crucially though, both force fields are
+independent of each other: the external force field rep-
+resents a prescribed and inert influence on the system.
+In contrast, the superadiabatic force field is an emer-
+gent phenomenon that arises due to interparticle inter-
+actions and, from the functional point of view, depends
+non-locally in position and causally in time on the one-
+body density and on the current profile.
+Although setting fsup(r, t) = 0 yields the DDFT (1),
+the superadiabatic force field fsup(r, t) was demonstrated
+to exist [54–60] and in general to play a major role in
+the dynamics on the one-body level and, based on test-
+particle concepts [61–66] also for two-body correlation
+functions [67–69] and for active matter [70–74]. Both the
+flow properties as well as the spatial structure formation
+in the system are affected.
+To reveal additional physics, it is useful to split into
+“structural” and “flow” contributions. This was estab-
+lished e.g. for complex flow patterns that occur in driven
+BD [55, 59], for active Brownian particles which form
+a self-sustained interface at motility-induced phase co-
+existence [70–74], as well as very recently for a sheared
+
+5
+FIG. 1. Illustration of unidirectional compressional flow of a liquid. The three-dimensional system is set into motion (red
+arrows) by the action of an external force profile fext(x) (blue arrows) which acts along the x-axis. The system retains planar
+geometry such that spatial inhomogeneities only occur as a function of x. The density profile ρ(x) (orange curve) and the
+velocity profile v(x) (red curve) are both stationary in time but inhomogeneous in position.
+The local one-body current
+J(x) = ρ(x)v(x) = const and as a result the system is in a nonequilibrium steady state. The corresponding adiabatic system
+is in equilibrium (it has no mean flow) and it has by construction an unchanged density profile ρ(x). In the adiabatic system
+the spatial variation of ρ(x) is stabilized by the action of an external force field −∇Vad(x) (olive arrows), which acts solely in
+the adiabatic system.
+three-body colloidal gel former [60]. Before we demon-
+strate these concepts for an example of steady nonequi-
+librium below, we first describe two simple model power
+functionals that respectively generate structure and vis-
+cously dampen the motion and that, as we will see, give
+a good account of the nonequilibrium flow considered be-
+low.
+We concentrate on the low-order terms that are rel-
+evant for compressional/extensional flow, i.e., for situa-
+tions where ∇ · v(r, t) ̸= 0.
+We focus on cases where
+there is no rotational motion (such as shearing) and hence
+∇ × v(r, t) = 0.
+The velocity gradient superadiabatic
+power functional consists of a sum,
+P exc
+t
+[ρ, v] = P flow
+t
+[ρ, v] + P str
+t
+[ρ, v].
+(15)
+Here the flow and structural [55, 59] contributions are
+approximated, respectively, by the following time-local
+(Markovian) and space-semilocal (i.e. involving ∇) forms
+P flow
+t
+[ρ, v] = η
+2
+�
+dr[ρ(r, t)∇ · v(r, t)]2,
+(16)
+P str
+t
+[ρ, v] = −χ
+3
+�
+dr[ρ(r, t)∇ · v(r, t)]3,
+(17)
+where the overall prefactors η and χ control the respec-
+tive magnitude.
+The flow functional (16) is quadratic
+both in density and in the velocity field; the structural
+functional (17) is of cubic order in each of these variables.
+Explicit higher-order functionals exist [59] and they be-
+come relevant when driving the system strongly. We will
+return to the consequences of Eqs. (16) and (17) after
+laying out in Sec. V the actual flow situation that we use
+as a model to exemplify the implications for the physics.
+Before doing so, we briefly describe several further key
+aspects of the power functional framework.
+Power functional theory provides a formal framework
+for the inclusion of time- and space-nonlocal dynamics
+[56, 68, 79]. While Eq. (12) applies to overdamped dy-
+namics, the acceleration field becomes a further relevant
+degree of freedom if inertia are relevant [78–81] whether
+classically in molecular dynamics [78, 79] or in quantum
+dynamics [80, 81].
+Here the memory functions act as
+convolution kernels on specific kinematic fields and rota-
+tional and compressional contributions to the dynamics
+are genuinely built in. As laid out above, the framework
+is based on an exact variational concept [42, 53], and the
+resulting functional mapping was shown to be explicitly
+accessible in many-body simulation via the custom flow
+computer simulation method [51, 52].
+Even
+simple
+mathematical
+model
+forms
+for
+the
+nonequilibrium contribution to the power functional,
+such as Eqs. (16) and (17), already capture essential
+physics (as we demonstrate below) and dynamical two-
+body correlation functions are accessible via test particle
+dynamics [8, 9, 61–69]. The power functional is thereby
+not to be confused with the often vague concept of a
+
+v(C
+p(α)
+noneguilibrium
+fext(α)
+(α) PeA△
+equilibrium6
+“nonequilibrium free energy”.
+The proper equilibrium
+free energy functional does play a central role in power
+functional theory though, via providing the description
+of the adiabatic reference state [42], see the generation
+of the force density distribution via functional differenti-
+ation (10), as is relevant for the interparticle force split-
+ting (11), and the full density equation of motion (14).
+The relevance of superadiabatic contributions to the
+dynamics, i.e. of those effects that lie beyond Eq. (1), has
+been amply demonstrated in the literature [54–59, 67–
+69]. Both adiabatic and superadiabatic effects arise from
+integrating out the dynamical degrees of freedom of the
+many-body problem.
+Ensemble differences between canonical dynamics and
+grand canonical equilibrium have been systematically ad-
+dressed [75–77] and these do not account for the observed
+differences between adiabatic and superadiabatic dynam-
+ics.
+The kinematic dependence on the motion of the
+system arises formally [42], it can be explicitly traced
+in many-body computer simulation work [59], and it
+is amenable to machine learning, as we demonstrate in
+Sec. VII. Before doing so, we first formulate the represen-
+tative flow problem that we will use to apply the above
+concepts.
+V.
+NONEQUILIBRIUM STEADY STATES
+We restrict ourselves to flow situations with one-body
+fields that are inhomogeneous in position but indepen-
+dent of time, i.e. ρ(r) and v(r). Then trivially ∂ρ(r)/∂t =
+0 and the continuity equation (5) constrains both fields
+to satisfy ∇ · [ρ(r)v(r)] = 0. As a representative case
+we illustrate in Fig. 1 a nonequilibrium steady state of a
+three-dimensional liquid undergoing unidirectional com-
+pressional flow. Flow along a single given direction occurs
+e.g. under the influence of gravity, where sedimentation
+of colloids leads to both compression in the lower parts of
+the sample and expansion in the upper parts of the sam-
+ple. Here we disregard transient phenomena and investi-
+gate an idealized periodic system, where flowing steady
+states can form.
+In order to elucidate the physics in such setups, we fol-
+low the splitting (15) of the superadiabatic power func-
+tional into structural and flow contributions and hence
+decompose the superadiabatic force field accordingly as
+fsup(r) = fstr(r) + fflow(r),
+(18)
+where the right hand side consists of the nonequilib-
+rium structural force field fstr(r) and the flow force
+field fflow(r).
+Both of these force contributions arise
+from the microscopic interparticle interactions, as coarse-
+grained in a microscopically sharp way to the one-body
+level.
+We lay out in the following the benefits of the
+structure-flow splitting (18) and its definition via flow
+reversal symmetry.
+First, on the more practical level, Eq. (18) allows to
+carry out a corresponding splitting of the force density
+balance (13) [we divide by ρ(r) to obtain force fields].
+The result is a set of two coupled equations of motion,
+with one of them depending explicitly on the velocity
+profile and the second one depending explicitly on the
+density profile:
+γv(r) = fflow(r) + fext,f(r),
+(19)
+0 = fstr(r) − kBT∇ ln ρ(r) + fad(r) + fext,s(r). (20)
+Building the sum of Eqs. (19) and (20) and multiplying
+by the density profile restores the full force density bal-
+ance (13). The external force field is split according to
+fext(r) = fext,f(r) + fext,s(r), where the two terms couple
+to the flow via fext,f(r) in Eq. (19) and to the structure
+via fext,s(r) in Eq. (20).
+On the superficial level the two equations (19) and
+(20) appear to be independent of each other, as no sin-
+gle field appears explicitly in both equations. However,
+the two equations are indeed intimately coupled to each
+other by the interparticle interactions, as represented by
+both the adiabatic and the two superadiabatic (flow and
+structural) force fields. These three intrinsic force con-
+tributions provide the physical representation of the true
+nonequilibrium steady state dynamics.
+The flow-structure splitting (18) is uniquely deter-
+mined by the symmetry properties of the forces upon
+motion reversal of the system [59].
+Motion reversal is
+a discrete symmetry operation, and hence different from
+continuous invariances where Noether’s theorem applies
+[44–50].
+One considers a “reversed” system, which is
+also in steady state and possesses an unchanged den-
+sity profile ρ(r). The flow, however, is directed against
+the velocity orientation in the original “forward” system.
+Hence the velocity profile in the reversed system is sim-
+ply −v(r). As a result the current also acquires a mi-
+nus sign, −ρ(r)v(r), which however does not affect the
+(vanishing) divergence, ∇ · [−ρ(r)v(r)] = 0. Thus the re-
+versed state indeed is stationary. The two superadiabatic
+contributions are then defined to be unchanged [fstr(r)]
+and inverted [−fflow(r)] in the reversed system. Conse-
+quentially, the superadiabatic force field in the reversed
+system is the difference fstr(r) − fflow(r).
+Analyzing the symmetry properties of the adiabatic
+force field is straightforward.
+We recall that fad(r) is
+a density functional via Eq. (10). The density profiles
+in the forward and in the reversed systems are identical
+though. Hence fad(r) is invariant under motion reversal.
+Motion reversal is a useful device in order to i) rationalize
+the nonequilibrium behaviour according to the split force
+balance (19) and (20), and to ii) classify the dependence
+of superadiabatic forces on the velocity field into even
+powers, which constitute fstr(r), and odd powers, which
+form fflow(r).
+We can demonstrate this mechanism explicitly on the
+basis of the above flow and structural power functionals
+(16) and (17). Superadiabatic force fields are generated
+via the functional derivative (12) with respect to the cur-
+rent or, analogously, by functionally deriving by v(r, t)
+
+7
+and dividing the result by ρ(r, t). The resulting supera-
+diabatic one-body force field consists of two components.
+The viscous flow force and [55, 58] and the structural
+force follow respectively as
+fflow(r) =
+η
+ρ(r)∇[ρ(r)2∇ · v(r)],
+(21)
+fstr(r) = − χ
+ρ(r)∇{ρ(r)3[∇ · v(r)]2},
+(22)
+where Eq. (21) is odd (linear) and Eq. (22) is even
+(quadratic) in the derivatives of the velocity field, as de-
+sired.
+One might wonder where all this genuine nonequilib-
+rium physics leaves the DDFT! Some readers will find the
+instantaneous dynamics, as generated from an adiabatic
+free energy according to (1), to be more appealing and in-
+tuitive than the thinking in terms of the above described
+apparently intricate functional relationships.
+Why not
+live with Eq. (1), use it, and simply accept its defects?
+In order to address this question and to demonstrate why
+this path is severely restricted from the outset, we turn
+in Sec. VII to an explicit demonstration of the functional
+relationship that governs the nonequilibrium physics, i.e.
+the kinematic functional map from the one-body mean
+motion to the internal force field. Before doing so, we
+demonstrate that Noether’s theorem of invariant varia-
+tions has much to say about our present setup.
+VI.
+NOETHER FORCE SUM RULES
+We discuss one of the arguably simplest cases of ex-
+ploitation of the inherent symmetries of a thermal many-
+body system, that of global translational invariance of its
+statistical mechanics [44, 45]. We consider a “shifting”
+transformation, where all particle coordinates change ac-
+cording to the map ri → ri + ϵ, where ϵ = const. This
+uniform shifting operation leaves all interparticle dis-
+tance unchanged, ri−rj → (ri+ϵ)−(rj+ϵ) ≡ ri−rj. As
+a consequence the interparticle potential is invariant un-
+der the transformation, which we can express as the iden-
+tity u(r1, . . . , rN) = u(r1 + ϵ, . . . , rN + ϵ). Here equality
+holds irrespectively of the magnitude and the direction
+of the shifting vector ϵ.
+The Noether argument proceeds with a twist.
+De-
+spite the absence of dependence on ϵ, we can neverthe-
+less differentiate both sides of the equation with respect
+to ϵ and the result will be a valid identity. We obtain
+0 = ∂u(ri + ϵ, . . . , rN + ϵ)/∂ϵ = �
+i ∇iu(r1, . . . , rN),
+where we have set ϵ = 0 after taking the derivative. We
+multiply by −1 and insert 1 =
+�
+drδ(r−ri), which yields
+−
+�
+dr
+�
+i
+δ(r − ri)∇iu(rN) = 0.
+(23)
+The expression on the left hand side allows to identify
+the locally resolved interparticle force operator ˆFint(r) =
+− �
+i δ(r − ri)∇iu(rN), such that Eq. (23) attains the
+form
+�
+drˆFint(r) = 0. This identity holds for each mi-
+crostate rN and hence it remains trivially valid upon av-
+eraging over the many-body distribution function, irre-
+spective of whether this is in- or out-of-equilibrium. We
+can hence conclude the vanishing of the global interpar-
+ticle force, expressed as the integral over the mean force
+density Fint(r) = ⟨ˆFint(r)⟩ as
+�
+drFint(r, t) = 0.
+(24)
+Equation (24) holds at all times t and it can be viewed as
+a consequence of Newton’s third law, see the discussion in
+Ref. [44]. Using the adiabatic-superadiabatic force split-
+ting (11) one can further conclude that the both global
+contributions need to vanish individually,
+�
+drFad(r, t) = 0,
+(25)
+�
+drFsup(r, t) = 0.
+(26)
+The proof can either be based on the fact that Eq. (25)
+is merely Eq. (24) for the special case of an equilibrium
+system, from which then Eq. (26) follows from the force
+splitting (11).
+Alternatively and starting from a very
+fundamental point of view, the global translational in-
+variance of the excess free energy functional Fexc[ρ] and
+of the superadiabatic free power functional P exc
+t
+[ρ, v],
+here considered instantaneously at time t, lead directly
+to Eqs. (25) and (26), see Refs. [44, 45] for the detailed
+account.
+It is interesting to apply the Noether concept to the
+flow-structure splitting Eq. (18) of the superadiabatic
+force field. One can see straightforwardly, from the sym-
+metry upon motion reversal, that both the global struc-
+tural force and the global flow force need to vanish indi-
+vidually:
+�
+drρ(r)fflow(r) = 0,
+(27)
+�
+drρ(r)fstr(r) = 0.
+(28)
+We prove by contradiction and assume that it is not
+the case, i.e. that each integral gives the same global
+force, but with opposite sign, such that the sum vanishes
+and Eq. (26) remains valid.
+Per construction, fflow(r)
+changes sign in the motion reversed system, but fstr(r)
+does not.
+Hence Eq. (26) can only be satisfied in the
+motion-reversed system provided that both the flow and
+structural contribution vanish separately.
+We
+can
+explicitly
+test
+the
+validity
+of
+the
+sum
+rules (27) and (28) for the above analytical force ap-
+proximations (21) and (22).
+The respective integrals
+are η
+�
+dr∇[ρ(r)2∇ · v(r)] = 0 and χ
+�
+dr∇{ρ(r)3[∇ ·
+v(r)]2} = 0, which follows from the divergence theorem,
+as boundary terms vanish. Hence the simple non-local
+velocity gradient power functional approximations (16)
+
+8
+density
+current
+external force field
+interparticle 
+force field
+Mermin 
+Evans 
+map 
+(DFT)
+kinematic fields
+kinematic 
+map
+adiabatic-superadiabatic 
+splitting
+structure-flow splitting
+superadiabatic 
+force field
+adiabatic 
+force field
+flow 
+force
+structural 
+force
+adaptive 
+BD
+super- 
+adiabatic 
+map 
+(PFT)
+ 0.3
+ 0.4
+ 0.5
+ 0.6
+ 0.7
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+��3
+x/�
+ 0
+ 1
+ 2
+ 3
+ 4
+ 5
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+J�2�
+x/�
+-1.5
+-1
+-0.5
+ 0
+ 0.5
+ 1
+ 1.5
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+fint�/�
+x/�
+-1.5
+-1
+-0.5
+ 0
+ 0.5
+ 1
+ 1.5
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+fad�/�
+x/�
+ 0
+ 4
+ 8
+ 12
+ 16
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+fext�/�
+x/�
+-0.6
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+fsup�/�
+x/�
+-0.6
+-0.4
+-0.2
+ 0
+ 0.2
+ 0.4
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+f�ow�/�
+x/�
+-0.2
+-0.1
+ 0
+ 0.1
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+fstr�/�
+x/�
+FIG. 2. Kinematic profiles and force fields for uniaxial compressional flow of the LJ fluid. Results are shown from machine
+learning (lines) and from direct adaptive BD simulations (symbols). Functional relationships are represented by vertical arrows.
+Shown are the density profile ρ(x), the one-body current J(x) and the external force field fext(x) (top row) as a function of the
+scaled distance x/σ, where σ is the LJ length scale. The density and the current functionally determine both the interparticle
+force field fint(x) via the kinematic map and the superadiabatic force field fsup(x) via the superadiabatic kinematic map (middle
+row). The internal force field fint(x) splits into superadiabatic and adiabatic force contributions. The adiabatic force field fad(x)
+is a density functional via the Mermin-Evans map of density functional theory. The structural and flow force fields are split
+according to their symmetry upon motion reversal. The colour code represents different values of the current J0 = 0, 1, 2, 3, 4, 5
+(from violet to yellow, see the center panel in the top row); the two insets show the predictions from the analytical velocity
+gradient functionals (21) and (22). The system with J0 = 0 is at rest in equilibrium and it doubles as the adiabatic state as its
+density profile is identical to that of the flowing systems (first panel).
+and (17) have passed the global Noether validation test.
+This is nontrivial, as the proof rests on the specific struc-
+ture of the integrands being gradients, which for more
+general analytical forms will not be the case. This exem-
+plifies the merits of Noether sum rules for assessing and
+by extension also constructing theoretical nonequilibrium
+force approximations.
+The Noether concept carries much further. Reference
+[44] presents memory sum rules for so-called time di-
+rect correlation functions. These are defined via func-
+tional derivatives of the superadiabatic power functional,
+in generalization of the superadiabatic force density as
+generated via the derivative (12) with respect to the cur-
+rent distribution. We expect the corresponding identities
+to be helpful in the study of temporal nonlocality. Fur-
+ther work was addressed at the variance of global fluctu-
+ations, which were shown to be constrained by Noether
+invariance at the second order global level [49]. Noether’s
+theorem also yields the locally resolved force balance re-
+lationship in quantum mechanical many-body systems
+[48].
+Very recently, striking two-body force-force and
+force-gradient correlation functions for the precise and
+novel characterization of disordered (liquid and gel) sys-
+tems [50] were revealed. Exploiting Noether’s concept in
+a stastical mechanical setting is robust against changes of
+ensemble, Ref. [45] presents the transfer of the grand en-
+semble formalism [44] to canonical systems. Considering
+global rotational invariance leads to (classical) spin-orbit
+
+J&pJ&p9
+coupling of torque identities [44].
+We return to steady states and demonstrate that the
+seemingly entirely formal functional relationships do in
+fact apply to real systems. We present in the following
+new computational methodology that we use to demon-
+strate the functional point of view. We will also demon-
+strate that the sum rules (26) and (27) are highly valuable
+in providing checks for numerical results.
+VII.
+MACHINE LEARNING THE KINEMATIC
+MAP
+Machine learning proves itself to be an increasingly
+useful tool in a variety of settings in soft matter, rang-
+ing from soft matter characterization [82], engineering of
+colloidal self-assembly [83], to the inverse design of soft
+materials [84]. Pivotal studies were addressed at colloidal
+structure detection [85], the identification of combinato-
+rial rules in mechanical metamaterials [86], the learning
+of many-body interaction potentials for spherical [87] and
+for anisotropic particles [88], and the prediction of the
+dynamics of supercooled liquids from their static proper-
+ties [89].
+More specifically, in the context of classical density
+functional theory, an early and pioneering study formu-
+lated a neural-network approach to liquid crystal order-
+ing in confinement [90].
+Free energy density function-
+als were obtained for one-dimensional fluids from a con-
+volutional neural network [91] and an analytical form
+of an excess free energy functional was generated from
+an equation learning network [92]. Cats et al. [93] re-
+cently used machine learning to improve the standard
+mean-field approximation of the excess Helmholtz free-
+energy functional for a three-dimensional Lennard-Jones
+(LJ) system at a supercritical temperature. These signif-
+icant reserach efforts were devoted to tailoring analytical
+forms of model free energy functionals, by training cer-
+tain key components such as spatial convolution kernels,
+and much insight into the inner workings of excess free
+energy functionals was gained [91–93].
+However, here we proceed very differently and more-
+over do so out-of-equilibrium. We use the LJ model and
+the identical planar geometry as in Ref. [93], such that
+the density profile ρ(x) depends only on a single posi-
+tion coordinate x. We consider steady states and retain
+planar symmetry by considering flow that is directed in
+the x-direction, such that the current J(x) = J(x)ex,
+where J(x) is the magnitude of the current and ex is the
+unit vector in the x-direction. Both the density profile
+ρ(x) and the velocity field v(x) = J(x)/ρ(x) are indepen-
+dent of time. The continuity equation (5) then implies
+0 = ∂ρ(x)/∂t = −∂[v(x)ρ(x)]/∂x, from which one ob-
+tains by spatial integration ρ(x)v(x) = J0 = const. Here
+the value of J0 determines the intensity of the flow; we
+recall the illustration shown in Fig. 1.
+We base the machine learning procedure on a convolu-
+tional neural network, as was done e.g. in Ref. [91], and
+following Refs. [91–93] we use many-body computer sim-
+ulations to provide training, validation, and test data.
+In contrast to these equilibrium studies though, in or-
+der to address the nonequilibrium problem we need to
+represent the physical time evolution on the many-body
+trajectory level. We use the recently developed highly
+performant adaptive BD algorithm [6] and apply it to
+the three-dimensional LJ fluid.
+As laid out above, in
+order to address situations of planar symmetry we drive
+the system only along the ex-direction. The specific form
+of the driving force field fext(x)ex is however irrelevant,
+as the training data only serves to extract the intrinsic
+kinematic functional relationship.
+In order to cover a sufficiently broad range of flow sit-
+uations, we represent the external force field as a trun-
+cated Fourier series fext(x) = �nmax
+n=0 An cos(2πnx/L),
+where L is the size of the cubic simulation box with pe-
+riodic boundary conditions and An are random ampli-
+tudes with zero mean and uniform distribution inside of
+a given finite interval. We truncate at order nmax = 5
+such that the length scale L/(2πnmax) is comparable to
+the LJ molecular size σ. Ten percent of our simulation
+runs are carried out in equilibrium, i.e. for A0 = 0. We
+use N = 500 LJ particles inside of a cubic simulation
+box of size L = 10σ. The temporal duration of each run
+is 1000τ, where τ = σ2/D0 is the Brownian time scale.
+After initialization the system is randomized for 1τ at a
+very high temperature. Then we wait for 100τ to allow
+the system to reach a steady state and then collect data
+during the remaining time. In total we use 1000 such sim-
+ulation runs; these are subdivided for purposes of train-
+ing (520), validation (280) and testing (200).
+A more
+detailed account will be given elsewhere.
+Our aim is to machine-learn and hence to explicitly
+demonstrate the kinematic map, ρ(r), v(r) → fint(r) in
+steady state. We present the learning algorithm with in-
+puts ρ(x), v(x) and targets fint(x). The data for these
+three fields are from building steady state averages via
+the adaptive BD over the corresponding one-body oper-
+ators. We recall the microscopic definition of the inter-
+particle one-body force density Fint(r) via Eq. (7) and
+we refer the reader to Appendix A of Ref. [51] for a de-
+scription of several methods to sample the current in
+BD and hence obtain the overdamped velocity profile
+v(r). Finally, we use the standard counting method for
+the density profile ρ(r), although more efficient “force
+sampling” methods [94–97] exist. At this stage we nei-
+ther impose adiabatic-superadiabatic splitting (11), nor
+structure-flow splitting (18), nor do we use any analyti-
+cal model form of the functional relationship. We rather
+work on the level of the bare one-body simulation data,
+generated in the above described randomized uniaxial
+flow situations of the desired planar symmetry.
+We refer to the result of this procedure as the machine-
+learned internal force field f ⋆
+int(x, [ρ, v]). This represents
+a “surrogate model” in the sense of the terminology of the
+machine learning community. By construction this data
+structure depends functionally on the density profile and
+
+10
+on the velocity profile. Importantly the external force
+field fext(x), as given by the above described randomized
+Fourier series, has not been used in the training, which
+was rather based solely on the intrinsic force field and
+its kinematic dependence on the density profile and the
+velocity field.
+In order to test the validity of the functional relation-
+ship and to address the question whether f ⋆
+int(x, [ρ, v])
+indeed represents the true fint(r, t, [ρ, v]) of power func-
+tional theory restricted to the present planar and steady
+situation, we consider a toy flow situation as an appli-
+cation.
+We choose the density profile to consist of a
+single (co)sinusoidal deviation from the bulk, ρ(x) =
+[0.5 + 0.2 cos(2πx/L)]σ−3.
+In order for the system to
+be in steady state, the velocity then necessarily needs to
+satisfy v(x) = J0/ρ(x), where the strength of the current
+J0 = const is a free parameter.
+We proceed in two ways.
+First, we check for self-
+consistency. Therefore we solve the force density balance
+relationship (6) for the external force field, which yields
+the explicit result:
+fext(x) = kBTρ′(x) + γv(x) − f ⋆
+int(x, [ρ, v]),
+(29)
+where ρ′(x) = ∂ρ(x)/∂x. As is explicit in Eq. (29), in-
+putting the toy state ρ(x), v(x) on the right hand side
+yields a concrete machine learning prediction for the ex-
+ternal force field on the left hand side. We then input
+this result for fext(x) as the driving force field in a single
+adaptive BD simulation run and expect this procedure to
+reproduce the density and velocity profile of the toy state.
+The reproductive success will however materialize only
+provided that i) the functional kinematic dependence ac-
+tually exists and that ii) it is accurately represented by
+the neural network.
+The results, shown in Fig. 2, demonstrate the accom-
+plishment of the reconstruction of the toy state.
+This
+establishes that the machine learned functional provides
+a numerically very highly accurate representation of the
+true internal force functional. We take this validation via
+the machine learning to be a practical, data-science-level
+verification of the existence of the power functional kine-
+matic map. We recall the original formal construction
+[42, 53] and its subsequent confirmation via custom flow
+[51, 52].
+Turning to the physics of the compressional flow, we
+use the adiabatic-superadiabatic decomposition (11) to-
+gether with the flow-structure splitting (18) to analyze
+both the machine-learned functional f ⋆
+int(x, [ρ, v]) as well
+as the direct simulation results. As anticipated, both flow
+and structural force fields have nontrivial spatial varia-
+tion, see Fig. 2. The flow force primarily contains viscous
+effects that stem from the dissipation that the compres-
+sional and extensional regions of the flow pattern gener-
+ate. The structural force field becomes more strongly in-
+homogeneous and also larger in magnitude upon increas-
+ing the amplitude of the flow. This trend is necessary
+to provide a balance for the increasingly asymmetric and
+growing external force field, which in turn is required to
+keep the density profile unchanged upon increasing the
+throughput through the prescribed density wave.
+The
+power functional predictions (21) and (22) capture these
+effects reasonably well given the simplicity of the ana-
+lytical expressions, see the insets in Fig. 2. We find our
+numerical results to satisfy the Noether sum rules (26)
+and (27) to very good accuracy.
+It remains to point out the stark contrast with the
+standard DDFT (1), which gives a trivial null result in
+the present setup by construction: the density profile re-
+mains unchanged upon increasing flow, and so does the
+adiabatic force field. So the DDFT provides no mecha-
+nism to account for the nonequilibrium physics.
+VIII.
+CONCLUSIONS
+For the purpose of assessing the status of the DDFT
+equation of motion (1) we have first described two exact
+limits that this approximation reproduces: the dynamics
+of the noninteracting diffusive ideal gas [see Eq. (4)] and
+the spatially inhomogeneous static equilibrium limit [see
+Eq. (2)]. On general grounds one expects the DDFT to
+perform well when the situation under consideration is
+close to one of these limits. In particular near the static
+case this is nontrivial, as the system might be dense and
+spatially highly structured, as evident by a strongly inho-
+mogeneous density profile. Provided that the dynamics
+are driven weakly enough via a time-dependent external
+potential then the DDFT can be a highly useful device,
+which enables one to describe the temporal evolution as
+a chain of equilibrium states, labelled by time.
+In general the contributions beyond the equilibrium
+physics will however be relevant.
+On the level of the
+formally exact one-body equation of motion (14), the su-
+peradiabatic force field fsup(r, t) will then contribute and
+potentially very significantly so. Together with the adi-
+abatic force field, which follows from the equilibrium ex-
+cess free energy functional via −∇δFexc[ρ]/δρ(r, t), their
+sum constitutes the full interparticle forces. These are
+coarse-grained, in a microscopically sharp way, to the
+one-body level of dynamical correlation functions. We
+have argued i) that power functional theory is a con-
+crete formal structure that allows to obtain fsup(r, t)
+from a generating functional and ii) that simple approx-
+imate forms already capture much relevant nonequilib-
+rium physics and they do so in a transparent and sys-
+tematic way.
+We have described and exemplified for uniaxial steady
+compressional flow of the three-dimensional Lennard-
+Jones fluid the kinematic functional map that governs
+the exact nonequilibrium dynamics on the one-body level
+of dynamic correlation functions. As this description is
+based on a single position coordinate and a single time
+variable, it is of both conceptual and practical simplicity.
+As described by power functional theory the superadia-
+batic interparticle force field functionally depends on the
+density and the velocity field, i.e. fsup(r, t, [ρ, v]), for over-
+
+11
+damped Brownian motion. The functional dependence is
+causal, i.e. on the values of the density profile and velocity
+field at previous times, in general up to an initial state.
+The superadiabatic force field carries this kinematic de-
+pendence, i.e. on the history of ρ(r, t) and v(r, t), but
+crucially it is independent of the external force field that
+drives the system.
+We have explicitly demonstrated the functional map
+ρ(r, t), v(r, t) → fint(r, t) by establishing this functional
+relationship via machine learning the intrinsic force field.
+Using the force balance then gives direct access to the
+form of the required external force field via Eq. (29). The
+machine-learned model of the functional map hence en-
+ables “instant custom flow” at negligible computational
+cost at the time of use. We recall that the custom flow
+method [51, 52] is based on the kinematic functional map,
+such that from knowing the kinematic one-body fields,
+the external force field that is necessary to generate the
+given time evolution follows straightforwardly from the
+exact force balance (6).
+An analytical approach to one-body functional maps
+leads to the simple structure of velocity gradient forms
+for the viscous and structural superadiabatic forces, as
+exemplified in Eqs. (16) and (17) for compressional flow,
+i.e. for velocity fields with nonvanishing divergence. As
+we have shown, the resulting predictions for the flow
+force (21) and for the structural force field (22) represent
+a reasonable description of the simulation data and its
+representation via the machine-learned functional. We
+attribute the remaining differences to higher-order terms
+[59] which we have not addressed here for simplicity. As
+we have shown, our results from direct simulation, from
+machine learning, and from the analytical approxima-
+tions, satisfy exact global Noether sum rules.
+We have restricted our discussion to a single and rela-
+tively easily accessible type of nonequilibrium dynamics,
+that of stationary uniaxial compressional flow that rep-
+resents a model steady (batch) sedimentation situation.
+The power functional approach allows to go much fur-
+ther, including the treatment of viscoelasticity [56], as
+arising from superadiabatic memory, deconfinement un-
+der shear [57], the dynamic decay of the van Hove pair
+correlation function as governed by drag, viscous and
+structural forces [68, 69], and the complex forms of both
+flow and structural forces that arise under spatially com-
+plex forms of driving [59]. Time-dependent uniaxial flow
+is relevant in a variety of situations, including colloidal
+stratification [98, 99] and sedimentation [100].
+Although power functional theory operates on the one-
+body level of dynamical correlation functions, two-body
+correlation functions are accessible both formally via the
+nonequilibrium Ornstein-Zernike route [42] and explic-
+itly by the dynamical test particle limit. The latter is
+the dynamic generalization of Percus’ static test parti-
+cle limit [61], which identifies two-point correlation func-
+tions, such as g(r) as also recently shown to be intimat-
+edly related to thermal Noether invariance at second or-
+der [50], with one-body density profiles in an external
+potential. This is set equal to the interparticle pair po-
+tential.
+The dynamical test-particle limit goes further
+in that it describes the test particle via its own dynami-
+cal degrees of freedom, which are coupled to those of all
+other particles in the system. The concept was originally
+formulated as an approximation within DDFT [62, 63]
+and formally exactly within power functional theory [66].
+Two-body superadiabatic effects were shown via simula-
+tion work to be significant [67–69] and they arise natu-
+rally in an exact formulation of the test particle dynamics
+[66]. The test particle limit allowed for a rationalization
+of the dynamical pair structure as e.g. experimentally
+observed in two-dimensional colloids [9]. Recently an ap-
+proach to DDFT based on the two-body level was formu-
+lated [101].
+In event-driven BD simulations superadiabatic forces
+were shown to consist of drag, viscous, and structural
+contributions [68, 69]; see Ref. [42] for an extended dis-
+cussion. The physics of active particles [70–74] is very
+significantly governed by a vigorous interplay between su-
+peradiabatic and adiabatic forces, both of which are very
+strong, as the tendency of these systems to self-compress
+leads naturally to very high local densities.
+Furthermore,
+relevant and interesting microscopic
+models that go beyond the simple fluid paradigm of a
+pair potential, such as the monatomic water model by
+Molinero and Moore [102, 103] and the three-body gel
+by Saw et al. [104, 105], are accessible. Despite the com-
+plexity of both its defining Hamiltonian and the intricate
+transient network structure, the inhomogeneous viscous
+response of the three-body gel was recently demonstrated
+[60] to be surprisingly well captured by a simple power
+functional flow approximation.
+We finally recall that
+superadiabatic effects transcend overdamped dynamics,
+and are relevant both in quantum dynamics [42, 80, 81]
+and in classical molecular dynamics [42, 78, 79].
+While we have restricted ourselves to discussing the
+point of view of functional relationships, it would be in-
+teresting to explore in future work possible cross connec-
+tions to other theoretical approaches, such as Onsager’s
+variational principle for soft matter [106–109], stochastic
+thermodynamics [110], large deviation theory [111, 112],
+mode-coupling theory [113, 114], generalized hydrody-
+namics [115], as well as to the physics of nonequilibrium
+phase transitions [116] and of Brownian solitons [117].
+ACKNOWLEDGMENTS
+This work is supported by the German Research Foun-
+dation (DFG) via Project No. 436306241.
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf,len=1342
+page_content='Perspective: How to overcome dynamical density functional theory  Daniel de las Heras,1 Toni Zimmermann,1 Florian Samm¨uller,1 Sophie Hermann,1 and Matthias Schmidt1  1Theoretische Physik II, Physikalisches Institut, Universit¨at Bayreuth, D-95447 Bayreuth, Germany  (Dated: 28 January 2023)  We argue in favour of developing a comprehensive dynamical theory for rationalizing, predicting,  and machine learning nonequilibrium phenomena that occur in soft matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' To give guidance for  navigating the theoretical and practical challenges that lie ahead, we discuss and exemplify the  limitations of dynamical density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Instead of the implied adiabatic sequence of  equilibrium states that this approach provides as a makeshift for the true time evolution, we posit  that the pending theoretical tasks lie in developing a systematic understanding of the dynamical  functional relationships that govern the genuine nonequilibrium physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' While static density func-  tional theory gives a comprehensive account of the equilibrium properties of many-body systems,  we argue that power functional theory is the only present contender to shed similar insights into  nonequilibrium dynamics, including the recognition and implementation of exact sum rules that  result from the Noether theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' As a demonstration of the power functional point of view, we  consider an idealized steady sedimentation flow of the three-dimensional Lennard-Jones fluid and  machine-learn the kinematic map from the mean motion to the internal force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' This proof of con-  cept demonstrates the significant potential of machine learning the inherent functional relationships  that govern nonequilibrium many-body physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  INTRODUCTION  The coupled dynamics of the microscopic degrees of  freedom in typical soft matter systems generates a wide  array of relevant and also often unsolved nonequilibrium  phenomena [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  One central quantity for the char-  acterization of self-assembly and structure formation in  complex systems is the microscopically resolved one-body  density distribution ρ(r, t), where r indicates position  and t denotes time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The “density profile” ρ(r, t) acts as a  central order parameter both due to its intuitive physical  interpretation and clearcut mathematical definition [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  According to the dynamical density functional theory  (DDFT), as originally proposed by Evans in 1979 [4],  the time evolution of the microscopic density profile is  assumed to be determined by the following partial differ-  ential equation:  ∂ρ(r, t)  ∂t  = γ−1∇ · ρ(r, t)∇  � δF[ρ]  δρ(r, t) + Vext(r, t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (1)  Here γ is a friction constant, F[ρ] is an intrinsic free  energy functional that depends functionally on the den-  sity profile, and the external potential Vext(r, t) repre-  sents interactions of the system with the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The system is set into motion by a temporal variation  of Vext(r, t), such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' step-like switching at an initial  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The time evolution according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (1) conserves the  particle number locally and hence it constitutes dynam-  ics of model B type [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  In standard applications one  starts with an equilibrium state of the system and then  the dynamics are monitored on the basis of numerical  time integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' In order to provide reference  data and to allow for the generation of benchmark results  to assess the quality of the theory, resorting to many-  body computer simulations is common, with overdamped  Brownian dynamics (BD) being a popular choice (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' [6]  describes a modern and stable algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Comparison  of DDFT data with experimental results are more scarce,  but notable exceptions include non-equilibrium sedimen-  tation of colloids [7], the self-diffusion of particles in com-  plex fluids [8], and the bulk dynamics of Brownian hard  disks [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The DDFT time evolution reaches a stationary state  if the gradient on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (1) vanishes,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' provided that the expression inside of the parentheses  is constant:  δF[ρ]  δρ(r) + Vext(r) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (2)  Here we have dropped the dependence on time in the  notation, as the situation is now static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The constant µ  can be identified with the chemical potential, which in a  grand canonical statistical mechanical setting is the con-  jugate control parameter of the mean particle number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Equation (2) is exact in equilibrium, as was shown by  Evans [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' He proved the equilibrium intrinsic free en-  ergy functional F[ρ] to exist, to be unique, and to form  the starting point for a modern equilibrium theory of  spatially inhomogeneous liquids and crystals [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  In practice one needs to rely on approximations  for F[ρ], given a microscopic fluid model under consid-  eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Once one has solved Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (2) for given values  of µ and temperature T (the dependence of F[ρ] on T  is suppressed in the notation), then in principle com-  plete knowledge of the thermal system is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The  value of the density functional F[ρ] is the true intrinsic  free energy, and higher-order correlation functions are  determined via higher-order derivatives of the free en-  ergy functional or via test-particle procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' In par-  ticular two-body correlations functions, such as the bulk  pair correlation function g(r) as well as its generalization  to inhomogeneous systems are accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' These exhibit  defining characteristics of liquids and more general soft  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='12156v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='soft]  28 Jan 2023  2  matter systems and they are formally fully contained in  the static density functional theory framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Together with a number of available reliable approxi-  mate free energy functionals, density functional theory is  a powerful theoretical framework that has been used to  elucidate much intricate and complex behaviour in soft  matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Recent representative highlights include trac-  ing hydrophobicity to critical drying at substrates [12–  14], resolving three-dimensional structures of electrolyte  aqueous solutions near surfaces [15, 16], and addressing  the magnitude of the decay lengths in electrolytes [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Rosenfeld’s celebrated hard sphere fundamental measure  free energy functional [18–21] is at the core of much of  this research activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  In the following we wish to address whether or not  the DDFT has the prowess to play a similar role in  nonequilibrium, as is often at least implicitly assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We demonstrate on the basis of an explicit and generic  example, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=', that of uniaxial compressional flow of the  three-dimensional Lennard-Jones fluid, that the DDFT is  fundamentally flawed and that in reality, as represented  by many-body simulations, recognizing the flow field as  a further relevant degree of freedom is required to rep-  resent true nonequilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' These conclusions are based  on analytical power functional approximations, adaptive  BD simulation data, and explicit machine learning of the  power functional map from motion to the interparticle  one-body force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  This Perspective is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We first make  some key aspects of DDFT explicit in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' II and describe  several prominent shortcomings of this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We then  give an account of how to go towards the formally exact  one-body dynamics in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' III and provide in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' IV a  description of key aspects of the power functional frame-  work, which as we wish to argue overcomes the funda-  mental defects of DDFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We describe the exemplary sta-  tionary compressional flow situation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' V and lay  put the application of Noether’s theorem in this statis-  tical mechanical setting in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We present machine  learning results for the kinematic functional relationships  of the streaming Lennard-Jones fluid in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We give  conclusion and an outlook in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  LIMITS AND LIMITATIONS OF  ADIABATIC DYNAMICS  We go into some detail and describe why the DDFT  represents adiabatic dynamics in the sense of a temporal  sequence of spatially inhomogeneous equilibrium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The equilibrium intrinsic free energy functional splits into  ideal and excess (over ideal gas) contributions according  to F[ρ] = Fid[ρ] + Fexc[ρ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Here the excess free energy  functional Fexc[ρ] accounts for the effects of the inter-  particle interactions on the equilibrium properties of the  system and it is in general unknown and requires approx-  imations to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The ideal gas free energy functional  however is exactly given by  Fid[ρ] = kBT  �  drρ(r) ln(ρ(r)Λ3) − 1],  (3)  where kB denotes the Boltzmann constant, Λ is the  thermal de Broglie wavelength, and we consider three-  dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The functional derivative, as it is  relevant for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (1), is δFid[ρ]/δρ(r) = kBT ln(ρ(r)Λ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  When disregarding the excess contribution and in-  serting this result alone into the DDFT equation  of motion (1), its right hand side becomes γ−1∇ ·  ρ(r, t)∇[kBT ln(ρ(r, t)Λ3) + Vext(r, t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' This can be re-  written further such that for the case of the ideal gas,  where Fexc[ρ] = 0 and F[ρ] = Fid[ρ], the equation of  motion (1) attains the following form:  ∂ρ(r, t)  ∂t  = D0∇2ρ(r, t) − ∇ · ρ(r, t)fext(r, t)/γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (4)  Here D0 = kBT/γ is the diffusion constant, ∇2 is the  Laplace operator and the external force field is given  (here) as fext(r, t) = −∇Vext(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Equation (4) is the  exact drift-diffusion equation for overdamped motion of  a mutually noninteracting system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=', the ideal gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Besides Evans’ original proposal [4] based on the con-  tinuity equation and undoubtedly his physical intuition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  derivations of the DDFT (1) were founded much more  recently on Dean’s equation of motion for the density op-  erator [22],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' the Smoluchowski equation [23],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' a stationary  action principle for the density [24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' the projection op-  erator formalism [25],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' a phase-space approach [26],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' the  mean-field approximation [27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' a local equilibrium as-  sumption [28],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' and a non-equilibrium free energy [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The question of the well-posedness of the DDFT was ad-  dressed [30] and several extensions beyond overdamped  Brownian dynamics were formulated, such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' for dy-  namics including inertia [31–34] and for particles that ex-  perience hydrodynamic interactions [34, 35] or undergo  chemical reactions [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The DDFT was also used beyond the description of flu-  ids, such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' for opinion dynamics [38] and epidemic  spreading [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Recent reviews of DDFT are given in  Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The theory is put into a wider perspective,  together with much background pedagogical material in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' A modern and well-accessible account of the  general strategy of dynamical coarse-graining in statisti-  cal physics, of which the DDFT can be viewed as being a  representative, has recently been given by Schilling [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The fact that both the static limit for the fully in-  teracting system (2) as well as the full dynamics of the  noninteracting system (4) are exact, taken together with  the heft of the DDFT literature, appears to give much  credibility to the equation of motion (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' However, de-  spite the range of theoretical techniques employed [22–29]  neither of these approaches has provided us with a con-  crete way of going beyond Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Apart from several  case-by-case and rather ad hoc modifications, no system-  atic or even only practical identification of what is miss-  ing has been formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (We turn to power functional  3  theory in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=') This is a problematic situation as  two defects of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (1) are immediately obvious upon in-  spection: i) the description is local in time and there is  no natural mechanism for the inclusion of memory while  time-locality is not sufficient for general nonequilibrium  situations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' ii) only flow that leads to direct changes in  the density profile is captured and hence effects of rota-  tional flow, such as shearing, as well as of nonequilibrium  effects in compression and expansion are lost (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Here we argue that these defects are indicative of a  broader failure of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (1) to describe nonequilibrium  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We show that the DDFT is only fit to describe  situations in which the dynamics follow an adiabatic path  through a sequence of equilibrium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The description  of genuine nonequilibrium dynamics in a functional set-  ting on the one-body level rather requires recognition of  the local velocity field as a further relevant physical vari-  able besides the density profile, and this is provided by  power functional theory [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Before laying out key prin-  ciples of this approach in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' IV, we first describe the mi-  croscopically sharp coarse-graining on the one-body level  of correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  TOWARDS EXACT ONE-BODY  DYNAMICS  Evans based his original derivation [4] of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (1) on the  continuity equation,  ∂ρ(r, t)  ∂t  = −∇ · J(r, t),  (5)  where J(r, t) is the microscopically resolved one-body  current distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Equation (5) is exact in a variety of  contexts, including overdamped Brownian dynamics, as  described either on the Fokker-Planck level by the Smolu-  chowski equation or by the corresponding overdamped  Langevin equation that governs the trajectories, as they  are realized in simulation work [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' For BD the one-body  current distribution is given exactly by [42]:  γJ(r, t) = −kBT∇ρ(r, t) + Fint(r, t) + ρ(r, t)fext(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (6)  This identity expresses the force density balance of the  negative friction force density (left hand side) with the  force densities due to ideal thermal diffusion, interparti-  cle interactions, and external influence (three contribu-  tions on the right hand side).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Here the interparticle force  density distribution is given by the statistical average  Fint(r, t) = −  � �  i  δ(r − ri)∇iu(rN)  ����  t,  (7)  where the angular brackets indicate an average at fixed  time t over the nonequilibrium many-body distribu-  tion, u(rN) is the interparticle interaction potential  that depends on all particle position coordinates rN ≡  r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' , rN and ∇i indicates the derivative with respect to  the position ri of particle i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The formulation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (7) is  based on the concept of static operators and a dynami-  cally evolving probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' This is analogous  to the Schr¨odinger picture of quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The  Heisenberg picture is more closely related to simulation  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Here the probability distribution is that of the ini-  tial microstates and the operators move forward in time,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=', the position ri(t) of particle i changes over the course  of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Then the Dirac distribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (7) becomes  δ(r − ri(t)), with the generic position variable r however  remaining static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The forces are those that act in the  given microstate rN(t) at time t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=', the interparticle  force on particle i at time t is −∇iu(rN(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  In practice, using BD simulations, carrying out the  average in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (7) requires to build the mean over suf-  ficiently many separate realizations of the microscopic  evolution of the many-body system that differ in the ini-  tial state and in the realization of the thermal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' As  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (7) measures both the probability to find particle i at  position r (via the delta function) and the interparticle  force that acts via the negative gradient −∇iu(rN), we  refer to Fint(r, t) as a force density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The corresponding  force field fint(r, t) is obtained by simple normalization  with the density profile, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' fint(r, t) = Fint(r, t)/ρ(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Building this ratio scales out the probability effect and  the force field then carries physical units of force, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  energy per length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  In equilibrium the definition (7) remains intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Com-  plementing the statistical average, static density func-  tional theory allows to express the equilibrium force den-  sity as being functionally dependent on the density pro-  file via the functional derivative of the excess free energy  functional according to:  Fint(r)  ��  eq = −ρ(r)∇δFexc[ρ]  δρ(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (8)  Crucially, and in contrast to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (7), here the internal  force density is directly expressed as a density functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  This dependence has superseded the original dependence  on the external potential, as is manifest in the probability  distribution for building the average (7) in equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  As a self-consistency check we insert the force density  functional (8) into the equilibrium limit of the force den-  sity balance (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The current vanishes in the equilibrium  case, J(r, t) ≡ 0, and we obtain  −kBT∇ρ(r) + Fint(r)|eq + ρ(r)fext(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (9)  This result is independent of time and it consti-  tutes the gradient of the static Euler-Lagrange equa-  tion (2) when divided by the density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (Insert  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (8), identify the ideal gas contribution −kBT∇ρ(r) =  −ρ(r)δFid[ρ]/δρ(r), and divide by ρ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=')  The classical  force density balance result (9) by Yvon, Born and Green  [3] has recently been derived from systematically address-  ing thermal Noether invariance [44, 45] against locally  resolved spatial deformations of the statistical ensemble  [46–48], as also valid quantum mechanically [48] and at  4  second order in the displacement field [49, 50];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' we give a  brief account of this theory in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' VI below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  A naive transfer of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (8) to nonequilibrium lets  one simply evaluate the equilibrium excess free energy  functional at the instantaneous nonequilibrium density  ρ(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' In order to separate this contribution from true  static equilibrium, we refer to this force density as being  adiabatic (subscript “ad”) and to be defined as  Fad(r, t) = −ρ(r, t)∇δFexc[ρ]  δρ(r, t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (10)  We recall that the right hand side offers a concrete com-  putational structure that is of practical usefulness in ac-  tual applications, as considerable knowledge about ap-  proximative forms of the excess free energy density func-  tional Fexc[ρ] is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Using the adiabatic force den-  sity as a proxy for the true nonequilibrium intrinsic force  density distribution (7), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' setting Fint(r, t) = Fad(r, t)  in the force density balance (6) together with the conti-  nuity equation (5) leads to the DDFT equation of mo-  tion (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The adiabatic force density approximation is  uncontrolled though and the theory inherently yields the  dynamics as an adiabatic sequence of equilibrium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Surely, more than 40 years after the conception of the  DDFT [4], we have to be able to do better!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  POWER FUNCTIONAL TECHNIQUES  Power functional theory [42] offers a concrete math-  ematical structure to go forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We describe the es-  sential steps that enable one to go beyond the DDFT  and to hence address a significantly expanded realm of  nonequilibrium physics which Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (1) is oblivious of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The interparticle force density profile (7) is identified  to consist of two contributions according to:  Fint(r, t) = Fad(r, t) + Fsup(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (11)  Here Fad(r, t) is the adiabatic force density profile, as  given formally via the explicit equilibrium free energy  derivative (10) and directly accessible in simulations via  the custom flow method [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The custom flow al-  gorithm allows to systematically construct a hypotheti-  cal adiabatic (equilibrium) system that shares its density  profile with the nonequilibrium system at the given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Then sampling the internal force density in the adiabatic  system yields results for Fad(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The second, superadiabatic contribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (11),  Fsup(r, t), contains all effects that are not expressible  as an instantaneous density functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  This includes  forces that lead to viscous and to nonequilibrium struc-  ture forming phenomena, as we exemplify below in a con-  crete model compressional flow situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Formally, the  superadiabatic force density is generated from the su-  peradiabatic excess free power functional P exc  t  [ρ, J] upon  functional differentiation with respect to the one-body  current via [42, 53]:  Fsup(r, t) = −ρ(r, t)δP exc  t  [ρ, J]  δJ(r, t)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (12)  The functional dependence of P exc  t  [ρ, J] on the density  and current is causal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' on the values of these fields  at prior times to t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' density and current need to satisfy  the continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Upon using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (11) the force  density balance (6) attains the following form:  γJ(r, t) = −kBT∇ρ(r, t) + Fad(r, t)  + Fsup(r, t) + ρ(r, t)fext(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (13)  This  relationship  holds  beyond  gradient  forms  of  fext(r, t), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' for external force fields that contain non-  conservative contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Crucially Fsup(r, t) will in  general also acquire nonconservative contributions, such  as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' damping effects that represent viscous behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Moreover, nonequilibrium structure-forming effects will  also arise in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' These affect directly the shape of  the density profile, whether this evolves in time or per-  sists in a nonequilibrium steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  If one wishes to eliminate the explicit occurrence of the  current from the dynamics, then inputting the force den-  sity balance (13) into the continuity equation (5) leads  to the following formally exact form of the equation of  motion for the density profile:  ∂ρ(r, t)  ∂t  = D0∇2ρ(r, t) + ∇ · ρ(r, t)  γ  ∇δFexc[ρ]  δρ(r, t)  − ∇ · ρ(r, t)  γ  [fsup(r, t) + fext(r, t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (14)  Here it is apparent that the superadiabatic force field  fsup(r, t) = Fsup(r, t)/ρ(r, t) has a direct effect on the  system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The effect is similar to that of the ex-  ternal force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Crucially though, both force fields are  independent of each other: the external force field rep-  resents a prescribed and inert influence on the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  In contrast, the superadiabatic force field is an emer-  gent phenomenon that arises due to interparticle inter-  actions and, from the functional point of view, depends  non-locally in position and causally in time on the one-  body density and on the current profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Although setting fsup(r, t) = 0 yields the DDFT (1),  the superadiabatic force field fsup(r, t) was demonstrated  to exist [54–60] and in general to play a major role in  the dynamics on the one-body level and, based on test-  particle concepts [61–66] also for two-body correlation  functions [67–69] and for active matter [70–74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Both the  flow properties as well as the spatial structure formation  in the system are affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  To reveal additional physics, it is useful to split into  “structural” and “flow” contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' This was estab-  lished e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' for complex flow patterns that occur in driven  BD [55, 59], for active Brownian particles which form  a self-sustained interface at motility-induced phase co-  existence [70–74], as well as very recently for a sheared  5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Illustration of unidirectional compressional flow of a liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The three-dimensional system is set into motion (red  arrows) by the action of an external force profile fext(x) (blue arrows) which acts along the x-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The system retains planar  geometry such that spatial inhomogeneities only occur as a function of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The density profile ρ(x) (orange curve) and the  velocity profile v(x) (red curve) are both stationary in time but inhomogeneous in position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The local one-body current  J(x) = ρ(x)v(x) = const and as a result the system is in a nonequilibrium steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The corresponding adiabatic system  is in equilibrium (it has no mean flow) and it has by construction an unchanged density profile ρ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' In the adiabatic system  the spatial variation of ρ(x) is stabilized by the action of an external force field −∇Vad(x) (olive arrows), which acts solely in  the adiabatic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  three-body colloidal gel former [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Before we demon-  strate these concepts for an example of steady nonequi-  librium below, we first describe two simple model power  functionals that respectively generate structure and vis-  cously dampen the motion and that, as we will see, give  a good account of the nonequilibrium flow considered be-  low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We concentrate on the low-order terms that are rel-  evant for compressional/extensional flow, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=', for situa-  tions where ∇ · v(r, t) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We focus on cases where  there is no rotational motion (such as shearing) and hence  ∇ × v(r, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The velocity gradient superadiabatic  power functional consists of a sum,  P exc  t  [ρ, v] = P flow  t  [ρ, v] + P str  t  [ρ, v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (15)  Here the flow and structural [55, 59] contributions are  approximated, respectively, by the following time-local  (Markovian) and space-semilocal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' involving ∇) forms  P flow  t  [ρ, v] = η  2  �  dr[ρ(r, t)∇ · v(r, t)]2,  (16)  P str  t  [ρ, v] = −χ  3  �  dr[ρ(r, t)∇ · v(r, t)]3,  (17)  where the overall prefactors η and χ control the respec-  tive magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The flow functional (16) is quadratic  both in density and in the velocity field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' the structural  functional (17) is of cubic order in each of these variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Explicit higher-order functionals exist [59] and they be-  come relevant when driving the system strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We will  return to the consequences of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (16) and (17) after  laying out in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' V the actual flow situation that we use  as a model to exemplify the implications for the physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Before doing so, we briefly describe several further key  aspects of the power functional framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Power functional theory provides a formal framework  for the inclusion of time- and space-nonlocal dynamics  [56, 68, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' While Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (12) applies to overdamped dy-  namics, the acceleration field becomes a further relevant  degree of freedom if inertia are relevant [78–81] whether  classically in molecular dynamics [78, 79] or in quantum  dynamics [80, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Here the memory functions act as  convolution kernels on specific kinematic fields and rota-  tional and compressional contributions to the dynamics  are genuinely built in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' As laid out above, the framework  is based on an exact variational concept [42, 53], and the  resulting functional mapping was shown to be explicitly  accessible in many-body simulation via the custom flow  computer simulation method [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Even  simple  mathematical  model  forms  for  the  nonequilibrium contribution to the power functional,  such as Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (16) and (17), already capture essential  physics (as we demonstrate below) and dynamical two-  body correlation functions are accessible via test particle  dynamics [8, 9, 61–69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The power functional is thereby  not to be confused with the often vague concept of a  v(C  p(α)  noneguilibrium  fext(α)  (α) PeA△  equilibrium6  “nonequilibrium free energy”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The proper equilibrium  free energy functional does play a central role in power  functional theory though, via providing the description  of the adiabatic reference state [42], see the generation  of the force density distribution via functional differenti-  ation (10), as is relevant for the interparticle force split-  ting (11), and the full density equation of motion (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The relevance of superadiabatic contributions to the  dynamics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' of those effects that lie beyond Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (1), has  been amply demonstrated in the literature [54–59, 67–  69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Both adiabatic and superadiabatic effects arise from  integrating out the dynamical degrees of freedom of the  many-body problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Ensemble differences between canonical dynamics and  grand canonical equilibrium have been systematically ad-  dressed [75–77] and these do not account for the observed  differences between adiabatic and superadiabatic dynam-  ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The kinematic dependence on the motion of the  system arises formally [42], it can be explicitly traced  in many-body computer simulation work [59], and it  is amenable to machine learning, as we demonstrate in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Before doing so, we first formulate the represen-  tative flow problem that we will use to apply the above  concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  NONEQUILIBRIUM STEADY STATES  We restrict ourselves to flow situations with one-body  fields that are inhomogeneous in position but indepen-  dent of time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' ρ(r) and v(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Then trivially ∂ρ(r)/∂t =  0 and the continuity equation (5) constrains both fields  to satisfy ∇ · [ρ(r)v(r)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' As a representative case  we illustrate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' 1 a nonequilibrium steady state of a  three-dimensional liquid undergoing unidirectional com-  pressional flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Flow along a single given direction occurs  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' under the influence of gravity, where sedimentation  of colloids leads to both compression in the lower parts of  the sample and expansion in the upper parts of the sam-  ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Here we disregard transient phenomena and investi-  gate an idealized periodic system, where flowing steady  states can form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  In order to elucidate the physics in such setups, we fol-  low the splitting (15) of the superadiabatic power func-  tional into structural and flow contributions and hence  decompose the superadiabatic force field accordingly as  fsup(r) = fstr(r) + fflow(r),  (18)  where the right hand side consists of the nonequilib-  rium structural force field fstr(r) and the flow force  field fflow(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Both of these force contributions arise  from the microscopic interparticle interactions, as coarse-  grained in a microscopically sharp way to the one-body  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We lay out in the following the benefits of the  structure-flow splitting (18) and its definition via flow  reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  First, on the more practical level, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (18) allows to  carry out a corresponding splitting of the force density  balance (13) [we divide by ρ(r) to obtain force fields].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The result is a set of two coupled equations of motion,  with one of them depending explicitly on the velocity  profile and the second one depending explicitly on the  density profile:  γv(r) = fflow(r) + fext,f(r),  (19)  0 = fstr(r) − kBT∇ ln ρ(r) + fad(r) + fext,s(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (20)  Building the sum of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (19) and (20) and multiplying  by the density profile restores the full force density bal-  ance (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The external force field is split according to  fext(r) = fext,f(r) + fext,s(r), where the two terms couple  to the flow via fext,f(r) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (19) and to the structure  via fext,s(r) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  On the superficial level the two equations (19) and  (20) appear to be independent of each other, as no sin-  gle field appears explicitly in both equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' However,  the two equations are indeed intimately coupled to each  other by the interparticle interactions, as represented by  both the adiabatic and the two superadiabatic (flow and  structural) force fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' These three intrinsic force con-  tributions provide the physical representation of the true  nonequilibrium steady state dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The flow-structure splitting (18) is uniquely deter-  mined by the symmetry properties of the forces upon  motion reversal of the system [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Motion reversal is  a discrete symmetry operation, and hence different from  continuous invariances where Noether’s theorem applies  [44–50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  One considers a “reversed” system, which is  also in steady state and possesses an unchanged den-  sity profile ρ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The flow, however, is directed against  the velocity orientation in the original “forward” system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Hence the velocity profile in the reversed system is sim-  ply −v(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' As a result the current also acquires a mi-  nus sign, −ρ(r)v(r), which however does not affect the  (vanishing) divergence, ∇ · [−ρ(r)v(r)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Thus the re-  versed state indeed is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The two superadiabatic  contributions are then defined to be unchanged [fstr(r)]  and inverted [−fflow(r)] in the reversed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Conse-  quentially, the superadiabatic force field in the reversed  system is the difference fstr(r) − fflow(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Analyzing the symmetry properties of the adiabatic  force field is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We recall that fad(r) is  a density functional via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The density profiles  in the forward and in the reversed systems are identical  though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Hence fad(r) is invariant under motion reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Motion reversal is a useful device in order to i) rationalize  the nonequilibrium behaviour according to the split force  balance (19) and (20), and to ii) classify the dependence  of superadiabatic forces on the velocity field into even  powers, which constitute fstr(r), and odd powers, which  form fflow(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We can demonstrate this mechanism explicitly on the  basis of the above flow and structural power functionals  (16) and (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Superadiabatic force fields are generated  via the functional derivative (12) with respect to the cur-  rent or, analogously, by functionally deriving by v(r, t)  7  and dividing the result by ρ(r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The resulting supera-  diabatic one-body force field consists of two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The viscous flow force and [55, 58] and the structural  force follow respectively as  fflow(r) =  η  ρ(r)∇[ρ(r)2∇ · v(r)],  (21)  fstr(r) = − χ  ρ(r)∇{ρ(r)3[∇ · v(r)]2},  (22)  where Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (21) is odd (linear) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (22) is even  (quadratic) in the derivatives of the velocity field, as de-  sired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  One might wonder where all this genuine nonequilib-  rium physics leaves the DDFT!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Some readers will find the  instantaneous dynamics, as generated from an adiabatic  free energy according to (1), to be more appealing and in-  tuitive than the thinking in terms of the above described  apparently intricate functional relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Why not  live with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (1), use it, and simply accept its defects?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  In order to address this question and to demonstrate why  this path is severely restricted from the outset, we turn  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' VII to an explicit demonstration of the functional  relationship that governs the nonequilibrium physics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  the kinematic functional map from the one-body mean  motion to the internal force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Before doing so, we  demonstrate that Noether’s theorem of invariant varia-  tions has much to say about our present setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  NOETHER FORCE SUM RULES  We discuss one of the arguably simplest cases of ex-  ploitation of the inherent symmetries of a thermal many-  body system, that of global translational invariance of its  statistical mechanics [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We consider a “shifting”  transformation, where all particle coordinates change ac-  cording to the map ri → ri + ϵ, where ϵ = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' This  uniform shifting operation leaves all interparticle dis-  tance unchanged, ri−rj → (ri+ϵ)−(rj+ϵ) ≡ ri−rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' As  a consequence the interparticle potential is invariant un-  der the transformation, which we can express as the iden-  tity u(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' , rN) = u(r1 + ϵ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' , rN + ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Here equality  holds irrespectively of the magnitude and the direction  of the shifting vector ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The Noether argument proceeds with a twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  De-  spite the absence of dependence on ϵ, we can neverthe-  less differentiate both sides of the equation with respect  to ϵ and the result will be a valid identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We obtain  0 = ∂u(ri + ϵ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' , rN + ϵ)/∂ϵ = �  i ∇iu(r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' , rN),  where we have set ϵ = 0 after taking the derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We  multiply by −1 and insert 1 =  �  drδ(r−ri), which yields  −  �  dr  �  i  δ(r − ri)∇iu(rN) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (23)  The expression on the left hand side allows to identify  the locally resolved interparticle force operator ˆFint(r) =  − �  i δ(r − ri)∇iu(rN), such that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (23) attains the  form  �  drˆFint(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' This identity holds for each mi-  crostate rN and hence it remains trivially valid upon av-  eraging over the many-body distribution function, irre-  spective of whether this is in- or out-of-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We  can hence conclude the vanishing of the global interpar-  ticle force, expressed as the integral over the mean force  density Fint(r) = ⟨ˆFint(r)⟩ as  �  drFint(r, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (24)  Equation (24) holds at all times t and it can be viewed as  a consequence of Newton’s third law, see the discussion in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Using the adiabatic-superadiabatic force split-  ting (11) one can further conclude that the both global  contributions need to vanish individually,  �  drFad(r, t) = 0,  (25)  �  drFsup(r, t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (26)  The proof can either be based on the fact that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (25)  is merely Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (24) for the special case of an equilibrium  system, from which then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (26) follows from the force  splitting (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Alternatively and starting from a very  fundamental point of view, the global translational in-  variance of the excess free energy functional Fexc[ρ] and  of the superadiabatic free power functional P exc  t  [ρ, v],  here considered instantaneously at time t, lead directly  to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (25) and (26), see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' [44, 45] for the detailed  account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  It is interesting to apply the Noether concept to the  flow-structure splitting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (18) of the superadiabatic  force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' One can see straightforwardly, from the sym-  metry upon motion reversal, that both the global struc-  tural force and the global flow force need to vanish indi-  vidually:  �  drρ(r)fflow(r) = 0,  (27)  �  drρ(r)fstr(r) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  (28)  We prove by contradiction and assume that it is not  the case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' that each integral gives the same global  force, but with opposite sign, such that the sum vanishes  and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (26) remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Per construction, fflow(r)  changes sign in the motion reversed system, but fstr(r)  does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Hence Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (26) can only be satisfied in the  motion-reversed system provided that both the flow and  structural contribution vanish separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We  can  explicitly  test  the  validity  of  the  sum  rules (27) and (28) for the above analytical force ap-  proximations (21) and (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The respective integrals  are η  �  dr∇[ρ(r)2∇ · v(r)] = 0 and χ  �  dr∇{ρ(r)3[∇ ·  v(r)]2} = 0, which follows from the divergence theorem,  as boundary terms vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Hence the simple non-local  velocity gradient power functional approximations (16)  8  density  current  external force field  interparticle  force field  Mermin  Evans  map  (DFT)  kinematic fields  kinematic  map  adiabatic-superadiabatic  splitting  structure-flow splitting  superadiabatic  force field  adiabatic  force field  flow  force  structural  force  adaptive  BD  super-  adiabatic  map  (PFT)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='7  0  2  4  6  8  10  ��3  x/�  0  1  2  3  4  5  0  2  4  6  8  10  J�2�  x/�  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='5  0  2  4  6  8  10  fint�/�  x/�  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='5  0  2  4  6  8  10  fad�/�  x/�  0  4  8  12  16  0  2  4  6  8  10  fext�/�  x/�  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='4  0  2  4  6  8  10  fsup�/�  x/�  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='4  0  2  4  6  8  10  f�ow�/�  x/�  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='1  0  2  4  6  8  10  fstr�/�  x/�  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Kinematic profiles and force fields for uniaxial compressional flow of the LJ fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Results are shown from machine  learning (lines) and from direct adaptive BD simulations (symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Functional relationships are represented by vertical arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Shown are the density profile ρ(x), the one-body current J(x) and the external force field fext(x) (top row) as a function of the  scaled distance x/σ, where σ is the LJ length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The density and the current functionally determine both the interparticle  force field fint(x) via the kinematic map and the superadiabatic force field fsup(x) via the superadiabatic kinematic map (middle  row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The internal force field fint(x) splits into superadiabatic and adiabatic force contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The adiabatic force field fad(x)  is a density functional via the Mermin-Evans map of density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The structural and flow force fields are split  according to their symmetry upon motion reversal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The colour code represents different values of the current J0 = 0, 1, 2, 3, 4, 5  (from violet to yellow, see the center panel in the top row);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' the two insets show the predictions from the analytical velocity  gradient functionals (21) and (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The system with J0 = 0 is at rest in equilibrium and it doubles as the adiabatic state as its  density profile is identical to that of the flowing systems (first panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  and (17) have passed the global Noether validation test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  This is nontrivial, as the proof rests on the specific struc-  ture of the integrands being gradients, which for more  general analytical forms will not be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' This exem-  plifies the merits of Noether sum rules for assessing and  by extension also constructing theoretical nonequilibrium  force approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The Noether concept carries much further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Reference  [44] presents memory sum rules for so-called time di-  rect correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' These are defined via func-  tional derivatives of the superadiabatic power functional,  in generalization of the superadiabatic force density as  generated via the derivative (12) with respect to the cur-  rent distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We expect the corresponding identities  to be helpful in the study of temporal nonlocality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Fur-  ther work was addressed at the variance of global fluctu-  ations, which were shown to be constrained by Noether  invariance at the second order global level [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Noether’s  theorem also yields the locally resolved force balance re-  lationship in quantum mechanical many-body systems  [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Very recently, striking two-body force-force and  force-gradient correlation functions for the precise and  novel characterization of disordered (liquid and gel) sys-  tems [50] were revealed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Exploiting Noether’s concept in  a stastical mechanical setting is robust against changes of  ensemble, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' [45] presents the transfer of the grand en-  semble formalism [44] to canonical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Considering  global rotational invariance leads to (classical) spin-orbit  J&pJ&p9  coupling of torque identities [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We return to steady states and demonstrate that the  seemingly entirely formal functional relationships do in  fact apply to real systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We present in the following  new computational methodology that we use to demon-  strate the functional point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We will also demon-  strate that the sum rules (26) and (27) are highly valuable  in providing checks for numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  MACHINE LEARNING THE KINEMATIC  MAP  Machine learning proves itself to be an increasingly  useful tool in a variety of settings in soft matter, rang-  ing from soft matter characterization [82], engineering of  colloidal self-assembly [83], to the inverse design of soft  materials [84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Pivotal studies were addressed at colloidal  structure detection [85], the identification of combinato-  rial rules in mechanical metamaterials [86], the learning  of many-body interaction potentials for spherical [87] and  for anisotropic particles [88], and the prediction of the  dynamics of supercooled liquids from their static proper-  ties [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  More specifically, in the context of classical density  functional theory, an early and pioneering study formu-  lated a neural-network approach to liquid crystal order-  ing in confinement [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Free energy density function-  als were obtained for one-dimensional fluids from a con-  volutional neural network [91] and an analytical form  of an excess free energy functional was generated from  an equation learning network [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Cats et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' [93] re-  cently used machine learning to improve the standard  mean-field approximation of the excess Helmholtz free-  energy functional for a three-dimensional Lennard-Jones  (LJ) system at a supercritical temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' These signif-  icant reserach efforts were devoted to tailoring analytical  forms of model free energy functionals, by training cer-  tain key components such as spatial convolution kernels,  and much insight into the inner workings of excess free  energy functionals was gained [91–93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  However, here we proceed very differently and more-  over do so out-of-equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We use the LJ model and  the identical planar geometry as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' [93], such that  the density profile ρ(x) depends only on a single posi-  tion coordinate x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We consider steady states and retain  planar symmetry by considering flow that is directed in  the x-direction, such that the current J(x) = J(x)ex,  where J(x) is the magnitude of the current and ex is the  unit vector in the x-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Both the density profile  ρ(x) and the velocity field v(x) = J(x)/ρ(x) are indepen-  dent of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The continuity equation (5) then implies  0 = ∂ρ(x)/∂t = −∂[v(x)ρ(x)]/∂x, from which one ob-  tains by spatial integration ρ(x)v(x) = J0 = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Here  the value of J0 determines the intensity of the flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' we  recall the illustration shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We base the machine learning procedure on a convolu-  tional neural network, as was done e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' [91], and  following Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' [91–93] we use many-body computer sim-  ulations to provide training, validation, and test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  In contrast to these equilibrium studies though, in or-  der to address the nonequilibrium problem we need to  represent the physical time evolution on the many-body  trajectory level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We use the recently developed highly  performant adaptive BD algorithm [6] and apply it to  the three-dimensional LJ fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  As laid out above, in  order to address situations of planar symmetry we drive  the system only along the ex-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The specific form  of the driving force field fext(x)ex is however irrelevant,  as the training data only serves to extract the intrinsic  kinematic functional relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  In order to cover a sufficiently broad range of flow sit-  uations, we represent the external force field as a trun-  cated Fourier series fext(x) = �nmax  n=0 An cos(2πnx/L),  where L is the size of the cubic simulation box with pe-  riodic boundary conditions and An are random ampli-  tudes with zero mean and uniform distribution inside of  a given finite interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We truncate at order nmax = 5  such that the length scale L/(2πnmax) is comparable to  the LJ molecular size σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Ten percent of our simulation  runs are carried out in equilibrium, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' for A0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We  use N = 500 LJ particles inside of a cubic simulation  box of size L = 10σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The temporal duration of each run  is 1000τ, where τ = σ2/D0 is the Brownian time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  After initialization the system is randomized for 1τ at a  very high temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Then we wait for 100τ to allow  the system to reach a steady state and then collect data  during the remaining time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' In total we use 1000 such sim-  ulation runs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' these are subdivided for purposes of train-  ing (520), validation (280) and testing (200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  A more  detailed account will be given elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Our aim is to machine-learn and hence to explicitly  demonstrate the kinematic map, ρ(r), v(r) → fint(r) in  steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We present the learning algorithm with in-  puts ρ(x), v(x) and targets fint(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The data for these  three fields are from building steady state averages via  the adaptive BD over the corresponding one-body oper-  ators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We recall the microscopic definition of the inter-  particle one-body force density Fint(r) via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (7) and  we refer the reader to Appendix A of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' [51] for a de-  scription of several methods to sample the current in  BD and hence obtain the overdamped velocity profile  v(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Finally, we use the standard counting method for  the density profile ρ(r), although more efficient “force  sampling” methods [94–97] exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' At this stage we nei-  ther impose adiabatic-superadiabatic splitting (11), nor  structure-flow splitting (18), nor do we use any analyti-  cal model form of the functional relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We rather  work on the level of the bare one-body simulation data,  generated in the above described randomized uniaxial  flow situations of the desired planar symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We refer to the result of this procedure as the machine-  learned internal force field f ⋆  int(x, [ρ, v]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' This represents  a “surrogate model” in the sense of the terminology of the  machine learning community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' By construction this data  structure depends functionally on the density profile and  10  on the velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Importantly the external force  field fext(x), as given by the above described randomized  Fourier series, has not been used in the training, which  was rather based solely on the intrinsic force field and  its kinematic dependence on the density profile and the  velocity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  In order to test the validity of the functional relation-  ship and to address the question whether f ⋆  int(x, [ρ, v])  indeed represents the true fint(r, t, [ρ, v]) of power func-  tional theory restricted to the present planar and steady  situation, we consider a toy flow situation as an appli-  cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We choose the density profile to consist of a  single (co)sinusoidal deviation from the bulk, ρ(x) =  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='5 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='2 cos(2πx/L)]σ−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  In order for the system to  be in steady state, the velocity then necessarily needs to  satisfy v(x) = J0/ρ(x), where the strength of the current  J0 = const is a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We proceed in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  First, we check for self-  consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Therefore we solve the force density balance  relationship (6) for the external force field, which yields  the explicit result:  fext(x) = kBTρ′(x) + γv(x) − f ⋆  int(x, [ρ, v]),  (29)  where ρ′(x) = ∂ρ(x)/∂x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' As is explicit in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (29), in-  putting the toy state ρ(x), v(x) on the right hand side  yields a concrete machine learning prediction for the ex-  ternal force field on the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We then input  this result for fext(x) as the driving force field in a single  adaptive BD simulation run and expect this procedure to  reproduce the density and velocity profile of the toy state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The reproductive success will however materialize only  provided that i) the functional kinematic dependence ac-  tually exists and that ii) it is accurately represented by  the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The results, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' 2, demonstrate the accom-  plishment of the reconstruction of the toy state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  This  establishes that the machine learned functional provides  a numerically very highly accurate representation of the  true internal force functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We take this validation via  the machine learning to be a practical, data-science-level  verification of the existence of the power functional kine-  matic map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We recall the original formal construction  [42, 53] and its subsequent confirmation via custom flow  [51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Turning to the physics of the compressional flow, we  use the adiabatic-superadiabatic decomposition (11) to-  gether with the flow-structure splitting (18) to analyze  both the machine-learned functional f ⋆  int(x, [ρ, v]) as well  as the direct simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' As anticipated, both flow  and structural force fields have nontrivial spatial varia-  tion, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The flow force primarily contains viscous  effects that stem from the dissipation that the compres-  sional and extensional regions of the flow pattern gener-  ate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The structural force field becomes more strongly in-  homogeneous and also larger in magnitude upon increas-  ing the amplitude of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' This trend is necessary  to provide a balance for the increasingly asymmetric and  growing external force field, which in turn is required to  keep the density profile unchanged upon increasing the  throughput through the prescribed density wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The  power functional predictions (21) and (22) capture these  effects reasonably well given the simplicity of the ana-  lytical expressions, see the insets in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We find our  numerical results to satisfy the Noether sum rules (26)  and (27) to very good accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  It remains to point out the stark contrast with the  standard DDFT (1), which gives a trivial null result in  the present setup by construction: the density profile re-  mains unchanged upon increasing flow, and so does the  adiabatic force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' So the DDFT provides no mecha-  nism to account for the nonequilibrium physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  CONCLUSIONS  For the purpose of assessing the status of the DDFT  equation of motion (1) we have first described two exact  limits that this approximation reproduces: the dynamics  of the noninteracting diffusive ideal gas [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (4)] and  the spatially inhomogeneous static equilibrium limit [see  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' On general grounds one expects the DDFT to  perform well when the situation under consideration is  close to one of these limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' In particular near the static  case this is nontrivial, as the system might be dense and  spatially highly structured, as evident by a strongly inho-  mogeneous density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Provided that the dynamics  are driven weakly enough via a time-dependent external  potential then the DDFT can be a highly useful device,  which enables one to describe the temporal evolution as  a chain of equilibrium states, labelled by time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  In general the contributions beyond the equilibrium  physics will however be relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  On the level of the  formally exact one-body equation of motion (14), the su-  peradiabatic force field fsup(r, t) will then contribute and  potentially very significantly so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Together with the adi-  abatic force field, which follows from the equilibrium ex-  cess free energy functional via −∇δFexc[ρ]/δρ(r, t), their  sum constitutes the full interparticle forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' These are  coarse-grained, in a microscopically sharp way, to the  one-body level of dynamical correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We  have argued i) that power functional theory is a con-  crete formal structure that allows to obtain fsup(r, t)  from a generating functional and ii) that simple approx-  imate forms already capture much relevant nonequilib-  rium physics and they do so in a transparent and sys-  tematic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We have described and exemplified for uniaxial steady  compressional flow of the three-dimensional Lennard-  Jones fluid the kinematic functional map that governs  the exact nonequilibrium dynamics on the one-body level  of dynamic correlation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' As this description is  based on a single position coordinate and a single time  variable, it is of both conceptual and practical simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  As described by power functional theory the superadia-  batic interparticle force field functionally depends on the  density and the velocity field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' fsup(r, t, [ρ, v]), for over-  11  damped Brownian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The functional dependence is  causal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' on the values of the density profile and velocity  field at previous times, in general up to an initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The superadiabatic force field carries this kinematic de-  pendence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' on the history of ρ(r, t) and v(r, t), but  crucially it is independent of the external force field that  drives the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We have explicitly demonstrated the functional map  ρ(r, t), v(r, t) → fint(r, t) by establishing this functional  relationship via machine learning the intrinsic force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Using the force balance then gives direct access to the  form of the required external force field via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The  machine-learned model of the functional map hence en-  ables “instant custom flow” at negligible computational  cost at the time of use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We recall that the custom flow  method [51, 52] is based on the kinematic functional map,  such that from knowing the kinematic one-body fields,  the external force field that is necessary to generate the  given time evolution follows straightforwardly from the  exact force balance (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  An analytical approach to one-body functional maps  leads to the simple structure of velocity gradient forms  for the viscous and structural superadiabatic forces, as  exemplified in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' (16) and (17) for compressional flow,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' for velocity fields with nonvanishing divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' As  we have shown, the resulting predictions for the flow  force (21) and for the structural force field (22) represent  a reasonable description of the simulation data and its  representation via the machine-learned functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' We  attribute the remaining differences to higher-order terms  [59] which we have not addressed here for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' As  we have shown, our results from direct simulation, from  machine learning, and from the analytical approxima-  tions, satisfy exact global Noether sum rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We have restricted our discussion to a single and rela-  tively easily accessible type of nonequilibrium dynamics,  that of stationary uniaxial compressional flow that rep-  resents a model steady (batch) sedimentation situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The power functional approach allows to go much fur-  ther, including the treatment of viscoelasticity [56], as  arising from superadiabatic memory, deconfinement un-  der shear [57], the dynamic decay of the van Hove pair  correlation function as governed by drag, viscous and  structural forces [68, 69], and the complex forms of both  flow and structural forces that arise under spatially com-  plex forms of driving [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Time-dependent uniaxial flow  is relevant in a variety of situations, including colloidal  stratification [98, 99] and sedimentation [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Although power functional theory operates on the one-  body level of dynamical correlation functions, two-body  correlation functions are accessible both formally via the  nonequilibrium Ornstein-Zernike route [42] and explic-  itly by the dynamical test particle limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The latter is  the dynamic generalization of Percus’ static test parti-  cle limit [61], which identifies two-point correlation func-  tions, such as g(r) as also recently shown to be intimat-  edly related to thermal Noether invariance at second or-  der [50], with one-body density profiles in an external  potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' This is set equal to the interparticle pair po-  tential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  The dynamical test-particle limit goes further  in that it describes the test particle via its own dynami-  cal degrees of freedom, which are coupled to those of all  other particles in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The concept was originally  formulated as an approximation within DDFT [62, 63]  and formally exactly within power functional theory [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Two-body superadiabatic effects were shown via simula-  tion work to be significant [67–69] and they arise natu-  rally in an exact formulation of the test particle dynamics  [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The test particle limit allowed for a rationalization  of the dynamical pair structure as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' experimentally  observed in two-dimensional colloids [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Recently an ap-  proach to DDFT based on the two-body level was formu-  lated [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  In event-driven BD simulations superadiabatic forces  were shown to consist of drag, viscous, and structural  contributions [68, 69];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' [42] for an extended dis-  cussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' The physics of active particles [70–74] is very  significantly governed by a vigorous interplay between su-  peradiabatic and adiabatic forces, both of which are very  strong, as the tendency of these systems to self-compress  leads naturally to very high local densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  Furthermore,  relevant and interesting microscopic  models that go beyond the simple fluid paradigm of a  pair potential, such as the monatomic water model by  Molinero and Moore [102, 103] and the three-body gel  by Saw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' [104, 105], are accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' Despite the com-  plexity of both its defining Hamiltonian and the intricate  transient network structure, the inhomogeneous viscous  response of the three-body gel was recently demonstrated  [60] to be surprisingly well captured by a simple power  functional flow approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  We finally recall that  superadiabatic effects transcend overdamped dynamics,  and are relevant both in quantum dynamics [42, 80, 81]  and in classical molecular dynamics [42, 78, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  While we have restricted ourselves to discussing the  point of view of functional relationships,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' it would be in-  teresting to explore in future work possible cross connec-  tions to other theoretical approaches,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' such as Onsager’s  variational principle for soft matter [106–109],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' stochastic  thermodynamics [110],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' large deviation theory [111,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' 112],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  mode-coupling theory [113,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' 114],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' generalized hydrody-  namics [115],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content=' as well as to the physics of nonequilibrium  phase transitions [116] and of Brownian solitons [117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work is supported by the German Research Foun-  dation (DFG) via Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FLT4oBgHgl3EQfui_z/content/2301.12156v1.pdf'}
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diff --git a/9tFAT4oBgHgl3EQfpx3L/content/tmp_files/2301.08642v1.pdf.txt b/9tFAT4oBgHgl3EQfpx3L/content/tmp_files/2301.08642v1.pdf.txt
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index 0000000000000000000000000000000000000000..692735858130dada471c0cf70571fdcb9493888c
--- /dev/null
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@@ -0,0 +1,2672 @@
+Optimal multiple FSO transceiver configuration for using on High-altitude
+platforms
+Dieu Linh Truong ∗1 and The Ngoc Dang †2
+1School of Information and Communication Technology, Hanoi University of Science and Technology,
+Vietnam
+2Department of Wireless Communications, Posts and Telecommunication Institute of Technology,
+Vietnam
+January 23, 2023
+Abstract
+Free-space optical (FSO) communication requires light of
+sight (LoS) between the transmitter and the receiver.
+For
+long-distance communication, many research projects have
+been conducted towards using a network composed of high-
+altitude platforms (HAPs) flying at an elevation of 20 km to
+carry intermediate FSO transceivers that forward data be-
+tween ground stations.
+The clear environment at high el-
+evations prevents terrestrial obstacles from cutting the LoS
+between the transceivers.
+An FSO transceiver on a HAP
+can communicate with ground stations within a small area
+owing to its limited beam size. We suggest using multiple
+FSO transceivers on a HAP to extend its ground coverage.
+However, the use of too many FSO transceivers may quickly
+exhaust the onboard energy of the HAP. As a result, HAP
+must be lowered to recharge frequently.
+In this study, we first propose a configuration of multiple
+FSO transceivers to widen the ground coverage of a HAP.
+We then propose a set of closed-form expressions to calculate
+the extended coverage. Finally, to implement a HAP network
+using multiple FSO transceivers, we seek the optimal config-
+uration of multiple FSO transceivers that minimizes the to-
+tal cost of the HAP network, including amortization, energy,
+and maintenance costs. The simulation results show that the
+proposed multiple FSO transceiver configuration clearly in-
+creases the ground coverage of a HAP and significantly re-
+duces the cost of the HAP network.
+Keywords— Free Space Optics, High-altitude platform, Beam
+size optimization, HAP based FSO network
+1
+Introduction
+Free-space optical (FSO) communication uses light propagation
+in free space to transmit data. In recent years, this technology
+has emerged as a promising choice for short-distance high-speed
+communication between endpoints with a clear light of sight (LoS).
+∗linhtd@soict.hust.edu.vn
+†ngocdt@ptit.edu.vn
+Commercial FSO transmitters available in the market at prices of
+thousands of dollars can operate at 1.25 − 10 Gbps over 1 − 2
+kilometers, for example, the SONABeam series of fSona [1].
+To reach a long distance, a multi-hop FSO system can be used,
+where data are transmitted through intermediate FSO transceivers
+[2], [3]. To avoid obstacles that cut the LoS between terrestrial
+FSO transceivers, researchers from academia and industry have
+proposed placing intermediate FSO transceivers of the multi-hop
+FSO system on high-altitude platforms (HAPs).
+High-altitude
+platforms are flying objects that operate at altitudes of 17–24 km
+in the stratosphere. Several HAP models have been proposed and
+piloted previously. Some projects continue until recently, such as
+the Loon Project of Google [4], the UAV project of Facebook [5],
+and the Stratobus project of Thales Alenia Space [6].
+A multi-hop FSO system using a HAP network is described
+in [7] and illustrated in Figure 1. According to this model, FSO
+transceivers on the ground (so-called ground FSO nodes) are re-
+grouped into clusters to become the serving zones of HAPs.
+A
+HAP has an FSO transceiver looking down to exchange data
+with the ground FSO nodes of the cluster under it.
+This FSO
+transceiver is called serving FSO transceiver. A HAP also carries
+several FSO transceivers pointing towards other HAPs for inter-
+HAP communication. These FSO transceivers are known as inter-
+HAP FSO transceivers.
+Although the ITU recommends a HAP footprint width of ap-
+proximately 500 km in radius, experimental projects show much
+smaller coverage areas [8].
+Nevertheless, a network of multiple
+HAPs can cover a country entirely. For example, a constellation
+of 16 HAPs with multiple radio frequency antennas was considered
+to cover Japan [9].
+An end-to-end data-switching scheme for a multi-hop FSO sys-
+tem using HAP was proposed in [7].
+Since the communication
+between a HAP and the ground is point-to-multipoint, the serv-
+ing FSO transceiver on the HAP controls multiple accesses from
+ground FSO nodes under it using the WDM technique.
+Each
+ground node is assigned a separate wavelength for up and down
+communication. An IP router on the HAP aggregates IP packets
+heading toward a common cluster within a single flow. The flow
+will be carried by one or more continuous lightpaths between the
+source and destination HAPs. The number of lightpaths is deter-
+mined according to the size of the flow and the transport capacity
+of a wavelength. A WDM switch is installed on each HAP to route
+1
+arXiv:2301.08642v1  [cs.NI]  20 Jan 2023
+
+these lightpaths over the HAP network on a wavelength-switched
+basis. In Figure 1, the blue path HAP1-HAP2-HAP4-HAP5 and
+the red path HAP1-HAP2-HAP3 are two flows.
+1
+2
+2
+1
+1
+1
+1
+2
+3
+3
+3
+3
+3
+WDM switch
+IP router 
+IP router
+WDM switch
+1
+2
+2
+p-HAP-2
+1
+3
+3
+p-HAP-1
+toward HAP-2 of cluster-2
+toward HAP-1 of cluster-1 
+ HAP2
+Inter-HAP FSO transceiver
+Ground FSO node
+Serving FSO
+3
+1
+2
+A cluster
+A cluster
+transceiver
+ HAP1
+ HAP3
+ HAP4
+ HAP5
+Serving zone of HAP 1
+Serving zone of HAP 2
+inter-HAP link
+inter-HAP link
+Figure 1: Multi-hop FSO communication system using HAP.
+In terrestrial FSO communications, the light beams are usually
+set to be very narrow for low transmission energies. However, for
+HAP and ground communication, the serving FSO transceiver of
+the HAP must project a sufficiently wide laser beam for covering
+distributed ground FSO nodes.
+A single serving FSO transceiver has a relatively small foot-
+print owing to the low capacity of the current laser source, and
+the limited sensibility and aperture sizes of ground receivers. The
+calculation in Section 2.1 shows that with a laser source of 1 Watt,
+required received power at receivers of -41.1dBm, and receiver
+aperture radius of 2 m, a single serving FSO transceiver at an
+elevation of 20 km can cover a ground area of 6.691 km radius
+only (see Table 3).
+To extend the coverage of a HAP, we propose using multiple
+serving FSO transceivers arranged in a bundle, as shown in Fig-
+ure 2. Each serving FSO transceiver points in a slightly different
+direction to cover a particular ground area that overlaps other ar-
+eas to create a continuous coverage region. Given a ground region
+to be served, using HAPs with multiple serving FSO transceivers
+reduces the number of required HAPs compared to using HAPs
+with a single serving FSO transceiver. However, the expenditure
+for serving FSO transceivers increases. Therefore, the number of
+serving FSO transceivers to be used on a HAP should be carefully
+considered.
+Regarding the communication between ground nodes and a
+HAP, the multiple serving FSO transceiver model still uses the
+WDM technique, where each ground node is assigned a unique
+wavelength within its cluster to communicate with its HAP. The
+number of ground nodes to be served by a HAP is restricted by
+the number of wavelengths offered by the WDM technique.
+In this study, we focus on identifying the optimal configuration
+of multiple serving FSO transceivers to achieve a minimal-cost
+HAP network for serving a set of ground FSO nodes. The optimal
+configuration should define the number of serving FSO transceivers
+to be set up on a HAP and the beam width for each transceiver.
+The cost of the HAP network includes the investment, energy, and
+maintenance costs.
+Compared with the previous study in reference [7], the current
+research differs in two aspects.
+First, the current research pro-
+Figure 2: A HAP with multiple serving FSO transceivers and
+its footprint.
+poses the use of multiple serving FSO transceivers on each HAP
+instead of a single serving FSO transceiver, as in [7]. Second, the
+current research identifies the optimal beam widths for serving
+FSO transceivers, whereas in [7], the beam widths are predefined.
+The current study also differs from that in [10], where beam size
+was optimized for an inter-HAP link, which is a point-to-point link.
+The remainder of this paper is organized as follows. First, we
+analyze the single and multiple serving FSO transceivers configu-
+rations in Section 2 to determine their ground coverage sizes and
+constraints on transmitter beams. In Section 3, we state the prob-
+lem of designing a minimal-cost HAP-based FSO network, which is
+the target of the optimization of multiple serving FSO transceiver
+configuration. Then, in Section 4, we define a HAP energy con-
+sumption formula and show that solar energy is necessary for keep-
+ing the HAP working in space for a long period. We also present
+a constraint that a HAP must respect to relying uniquely on so-
+lar energy. In Section 5, we present the algorithms for identifying
+the optimal multiple serving FSO transceiver configuration and
+its footprint radius. Section 6 presents the process designing the
+minimal cost HAP-based FSO network using the optimal multi-
+ple serving FSO transceiver configuration. Section 7 presents the
+simulation results. Finally, Section 8 concludes the paper.
+2
+Serving FSO transceiver configu-
+rations
+2.1
+Single serving FSO transceiver configu-
+ration
+In this section, the allowable beam width and ground coverage of
+a single serving FSO transceiver are determined. The beam size
+is restricted to ensure that the received power at a ground point
+within the beam footprint is detectable by receivers.
+2
+
+Figure 3: Surface of the part of sphere blocked by solid angle
+α is calculated as the sum of the surface of all ribbons around
+the sphere when the solid angle varies from α to 0.
+Assume that the transmitter source radiates within a solid angle
+α and that the radiation density is uniform in all directions within
+the solid angle at a distance r from the source. The radiation den-
+sity at distance r is inversely proportional to the surface of the part
+of the sphere radius r blocked by the solid angle α. To calculate
+this surface, we divide the sphere into thin ribbons corresponding
+to open angles of d(α/2). The width of a ribbon is rd(α/2), as
+shown in Figure 3. The radius of the ribbon at zenith angle α/2 is
+r sin(α/2). Thus, the ribbon surface is 2πr sin(α/2)rd(α/2). The
+surface of the part of the sphere blocked by the solid angle α is the
+sum of the surfaces of all ribbons when zenith angle varies from α
+to 0, as follows:
+� 0
+α
+2πr sin (α
+2 )rd(α
+2 ) = 2πr2(1 − cos (α
+2 ))
+Let Ur be the radiation density at distance r and Ptx be the
+transmitted power at the source. We deduce:
+Ur =
+Ptx
+2πr2(1 − cos (α/2))
+(1)
+Let P rx
+j
+be the received power at ground FSO node j.
+The
+received power is proportional to the radiation density and the
+received aperture of the ground node. It is:
+P rx
+j
+=
+e−σLjULjAR
+(2)
+where
+• Lj is the distance between ground FSO node j and its serving
+HAP Hi (see Figure 4),
+• σ is the attenuation coefficient of the links between the HAP
+and ground,
+• ULj is radiation density at distance Lj from the source,
+• AR is the aperture area of the receiver. Let Rrx be the receiver
+aperture radius, then, AR = πR2
+rx.
+In (2), the first term represents the attenuation of laser power
+through the atmosphere, which is described by the exponential
+Beer–Lambert Law [11].
+Figure 4: Received power on border nodes of a coverage area
+is the smallest amongst all nodes in the area.
+By substituting ULj from (1) into (2), we obtain the received
+power at node j as follows:
+P rx
+j
+= e−σLj × Ptx × R2
+rx
+2L2
+j
+×
+1
+1 − cos (α/2)
+(3)
+The power received at node j must not be less than the required
+level of the receiver, denoted by ρrx. It is obvious that point j at
+the border of the ground coverage area receives the least power
+because it is the furthest from the source (see Figure 4). Hence,
+all points in the coverage areas of HAP Hi receive sufficient power
+if and only if the border points receive at least the required power;
+that is,
+P rx
+j
+= e−σH/ cos ( α
+2 ) PtxR2
+rx cos2 ( α
+2 )
+2H2(1 − cos ( α
+2 )) ≥ ρrx
+(4)
+where Lj is substituted by H/ cos( α
+2 ) for border node j.
+Solving inequation (4) yields the beam width of the single serv-
+ing FSO transceiver configuration. Corresponding to beam width
+α, the ground coverage radius of the configuration is:
+Ri = H tan(α
+2 )
+(5)
+Lemma 1. Function P rx
+j
+decreases with α ∈ [0..π].
+Proof of Lemma 1 is given in Appendix A.
+Figure 5 shows the received power at the border of the cover-
+age area with different receiver aperture radius Rrx. This figure
+confirms that P rx
+j
+decrease with an increase in α.
+Let αmax be the value for α that makes P rx
+j (αmax) = ρrx; then
+according to Lemma 1,
+P rx
+j (α) ≥ P rx
+j (αmax) = ρrx, ∀α ∈ [0..αmax]
+thus all α ∈ [0..αmax] satisfy constraint (4).
+Calculations using the parameters given in Table 1 show that
+when Rrx = 2 m, αmax = 37° and the coverage radius is 6.691 km.
+When Rrx = 4 m, αmax
+= 67° and the coverage radius is
+13.237 km.
+3
+
+Ribbon surface= 2πr sin(α/2) r d(α/2)
+Kd(a/2)
+r.sin(a/2)
+a/2
+SourceHAP H;
+α
+H
+Received power Prx
+R;
+Nodej
+coverage area of HAP H 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 140
+ 160
+ 180
+Received power at coverage border (10-8 W)
+Beam size α(degree)
+Rrx=0.125m
+Rrx=1m
+Rrx=2m
+Rrx=4m
+Required at receiver (Prx)
+Figure 5: Received power at the coverage border of the single
+serving FSO transceiver configuration with different receiver
+apertures.
+2.2
+Multiple serving FSO transceiver config-
+uration
+The ground coverage of a HAP can be widened by combining sev-
+eral serving FSO transceivers. Different combinations are possible.
+In this research, we study a straightforward configuration in which
+a principal serving FSO transceiver is in the center projecting
+light perpendicular to the ground, and several identical supplemen-
+tary serving FSO transceivers are set evenly around the principal
+one (Figure 6). Each supplementary transceiver projects slanted
+beams to extend the coverage in one direction. This arrangement
+is referred to as mFSO configuration. Usually, the transmitters in
+a bundle are considered to project signals in parallel. However,
+because of the large principal beam, the supplementary serving
+FSO transceiver projection directions are far from being perpen-
+dicular to the ground, and their footprints are ellipses instead of
+circles.
+To create a continuous coverage region, the footprint of the
+principal serving FSO transceiver and those of the supplemen-
+tary serving FSO transceivers should overlap.
+Therefore, there
+should be a sufficiently large number of supplementary serving
+FSO transceivers to cover entirely the contour of the principal foot-
+print. The extended coverage area is defined as the largest circle
+covered by these footprints (Figure 6). The principal transceiver is
+responsible for the region defined by its footprint. A supplemen-
+tary serving FSO transceiver is responsible for the part limited
+by its footprint, principal coverage circle, and extended coverage
+circle.
+Let α be always the beam width of the principal serving
+FSO transceiver.
+To ensure that ground nodes under principal
+coverage receive sufficient power, α should still respect constraint
+(4), as in the single serving FSO transceiver configuration.
+Let the beam width of a supplementary serving FSO transceiver
+be β. In the responsible area of the supplementary transceiver, the
+points on the extended coverage circle are the farthest from the
+supplementary transceiver; thus, they receive the least power. If
+these points receive at least ρrx, all other points receive sufficient
+power.
+It is easy to note that the footprints of the neighboring supple-
+Figure 6: footprint of multiple FSO transceiver (mFSO) con-
+figuration.
+mentary serving FSO transceivers join each other on the extended
+coverage circle. Let J be such a joint point, the power J receives
+from the supplementary FSO transceiver is defined similar to (3)
+but with beam width β, which is
+P rx
+J
+= e−σ×LJ × Ptx × R2
+rx
+4L2
+J
+×
+2
+1 − cos (β/2)
+(6)
+Thus, β is constrained by the condition P rx
+J
+≥ ρrx, which gives:
+e−σ×LJ
+PtxR2
+rx
+2L2
+J(1 − cos (β/2)) ≥ ρrx
+(7)
+Let us denote the extended coverage radius by Rext then
+LJ =
+�
+H2 + R2
+ext
+(8)
+Appendix B presents detailed calculations of LJ and Rext. The
+calculations yielded the following results
+Rext = H2 tan( ξ+α
+2 ) − tan( α
+2 )(1 − tan2( ξ+α
+2 ))
+1 − tan2( ξ+α
+2 ) + 2 tan( ξ+α
+2 ). tan( α
+2 )
+(9)
+where
+tan(ξ + α
+2
+) = tan(γ) + tan(θ)
+1 − tan(γ). tan(θ). cos( π
+m)
+tan(γ) = tan(α
+2 ). cos( π
+m)
+tan(θ) =
+�
+sin2( β
+2 ) − sin2( α
+2 ). sin2( π
+m)
+cos( β
+2 )
+(10)
+(11)
+(12)
+and m is the number of supplementary FSO transceivers set
+around the principal one.
+We can remark that Rext and thus LJ depend on α, β and m.
+Hereafter, Rext is sometimes denoted by Rext(α, m, β) and LJ by
+LJ(α, m, β) to express these dependencies.
+4
+
+Principle coverage circle
+0
+K'
+K
+2T
+m
+Extended coverage circle3
+Problem of designing minimal cost
+HAP network
+There are several costs in a HAP network, such as investment, en-
+ergy, and maintenance costs. Based on the expected life duration
+and maintenance cycle of a HAP, these costs can be distributed
+by day as 1) daily amortization cost representing investment cost,
+2) average daily maintenance cost, and 3) daily energy cost. Con-
+sequently, the problem of minimizing network cost becomes min-
+imizing the daily network cost, which comprises these three com-
+ponents.
+Following variables are introduced for formulating mathemati-
+cally the daily network cost:
+• K: Number of HAPs in the network. The HAPs are indexed
+by i ∈ 1..K.
+• niF
+i : Number of FSO transceivers used on HAPi for inter-
+HAP communications.
+• nsF
+i : Number of serving FSO transceiver of HAPi.
+Let ζday
+H
+and ζday
+F
+be constants that express the daily amortiza-
+tion costs of a HAP and an FSO transceiver, respectively. These
+costs are defined as the ratio of the prices of the HAP or FSO
+transceiver to their expected lifetime duration. Then, the overall
+daily amortization cost of the HAP network is:
+Kζday
+H
++ (
+K
+�
+i=1
+nsF
+i
++
+K
+�
+i=1
+niF
+i )ζday
+F
+(13)
+To evaluate the daily maintenance and energy costs, we need to
+consider the HAP design. HAPs are classified into two categories
+based on the underlying physical principle that provides the lifting
+force for the HAPs: aerodynamic (the HAP is heavier than air)
+and aerostatic (the HAP is lighter than air).
+While aerostatic
+platforms use buoyancy to float in the air, aerodynamic platforms
+use dynamic forces created by movement through the air [8]. In
+general, both aerostatic and aerodynamic systems require a “flying
+energy” to keep the HAP relatively stable for maintaining FSO
+communication between HAPs and that between HAPs and FSO
+ground nodes. An aerodynamic system requires a large propulsion
+power to move. Aerostatic systems typically consume less energy
+than aerodynamic systems do. To be able to operate for a long
+duration in space, HAPs are mainly unmanned.
+HAPs are equipped with different energy resources such as on-
+site production (e.g., solar energy harvested by solar panels) or
+rechargeable energy (e.g., batteries or fuel cells brought from the
+ground).
+Solar energy-based HAPs can operate continuously in
+space until they are lowered for maintenance purpose. Recharge-
+able energy-based HAPs are lowered once the reserved energy is
+depleted. In brief, the continuous in-space working duration of a
+HAP is limited by its available energy, which is relatively fixed by
+the HAP design, its energy consumption level, which varies de-
+pending on the payload weight and communication of the HAP,
+and its maintenance cycle.
+We define the maintenance cost of a HAP as the expense of low-
+ering the HAP to perform technical maintenance, energy recharge
+on the ground, and then reinstall it in space.
+Let di be the number of days on which HAPi can operate con-
+tinuously in space. Let ζmtn be constant expressing the cost of
+one time lowering a HAP, maintaining it, recharging it, and then
+reinstalling it in space. The daily maintenance cost of the HAP
+network is
+K
+�
+i=1
+ζmtn
+di
+(14)
+Regarding the daily energy cost, we consider solar energy to be
+free, whereas the solar panel cost is counted in the cost of the HAP.
+The cost of rechargeable energy is part the maintenance cost. As a
+result, the energy cost does not explicitly represent the total cost.
+Nonetheless, the energy consumption level of a HAP affects its
+in-space working duration di; therefore, we analyze this in Section
+4.
+Combining (13) and (14), we obtain the following overall daily
+cost of the HAP network:
+Cost = Kζday
+H
++ (
+K
+�
+i=1
+nsF
+i
++
+K
+�
+i=1
+niF
+i )ζday
+F
++
+K
+�
+i=1
+ζmtn
+di
+(15)
+The problem of minimizing daily cost of the HAP network is
+stated as follows.
+• Given input parameters including
+– NFSO: Set of ground FSO nodes and their coordinates.
+The number of nodes in the set is denoted as |NFSO|,
+– M:
+Data traffic to be carried between ground FSO
+nodes. This is the list of bandwidth demands between
+the ground nodes.
+• Outputs to seek are
+– A HAP network with HAP locations and inter-HAP
+links,
+– Beam width to set to each serving FSO transceiver.
+• Optimization objective is
+– Minimizing the daily cost expressed in (15) of the HAP
+network.
+The following two remarks drive us to conduct further analyses
+in subsequent sections.
+First, if a HAP has self-sufficient solar
+energy, its in-space working duration di is not limited by its energy
+consumption but depends uniquely on the maintenance cycle of the
+HAP, which is usually constant. In Section 4, we show the daily
+energy consumption of a HAP and the constraint that a HAP
+needs to respect to rely solely on solar energy.
+Second, the cost of the HAP network increases with an increase
+in the number of FSO transceivers and HAPs.
+The number of
+HAPs can be reduced by increasing ground coverage. To increase
+ground coverage, more serving FSO transceivers can be used on
+each HAP, but this introduces greater energy consumption and ex-
+tra amortization cost. Section 5 focuses on identifying the optimal
+configuration for serving FSO transceivers on a HAP to achieve a
+minimal HAP network cost.
+4
+Daily energy consumption of a
+HAP with payload
+Several parameters affect the power consumption of a HAP. The
+descriptions and notations of these parameters are listed in section
+Energy parameters of Table 1. Most parameters were set based on
+industrial experimental projects such as the Loon project [4], Stra-
+tobus project [6], and other studies listed in the reference column.
+Section 7.1 presents the choice of parameter values in detail.
+5
+
+Param.
+nota-
+tions
+Descriptions
+Values
+References
+Cost related parameters
+ζday
+H
+Daily amortization cost of a HAP.
+100
+ζday
+F
+Daily amortization cost of an FSO transceiver on HAP.
+10
+ζmtn
+Cost of one-time maintenance of a HAP including lowing it down,
+1000
+maintenance, charging and reinstall it in the stratosphere.
+Dm
+Maintenance cycle.
+365 days
+[6]
+Energy parameters
+Esolar
+Minimum daily harvested solar energy by a HAP.
+42 - 290 kWh
+[12]
+ρavion
+Power consumed by the avionic part of a HAP to carry an unit of mass.
+2 W/kg
+ρHCM
+F
+Power for heating, cooling, and management for each FSO on HAP.
+20 W
+[4]
+ρPAT
+Power consumed by a PAT system.
+15W
+[13]
+ρinter
+F
+Power consumed by inter-HAP FSO transceivers for laser source (0.1 W),
+heating/cooling/management (20 W) and PAT (15 W).
+35.1 W
+[4]
+Inter-HAP FSO link parameters
+C2
+n
+Atmosphere structure parameter.
+5.0 × 10−18m−2/3
+-
+Attenuation coefficient.
+3.5 × 10−6m−1
+[4]
+-
+Coupling loss.
+45 dBm
+-
+Transmitted power of an inter-HAP FSO transceiver.
+0.1 W
+[4]
+-
+Receiver aperture diameter of an inter-HAP FSO transceiver.
+0.037 m
+[4]
+-
+Beam width of an inter-HAP FSO transmitter.
+280 µrad
+[4]
+HAP-ground link parameters and variables
+σ
+Attenuation coefficient.
+3.5 × 10−6m−1
+ρFSO
+tx
+Transmitted power of the laser source of a serving FSO transceiver.
+1 Watt
+Rrx
+Receiver aperture radius of a ground FSO transceiver.
+0.05 m
+SONABeam [1]
+ρrx
+Required received power at a ground FSO transceiver.
+7.76.10−8 W
+-41.1 dBm in [4]
+Other parameters
+H
+Elevation of HAPs.
+20 km
+LHH
+Maximum length of an inter-HAP link so that its BER is under δ.
+88 km
+δ
+BER threshold for inter-HAP links and lightpaths between HAPs.
+W
+The number of wavelengths in WDM technique.
+40; 80
+µH
+Platform mass excluding FSO transceivers.
+28.5 kg; 500 kg
+[4]
+µF
+FSO transceiver mass.
+6.3 kg
+[4]
+Table 1: Parameters. Greek characters are used for denoting constant parameters.
+Let us consider the power consumption of a single HAP Hi
+that has m serving FSO transceiver and niF
+i
+inter-HAP FSO
+transceivers. The power consumption includes:
+• P avion
+Hi
+: Power draw of avionic part for maintaining Hi with
+payload in space.
+• P down
+Hi
+: Power draw of all serving FSO transceivers on HAP
+Hi.
+This power includes the heating/cooling/management
+power, laser transmitted power of all serving FSO transceivers
+on the HAP, and the power consumed by the Pointing Acqui-
+sition and Tracking (PAT) system of the HAP.
+• P inter
+Hi
+: Power draw of all inter-HAP FSO transceivers on
+HAP Hi for inter-HAP communication. The power includes
+the heating/cooling/management, and PAT power for each
+inter-HAP FSO transceiver.
+Inter-HAP FSO transceivers
+are oriented towards different remote HAPs; therefore, each
+transceiver must have a PAT system.
+The total daily energy consumption (by 24 hours) of Hi is
+Econsum = (P avion
+Hi
++ P down
+Hi
++ P inter
+Hi
+) × 24
+(16)
+To breakdown further P avion
+Hi
+, P down
+Hi
+, and P inter
+Hi
+, we introduce
+following parameters:
+• ρavion: Power consumed by the avionic part of the HAP to
+carry a unit of mass.
+• ρFSO
+tx
+: Transmitted power of each serving FSO transceiver.
+Because the current power of laser source is limited to 1 W,
+which is very small in comparison with the power consumed
+by other factors on the HAP, we consider that ρFSO
+tx
+= 1 W,
+regardless of the beam width of the serving FSO transceiver.
+• ρHCM
+F
+: Power draw for heating, cooling, and management. It
+is also considered constant for each serving FSO transceiver
+and is set to ρHCM
+F
+= 20 W, according to reference [4].
+• ρPAT : Power draw for Pointing, Acquisition and Tracking
+activity; it is another constant and is set to ρPAT = 15 W
+[13]. A HAP system uses a single PAT for its set of serving
+FSO transceivers.
+• ρinter
+F
+: Power draw of a single inter-HAP FSO transceiver
+including communication, heating, cooling, management, and
+6
+
+PAT. According to [4], 0.1 W laser power is sufficient for an
+inter-HAP communication of 100 km distance. In this study,
+we limited the inter-HAP link length to less than 100 km and
+considered the laser power constantly 0.1 W regardless of the
+distance. Therefore, ρinter
+F
+= ρHCM
+F
++ ρPAT + 0.1.
+• µH: Mass of the HAP.
+• µF: Mass of an FSO on the HAP.
+Assuming that P avion
+Hi
+is linearly proportional to the weight of
+the HAP by ρavion,
+P avion
+Hi
+= [µH + (nsF
+i
++ niF
+i )µF]ρavion
+(17)
+P down
+Hi
+is
+the
+sum
+of
+the
+power
+consumed
+by
+serving
+FSO transceivers and PAT activity of the HAP; thus,
+P down
+Hi
+= nsF
+i (ρFSO
+tx
++ ρHCM
+F
+) + ρPAT
+(18)
+P inter
+Hi
+is the sum of the power consumed by inter-HAP FSO
+transceivers; thus,
+P inter
+Hi
+= ρinter
+F
+.niF
+i
+(19)
+Substituting (17), (18), and (19) into (16), we obtain the daily
+power consumption of a HAP as
+Econsum = {[µH + (nsF
+i
++ niF
+i )µF]ρavion
++ nsF
+i (ρFSO
+tx
++ ρHCM
+F
+) + ρPAT
++ ρinter
+F
+niF
+i } × 24
+(20)
+4.1
+Necessity of solar energy and utilization
+constraint
+Current HAPs mainly use energy from solar panels mounted on
+HAP wings and/or energy from batteries or hydrogen fuel cells
+(HFC) onboard. Solar energy can be harvested and charged into
+batteries during the day for nighttime use. Harvested solar energy
+varies with year time and location. According to the experiments
+in [12], in York, UK, the harvested solar power is 42–80 kWh/day,
+and in Enugu, Nigeria, it is 290–545 kWh/day, depending on the
+size of the solar panel.
+Figure
+7
+depicts
+the
+total
+daily
+energy
+consumption
+of
+a HAP, calculated from (20), versus the number of serving
+FSO transceivers.
+Parameters were ρavion = 2 W/kg, ρPAT =
+15 W, HAP weights µH = 28.5 kg or 500 kg. The HAP carried
+10 inter-HAP FSO transceivers.
+The referenced daily solar en-
+ergy levels were the minimum daily solar energy levels in York
+and Enugu. From a certain number of serving FSO transceivers,
+a HAP consumes more energy than the harvested solar energy
+in York, and an HFC would be necessary. Owing to the limited
+payload capacity of a HAP, its HFC capacity is also very limited.
+According to [8], the current state-of-the-art fuel-cell density is
+approximately 1600 Wh/kg. A lightweight HAP, such as a Google
+balloon weights 28.5 kg, cannot carry heavy long-lasting fuel cells
+on board. The larger HAP Stratobus can carry up to 450 kg, but
+it weights already 7 tons leading to high energy consumption for
+flying. Even if the Stratobus payload capacity is reserved for the
+HFC, its energy would quickly run out within a few days.
+Based on this observation, we believe that long-duration flights
+should consider solar energy as the principal energy source. In this
+case, the power consumption of a HAP with payload must not
+ 0
+ 50000
+ 100000
+ 150000
+ 200000
+ 250000
+ 300000
+ 350000
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+Total daily energy consumption (W-hr)
+Number of serving FSO transceivers
+µΗ=28.5 kg
+µΗ=500 kg
+Min solar energy at York
+Min solar energy at Enugu
+Figure 7: Energy consumption by a HAP with different num-
+ber of serving FSO transceivers in comparison with the min-
+imum harvested solar energy at York and Enugu. ρavion =
+2/kg W and ρPAT = 15 W.
+exceed the daily harvested solar energy. Let the daily harvested
+solar energy be Esolar; then,
+�
+[µH + (nsF
+i
++ niF
+i )µF]ρavion + ρPAT
++ nsF
+i (ρFSO
+tx
++ ρHCM
+F
+) + ρinter
+F
+niF
+i
+�
+≤ Esolar
+24
+(21)
+According to Figure 7, solar energy provision does not need to
+be very large. A solar energy level between the minimum harvested
+in York and Enugu allows a 500 kg HAP to carry at least 6 serving
+FSO transceivers. A HAP can carry hundreds FSO transceivers
+with more than 125 kWh solar energy. Therefore, it is realistic to
+rely on the solar energy. Hereafter, we consider that HAPs solely
+use solar energy.
+Despite self-sufficient solar energy, HAPs still need to be lowered
+periodically for maintenance, for example, after one year in the
+case of Stratobus [6]. Let us denote the maintenance cycle as a
+constant Dm. Then
+di = Dm,
+∀i ∈ 1..K
+(22)
+5
+Optimal mFSO configuration
+Using multiple serving FSO transceivers increases the expense of
+FSO transceivers, although it can reduce the expense of HAPs.
+This section aims to determine the mFSO configuration that min-
+imizes the HAP network cost defined in (15). We assume that all
+HAPs use identical mFSO configurations, that is, identical prin-
+cipal beam width α, supplementary beam width β and number of
+supplementary serving FSO transceivers m.
+Let us now consider the dependence of the HAP network cost on
+mFSO configuration. As each HAP has m supplementary serving
+FSO transceivers and uses only solar energy, the cost (15) becomes
+Cost = Kζday
+H
++ (Km +
+K
+�
+i=1
+niF
+i )ζday
+F
++ Kζmtn
+Dm
+Cost is a function of K, m and niF
+i . K depends on the coverage
+radius Rext(α, m, β) of the mFSO configuration. niF
+i , as the num-
+ber of inter-HAP links of HAP i, depends on the traffic demand
+7
+
+set M. Hence, Cost depends on mFSO configuration and M. It
+is difficult to determine the optimal mFSO configuration without
+considering M. To relax the dependance on M, we estimate Cost
+by a function that depends solely on mFSO configuration, that is,
+tuple (α, m, β); then try to find an instance (α, m, β) minimizing
+the estimated cost in expecting that the instance also drives the
+real cost to a minimum.
+5.1
+Cost estimation
+Figure 8: A ground area is divided into grid of square cells;
+each cell is circumscribed by a circle representing a serving
+zone of a HAP.
+First, we estimate the number of HAPs K.
+Samples of the
+estimation are datasets with uniformly distributed ground nodes.
+Let S be the surface of the ground area containing those nodes,
+and W the number of wavelengths in the WDM technique. We
+divide the ground zone S into a grid of square cells of size ℓ × ℓ,
+each one will be covered by a HAP (see Figure 8). To be served
+by a HAP, a cell must satisfy the following two conditions:
+1. A cell can contain at most W ground nodes because a HAP
+can use at most W wavelengths to serve ground nodes. Owing
+to the uniform distribution of ground nodes, we have
+ℓ2
+S |NFSO| ≤ W
+2. A cell must be contained inside by a circle radius equivalent
+to the extended radius Rext of a HAP
+ℓ ≤
+√
+2Rext
+The maximum number of HAPs required to cover region S is the
+number of cells. Let this number be ˆK, then,
+ˆK = S
+ℓ2 = ⌈max {|NFSO|
+W
+,
+S
+2R2
+ext
+}⌉
+(23)
+Hence, ˆK is an overestimation of the number of HAPs.
+Next, we estimate the value of niF
+i .
+Let V be the maximum
+number of inter-HAP links that a HAP may have. Then
+niF
+i
+≤ V, ∀i.
+Finally, Cost can be overestimated as:
+�
+Cost = ˆK
+�
+ζday
+H
++ (m + V + 1)ζday
+F
++ ζmtn
+Dm
+�
+(24)
+�
+Cost is a function of Rext(α, m, β) and m while V is a parameter
+of the estimator. The estimation is more precise when V is set close
+to the actual number of inter-HAP links of a HAP, and coarser
+otherwise.
+5.2
+Algorithms finding optimal configura-
+tion
+Given α and m, a larger β results in a larger Rext, and thus a
+smaller ˆK and �
+Cost. Therefore, β should be set to the largest value
+according to (7) for a given α and m. It is worth noting that the
+value of β does not affect the solar energy consumption because the
+laser power ρFSO
+tx
+is small and is considered constant. Determining
+the optimal configuration becomes finding the optimal values of α
+and m.
+Algorithm 1 Find the optimal mFSO configuration
+1: function Find-optimal-mFSO
+2:
+niF
+i ← V
+3:
+cMin ← ∞
+▷ cost min
+4:
+αMax ← maximum α by (4)
+5:
+for α = αMax . . . 0 do
+6:
+mMax ← calculated by (25)
+▷ max m
+7:
+mOpt ← 0
+▷ optimal m
+8:
+for m = 0 . . . mMax do
+9:
+β ← Beta-max(α, m)
+▷ max β
+10:
+Calculate Rext(α, m, β) using (9),(10),(11) (12)
+11:
+Calculate �
+Cost(α, m, β) using (24)
+12:
+if �
+Cost < cmin then
+13:
+cmin ← �
+Cost
+14:
+αOpt ← α
+▷ optimal α
+15:
+mOpt ← m
+▷ optimal m
+16:
+βOpt ← β
+▷ optimal β
+17:
+end if
+18:
+end for
+19:
+end for
+20:
+return αOpt, mOpt, βOpt
+21: end function
+Algorithm 2 Find the maximum β given α, m
+1: function Beta-max(α, m)
+2:
+for β = 0 . . . 180 do
+3:
+Calculate Rext(α, m, β) using (9),(10),(11) (12)
+4:
+Calculate LJ using (8)
+5:
+Calculate P rx
+J
+using (6)
+6:
+if P rx
+J
+¡ρrx then ▷ Looking for the first β violate
+constraint (7)
+7:
+return β-1
+▷ the previous trial β was the
+maximum
+8:
+end if
+9:
+end for
+10: end function
+Following an exhaustive search approach, we examine all possi-
+ble values of α and m to seek for the pair that minimizes �
+Cost
+in (24).
+The search range of α is from 0° to the maximum
+8
+
+Serving zone
+of a HAPvalue set by constraint (4). The number of supplementary serving
+FSO transceivers m is also limited. Indeed, since the number of
+inter-HAP links of a HAP can go up to V as set in Section 5.1, and
+nsF
+i
+= m + 1, ∀i, then from the energy constraint (21), we deduce
+the upper bound for m:
+m ≤
+Esolar
+24
+− (VµFρavion + Vρinter
+F
++ µHρavion + ρPAT)
+µFρavion + ρHCM
+F
++ ρFSO
+tx
+− 1 (25)
+Algorithm 1 implements the exhaustive search idea. First, two
+nested loops scan all possible values of α satisfying constraint (4)
+and all possible values of m satisfying (25) to find the pair that
+minimizes �
+Cost in (24). For each pair (α, m), the largest value
+of β according to constraint (7) is selected using Algorithm 2.
+The optimal mFSO configuration is reported by the algorithms as
+(αOpt, mOpt, βOpt).
+Algorithm 2 finds the maximum β that satisfies constraint (7)
+for a given pair of (α, m) by testing the possible values of β in-
+creasingly from 0 until the received power P rx
+J
+at the border of
+the extended coverage area reaches the required received power
+ρrx. The received power P rx
+J
+is calculated using the set of equa-
+tions (6), (8), (9),(10),(11), and (12).
+In the implementation of both algorithms, α and β step by 1°
+after each iteration. Finer stepping allows obtaining more accu-
+rate results. However, even with 1° stepping, the variation in the
+optimal Rext is only a few hundred meters, which is negligible in
+comparison to the absolute value of Rext which is in the range of
+6-30 kilometers.
+The complexity of Algorithm 1 is O(m) because α ≤ π. The
+complexity of Algorithm 2 is constant because β ≤ π.
+6
+Design HAP network topology
+This section presents the HAP network design using the optimal
+configuration identified above. Let denote Linter as the number
+of inter-HAP links. Since �K
+i=1 niF
+i
+is the total number of inter-
+HAP FSO transceivers, it is equal to 2Linter. The network cost
+becomes:
+Cost = Kζday
+H
++ (K(m + 1) + 2Linter)ζday
+F
++ K ζmtn
+Dm
+and is equivalent to
+Cost = K
+�
+ζday
+H
++ (m + 1)ζday
+F
++ ζmtn
+Dm
+�
++ 2Linterζday
+F
+(26)
+The cost is proportional to the number of HAPs K and the
+number of inter-HAP links Linter.
+We consider that the daily
+amortization cost of a HAP is much greater than that of an FSO
+transceiver; thus, the coefficient of K is much greater than the
+coefficient of Linter in Cost. Consequently, K should be prioritized
+to minimize over Linter. Therefore, the topology design is broken
+into following two steps:
+i) ground nodes are clustered into equal radius circles that will
+become serving zones of HAPs in such a way that the number
+of clusters is the smallest for minimizing K;
+ii) corresponding HAPs are located at the centers of clusters but
+at an elevation of 20 km and are interconnected by the fewest
+number of inter-HAP links, Linter.
+A HAP network topology design algorithm was proposed in [7]
+following these two steps. In this algorithm, the clustering radius
+was not determined but was left as an input of the algorithm. In
+the current study, we set the clustering radius as the extended
+coverage radius Rext of the optimal mFSO configuration to drive
+towards a HAP network with minimal Cost. The main steps of the
+Figure 9: HAP network design flowchart.
+HAP network design process are presented in Figure 9, where the
+steps taken from [7] are shown in color. The process is explained
+as follows:
+• Initialize V, the maximum number of inter-HAP links of a
+HAP, by a constant.
+• Calculate the optimal mFSO configuration using Algorithm
+1, and set the clustering radius as its Rext.
+• Apply the clustering algorithm proposed in [7] to distribute
+ground nodes into clusters of radius Rext while keeping the
+number of ground nodes in each cluster under W. Each cluster
+becomes a serving zone of a HAP. The HAP is located at the
+center of the cluster but at an elevation of 20 km.
+• Bandwidth demands between ground nodes belonging to dif-
+ferent serving zones are bundled into lightpaths between
+corresponding HAPs, creating the inter-HAP traffic matrix
+MHAP .
+• Apply HAP topology design algorithm proposed in [7] to build
+the HAP topology.
+The algorithm begins with an empty
+topology. It finds a route for each lightpath demand of MHAP
+9
+
+Init V
+Find optimal conf.
+(α, m, β) and Rext
+Clustering ground nodes
+with Rext radius [7]
+V=V+1
+Calculate inter-HAP
+Design HAP topo [7]
+M
+HAF
+No
+is
+routed entirely
+Report HAP topofrom a full-mesh graph linking all HAPs within communica-
+tion distance limit LHH. Each time a lightpath uses an inter-
+HAP link that has not yet been included in the current HAP
+topology, the link is incorporated into the topology. The link
+in the topology is prioritized for use in building routes for the
+next lightpath demands.
+• Once all lightpath demands in MHAP are routed, the final
+topology is achieved. Otherwise, routing may fail due to the
+low connectivity between HAPs. In this case, V is increased
+by one, and the process is repeated until all lightpath de-
+mands in MHAP are routed.
+7
+Simulation results
+The algorithms for finding the optimal mFSO configuration were
+implemented and integrated with the topology designed algorithm
+described in Section 6. We performed simulations with practical
+parameters and evaluated the efficiency of mFSO configuration
+compared to the single serving FSO transceiver configuration.
+7.1
+Parameter values
+The simulation parameters are listed in Table 1. The values of
+these parameters were chosen according to experiments reported
+in the literature. This subsection explains the choices of the pa-
+rameter values.
+Cost-related parameters:
+The cost-related parameters are set
+such that the daily amortization cost of a HAP is significantly
+greater than that of an FSO transceiver, and the one-time mainte-
+nance cost is significantly higher than the daily amortization cost
+of a HAP. The maintenance cycle of a HAP is set as Dm = 1 year
+according to published information on Stratobus [6].
+Energy-related parameters:
+• Esolar - daily harvested solar energy.
+We considered daily
+solar energy levels between the minimum daily solar energy
+values in York and Enugu reported in [12], which were 42
+kWh and 290 kWh, respectively.
+• ρavion - power consumed by the avionic part of a HAP to
+carry a unit of mass. Although the power-to-mass ratio can
+be estimated as 6 W/kg according to [12], the published power
+rates of real systems are smaller. For aerodynamic systems
+such as Zephir-S, Zephir-T [14], and Phasa-35 [15], ρavion
+varies from 2.68 -3.04 W/kg. Indeed, Zephir-S weighs 80 kg
+(75 kg platform and 5 kg payload) and consumes 243 W,
+Zephir-T weighs 160 kg (140 kg platform and 20 kg payload)
+and consumes 429 W, and Phasa-35 weighs 165 kg (150 kg
+platform and 15 kg payload) and consumes 459 W. Aerostatic
+systems consume even less power. The Stratobus weighs 7000
+kg and consumes 5 kW when it carries a 250 kg payload
+and 8 kW when it carries 450 kg [6]. Thus, the power-to-
+mass ratio of Stratobus is between 0.69 and 1.07 W/kg only.
+Therefore, in this simulation ρavion was set to 2 W/kg.
+• ρPAT - power consumed by a PAT system; it was set to 15 W
+according to [13].
+• ρHCM
+F
+- power consumed for heating, cooling, and manage-
+ment; it was set to 20 W according to [4].
+• ρinter
+F
+- power consumed by an inter-HAP FSO transceiver;
+it was set to 35.1 W including laser power, ρHCM
+F
+and ρPAT.
+Inter-HAP link parameters: These parameters were set to values
+similar to those provided in the Loon project [4].
+HAP-ground FSO link parameters: The attenuation coefficient
+of an FSO link between a HAP and a ground node is set identical
+to that of inter-HAP links. The required received power ρrx at a
+ground node was set according to [4]. The aperture radius Rrx of
+a ground FSO receiver was set according to the commercial FSO
+transceiver SONABeam [1].
+Other parameters:
+• δ - BER threshold for inter-HAP links and lightpaths. We
+set δ = 10−3 because errors with that BER can be corrected
+using current Forward Error Correction (FEC) techniques.
+• LHH - the maximum allowable distance between two HAPs
+such that the BER of an inter-HAP link is less than δ = 10−3.
+Using the inter-HAP FSO link parameters listed in Table 1,
+the calculation yielded LHH = 88 km.
+• µH - platform mass; it varies significantly from one design
+to another. The Loon balloon weighs just 28.5 kg while the
+Stratobus weighs 7000 kg. With ρavion = 2 W/kg, a HAP
+weighing more than 7000 kg already consumes 326 kWh/day
+to carry itself, which is more than the maximum harvested
+solar energy, leading to no remaining energy to carry FSO
+transceivers. Therefore, µH = 500 kg was used in the simula-
+tions.
+• µF - mass of an FSO transceiver on HAPs. It was set accord-
+ing to the FSO transceiver used in the Loon project, which
+weighs 6.3 kg [4]. This value is consistent with the weights
+between 8 and 10 kg of commercial terrestrial SONABeam
+FSO transceivers [1].
+• W - the number of wavelengths per FSO link. It was set to
+40 or 80 according to the current WDM technique.
+The test dataset contained 19 test cases, each with 400 – 2800
+ground nodes.
+The ground FSO node locations were randomly
+generated on a square surface of 100 × 100 km, which is the size
+of a large metropolis.
+The test cases had different numbers of
+ground nodes, reflecting different ground node densities. The traf-
+fic requirement M contained demands randomly generated between
+ground FSO nodes such that the total incoming or outgoing traffic
+of a ground FSO node did not exceed 1 Gbps, which is the capacity
+of a single wavelength.
+Initially, V was set to 10.
+The optimal multiple serving
+FSO transceiver configuration (α, m, β) was calculated using Algo-
+rithms 1 and 2. The extended radius Rext of the optimal configura-
+tions was calculated using (9) and was then used as the clustering
+radius in the HAP topology design step.
+With Esolar = 42 kWh and W = 40, V must be increased to 12
+to get all demands in MHAP routed successfully for all test cases.
+With all other Esolar and W values, the topology design algorithm
+successfully routed all demands in MHAP for all test cases right
+with initial V = 10.
+Figure 10 illustrates the HAP locations and their footprints
+calculated using the proposed algorithms for a test case of 1005
+ground FSO nodes, Esolar = 75 kWh, and W = 80.
+7.2
+mFSO configuration versus single serv-
+ing FSO transceiver configuration
+Table 3 lists the maximum beam width αmax according to (4)
+and the maximum ground coverage radius of the single serving
+FSO transceiver configuration when the receiver aperture radius
+10
+
+Esolar = 42 kWh
+Esolar = 50 ∼ 290 kWh
+W = 40, V = 12
+W = 80, V = 10
+W = 40, V = 10
+W = 80, V = 10
+|NFSO|
+α
+m
+β
+Rext
+Cost
+α
+m
+β
+Rext
+Cost
+α
+m
+β
+Rext
+Cost
+α
+m
+β
+Rext
+Cost
+(1)
+(2)
+(3)
+(4)
+(5)
+(6)
+(7)
+(8)
+(9)
+(10)
+(11)
+(12)
+(13)
+(14)
+(15)
+(16)
+(17)
+(18)
+(19)
+(20)
+(21)
+480
+37
+0
+-
+6691
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+0
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+0
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+16
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+763
+37
+0
+-
+6691
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+-
+6691
+17612
+37
+12
+16
+11539
+13115
+37
+13
+16
+11929
+10335
+1523
+37
+0
+-
+6691
+21895
+37
+0
+-
+6691
+18595
+37
+12
+16
+11539
+14053
+37
+13
+16
+11929
+10375
+1675
+37
+0
+-
+6691
+21735
+37
+0
+-
+6691
+18535
+37
+11
+16
+11042
+14128
+37
+13
+16
+11929
+10495
+1736
+37
+0
+-
+6691
+22461
+37
+0
+-
+6691
+19301
+37
+11
+16
+11042
+14874
+37
+13
+16
+11929
+10455
+1911
+37
+0
+-
+6691
+22481
+37
+0
+-
+6691
+19021
+37
+10
+16
+10395
+14869
+37
+13
+16
+11929
+10495
+2009
+37
+0
+-
+6691
+22641
+37
+0
+-
+6691
+19321
+37
+10
+16
+10395
+15575
+37
+13
+16
+11929
+10595
+2135
+37
+0
+-
+6691
+22761
+37
+0
+-
+6691
+19221
+37
+10
+16
+10395
+16493
+37
+13
+16
+11929
+10575
+2304
+37
+0
+-
+6691
+22881
+37
+0
+-
+6691
+19301
+37
+9
+16
+9524
+18192
+37
+13
+16
+11929
+10655
+2325
+37
+0
+-
+6691
+22368
+37
+0
+-
+6691
+18948
+37
+9
+16
+9524
+18555
+37
+13
+16
+11929
+10675
+2491
+37
+0
+-
+6691
+22761
+37
+0
+-
+6691
+19401
+37
+8
+16
+8946
+18660
+37
+13
+16
+11929
+10655
+2753
+37
+0
+-
+6691
+23346
+37
+0
+-
+6691
+19926
+37
+8
+16
+8946
+20284
+37
+13
+16
+11929
+11178
+Table 2: Optimal configurations and costs of all test cases with Rrx = 2 m.
+Receiver aperture
+Maximum beam
+Maximum
+radius Rrx (m)
+width αmax
+coverage radius (m)
+2
+37 °
+6691
+4
+67 °
+13237
+Table 3: Maximum beam width and coverage radius of single
+serving FSO transceiver configuration.
+was varied. Table 4 lists the extended coverage radius of the max-
+imum mFSO configuration for different solar energy levels and
+receiver aperture radii. The maximum mFSO configuration was
+obtained using the largest principal beam αmax, largest m accord-
+ing to (25), and largest β according to (7), given αmax and m.
+The coverage radius of the maximum mFSO configuration was ex-
+tended approximately twice in comparison with that of single FSO
+transceiver configuration, except for Esolar = 42kWh. When solar
+energy level increased, the maximum m increased; thus, the ex-
+tended coverage radius increased. However, when m was already
+large, the extention increased slowly with m.
+Additionally, the
+maximum extended coverage was much larger when Rrx = 4 than
+Rrx = 2m because a receiver can accept weaker signals with larger
+apertures.
+To compare the network costs incurred by the two configura-
+tions, we examined the detailed results in Table 2.
+The table
+lists the optimal mFSO configurations and network costs. When
+Esolar = 42kWh, the optimal number of supplementary serving
+FSO transceivers is m = 0; thus, the configuration uses a single
+serving FSO transceiver. Therefore, these cases were used as ref-
+erences for single serving FSO transceiver configuration.
+When
+Esolar > 50kWh, all optimal configurations were truly mFSO,
+and the results were identical for all solar energy levels.
+The
+numbers indicate that mFSO configuration offered significantly
+Esolar
+Max Rext (m)
+(kWh)
+Max m
+Rrx = 2 (m)
+Rrx = 4 (m)
+42
+6
+6691
+13237
+50
+16
+12174
+25582
+75
+47
+13559
+28403
+100
+78
+13678
+28845
+125
+109
+13711
+28969
+150
+140
+13724
+29020
+175
+171
+13731
+29047
+200
+202
+13735
+29062
+225
+233
+13738
+29071
+250
+264
+13739
+29077
+275
+295
+13740
+29082
+290
+314
+13741
+29084
+Table 4: Maximum extended coverage radius of mFSO con-
+figuration when V = 10.
+lower costs (listed in columns 16th and 21th) than those of single
+serving FSO transceiver configuration (listed in columns 6th and
+11th) for the same test cases and number of wavelengths W. The
+costs resulting from mFSO configuration were as low as 54–87% of
+those resulting from single serving FSO transceiver configuration.
+These numbers confirm that when there is sufficient solar energy,
+mFSO configuration is definitively a better choice than single serv-
+ing FSO configuration.
+11
+
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+y-axis
+x-axis
+Figure 10: Footprints of HAPs with mFSO configuration ob-
+tained from the topology design for a test case of 1005 ground
+FSO nodes when Esolar = 75 kwh, W = 80. A circle repre-
+sents an extended coverage area of a HAP. Small points in-
+side the circle are ground nodes and the dot at the center of
+the circle is the projected location of its serving HAP on the
+ground.
+7.3
+Factors impact optimal mFSO configu-
+ration
+Comparing the values of the optimal extended coverage radius in
+Table 2 and the maximum extended coverage radius in Table 4,
+we can see that the optimal extended coverage radius was gener-
+ally not the maximum. This is reasonable because the maximum
+configuration uses an excessive number of supplementary serving
+FSO transceivers.
+Low solar energy may render mFSO configuration impossible.
+Indeed, Esolar = 42 kWh could afford maximally 6 supplemen-
+tary serving FSO transceivers (see Table 4), which was too few to
+entirely cover the contour of the principal coverage area. Thus,
+single FSO transceiver configuration was the unique choice.
+When the solar energy level exceeds 50 kWh, its exact value does
+not affect the optimal configuration. The simulation showed that
+the optimal configurations were identical for all solar energy levels
+from 50 kWh/day and above. This is explained by the fact that
+a greater solar energy level allows to accept configurations with
+large coverage but may be more expensive because of using more
+supplementary serving FSO transceivers. As a result, large con-
+figurations were not selected as optimal configurations. In other
+words, increasing solar energy does not necessarily improve the
+HAP network cost.
+Since the optimal multiple serving FSO transceiver configura-
+tions were identical for all Esolar ≥ 50 kWh, all other numerical
+results related to topology design and routing with these solar en-
+ergy levels were identical and are presented as single results in
+subsequent figures.
+The coverage of the optimal configurations decreased when the
+ground nodes became denser. Indeed, test cases with large num-
+bers of ground nodes had greater ground node densities, and
+columns 13th and 16th of Table 2 shows that the optimal m and
+Rext decreased when the density increased. The reason is that,
+with a greater ground node density, is a small ground region al-
+ready contains W ground nodes, which is the maximum serving
+capacity of a HAP. Therefore, a HAP could serve only a small
+zone and required only a few supplementary FSO transceivers to
+cover the zone.
+7.4
+Numbers of HAPs and inter-HAP links
+0
+20
+40
+60
+80
+100
+120
+140
+160
+0
+500
+1000
+1500
+2000
+2500
+3000
+Number of HAPs
+Number of ground FSO nodes
+ˆK for Esolar=42 kWh
+K for Esolar=42 kWh
+ˆK for Esolar ≥ 50 kWh kWh
+K for Esolar ≥ 50 kWh
+Lower bound
+(a) W=40
+0
+20
+40
+60
+80
+100
+120
+140
+160
+0
+500
+1000
+1500
+2000
+2500
+3000
+Number of HAPs
+Number of ground FSO nodes
+ˆK for Esolar=42 kWh
+K for Esolar=42 kWh
+ˆK for Esolar ≥ 50 kWh kWh
+K for Esolar ≥ 50 kWh
+Lower bound
+(b) W=80
+Figure 11: Number of HAPs and lower bound with (a) W =
+40 and (b) W = 80 in different solar energy levels.
+Since each HAP can serve at most W ground FSO nodes, a lower
+bound for the number of HAPs is:
+nLB
+HAP = |NFSO|
+W
+(27)
+Figure 11 shows the number of HAPs, the estimated number of
+HAPs ˆK and lower bound nLB
+HAP when (a) W = 40 and (b) W = 80.
+With Esolar ≥ 50 kWh, the actual number of HAPs was almost
+identical to ˆK in both subfigures.
+Furthermore, when W = 40
+12
+
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+ 0
+ 500
+ 1000
+ 1500
+ 2000
+ 2500
+ 3000
+Number of inter-HAP links
+Number of ground FSO nodes
+Esolar=42 kWh
+Esolar ≥ 50 kWh
+(a) W=40
+ 0
+ 100
+ 200
+ 300
+ 400
+ 500
+ 600
+ 0
+ 500
+ 1000
+ 1500
+ 2000
+ 2500
+ 3000
+Number of inter-HAP links
+Number of ground FSO nodes
+Esolar=42 kWh
+Esolar ≥ 50 kWh
+(b) W=80
+Figure 12: Number of inter-HAP links when (a) W = 40 and
+(b) W = 80 for different solar energy levels.
+and Esolar ≥ 50 kWh, the number of HAPs approached the lower
+bound starting from test cases with 1000 ground nodes or above.
+This implies that the number of HAPs was almost optimal.
+Figure 12 presents the absolute numbers of inter-HAP links.
+The number of inter-HAP links increased with the number of
+ground nodes, because the network size and traffic demand in-
+creased. The number of inter-HAP links clearly decreased when
+the wavelength density increased from W = 40 to W = 80. In
+other words, denser WDM technique helps reduce the number of
+inter-HAP FSO transceivers and consequently the network cost.
+mFSO configuration allows reducing significantly both the num-
+bers of HAPs and inter-HAP links. Indeed, according to Figure
+11, the number of HAPs was much smaller with Esolar ≥ 50
+kWh where mFSO configuration was used, in comparison with
+Esolar = 42 kWh, where single serving FSO configuration was
+used. A similar phenomenon is observed in Figure 12 for the num-
+ber of inter-HAP links.
+0
+5000
+10000
+15000
+20000
+25000
+30000
+35000
+40000
+0
+500
+1000
+1500
+2000
+2500
+3000
+Cost
+Number of ground FSO nodes
+�
+Cost of Esolar = 42 kWh
+Cost of Esolar = 42 kWh
+�
+Cost of Esolar ≥ 50 kWh
+Cost of Esolar ≥ 50 kWh
+(a) W=40
+0
+5000
+10000
+15000
+20000
+25000
+30000
+35000
+40000
+0
+500
+1000
+1500
+2000
+2500
+3000
+Cost
+Number of ground FSO nodes
+�
+Cost of Esolar = 42 kWh
+Cost of Esolar = 42 kWh
+�
+Cost of Esolar ≥ 50 kWh
+Cost of Esolar ≥ 50 kWh
+(b) W=80
+Figure 13: Real costs and overestimated costs with W = 40
+and W = 80.
+7.5
+Quality of cost estimation
+Figure 13 presents the estimated and actual costs for different
+solar energy levels and wavelength densities. The estimated cost
+was very close to the actual cost, mostly for Esolar ≥ 50kWh and
+W = 40.
+Parameter V, the threshold of the number of inter-HAP links
+of a HAP, affects the quality of the cost estimation. To evaluate
+the choice of V, we compared it with the number of inter-HAP
+links that a HAP finally has. Figure 14 shows the average number
+of inter-HAP links per HAP. When there were 40 wavelengths
+per link, the average number of inter-HAP links per HAP varied
+between 5.7 and 9.3 for Esolar ≥ 50 kWh and V = 10, and between
+8.8 and 11.8 for Esolar = 42 kWh while V raised up to 12. Hence,
+the value of V was close to the actual number of inter-HAP links
+required by a HAP. However, when there were 80 wavelengths per
+link, the average number of Inter-HAP links per HAP was reduced
+to between 4.4 and 8.4, which is slightly far from the threshold
+V = 10. A smaller V may help better estimate of the optimal cost
+in these cases.
+13
+
+ 0
+ 5
+ 10
+ 15
+ 20
+ 0
+ 500
+ 1000
+ 1500
+ 2000
+ 2500
+ 3000
+Average number of inter-HAP links per HAP
+Number of ground FSO nodes
+Esolar=42 kWh
+Esolar ≥ 50 kWh
+(a) W=40
+ 0
+ 5
+ 10
+ 15
+ 20
+ 0
+ 500
+ 1000
+ 1500
+ 2000
+ 2500
+ 3000
+Average number of inter-HAP links per HAP
+Number of ground FSO nodes
+Esolar=42 kWh
+Esolar ≥ 50 kWh
+(b) W=80
+Figure 14: Number of inter-HAP links per HAP when (a)
+W = 40 and (b) W = 80 for different solar energy levels.
+8
+Conclusions
+Using mFSO configuration widens a HAP footprint, however, its
+application is constrained by the available solar energy of the HAP.
+Moreover, mFSO configuration may imply an extra investment
+cost due to additional serving FSO transceivers in comparison with
+single FSO transceiver configuration. This study focused on de-
+termining the optimal mFSO configuration. First, we proposed
+a set of closed-form expressions for computing the coverage of
+an mFSO configuration in terms of beam widths of the princi-
+pal and supplementary transceivers and number of supplementary
+FSO transceivers. Second, we proposed an algorithm to determine
+the optimal mFSO configuration that minimizes the total HAP
+network cost. Third, we designed a HAP network topology using
+the optimal configuration to achieve a minimal final cost.
+The simulation results showed that mFSO significantly ex-
+tended the HAP footprint. With the testing dataset, the extended
+footprint radii were generally two times larger than the single FSO
+transceiver footprint radii, leading to a four-fold larger coverage
+surface. The network cost with the optimal mFSO configuration
+was as low as 54% of the network cost when using a single serving
+FSO transceiver on a HAP.
+Acknowledgements
+This research was funded by the Vietnam National Foundation for
+Science and Technology Development (NAFOSTED) under grant
+number 102.02-2018.305.
+References
+[1] fSONA,
+“SONABeam
+2500-E+
+model
+specifications.”
+http://fsona.com. Accessed Jan. 2022.
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+access network based on mesh-connected free-space optical
+links,” IEEE Personal Communications, vol. 6, no. 5, pp. 62–
+65, 1999.
+[3] J. Zhang, “Proposal of free space optical mesh network ar-
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+vol.4, 2002.
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+ong Kim, J. Moody, T. Tu, and W. Vermeer, “Demonstration
+of free-space optical communication for long-range data links
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+munication and Atmospheric Propagation XXIX (H. Hem-
+mati and D. M. Boroson, eds.), vol. 10096, pp. 259 – 272,
+International Society for Optics and Photonics, SPIE, 2017.
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+M. Hunwardsen, J. Demers, K. Kudinov, G. Mak, B. Smith,
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+D. Leang, J. Gin, M. Lewis, D. Nguyen, and K. Quirk,
+“High-speed optical links for UAV applications,” in Free-
+Space Laser Communication and Atmospheric Propagation
+XXIX (H. Hemmati and D. M. Boroson, eds.), vol. 10096,
+pp. 316 – 324, International Society for Optics and Photon-
+ics, SPIE, 2017.
+[6] Thales
+group,
+“What’s
+up
+with
+stratobus.”
+https://www.thalesgroup.com/en/worldwide/space/news/whats-
+stratobus, 2017. Accessed Jan. 2022.
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+space optical mesh network using high-altitude platforms,”
+Optical Switching and Networking, vol. 47, p. 100716, 2023.
+[8] G. Karabulut Kurt,
+M. G. Khoshkholgh,
+S. Alfattani,
+A. Ibrahim, T. S. J. Darwish, M. S. Alam, H. Yanikomeroglu,
+and A. Yongacoglu, “A Vision and Framework for the High
+Altitude Platform Station (HAPS) Networks of the Future,”
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+pp. 729–779, 2021.
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+14
+
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+A
+Proof of Lemma 1
+Proof. Let x = cos(α/2), a = σH, and b =
+PtxR2
+rx
+2H2
+then
+P rx
+j (x) = e−a/x
+bx2
+(1 − x)
+(28)
+Calculate the derivative of P rx
+j (x) we get
+P
+′rx
+j
+(x) = e−a/x
+�
+a
+1 − x + 2x − x2
+(1 − x)2
+�
+b
+(29)
+Thus, the derivative of P rx
+j (α) is
+P
+′rx
+j
+(α) = P
+′rx
+j
+(x).(− sin(α))
+(30)
+Beam α is limited between [0..π] because it orients to the ground.
+Thus, x ∈ [0..1].
+Consequently, 1 − x > 0 and 2x − x2 > 0.
+In addition, a, b > 0, then P
+′rx
+j
+(x) > 0 for all x ∈ [0..1].
+Be-
+cause − sin(α) < 0, ∀α ∈ [0..π], thus, P
+′rx
+j
+(α) < 0. Consequently,
+P rx
+j (α) decreases with α.
+B
+Calculation of extended coverage
+radius of mFSO configuration
+This section identifies formulas that calculate the extended cover-
+age radius of an mFSO configuration characterized by the princi-
+pal beam width α, supplementary beam width β and number of
+supplementary beams m.
+Conventionally, the coverage provided by a bundle of transmit-
+ters is calculated as if the transmitters project perpendicular to the
+ground. In mFSO configuration, the principal beam in the center
+is large, and it pushes the supplementary serving FSO transceiver
+projection directions far from perpendicular to the ground. These
+supplementary beams form oblique cones that intersect with the
+ground plane in ellipses. Considering of the elliptical form adds
+more complexity to the calculation.
+In Figure 15, H denotes the position of a HAP, and its projec-
+tion on the ground plane is O, thus HO = H. The principal beam
+forms a right circular cone whose axis is HO. The cone intersects
+the ground plane by a circle of radius Rα, which defines the prin-
+cipal footprint. The beam of a supplementary FSO transceiver is
+an oblique cone intersecting the ground plane by an ellipse that
+defines the corresponding supplementary footprint. The cone of
+the supplementary beam intersects with the cone of the principal
+beam by two lines: HK and HK′ where K and K′ are the two
+intersection points of the principal and supplementary footprints.
+Thus, OK = OK′ = Rα.
+m supplementary FSO transceivers are arranged evenly around
+the principal transceiver, each of which is responsible for extending
+the coverage within an angle of 2π/m from the center O.
+The
+responsible angle of the supplementary FSO transceiver in Figure
+15 is defined by rays −−→
+OK and −−→
+OK′. Thus, �
+KOK′ = 2π/m.
+Ray −−→
+OK intersects with the supplementary beam cone at J,
+then OJ is the radius of the extended coverage region. Readers
+refer to Figure 6 for a complete view of the extended coverage
+circle and the positions of K, K′ and J on the ground.
+Figure 15: Computation of the distance from supplementary
+FSO transceivers and the border of extended coverage area
+LJ in function of Beta.
+Since the principal beam width is α, then �
+OHK = α/2.
+Let the base plane containing K and K′ of the supplementary
+beam cone cuts the cone axis at T, the primary cone axis HO at P,
+and HJ at J1. Then �
+THK = β/2. In addition, the supplementary
+cone intersects with this base plane by a circle containing K, K′
+with center T. Let Rβ be the radius of the circle, then TK =
+TK′ = Rβ.
+Let M be the midpoint of KK′ then H, O, T, M belong to the
+same plane.
+Let ξ = �
+KHJ. The extended coverage radius is Rext = OJ =
+15
+
+H
+β/2
+LY
+a/2
+Supplementarycone base plane
+K'
+R
+Pilm
+a
+Supplementary foot print
+Principal
+foot print
+GroundHO tan(�
+OHJ) = H tan( α
+2 + ξ). Thus,
+Rext = H. tan
+�
+2(ξ + α
+2
+) − α
+2
+�
+Rext = H2 tan( ξ+α
+2 ) − tan( α
+2 )(1 − tan2( ξ+α
+2 ))
+1 − tan2( ξ+α
+2 ) + 2 tan( ξ+α
+2 ). tan( α
+2 )
+(31)
+B.1
+Calculation of tan( ξ+α
+2 )
+Let N be the midpoint of KJ1. As K and J1 are at the intersection
+of the supplementary cone and its base plane, HK = HJ1, HN ⊥
+KJ1, and HN is the angle bisector of �
+KHJ1. Therefore, �
+NHK =
+ξ/2, thus �
+NHP = ξ+α
+2 . In addition, since KO is on the base plane
+of the principal cone, HO ⊥ KO.
+Thus, △PNH and △POK
+are similar right triangles. Consequently, �
+OKP = �
+NHP = ξ+α
+2 .
+Furthermore,
+tan(ξ + α
+2
+) = OP
+OK = OP
+Rα
+(32)
+Let �
+OHM = γ and �
+THM = θ Then �
+OHT = θ + γ.
+Because MO is on the base plan of the principal cone, MO ⊥
+HO. In addition, as PT is on the base plane of the supplementary
+cone whose axis is HT then HT ⊥ PT. Consequently, △PTH and
+△POM are similar right triangles. We can deduce that �
+PMO =
+�
+PHT = θ + γ. Therefore,
+tan(θ + γ) = OP
+OM =
+OP
+Rα. cos( π
+m)
+Combining with (32) we deduce :
+tan(ξ + α
+2
+) = tan(θ + γ). cos( π
+m)
+(33)
+Thus
+tan(ξ + α
+2
+) =
+tan(γ) + tan(θ)
+1 − tan(γ). tan(θ). cos( π
+m)
+(34)
+Since γ = �
+OHM then, tan(γ) = MO
+HO .
+From right triangle △OMK we have MO = OK. cos( π
+m).
+From right triangle △HOK we have HO = OK/ tan( α
+2 ).
+Thus
+tan(γ) = tan(α
+2 ). cos( π
+m)
+(35)
+It remains to calculate tan (θ).
+B.2
+Calculation of tan (θ)
+Look at the right triangle △HTM, we can see that:
+tan(θ) = TM
+TH
+(36)
+Since K and K′ are on a circle centered at T, and M is the
+midpoint of KK′ then △TMK is a right triangle, then
+TM =
+�
+TK2 − KM 2 =
+�
+R2
+β − R2α. sin2( π
+m)
+(37)
+Easy to find that △THK is another right triangle then
+TH = TK/ tan(β
+2 ) = Rβ/ tan(β
+2 )
+(38)
+Replacing (37) and (38) in to (36) we get
+tan(θ)
+=
+�
+R2
+β − R2α. sin2( π
+m)
+Rβ/ tan( β
+2 )
+=
+tan(β
+2 )
+�
+1 − (Rα
+Rβ )2. sin2( π
+m)
+(39)
+From right triangle △HTK we obtain Rβ = HK sin( β
+2 ).
+From right triangle △HOK we obtain Rα = HK sin( α
+2 ).
+Replacing these values to (39), we obtain:
+tan(θ) =
+�
+sin2( β
+2 ) − sin2( α
+2 ). sin2( π
+m)
+cos( β
+2 )
+(40)
+Substituting the values of tan(γ) in (35) and tan(θ) in (40) into
+(34), we obtain tan( ξ+α
+2 ). Subsequently, replacing the obtained
+tan( ξ+α
+2 ) to (31) we get Rext.
+16
+
diff --git a/9tFAT4oBgHgl3EQfpx3L/content/tmp_files/load_file.txt b/9tFAT4oBgHgl3EQfpx3L/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..3728c07b9b5cb62fdd27c8b90153a6e14ce80af7
--- /dev/null
+++ b/9tFAT4oBgHgl3EQfpx3L/content/tmp_files/load_file.txt
@@ -0,0 +1,1221 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf,len=1220
+page_content='Optimal multiple FSO transceiver configuration for using on High-altitude  platforms  Dieu Linh Truong ∗1 and The Ngoc Dang †2  1School of Information and Communication Technology, Hanoi University of Science and Technology,  Vietnam  2Department of Wireless Communications, Posts and Telecommunication Institute of Technology,  Vietnam  January 23, 2023  Abstract  Free-space optical (FSO) communication requires light of  sight (LoS) between the transmitter and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  For  long-distance communication, many research projects have  been conducted towards using a network composed of high-  altitude platforms (HAPs) flying at an elevation of 20 km to  carry intermediate FSO transceivers that forward data be-  tween ground stations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The clear environment at high el-  evations prevents terrestrial obstacles from cutting the LoS  between the transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  An FSO transceiver on a HAP  can communicate with ground stations within a small area  owing to its limited beam size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' We suggest using multiple  FSO transceivers on a HAP to extend its ground coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  However, the use of too many FSO transceivers may quickly  exhaust the onboard energy of the HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' As a result, HAP  must be lowered to recharge frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  In this study, we first propose a configuration of multiple  FSO transceivers to widen the ground coverage of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  We then propose a set of closed-form expressions to calculate  the extended coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Finally, to implement a HAP network  using multiple FSO transceivers, we seek the optimal config-  uration of multiple FSO transceivers that minimizes the to-  tal cost of the HAP network, including amortization, energy,  and maintenance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The simulation results show that the  proposed multiple FSO transceiver configuration clearly in-  creases the ground coverage of a HAP and significantly re-  duces the cost of the HAP network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Keywords— Free Space Optics, High-altitude platform, Beam  size optimization, HAP based FSO network  1  Introduction  Free-space optical (FSO) communication uses light propagation  in free space to transmit data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In recent years, this technology  has emerged as a promising choice for short-distance high-speed  communication between endpoints with a clear light of sight (LoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  ∗linhtd@soict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='vn  †ngocdt@ptit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='vn  Commercial FSO transmitters available in the market at prices of  thousands of dollars can operate at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='25 − 10 Gbps over 1 − 2  kilometers, for example, the SONABeam series of fSona [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  To reach a long distance, a multi-hop FSO system can be used,  where data are transmitted through intermediate FSO transceivers  [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' To avoid obstacles that cut the LoS between terrestrial  FSO transceivers, researchers from academia and industry have  proposed placing intermediate FSO transceivers of the multi-hop  FSO system on high-altitude platforms (HAPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  High-altitude  platforms are flying objects that operate at altitudes of 17–24 km  in the stratosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Several HAP models have been proposed and  piloted previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Some projects continue until recently, such as  the Loon Project of Google [4], the UAV project of Facebook [5],  and the Stratobus project of Thales Alenia Space [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  A multi-hop FSO system using a HAP network is described  in [7] and illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' According to this model, FSO  transceivers on the ground (so-called ground FSO nodes) are re-  grouped into clusters to become the serving zones of HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  A  HAP has an FSO transceiver looking down to exchange data  with the ground FSO nodes of the cluster under it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  This FSO  transceiver is called serving FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' A HAP also carries  several FSO transceivers pointing towards other HAPs for inter-  HAP communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' These FSO transceivers are known as inter-  HAP FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Although the ITU recommends a HAP footprint width of ap-  proximately 500 km in radius, experimental projects show much  smaller coverage areas [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Nevertheless, a network of multiple  HAPs can cover a country entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' For example, a constellation  of 16 HAPs with multiple radio frequency antennas was considered  to cover Japan [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  An end-to-end data-switching scheme for a multi-hop FSO sys-  tem using HAP was proposed in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Since the communication  between a HAP and the ground is point-to-multipoint, the serv-  ing FSO transceiver on the HAP controls multiple accesses from  ground FSO nodes under it using the WDM technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Each  ground node is assigned a separate wavelength for up and down  communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' An IP router on the HAP aggregates IP packets  heading toward a common cluster within a single flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The flow  will be carried by one or more continuous lightpaths between the  source and destination HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The number of lightpaths is deter-  mined according to the size of the flow and the transport capacity  of a wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' A WDM switch is installed on each HAP to route  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='08642v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='NI]  20 Jan 2023  these lightpaths over the HAP network on a wavelength-switched  basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In Figure 1, the blue path HAP1-HAP2-HAP4-HAP5 and  the red path HAP1-HAP2-HAP3 are two flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  1  2  2  1  1  1  1  2  3  3  3  3  3  WDM switch  IP router  IP router  WDM switch  1  2  2  p-HAP-2  1  3  3  p-HAP-1  toward HAP-2 of cluster-2  toward HAP-1 of cluster-1  HAP2  Inter-HAP FSO transceiver  Ground FSO node  Serving FSO  3  1  2  A cluster  A cluster  transceiver  HAP1  HAP3  HAP4  HAP5  Serving zone of HAP 1  Serving zone of HAP 2  inter-HAP link  inter-HAP link  Figure 1: Multi-hop FSO communication system using HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  In terrestrial FSO communications, the light beams are usually  set to be very narrow for low transmission energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' However, for  HAP and ground communication, the serving FSO transceiver of  the HAP must project a sufficiently wide laser beam for covering  distributed ground FSO nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  A single serving FSO transceiver has a relatively small foot-  print owing to the low capacity of the current laser source, and  the limited sensibility and aperture sizes of ground receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The  calculation in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1 shows that with a laser source of 1 Watt,  required received power at receivers of -41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1dBm, and receiver  aperture radius of 2 m, a single serving FSO transceiver at an  elevation of 20 km can cover a ground area of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='691 km radius  only (see Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  To extend the coverage of a HAP, we propose using multiple  serving FSO transceivers arranged in a bundle, as shown in Fig-  ure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Each serving FSO transceiver points in a slightly different  direction to cover a particular ground area that overlaps other ar-  eas to create a continuous coverage region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Given a ground region  to be served, using HAPs with multiple serving FSO transceivers  reduces the number of required HAPs compared to using HAPs  with a single serving FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' However, the expenditure  for serving FSO transceivers increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Therefore, the number of  serving FSO transceivers to be used on a HAP should be carefully  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Regarding the communication between ground nodes and a  HAP, the multiple serving FSO transceiver model still uses the  WDM technique, where each ground node is assigned a unique  wavelength within its cluster to communicate with its HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The  number of ground nodes to be served by a HAP is restricted by  the number of wavelengths offered by the WDM technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  In this study, we focus on identifying the optimal configuration  of multiple serving FSO transceivers to achieve a minimal-cost  HAP network for serving a set of ground FSO nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The optimal  configuration should define the number of serving FSO transceivers  to be set up on a HAP and the beam width for each transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The cost of the HAP network includes the investment, energy, and  maintenance costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Compared with the previous study in reference [7], the current  research differs in two aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  First, the current research pro-  Figure 2: A HAP with multiple serving FSO transceivers and  its footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  poses the use of multiple serving FSO transceivers on each HAP  instead of a single serving FSO transceiver, as in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Second, the  current research identifies the optimal beam widths for serving  FSO transceivers, whereas in [7], the beam widths are predefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The current study also differs from that in [10], where beam size  was optimized for an inter-HAP link, which is a point-to-point link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' First, we  analyze the single and multiple serving FSO transceivers configu-  rations in Section 2 to determine their ground coverage sizes and  constraints on transmitter beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In Section 3, we state the prob-  lem of designing a minimal-cost HAP-based FSO network, which is  the target of the optimization of multiple serving FSO transceiver  configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Then, in Section 4, we define a HAP energy con-  sumption formula and show that solar energy is necessary for keep-  ing the HAP working in space for a long period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' We also present  a constraint that a HAP must respect to relying uniquely on so-  lar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In Section 5, we present the algorithms for identifying  the optimal multiple serving FSO transceiver configuration and  its footprint radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Section 6 presents the process designing the  minimal cost HAP-based FSO network using the optimal multi-  ple serving FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Section 7 presents the  simulation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Finally, Section 8 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  2  Serving FSO transceiver configu-  rations  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1  Single serving FSO transceiver configu-  ration  In this section, the allowable beam width and ground coverage of  a single serving FSO transceiver are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The beam size  is restricted to ensure that the received power at a ground point  within the beam footprint is detectable by receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  2  Figure 3: Surface of the part of sphere blocked by solid angle  α is calculated as the sum of the surface of all ribbons around  the sphere when the solid angle varies from α to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Assume that the transmitter source radiates within a solid angle  α and that the radiation density is uniform in all directions within  the solid angle at a distance r from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The radiation den-  sity at distance r is inversely proportional to the surface of the part  of the sphere radius r blocked by the solid angle α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' To calculate  this surface, we divide the sphere into thin ribbons corresponding  to open angles of d(α/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The width of a ribbon is rd(α/2), as  shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The radius of the ribbon at zenith angle α/2 is  r sin(α/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Thus, the ribbon surface is 2πr sin(α/2)rd(α/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The  surface of the part of the sphere blocked by the solid angle α is the  sum of the surfaces of all ribbons when zenith angle varies from α  to 0, as follows:  � 0  α  2πr sin (α  2 )rd(α  2 ) = 2πr2(1 − cos (α  2 ))  Let Ur be the radiation density at distance r and Ptx be the  transmitted power at the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' We deduce:  Ur =  Ptx  2πr2(1 − cos (α/2))  (1)  Let P rx  j  be the received power at ground FSO node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The  received power is proportional to the radiation density and the  received aperture of the ground node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' It is:  P rx  j  =  e−σLjULjAR  (2)  where  Lj is the distance between ground FSO node j and its serving  HAP Hi (see Figure 4),  σ is the attenuation coefficient of the links between the HAP  and ground,  ULj is radiation density at distance Lj from the source,  AR is the aperture area of the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Let Rrx be the receiver  aperture radius, then, AR = πR2  rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  In (2), the first term represents the attenuation of laser power  through the atmosphere, which is described by the exponential  Beer–Lambert Law [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Figure 4: Received power on border nodes of a coverage area  is the smallest amongst all nodes in the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  By substituting ULj from (1) into (2), we obtain the received  power at node j as follows:  P rx  j  = e−σLj × Ptx × R2  rx  2L2  j  ×  1  1 − cos (α/2)  (3)  The power received at node j must not be less than the required  level of the receiver, denoted by ρrx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' It is obvious that point j at  the border of the ground coverage area receives the least power  because it is the furthest from the source (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Hence,  all points in the coverage areas of HAP Hi receive sufficient power  if and only if the border points receive at least the required power;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  that is,  P rx  j  = e−σH/ cos ( α  2 ) PtxR2  rx cos2 ( α  2 )  2H2(1 − cos ( α  2 )) ≥ ρrx  (4)  where Lj is substituted by H/ cos( α  2 ) for border node j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Solving inequation (4) yields the beam width of the single serv-  ing FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Corresponding to beam width  α, the ground coverage radius of the configuration is:  Ri = H tan(α  2 )  (5)  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Function P rx  j  decreases with α ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='.π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Proof of Lemma 1 is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Figure 5 shows the received power at the border of the cover-  age area with different receiver aperture radius Rrx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' This figure  confirms that P rx  j  decrease with an increase in α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Let αmax be the value for α that makes P rx  j (αmax) = ρrx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' then  according to Lemma 1,  P rx  j (α) ≥ P rx  j (αmax) = ρrx, ∀α ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='.αmax]  thus all α ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='.αmax] satisfy constraint (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Calculations using the parameters given in Table 1 show that  when Rrx = 2 m, αmax = 37° and the coverage radius is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='691 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  When Rrx = 4 m, αmax  = 67° and the coverage radius is  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='237 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  3  Ribbon surface= 2πr sin(α/2) r d(α/2)  Kd(a/2)  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='sin(a/2)  a/2  SourceHAP H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  α  H  Received power Prx  R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Nodej  coverage area of HAP H 0  2  4  6  8  10  0  20  40  60  80  100  120  140  160  180  Received power at coverage border (10-8 W)  Beam size α(degree)  Rrx=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='125m  Rrx=1m  Rrx=2m  Rrx=4m  Required at receiver (Prx)  Figure 5: Received power at the coverage border of the single  serving FSO transceiver configuration with different receiver  apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='2  Multiple serving FSO transceiver config-  uration  The ground coverage of a HAP can be widened by combining sev-  eral serving FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Different combinations are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  In this research, we study a straightforward configuration in which  a principal serving FSO transceiver is in the center projecting  light perpendicular to the ground, and several identical supplemen-  tary serving FSO transceivers are set evenly around the principal  one (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Each supplementary transceiver projects slanted  beams to extend the coverage in one direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' This arrangement  is referred to as mFSO configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Usually, the transmitters in  a bundle are considered to project signals in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' However,  because of the large principal beam, the supplementary serving  FSO transceiver projection directions are far from being perpen-  dicular to the ground, and their footprints are ellipses instead of  circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  To create a continuous coverage region, the footprint of the  principal serving FSO transceiver and those of the supplemen-  tary serving FSO transceivers should overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Therefore, there  should be a sufficiently large number of supplementary serving  FSO transceivers to cover entirely the contour of the principal foot-  print.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The extended coverage area is defined as the largest circle  covered by these footprints (Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The principal transceiver is  responsible for the region defined by its footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' A supplemen-  tary serving FSO transceiver is responsible for the part limited  by its footprint, principal coverage circle, and extended coverage  circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Let α be always the beam width of the principal serving  FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  To ensure that ground nodes under principal  coverage receive sufficient power, α should still respect constraint  (4), as in the single serving FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Let the beam width of a supplementary serving FSO transceiver  be β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In the responsible area of the supplementary transceiver, the  points on the extended coverage circle are the farthest from the  supplementary transceiver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' thus, they receive the least power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' If  these points receive at least ρrx, all other points receive sufficient  power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  It is easy to note that the footprints of the neighboring supple-  Figure 6: footprint of multiple FSO transceiver (mFSO) con-  figuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  mentary serving FSO transceivers join each other on the extended  coverage circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Let J be such a joint point, the power J receives  from the supplementary FSO transceiver is defined similar to (3)  but with beam width β, which is  P rx  J  = e−σ×LJ × Ptx × R2  rx  4L2  J  ×  2  1 − cos (β/2)  (6)  Thus, β is constrained by the condition P rx  J  ≥ ρrx, which gives:  e−σ×LJ  PtxR2  rx  2L2  J(1 − cos (β/2)) ≥ ρrx  (7)  Let us denote the extended coverage radius by Rext then  LJ =  �  H2 + R2  ext  (8)  Appendix B presents detailed calculations of LJ and Rext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The  calculations yielded the following results  Rext = H2 tan( ξ+α  2 ) − tan( α  2 )(1 − tan2( ξ+α  2 ))  1 − tan2( ξ+α  2 ) + 2 tan( ξ+α  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' tan( α  2 )  (9)  where  tan(ξ + α  2  ) = tan(γ) + tan(θ)  1 − tan(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' tan(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' cos( π  m)  tan(γ) = tan(α  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' cos( π  m)  tan(θ) =  �  sin2( β  2 ) − sin2( α  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' sin2( π  m)  cos( β  2 )  (10)  (11)  (12)  and m is the number of supplementary FSO transceivers set  around the principal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  We can remark that Rext and thus LJ depend on α, β and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Hereafter, Rext is sometimes denoted by Rext(α, m, β) and LJ by  LJ(α, m, β) to express these dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content="  4  Principle coverage circle  0  K'  K  2T  m  Extended coverage circle3  Problem of designing minimal cost  HAP network  There are several costs in a HAP network, such as investment, en-  ergy, and maintenance costs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Based on the expected life duration  and maintenance cycle of a HAP, these costs can be distributed  by day as 1) daily amortization cost representing investment cost,  2) average daily maintenance cost, and 3) daily energy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Con-  sequently, the problem of minimizing network cost becomes min-  imizing the daily network cost, which comprises these three com-  ponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Following variables are introduced for formulating mathemati-  cally the daily network cost:  K: Number of HAPs in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The HAPs are indexed  by i ∈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='.K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  niF  i : Number of FSO transceivers used on HAPi for inter-  HAP communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  nsF  i : Number of serving FSO transceiver of HAPi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Let ζday  H  and ζday  F  be constants that express the daily amortiza-  tion costs of a HAP and an FSO transceiver, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' These  costs are defined as the ratio of the prices of the HAP or FSO  transceiver to their expected lifetime duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Then, the overall  daily amortization cost of the HAP network is:  Kζday  H  + (  K  �  i=1  nsF  i  +  K  �  i=1  niF  i )ζday  F  (13)  To evaluate the daily maintenance and energy costs, we need to  consider the HAP design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' HAPs are classified into two categories  based on the underlying physical principle that provides the lifting  force for the HAPs: aerodynamic (the HAP is heavier than air)  and aerostatic (the HAP is lighter than air).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  While aerostatic  platforms use buoyancy to float in the air, aerodynamic platforms  use dynamic forces created by movement through the air [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In  general, both aerostatic and aerodynamic systems require a “flying  energy” to keep the HAP relatively stable for maintaining FSO  communication between HAPs and that between HAPs and FSO  ground nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' An aerodynamic system requires a large propulsion  power to move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Aerostatic systems typically consume less energy  than aerodynamic systems do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' To be able to operate for a long  duration in space, HAPs are mainly unmanned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  HAPs are equipped with different energy resources such as on-  site production (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=', solar energy harvested by solar panels) or  rechargeable energy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=', batteries or fuel cells brought from the  ground).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Solar energy-based HAPs can operate continuously in  space until they are lowered for maintenance purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Recharge-  able energy-based HAPs are lowered once the reserved energy is  depleted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In brief, the continuous in-space working duration of a  HAP is limited by its available energy, which is relatively fixed by  the HAP design, its energy consumption level, which varies de-  pending on the payload weight and communication of the HAP,  and its maintenance cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  We define the maintenance cost of a HAP as the expense of low-  ering the HAP to perform technical maintenance, energy recharge  on the ground, and then reinstall it in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Let di be the number of days on which HAPi can operate con-  tinuously in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Let ζmtn be constant expressing the cost of  one time lowering a HAP, maintaining it, recharging it, and then  reinstalling it in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The daily maintenance cost of the HAP  network is  K  �  i=1  ζmtn  di  (14)  Regarding the daily energy cost, we consider solar energy to be  free, whereas the solar panel cost is counted in the cost of the HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The cost of rechargeable energy is part the maintenance cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' As a  result, the energy cost does not explicitly represent the total cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Nonetheless, the energy consumption level of a HAP affects its  in-space working duration di;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' therefore, we analyze this in Section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Combining (13) and (14), we obtain the following overall daily  cost of the HAP network:  Cost = Kζday  H  + (  K  �  i=1  nsF  i  +  K  �  i=1  niF  i )ζday  F  +  K  �  i=1  ζmtn  di  (15)  The problem of minimizing daily cost of the HAP network is  stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Given input parameters including  – NFSO: Set of ground FSO nodes and their coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The number of nodes in the set is denoted as |NFSO|,  – M:  Data traffic to be carried between ground FSO  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' This is the list of bandwidth demands between  the ground nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Outputs to seek are  – A HAP network with HAP locations and inter-HAP  links,  – Beam width to set to each serving FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Optimization objective is  – Minimizing the daily cost expressed in (15) of the HAP  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The following two remarks drive us to conduct further analyses  in subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  First, if a HAP has self-sufficient solar  energy, its in-space working duration di is not limited by its energy  consumption but depends uniquely on the maintenance cycle of the  HAP, which is usually constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In Section 4, we show the daily  energy consumption of a HAP and the constraint that a HAP  needs to respect to rely solely on solar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Second, the cost of the HAP network increases with an increase  in the number of FSO transceivers and HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The number of  HAPs can be reduced by increasing ground coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' To increase  ground coverage, more serving FSO transceivers can be used on  each HAP, but this introduces greater energy consumption and ex-  tra amortization cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Section 5 focuses on identifying the optimal  configuration for serving FSO transceivers on a HAP to achieve a  minimal HAP network cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  4  Daily energy consumption of a  HAP with payload  Several parameters affect the power consumption of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The  descriptions and notations of these parameters are listed in section  Energy parameters of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Most parameters were set based on  industrial experimental projects such as the Loon project [4], Stra-  tobus project [6], and other studies listed in the reference column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1 presents the choice of parameter values in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  5  Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  nota-  tions  Descriptions  Values  References  Cost related parameters  ζday  H  Daily amortization cost of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  100  ζday  F  Daily amortization cost of an FSO transceiver on HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  10  ζmtn  Cost of one-time maintenance of a HAP including lowing it down,  1000  maintenance, charging and reinstall it in the stratosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Dm  Maintenance cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  365 days  [6]  Energy parameters  Esolar  Minimum daily harvested solar energy by a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  42 - 290 kWh  [12]  ρavion  Power consumed by the avionic part of a HAP to carry an unit of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  2 W/kg  ρHCM  F  Power for heating, cooling, and management for each FSO on HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  20 W  [4]  ρPAT  Power consumed by a PAT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  15W  [13]  ρinter  F  Power consumed by inter-HAP FSO transceivers for laser source (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1 W),  heating/cooling/management (20 W) and PAT (15 W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1 W  [4]  Inter-HAP FSO link parameters  C2  n  Atmosphere structure parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='0 × 10−18m−2/3 Attenuation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='5 × 10−6m−1  [4] Coupling loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  45 dBm Transmitted power of an inter-HAP FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1 W  [4] Receiver aperture diameter of an inter-HAP FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='037 m  [4] Beam width of an inter-HAP FSO transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  280 µrad  [4]  HAP-ground link parameters and variables  σ  Attenuation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='5 × 10−6m−1  ρFSO  tx  Transmitted power of the laser source of a serving FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  1 Watt  Rrx  Receiver aperture radius of a ground FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='05 m  SONABeam [1]  ρrx  Required received power at a ground FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='10−8 W  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1 dBm in [4]  Other parameters  H  Elevation of HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  20 km  LHH  Maximum length of an inter-HAP link so that its BER is under δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  88 km  δ  BER threshold for inter-HAP links and lightpaths between HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  W  The number of wavelengths in WDM technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' 80  µH  Platform mass excluding FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='5 kg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' 500 kg  [4]  µF  FSO transceiver mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='3 kg  [4]  Table 1: Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Greek characters are used for denoting constant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Let us consider the power consumption of a single HAP Hi  that has m serving FSO transceiver and niF  i  inter-HAP FSO  transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The power consumption includes:  P avion  Hi  : Power draw of avionic part for maintaining Hi with  payload in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  P down  Hi  : Power draw of all serving FSO transceivers on HAP  Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  This power includes the heating/cooling/management  power, laser transmitted power of all serving FSO transceivers  on the HAP, and the power consumed by the Pointing Acqui-  sition and Tracking (PAT) system of the HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  P inter  Hi  : Power draw of all inter-HAP FSO transceivers on  HAP Hi for inter-HAP communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The power includes  the heating/cooling/management, and PAT power for each  inter-HAP FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Inter-HAP FSO transceivers  are oriented towards different remote HAPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' therefore, each  transceiver must have a PAT system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The total daily energy consumption (by 24 hours) of Hi is  Econsum = (P avion  Hi  + P down  Hi  + P inter  Hi  ) × 24  (16)  To breakdown further P avion  Hi  , P down  Hi  , and P inter  Hi  , we introduce  following parameters:  ρavion: Power consumed by the avionic part of the HAP to  carry a unit of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  ρFSO  tx  : Transmitted power of each serving FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Because the current power of laser source is limited to 1 W,  which is very small in comparison with the power consumed  by other factors on the HAP, we consider that ρFSO  tx  = 1 W,  regardless of the beam width of the serving FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  ρHCM  F  : Power draw for heating, cooling, and management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' It  is also considered constant for each serving FSO transceiver  and is set to ρHCM  F  = 20 W, according to reference [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  ρPAT : Power draw for Pointing, Acquisition and Tracking  activity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' it is another constant and is set to ρPAT = 15 W  [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' A HAP system uses a single PAT for its set of serving  FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  ρinter  F  : Power draw of a single inter-HAP FSO transceiver  including communication, heating, cooling, management, and  6  PAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' According to [4], 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1 W laser power is sufficient for an  inter-HAP communication of 100 km distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In this study,  we limited the inter-HAP link length to less than 100 km and  considered the laser power constantly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1 W regardless of the  distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Therefore, ρinter  F  = ρHCM  F  + ρPAT + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  µH: Mass of the HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  µF: Mass of an FSO on the HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Assuming that P avion  Hi  is linearly proportional to the weight of  the HAP by ρavion,  P avion  Hi  = [µH + (nsF  i  + niF  i )µF]ρavion  (17)  P down  Hi  is  the  sum  of  the  power  consumed  by  serving  FSO transceivers and PAT activity of the HAP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' thus,  P down  Hi  = nsF  i (ρFSO  tx  + ρHCM  F  ) + ρPAT  (18)  P inter  Hi  is the sum of the power consumed by inter-HAP FSO  transceivers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' thus,  P inter  Hi  = ρinter  F  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='niF  i  (19)  Substituting (17), (18), and (19) into (16), we obtain the daily  power consumption of a HAP as  Econsum = {[µH + (nsF  i  + niF  i )µF]ρavion  + nsF  i (ρFSO  tx  + ρHCM  F  ) + ρPAT  + ρinter  F  niF  i } × 24  (20)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1  Necessity of solar energy and utilization  constraint  Current HAPs mainly use energy from solar panels mounted on  HAP wings and/or energy from batteries or hydrogen fuel cells  (HFC) onboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Solar energy can be harvested and charged into  batteries during the day for nighttime use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Harvested solar energy  varies with year time and location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' According to the experiments  in [12], in York, UK, the harvested solar power is 42–80 kWh/day,  and in Enugu, Nigeria, it is 290–545 kWh/day, depending on the  size of the solar panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Figure  7  depicts  the  total  daily  energy  consumption  of  a HAP, calculated from (20), versus the number of serving  FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Parameters were ρavion = 2 W/kg, ρPAT =  15 W, HAP weights µH = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='5 kg or 500 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The HAP carried  10 inter-HAP FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The referenced daily solar en-  ergy levels were the minimum daily solar energy levels in York  and Enugu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' From a certain number of serving FSO transceivers,  a HAP consumes more energy than the harvested solar energy  in York, and an HFC would be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Owing to the limited  payload capacity of a HAP, its HFC capacity is also very limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  According to [8], the current state-of-the-art fuel-cell density is  approximately 1600 Wh/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' A lightweight HAP, such as a Google  balloon weights 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='5 kg, cannot carry heavy long-lasting fuel cells  on board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The larger HAP Stratobus can carry up to 450 kg, but  it weights already 7 tons leading to high energy consumption for  flying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Even if the Stratobus payload capacity is reserved for the  HFC, its energy would quickly run out within a few days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Based on this observation, we believe that long-duration flights  should consider solar energy as the principal energy source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In this  case, the power consumption of a HAP with payload must not  0  50000  100000  150000  200000  250000  300000  350000  0  20  40  60  80  100  Total daily energy consumption (W-hr)  Number of serving FSO transceivers  µΗ=28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='5 kg  µΗ=500 kg  Min solar energy at York  Min solar energy at Enugu  Figure 7: Energy consumption by a HAP with different num-  ber of serving FSO transceivers in comparison with the min-  imum harvested solar energy at York and Enugu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' ρavion =  2/kg W and ρPAT = 15 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  exceed the daily harvested solar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Let the daily harvested  solar energy be Esolar;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' then,  �  [µH + (nsF  i  + niF  i )µF]ρavion + ρPAT  + nsF  i (ρFSO  tx  + ρHCM  F  ) + ρinter  F  niF  i  �  ≤ Esolar  24  (21)  According to Figure 7, solar energy provision does not need to  be very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' A solar energy level between the minimum harvested  in York and Enugu allows a 500 kg HAP to carry at least 6 serving  FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' A HAP can carry hundreds FSO transceivers  with more than 125 kWh solar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Therefore, it is realistic to  rely on the solar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Hereafter, we consider that HAPs solely  use solar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Despite self-sufficient solar energy, HAPs still need to be lowered  periodically for maintenance, for example, after one year in the  case of Stratobus [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Let us denote the maintenance cycle as a  constant Dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Then  di = Dm,  ∀i ∈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='.K  (22)  5  Optimal mFSO configuration  Using multiple serving FSO transceivers increases the expense of  FSO transceivers, although it can reduce the expense of HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  This section aims to determine the mFSO configuration that min-  imizes the HAP network cost defined in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' We assume that all  HAPs use identical mFSO configurations, that is, identical prin-  cipal beam width α, supplementary beam width β and number of  supplementary serving FSO transceivers m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Let us now consider the dependence of the HAP network cost on  mFSO configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' As each HAP has m supplementary serving  FSO transceivers and uses only solar energy, the cost (15) becomes  Cost = Kζday  H  + (Km +  K  �  i=1  niF  i )ζday  F  + Kζmtn  Dm  Cost is a function of K, m and niF  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' K depends on the coverage  radius Rext(α, m, β) of the mFSO configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' niF  i , as the num-  ber of inter-HAP links of HAP i, depends on the traffic demand  7  set M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Hence, Cost depends on mFSO configuration and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' It  is difficult to determine the optimal mFSO configuration without  considering M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' To relax the dependance on M, we estimate Cost  by a function that depends solely on mFSO configuration, that is,  tuple (α, m, β);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' then try to find an instance (α, m, β) minimizing  the estimated cost in expecting that the instance also drives the  real cost to a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1  Cost estimation  Figure 8: A ground area is divided into grid of square cells;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  each cell is circumscribed by a circle representing a serving  zone of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  First, we estimate the number of HAPs K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Samples of the  estimation are datasets with uniformly distributed ground nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Let S be the surface of the ground area containing those nodes,  and W the number of wavelengths in the WDM technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' We  divide the ground zone S into a grid of square cells of size ℓ × ℓ,  each one will be covered by a HAP (see Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' To be served  by a HAP, a cell must satisfy the following two conditions:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' A cell can contain at most W ground nodes because a HAP  can use at most W wavelengths to serve ground nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Owing  to the uniform distribution of ground nodes, we have  ℓ2  S |NFSO| ≤ W  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' A cell must be contained inside by a circle radius equivalent  to the extended radius Rext of a HAP  ℓ ≤  √  2Rext  The maximum number of HAPs required to cover region S is the  number of cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Let this number be ˆK, then,  ˆK = S  ℓ2 = ⌈max {|NFSO|  W  ,  S  2R2  ext  }⌉  (23)  Hence, ˆK is an overestimation of the number of HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Next, we estimate the value of niF  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Let V be the maximum  number of inter-HAP links that a HAP may have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Then  niF  i  ≤ V, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Finally, Cost can be overestimated as:  �  Cost = ˆK  �  ζday  H  + (m + V + 1)ζday  F  + ζmtn  Dm  �  (24)  �  Cost is a function of Rext(α, m, β) and m while V is a parameter  of the estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The estimation is more precise when V is set close  to the actual number of inter-HAP links of a HAP, and coarser  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='2  Algorithms finding optimal configura-  tion  Given α and m, a larger β results in a larger Rext, and thus a  smaller ˆK and �  Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Therefore, β should be set to the largest value  according to (7) for a given α and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' It is worth noting that the  value of β does not affect the solar energy consumption because the  laser power ρFSO  tx  is small and is considered constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Determining  the optimal configuration becomes finding the optimal values of α  and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Algorithm 1 Find the optimal mFSO configuration  1: function Find-optimal-mFSO  2:  niF  i ← V  3:  cMin ← ∞  ▷ cost min  4:  αMax ← maximum α by (4)  5:  for α = αMax .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' 0 do  6:  mMax ← calculated by (25)  ▷ max m  7:  mOpt ← 0  ▷ optimal m  8:  for m = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' mMax do  9:  β ← Beta-max(α, m)  ▷ max β  10:  Calculate Rext(α, m, β) using (9),(10),(11) (12)  11:  Calculate �  Cost(α, m, β) using (24)  12:  if �  Cost < cmin then  13:  cmin ← �  Cost  14:  αOpt ← α  ▷ optimal α  15:  mOpt ← m  ▷ optimal m  16:  βOpt ← β  ▷ optimal β  17:  end if  18:  end for  19:  end for  20:  return αOpt, mOpt, βOpt  21: end function  Algorithm 2 Find the maximum β given α, m  1: function Beta-max(α, m)  2:  for β = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' 180 do  3:  Calculate Rext(α, m, β) using (9),(10),(11) (12)  4:  Calculate LJ using (8)  5:  Calculate P rx  J  using (6)  6:  if P rx  J  ¡ρrx then ▷ Looking for the first β violate  constraint (7)  7:  return β-1  ▷ the previous trial β was the  maximum  8:  end if  9:  end for  10: end function  Following an exhaustive search approach, we examine all possi-  ble values of α and m to seek for the pair that minimizes �  Cost  in (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The search range of α is from 0° to the maximum  8  Serving zone  of a HAPvalue set by constraint (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The number of supplementary serving  FSO transceivers m is also limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Indeed, since the number of  inter-HAP links of a HAP can go up to V as set in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1, and  nsF  i  = m + 1, ∀i, then from the energy constraint (21), we deduce  the upper bound for m:  m ≤  Esolar  24  − (VµFρavion + Vρinter  F  + µHρavion + ρPAT)  µFρavion + ρHCM  F  + ρFSO  tx  − 1 (25)  Algorithm 1 implements the exhaustive search idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' First, two  nested loops scan all possible values of α satisfying constraint (4)  and all possible values of m satisfying (25) to find the pair that  minimizes �  Cost in (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' For each pair (α, m), the largest value  of β according to constraint (7) is selected using Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The optimal mFSO configuration is reported by the algorithms as  (αOpt, mOpt, βOpt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Algorithm 2 finds the maximum β that satisfies constraint (7)  for a given pair of (α, m) by testing the possible values of β in-  creasingly from 0 until the received power P rx  J  at the border of  the extended coverage area reaches the required received power  ρrx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The received power P rx  J  is calculated using the set of equa-  tions (6), (8), (9),(10),(11), and (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  In the implementation of both algorithms, α and β step by 1°  after each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Finer stepping allows obtaining more accu-  rate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' However, even with 1° stepping, the variation in the  optimal Rext is only a few hundred meters, which is negligible in  comparison to the absolute value of Rext which is in the range of  6-30 kilometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The complexity of Algorithm 1 is O(m) because α ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The  complexity of Algorithm 2 is constant because β ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  6  Design HAP network topology  This section presents the HAP network design using the optimal  configuration identified above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Let denote Linter as the number  of inter-HAP links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Since �K  i=1 niF  i  is the total number of inter-  HAP FSO transceivers, it is equal to 2Linter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The network cost  becomes:  Cost = Kζday  H  + (K(m + 1) + 2Linter)ζday  F  + K ζmtn  Dm  and is equivalent to  Cost = K  �  ζday  H  + (m + 1)ζday  F  + ζmtn  Dm  �  + 2Linterζday  F  (26)  The cost is proportional to the number of HAPs K and the  number of inter-HAP links Linter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  We consider that the daily  amortization cost of a HAP is much greater than that of an FSO  transceiver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' thus, the coefficient of K is much greater than the  coefficient of Linter in Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Consequently, K should be prioritized  to minimize over Linter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Therefore, the topology design is broken  into following two steps:  i) ground nodes are clustered into equal radius circles that will  become serving zones of HAPs in such a way that the number  of clusters is the smallest for minimizing K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  ii) corresponding HAPs are located at the centers of clusters but  at an elevation of 20 km and are interconnected by the fewest  number of inter-HAP links, Linter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  A HAP network topology design algorithm was proposed in [7]  following these two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In this algorithm, the clustering radius  was not determined but was left as an input of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In  the current study, we set the clustering radius as the extended  coverage radius Rext of the optimal mFSO configuration to drive  towards a HAP network with minimal Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The main steps of the  Figure 9: HAP network design flowchart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  HAP network design process are presented in Figure 9, where the  steps taken from [7] are shown in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The process is explained  as follows:  Initialize V, the maximum number of inter-HAP links of a  HAP, by a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Calculate the optimal mFSO configuration using Algorithm  1, and set the clustering radius as its Rext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Apply the clustering algorithm proposed in [7] to distribute  ground nodes into clusters of radius Rext while keeping the  number of ground nodes in each cluster under W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Each cluster  becomes a serving zone of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The HAP is located at the  center of the cluster but at an elevation of 20 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Bandwidth demands between ground nodes belonging to dif-  ferent serving zones are bundled into lightpaths between  corresponding HAPs, creating the inter-HAP traffic matrix  MHAP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Apply HAP topology design algorithm proposed in [7] to build  the HAP topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The algorithm begins with an empty  topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' It finds a route for each lightpath demand of MHAP  9  Init V  Find optimal conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  (α, m, β) and Rext  Clustering ground nodes  with Rext radius [7]  V=V+1  Calculate inter-HAP  Design HAP topo [7]  M  HAF  No  is  routed entirely  Report HAP topofrom a full-mesh graph linking all HAPs within communica-  tion distance limit LHH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Each time a lightpath uses an inter-  HAP link that has not yet been included in the current HAP  topology, the link is incorporated into the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The link  in the topology is prioritized for use in building routes for the  next lightpath demands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Once all lightpath demands in MHAP are routed, the final  topology is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Otherwise, routing may fail due to the  low connectivity between HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In this case, V is increased  by one, and the process is repeated until all lightpath de-  mands in MHAP are routed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  7  Simulation results  The algorithms for finding the optimal mFSO configuration were  implemented and integrated with the topology designed algorithm  described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' We performed simulations with practical  parameters and evaluated the efficiency of mFSO configuration  compared to the single serving FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1  Parameter values  The simulation parameters are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The values of  these parameters were chosen according to experiments reported  in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' This subsection explains the choices of the pa-  rameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Cost-related parameters:  The cost-related parameters are set  such that the daily amortization cost of a HAP is significantly  greater than that of an FSO transceiver, and the one-time mainte-  nance cost is significantly higher than the daily amortization cost  of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The maintenance cycle of a HAP is set as Dm = 1 year  according to published information on Stratobus [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Energy-related parameters:  Esolar - daily harvested solar energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  We considered daily  solar energy levels between the minimum daily solar energy  values in York and Enugu reported in [12], which were 42  kWh and 290 kWh, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  ρavion - power consumed by the avionic part of a HAP to  carry a unit of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Although the power-to-mass ratio can  be estimated as 6 W/kg according to [12], the published power  rates of real systems are smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' For aerodynamic systems  such as Zephir-S, Zephir-T [14], and Phasa-35 [15], ρavion  varies from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='68 -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='04 W/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Indeed, Zephir-S weighs 80 kg  (75 kg platform and 5 kg payload) and consumes 243 W,  Zephir-T weighs 160 kg (140 kg platform and 20 kg payload)  and consumes 429 W, and Phasa-35 weighs 165 kg (150 kg  platform and 15 kg payload) and consumes 459 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Aerostatic  systems consume even less power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The Stratobus weighs 7000  kg and consumes 5 kW when it carries a 250 kg payload  and 8 kW when it carries 450 kg [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Thus, the power-to-  mass ratio of Stratobus is between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='69 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='07 W/kg only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Therefore, in this simulation ρavion was set to 2 W/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  ρPAT - power consumed by a PAT system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' it was set to 15 W  according to [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  ρHCM  F  power consumed for heating, cooling, and manage-  ment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' it was set to 20 W according to [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  ρinter  F  power consumed by an inter-HAP FSO transceiver;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  it was set to 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1 W including laser power, ρHCM  F  and ρPAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Inter-HAP link parameters: These parameters were set to values  similar to those provided in the Loon project [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  HAP-ground FSO link parameters: The attenuation coefficient  of an FSO link between a HAP and a ground node is set identical  to that of inter-HAP links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The required received power ρrx at a  ground node was set according to [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The aperture radius Rrx of  a ground FSO receiver was set according to the commercial FSO  transceiver SONABeam [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Other parameters:  δ - BER threshold for inter-HAP links and lightpaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' We  set δ = 10−3 because errors with that BER can be corrected  using current Forward Error Correction (FEC) techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  LHH - the maximum allowable distance between two HAPs  such that the BER of an inter-HAP link is less than δ = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Using the inter-HAP FSO link parameters listed in Table 1,  the calculation yielded LHH = 88 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  µH - platform mass;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' it varies significantly from one design  to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The Loon balloon weighs just 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='5 kg while the  Stratobus weighs 7000 kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' With ρavion = 2 W/kg, a HAP  weighing more than 7000 kg already consumes 326 kWh/day  to carry itself, which is more than the maximum harvested  solar energy, leading to no remaining energy to carry FSO  transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Therefore, µH = 500 kg was used in the simula-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  µF - mass of an FSO transceiver on HAPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' It was set accord-  ing to the FSO transceiver used in the Loon project, which  weighs 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='3 kg [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' This value is consistent with the weights  between 8 and 10 kg of commercial terrestrial SONABeam  FSO transceivers [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  W - the number of wavelengths per FSO link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' It was set to  40 or 80 according to the current WDM technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The test dataset contained 19 test cases, each with 400 – 2800  ground nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The ground FSO node locations were randomly  generated on a square surface of 100 × 100 km, which is the size  of a large metropolis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The test cases had different numbers of  ground nodes, reflecting different ground node densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The traf-  fic requirement M contained demands randomly generated between  ground FSO nodes such that the total incoming or outgoing traffic  of a ground FSO node did not exceed 1 Gbps, which is the capacity  of a single wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Initially, V was set to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The optimal multiple serving  FSO transceiver configuration (α, m, β) was calculated using Algo-  rithms 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The extended radius Rext of the optimal configura-  tions was calculated using (9) and was then used as the clustering  radius in the HAP topology design step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  With Esolar = 42 kWh and W = 40, V must be increased to 12  to get all demands in MHAP routed successfully for all test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  With all other Esolar and W values, the topology design algorithm  successfully routed all demands in MHAP for all test cases right  with initial V = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Figure 10 illustrates the HAP locations and their footprints  calculated using the proposed algorithms for a test case of 1005  ground FSO nodes, Esolar = 75 kWh, and W = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='2  mFSO configuration versus single serv-  ing FSO transceiver configuration  Table 3 lists the maximum beam width αmax according to (4)  and the maximum ground coverage radius of the single serving  FSO transceiver configuration when the receiver aperture radius  10  Esolar = 42 kWh  Esolar = 50 ∼ 290 kWh  W = 40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' V = 12  W = 80,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' V = 10  W = 40,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' V = 10  W = 80,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' V = 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='|NFSO|  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
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+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='11929  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='11178  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Table 2: Optimal configurations and costs of all test cases with Rrx = 2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Receiver aperture  Maximum beam  Maximum  radius Rrx (m)  width αmax  coverage radius (m)  2  37 °  6691  4  67 °  13237  Table 3: Maximum beam width and coverage radius of single  serving FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  was varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Table 4 lists the extended coverage radius of the max-  imum mFSO configuration for different solar energy levels and  receiver aperture radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The maximum mFSO configuration was  obtained using the largest principal beam αmax, largest m accord-  ing to (25), and largest β according to (7), given αmax and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The coverage radius of the maximum mFSO configuration was ex-  tended approximately twice in comparison with that of single FSO  transceiver configuration, except for Esolar = 42kWh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' When solar  energy level increased, the maximum m increased;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' thus, the ex-  tended coverage radius increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' However, when m was already  large, the extention increased slowly with m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Additionally, the  maximum extended coverage was much larger when Rrx = 4 than  Rrx = 2m because a receiver can accept weaker signals with larger  apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  To compare the network costs incurred by the two configura-  tions, we examined the detailed results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The table  lists the optimal mFSO configurations and network costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' When  Esolar = 42kWh, the optimal number of supplementary serving  FSO transceivers is m = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' thus, the configuration uses a single  serving FSO transceiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Therefore, these cases were used as ref-  erences for single serving FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  When  Esolar > 50kWh, all optimal configurations were truly mFSO,  and the results were identical for all solar energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The  numbers indicate that mFSO configuration offered significantly  Esolar  Max Rext (m)  (kWh)  Max m  Rrx = 2 (m)  Rrx = 4 (m)  42  6  6691  13237  50  16  12174  25582  75  47  13559  28403  100  78  13678  28845  125  109  13711  28969  150  140  13724  29020  175  171  13731  29047  200  202  13735  29062  225  233  13738  29071  250  264  13739  29077  275  295  13740  29082  290  314  13741  29084  Table 4: Maximum extended coverage radius of mFSO con-  figuration when V = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  lower costs (listed in columns 16th and 21th) than those of single  serving FSO transceiver configuration (listed in columns 6th and  11th) for the same test cases and number of wavelengths W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The  costs resulting from mFSO configuration were as low as 54–87% of  those resulting from single serving FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  These numbers confirm that when there is sufficient solar energy,  mFSO configuration is definitively a better choice than single serv-  ing FSO configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  11  0  20  40  60  80  100  120  0  20  40  60  80  100  y-axis  x-axis  Figure 10: Footprints of HAPs with mFSO configuration ob-  tained from the topology design for a test case of 1005 ground  FSO nodes when Esolar = 75 kwh, W = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' A circle repre-  sents an extended coverage area of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Small points in-  side the circle are ground nodes and the dot at the center of  the circle is the projected location of its serving HAP on the  ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='3  Factors impact optimal mFSO configu-  ration  Comparing the values of the optimal extended coverage radius in  Table 2 and the maximum extended coverage radius in Table 4,  we can see that the optimal extended coverage radius was gener-  ally not the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' This is reasonable because the maximum  configuration uses an excessive number of supplementary serving  FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Low solar energy may render mFSO configuration impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Indeed, Esolar = 42 kWh could afford maximally 6 supplemen-  tary serving FSO transceivers (see Table 4), which was too few to  entirely cover the contour of the principal coverage area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Thus,  single FSO transceiver configuration was the unique choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  When the solar energy level exceeds 50 kWh, its exact value does  not affect the optimal configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The simulation showed that  the optimal configurations were identical for all solar energy levels  from 50 kWh/day and above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' This is explained by the fact that  a greater solar energy level allows to accept configurations with  large coverage but may be more expensive because of using more  supplementary serving FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' As a result, large con-  figurations were not selected as optimal configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In other  words, increasing solar energy does not necessarily improve the  HAP network cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Since the optimal multiple serving FSO transceiver configura-  tions were identical for all Esolar ≥ 50 kWh, all other numerical  results related to topology design and routing with these solar en-  ergy levels were identical and are presented as single results in  subsequent figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The coverage of the optimal configurations decreased when the  ground nodes became denser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Indeed, test cases with large num-  bers of ground nodes had greater ground node densities, and  columns 13th and 16th of Table 2 shows that the optimal m and  Rext decreased when the density increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The reason is that,  with a greater ground node density, is a small ground region al-  ready contains W ground nodes, which is the maximum serving  capacity of a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Therefore, a HAP could serve only a small  zone and required only a few supplementary FSO transceivers to  cover the zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Numbers of HAPs and inter-HAP links  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='120  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='140  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='160  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='2500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='3000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Number of HAPs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Number of ground FSO nodes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='ˆK for Esolar=42 kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='K for Esolar=42 kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='ˆK for Esolar ≥ 50 kWh kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='K for Esolar ≥ 50 kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Lower bound  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='(a) W=40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='120  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='140  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='160  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='2500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='3000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Number of HAPs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Number of ground FSO nodes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='ˆK for Esolar=42 kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='K for Esolar=42 kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='ˆK for Esolar ≥ 50 kWh kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='K for Esolar ≥ 50 kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Lower bound  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='(b) W=80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Figure 11: Number of HAPs and lower bound with (a) W =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='40 and (b) W = 80 in different solar energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Since each HAP can serve at most W ground FSO nodes, a lower  bound for the number of HAPs is:  nLB  HAP = |NFSO|  W  (27)  Figure 11 shows the number of HAPs, the estimated number of  HAPs ˆK and lower bound nLB  HAP when (a) W = 40 and (b) W = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  With Esolar ≥ 50 kWh, the actual number of HAPs was almost  identical to ˆK in both subfigures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Furthermore, when W = 40  12  0  100  200  300  400  500  600  0  500  1000  1500  2000  2500  3000  Number of inter-HAP links  Number of ground FSO nodes  Esolar=42 kWh  Esolar ≥ 50 kWh  (a) W=40  0  100  200  300  400  500  600  0  500  1000  1500  2000  2500  3000  Number of inter-HAP links  Number of ground FSO nodes  Esolar=42 kWh  Esolar ≥ 50 kWh  (b) W=80  Figure 12: Number of inter-HAP links when (a) W = 40 and  (b) W = 80 for different solar energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  and Esolar ≥ 50 kWh, the number of HAPs approached the lower  bound starting from test cases with 1000 ground nodes or above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  This implies that the number of HAPs was almost optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Figure 12 presents the absolute numbers of inter-HAP links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The number of inter-HAP links increased with the number of  ground nodes, because the network size and traffic demand in-  creased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The number of inter-HAP links clearly decreased when  the wavelength density increased from W = 40 to W = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In  other words, denser WDM technique helps reduce the number of  inter-HAP FSO transceivers and consequently the network cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  mFSO configuration allows reducing significantly both the num-  bers of HAPs and inter-HAP links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Indeed, according to Figure  11, the number of HAPs was much smaller with Esolar ≥ 50  kWh where mFSO configuration was used, in comparison with  Esolar = 42 kWh, where single serving FSO configuration was  used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' A similar phenomenon is observed in Figure 12 for the num-  ber of inter-HAP links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
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+page_content='35000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='40000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='2500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='3000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Cost  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Number of ground FSO nodes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Cost of Esolar = 42 kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Cost of Esolar = 42 kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Cost of Esolar ≥ 50 kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Cost of Esolar ≥ 50 kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='(a) W=40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='5000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='10000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='15000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='20000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='25000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='30000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
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+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='2000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='2500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='3000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Cost  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Number of ground FSO nodes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Cost of Esolar = 42 kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Cost of Esolar = 42 kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Cost of Esolar ≥ 50 kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Cost of Esolar ≥ 50 kWh  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='(b) W=80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='Figure 13: Real costs and overestimated costs with W = 40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='and W = 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='5  Quality of cost estimation  Figure 13 presents the estimated and actual costs for different  solar energy levels and wavelength densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The estimated cost  was very close to the actual cost, mostly for Esolar ≥ 50kWh and  W = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Parameter V, the threshold of the number of inter-HAP links  of a HAP, affects the quality of the cost estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' To evaluate  the choice of V, we compared it with the number of inter-HAP  links that a HAP finally has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Figure 14 shows the average number  of inter-HAP links per HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' When there were 40 wavelengths  per link, the average number of inter-HAP links per HAP varied  between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='7 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='3 for Esolar ≥ 50 kWh and V = 10, and between  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='8 and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='8 for Esolar = 42 kWh while V raised up to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Hence,  the value of V was close to the actual number of inter-HAP links  required by a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' However, when there were 80 wavelengths per  link, the average number of Inter-HAP links per HAP was reduced  to between 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='4 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='4, which is slightly far from the threshold  V = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' A smaller V may help better estimate of the optimal cost  in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  13  0  5  10  15  20  0  500  1000  1500  2000  2500  3000  Average number of inter-HAP links per HAP  Number of ground FSO nodes  Esolar=42 kWh  Esolar ≥ 50 kWh  (a) W=40  0  5  10  15  20  0  500  1000  1500  2000  2500  3000  Average number of inter-HAP links per HAP  Number of ground FSO nodes  Esolar=42 kWh  Esolar ≥ 50 kWh  (b) W=80  Figure 14: Number of inter-HAP links per HAP when (a)  W = 40 and (b) W = 80 for different solar energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  8  Conclusions  Using mFSO configuration widens a HAP footprint, however, its  application is constrained by the available solar energy of the HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Moreover, mFSO configuration may imply an extra investment  cost due to additional serving FSO transceivers in comparison with  single FSO transceiver configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' This study focused on de-  termining the optimal mFSO configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' First, we proposed  a set of closed-form expressions for computing the coverage of  an mFSO configuration in terms of beam widths of the princi-  pal and supplementary transceivers and number of supplementary  FSO transceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Second, we proposed an algorithm to determine  the optimal mFSO configuration that minimizes the total HAP  network cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Third, we designed a HAP network topology using  the optimal configuration to achieve a minimal final cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The simulation results showed that mFSO significantly ex-  tended the HAP footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' With the testing dataset, the extended  footprint radii were generally two times larger than the single FSO  transceiver footprint radii, leading to a four-fold larger coverage  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The network cost with the optimal mFSO configuration  was as low as 54% of the network cost when using a single serving  FSO transceiver on a HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Acknowledgements  This research was funded by the Vietnam National Foundation for  Science and Technology Development (NAFOSTED) under grant  number 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='02-2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  References  [1] fSONA,  “SONABeam  2500-E+  model  specifications.”  http://fsona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Accessed Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Acampora and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Krishnamurthy, “A broadband wireless  access network based on mesh-connected free-space optical  links,” IEEE Personal Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' 6, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' 62–  65, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Zhang, “Proposal of free space optical mesh network ar-  chitecture for broadband access,” in 2002 IEEE Interna-  tional Conference on Communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Conference Proceed-  ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' ICC 2002 (Cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='02CH37333), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' 2142–2145  vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='4, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  [4] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Moision, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Erkmen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
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+page_content='  [14] Airbus,  “Zephir:  Persistance  and  flexibility.”  https://lf5422.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='com/wp-  content/uploads/2018/08/0296 18 2 zephyr datasheet e horizontal a4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='pdf,  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Accessed Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  [15] BAE Systems, “Phasa-35.” http://prismaticltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='uk/products/phasa-  35/, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Accessed Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  A  Proof of Lemma 1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Let x = cos(α/2), a = σH, and b =  PtxR2  rx  2H2  then  P rx  j (x) = e−a/x  bx2  (1 − x)  (28)  Calculate the derivative of P rx  j (x) we get  P  ′rx  j  (x) = e−a/x  �  a  1 − x + 2x − x2  (1 − x)2  �  b  (29)  Thus, the derivative of P rx  j (α) is  P  ′rx  j  (α) = P  ′rx  j  (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' (− sin(α))  (30)  Beam α is limited between [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='.π] because it orients to the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Thus, x ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='.1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Consequently, 1 − x > 0 and 2x − x2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  In addition, a, b > 0, then P  ′rx  j  (x) > 0 for all x ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='.1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Be-  cause − sin(α) < 0, ∀α ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='.π], thus, P  ′rx  j  (α) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Consequently,  P rx  j (α) decreases with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  B  Calculation of extended coverage  radius of mFSO configuration  This section identifies formulas that calculate the extended cover-  age radius of an mFSO configuration characterized by the princi-  pal beam width α, supplementary beam width β and number of  supplementary beams m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Conventionally, the coverage provided by a bundle of transmit-  ters is calculated as if the transmitters project perpendicular to the  ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In mFSO configuration, the principal beam in the center  is large, and it pushes the supplementary serving FSO transceiver  projection directions far from perpendicular to the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' These  supplementary beams form oblique cones that intersect with the  ground plane in ellipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Considering of the elliptical form adds  more complexity to the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  In Figure 15, H denotes the position of a HAP, and its projec-  tion on the ground plane is O, thus HO = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The principal beam  forms a right circular cone whose axis is HO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The cone intersects  the ground plane by a circle of radius Rα, which defines the prin-  cipal footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The beam of a supplementary FSO transceiver is  an oblique cone intersecting the ground plane by an ellipse that  defines the corresponding supplementary footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' The cone of  the supplementary beam intersects with the cone of the principal  beam by two lines: HK and HK′ where K and K′ are the two  intersection points of the principal and supplementary footprints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Thus, OK = OK′ = Rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  m supplementary FSO transceivers are arranged evenly around  the principal transceiver, each of which is responsible for extending  the coverage within an angle of 2π/m from the center O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  The  responsible angle of the supplementary FSO transceiver in Figure  15 is defined by rays −−→  OK and −−→  OK′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Thus, �  KOK′ = 2π/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Ray −−→  OK intersects with the supplementary beam cone at J,  then OJ is the radius of the extended coverage region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Readers  refer to Figure 6 for a complete view of the extended coverage  circle and the positions of K, K′ and J on the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Figure 15: Computation of the distance from supplementary  FSO transceivers and the border of extended coverage area  LJ in function of Beta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Since the principal beam width is α, then �  OHK = α/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Let the base plane containing K and K′ of the supplementary  beam cone cuts the cone axis at T, the primary cone axis HO at P,  and HJ at J1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Then �  THK = β/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In addition, the supplementary  cone intersects with this base plane by a circle containing K, K′  with center T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Let Rβ be the radius of the circle, then TK =  TK′ = Rβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Let M be the midpoint of KK′ then H, O, T, M belong to the  same plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Let ξ = �  KHJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=" The extended coverage radius is Rext = OJ =  15  H  β/2  LY  a/2  Supplementarycone base plane  K'  R  Pilm  a  Supplementary foot print  Principal  foot print  GroundHO tan(�  OHJ) = H tan( α  2 + ξ)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Thus,  Rext = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' tan  �  2(ξ + α  2  ) − α  2  �  Rext = H2 tan( ξ+α  2 ) − tan( α  2 )(1 − tan2( ξ+α  2 ))  1 − tan2( ξ+α  2 ) + 2 tan( ξ+α  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' tan( α  2 )  (31)  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='1  Calculation of tan( ξ+α  2 )  Let N be the midpoint of KJ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' As K and J1 are at the intersection  of the supplementary cone and its base plane, HK = HJ1, HN ⊥  KJ1, and HN is the angle bisector of �  KHJ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Therefore, �  NHK =  ξ/2, thus �  NHP = ξ+α  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In addition, since KO is on the base plane  of the principal cone, HO ⊥ KO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Thus, △PNH and △POK  are similar right triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Consequently, �  OKP = �  NHP = ξ+α  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Furthermore,  tan(ξ + α  2  ) = OP  OK = OP  Rα  (32)  Let �  OHM = γ and �  THM = θ Then �  OHT = θ + γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Because MO is on the base plan of the principal cone, MO ⊥  HO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' In addition, as PT is on the base plane of the supplementary  cone whose axis is HT then HT ⊥ PT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Consequently, △PTH and  △POM are similar right triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' We can deduce that �  PMO =  �  PHT = θ + γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Therefore,  tan(θ + γ) = OP  OM =  OP  Rα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' cos( π  m)  Combining with (32) we deduce :  tan(ξ + α  2  ) = tan(θ + γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' cos( π  m)  (33)  Thus  tan(ξ + α  2  ) =  tan(γ) + tan(θ)  1 − tan(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' tan(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' cos( π  m)  (34)  Since γ = �  OHM then, tan(γ) = MO  HO .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  From right triangle △OMK we have MO = OK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' cos( π  m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  From right triangle △HOK we have HO = OK/ tan( α  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Thus  tan(γ) = tan(α  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' cos( π  m)  (35)  It remains to calculate tan (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='2  Calculation of tan (θ)  Look at the right triangle △HTM, we can see that:  tan(θ) = TM  TH  (36)  Since K and K′ are on a circle centered at T, and M is the  midpoint of KK′ then △TMK is a right triangle, then  TM =  �  TK2 − KM 2 =  �  R2  β − R2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' sin2( π  m)  (37)  Easy to find that △THK is another right triangle then  TH = TK/ tan(β  2 ) = Rβ/ tan(β  2 )  (38)  Replacing (37) and (38) in to (36) we get  tan(θ)  =  �  R2  β − R2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' sin2( π  m)  Rβ/ tan( β  2 )  =  tan(β  2 )  �  1 − (Rα  Rβ )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' sin2( π  m)  (39)  From right triangle △HTK we obtain Rβ = HK sin( β  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  From right triangle △HOK we obtain Rα = HK sin( α  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  Replacing these values to (39), we obtain:  tan(θ) =  �  sin2( β  2 ) − sin2( α  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' sin2( π  m)  cos( β  2 )  (40)  Substituting the values of tan(γ) in (35) and tan(θ) in (40) into  (34), we obtain tan( ξ+α  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content=' Subsequently, replacing the obtained  tan( ξ+α  2 ) to (31) we get Rext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
+page_content='  16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFAT4oBgHgl3EQfpx3L/content/2301.08642v1.pdf'}
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+Noname manuscript No.
+(will be inserted by the editor)
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for
+Few-shot Action Recognition
+Xiang Wang · Shiwei Zhang · Zhiwu Qing · Zhengrong Zuo · Changxin Gao ·
+Rong Jin · Nong Sang
+Received: date / Accepted: date
+Abstract Few-shot action recognition is a challenging but
+practical problem aiming to learn a model that can be eas-
+ily adapted to identify new action categories with only a few
+labeled samples. Recent attempts mainly focus on learning
+deep representations for each video individually under the
+episodic meta-learning regime and then performing tempo-
+ral alignment to match query and support videos. However,
+they still suffer from two drawbacks: (i) learning individ-
+ual features without considering the entire task may result
+in limited representation capability, and (ii) existing align-
+ment strategies are sensitive to noises and misaligned in-
+stances. To handle the two limitations, we propose a novel
+Hybrid Relation guided temporal Set Matching (HyRSM++)
+approach for few-shot action recognition. The core idea of
+HyRSM++ is to integrate all videos within the task to learn
+discriminative representations and involve a robust match-
+ing technique. To be specific, HyRSM++ consists of two key
+components, a hybrid relation module and a temporal set
+matching metric. Given the basic representations from the
+feature extractor, the hybrid relation module is introduced
+to fully exploit associated relations within and cross videos
+in an episodic task and thus can learn task-specific embed-
+dings. Subsequently, in the temporal set matching metric, we
+carry out the distance measure between query and support
+Xiang Wang · Zhiwu Qing · Zhengrong Zuo · Changxin Gao (Corre-
+sponding author) · Nong Sang
+Key Laboratory of Ministry of Education for Image Processing and
+Intelligent Control, School of Artificial Intelligence and Automation,
+Huazhong University of Science and Technology
+E-mail: {wxiang, qzw, zhrzuo, cgao, nsang}@hust.edu.cn
+Shiwei Zhang
+Alibaba Group
+E-mail: zhangjin.zsw@alibaba-inc.com
+Rong Jin
+Twitter
+E-mail: rongjinemail@gmail.com
+videos from a set matching perspective and design a bidi-
+rectional Mean Hausdorff Metric to improve the resilience
+to misaligned instances. In addition, we explicitly exploit
+the temporal coherence in videos to regularize the matching
+process. In this way, HyRSM++ facilitates informative cor-
+relation exchanged among videos and enables flexible pre-
+dictions under the data-limited scenario. Furthermore, we
+extend the proposed HyRSM++ to deal with the more chal-
+lenging semi-supervised few-shot action recognition and un-
+supervised few-shot action recognition tasks. Experimental
+results on multiple benchmarks demonstrate that our method
+consistently outperforms existing methods and achieves
+state-of-the-art performance under various few-shot set-
+tings. The source code is available at https://github.
+com/alibaba-mmai-research/HyRSMPlusPlus.
+Keywords Few-shot Action Recognition · Set Match-
+ing · Semi-supervised Few-shot Action Recognition ·
+Unsupervised Few-shot Action Recognition
+1 Introduction
+Recently, the development of large-scale video bench-
+marks [8, 23, 6, 13, 24] and deep networks [88, 51, 18,
+89, 65, 52] have significantly boosted the progress of ac-
+tion recognition. To achieve this success, we typically re-
+quire large amounts of manually labeled data. However, ac-
+quiring these labeled examples consumes a lot of manpower
+and time, which actually limits further applications of this
+task. In this case, researchers look to alternatives to achieve
+action classification without extensive costly labeling. Few-
+shot action recognition is a promising direction to reduce
+manual annotations and thus has attracted much attention
+recently [112, 105]. It aims at learning to classify unseen
+action classes with extremely few annotated examples.
+arXiv:2301.03330v1  [cs.CV]  9 Jan 2023
+
+2
+Xiang Wang et al.
+...
+CNN
+CNN
+CNN
+...
+0.8
+0.1
+Query video
+...
+Support set
+Hybrid relation module
+Support: make coffee
+Query: make coffee
+Support: make coffee
+Query: make coffee
+
+
+Metric space 
+(support)
+Metric space 
+(quey)
+“pour water”
+“pour coffee powder”
+Temporal alignment
+Temporal set matching
+(b) 
+Time line
+Matching line
+
+
+(a)
+Pull
+Push
+Pull
+Push
+Fig. 1 (a) Concept of the proposed hybrid relation module. We adaptively produce task-specific video embeddings by extracting relevant discrim-
+inative patterns cross videos in an episodic task. (b) Example of make coffee, the current temporal alignment metrics tend to be strict, resulting in
+an incorrect match on misaligned videos. In contrast, the proposed temporal set matching metric involving set matching technique and temporal
+coherence regularization is more flexible in finding the best correspondences.
+To solve the few-shot data-scarcity problem, popu-
+lar attempts [112, 7, 68, 106] are mainly based on the
+metric-based meta-learning technique [86], in which a com-
+mon embedding space is first learnt via episodic training
+and then an explicit or implicit alignment metric is em-
+ployed to calculate the distances between the query (test)
+videos and support (reference) videos for classification in
+an episodic task. Typically, Ordered Temporal Alignment
+Module (OTAM) [7] adopts a deep feature extractor to con-
+vert an input video into a frame feature sequence indepen-
+dently and explicitly explores the ordered temporal align-
+ment path between support and query videos in this feature
+space. Temporal-Relational CrossTransformer (TRX) [68]
+learns a deep embedding space and tries to exhaustively con-
+struct temporally-corresponding sub-sequences of actions to
+compare. Some recent works [33, 94, 108, 62] propose to
+design multi-level metrics for few-shot action recognition.
+Although these methods have achieved remarkable per-
+formance, there are still two limitations: individual feature
+learning and inflexible matching strategy. First, discrimina-
+tive interactive clues cross videos in an episode are ignored
+when each video is considered independently during repre-
+sentation learning. As a result, these methods actually as-
+sume the learned representations are equally effective on
+different episodic tasks and maintain a fixed set of video fea-
+tures for all test-time tasks, i.e., task-agnostic, which hence
+might overlook the most discriminative dimensions for the
+current task. Existing work also shows that the task-agnostic
+methods tend to suffer inferior generalization in other fields,
+such as image recognition [47, 101], NLP [66, 57], and in-
+formation retrieval [53]. Second, actions are usually com-
+plicated and involve many subactions with different orders
+and offsets, which may cause the failure of existing tempo-
+ral alignment metrics. For example, as shown in Figure 1(b),
+to make coffee, you can pour water before pour coffee pow-
+der, or in a reverse order, hence it is hard for recent temporal
+alignment strategies to find the right correspondences. Thus
+a more flexible metric is required to cope with the misalign-
+ment.
+Inspired by the above observations, we thus solve the
+few-shot action recognition problem by developing a novel
+Hybrid Relation guided temporal Set Matching algorithm,
+dubbed HyRSM++, which is architecturally composed of a
+hybrid relation module and a temporal set matching metric.
+In the hybrid relation module, we argue that the considerable
+relevant relations within and cross videos are beneficial to
+generate a set of customized features that are discriminative
+for a given task. To this end, we first apply an intra-relation
+function to strengthen structural patterns within a video via
+modeling long-range temporal dependencies. Then an inter-
+relation function operates on different videos to extract rich
+semantic information to reinforce the features which are
+more relevant to query predictions, as shown in Figure 1(a).
+By this means, we can learn task-specific embeddings for
+the few-shot task. On top of the hybrid relation module,
+we design a novel temporal set matching metric consist-
+ing of a bidirectional Mean Hausdorff Metric and a tem-
+poral coherence regularization to calculate the distances be-
+tween query and support videos, as shown in Figure 1(b).
+The objective of the bidirectional Mean Hausdorff Metric
+is to measure video distance from the set matching per-
+spective. Concretely, we treat each video as a set of frames
+and alleviate the strictly ordered constraints to acquire bet-
+ter query-support correspondences. Furthermore, to exploit
+long-range temporal order dependencies, we explicitly im-
+pose temporal coherence regularization on the input videos
+for more stable measurement without introducing extra net-
+work parameters. In this way, by combining the hybrid re-
+lation module and temporal set matching metric, the pro-
+posed HyRSM++ can sufficiently integrate semantically re-
+lational representations within the entire task and provide
+flexible video matching in an end-to-end manner. We evalu-
+ate the proposed HyRSM++ on six challenging benchmarks
+and achieve remarkable improvements again current state-
+of-the-art methods.
+Although the intuition of HyRSM++ is straightforward,
+it is elaborately designed for few-shot action recognition.
+Can our HyRSM++ be applied to the more challenging
+
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+3
+semi-supervised or unsupervised action recognition tasks
+even if the settings are entirely different? To answer this
+question, we extend HyRSM++ to the semi-supervised and
+unsupervised objectives with minor task adaptation modi-
+fications, and experimental results indicate that HyRSM++
+can be well adapted to different scenarios well and achieves
+impressive performance.
+In summary, we make the following four contributions:
+(1) We propose a novel hybrid relation module to cap-
+ture the intra- and inter-relations inside the episodic task,
+yielding task-specific representations for different tasks.
+(2) We reformulate the query-support video pair distance
+metric as a set matching problem and develop a bidirectional
+Mean Hausdorff Metric, which can be robust to complex ac-
+tions. To utilize long-term temporal order cues, we further
+design a new temporal coherence regularization on videos
+without adding network parameters.
+(3) We conduct extensive experiments on six challeng-
+ing datasets to verify that the proposed HyRSM++ achieves
+superior performance over the state-of-the-art methods.
+(4) We show that the proposed HyRSM++ can be di-
+rectly extended to the more challenging semi-supervised
+few-shot action recognition and unsupervised few-shot ac-
+tion recognition task with minor modifications.
+In this paper, we have extended our preliminary CVPR-
+2022 conference version [91] in the following aspects. i)
+We integrate the temporal coherence regularization and set
+matching strategy into a temporal set matching metric so
+that the proposed metric can explicitly leverage temporal
+order information in videos and match flexibly. Note that
+temporal coherence regularization does not introduce ad-
+ditional parameters and will not increase the burden of in-
+ference. ii) We conduct more comprehensive ablation stud-
+ies to verify the effectiveness and efficiency of the pro-
+posed HyRSM++. iii) We clearly improve the few-shot
+action recognition performance over the previous version.
+Experimental results also manifest that HyRSM++ signifi-
+cantly surpasses existing competitive methods and achieves
+state-of-the-art performance. iv) We show that the proposed
+HyRSM++ can be easily extended to the more challeng-
+ing semi-supervised few-shot recognition and unsupervised
+few-shot action recognition tasks.
+2 Related Work
+In the literature, there are some techniques related to this
+paper, mainly including few-shot image classification, set
+matching, temporal coherence, semi-supervised few-shot
+learning, unsupervised few-shot learning, and few-shot ac-
+tion recognition. In this section, we will briefly review them
+separately.
+Few-shot Image Classification.
+Recently, the research
+of few-shot learning [17, 55, 56] has proceeded roughly
+along with the following directions: data augmentation,
+optimization-based, and metric-based. Data augmentation is
+an intuitive method to increase the number of training sam-
+ples and improve the diversity of data. Mainstream strategies
+include spatial deformation [70, 67] and semantic feature
+augmentation [9, 100]. Optimization-based methods learn
+a meta-learner model that can quickly adopt to a new task
+given a few training examples. These algorithms include the
+LSTM-based meta-learner [74], learning efficient model ini-
+tialization [19], and learning stochastic gradient descent op-
+timizer [50]. Metric-based methods attempt to address the
+few-shot classification problem by ”learning to compare”.
+This family of approaches aims to learn a feature space and
+compare query and support images through Euclidean dis-
+tance [76, 101, 99], cosine similarity [86, 98], or learnable
+non-linear metric [80, 29, 47]. Our work is more closely re-
+lated to the metric-based methods [47, 101] that share the
+same spirit of learning task-specific features, whereas we fo-
+cus on solving the more challenging few-shot action recog-
+nition task with diverse spatio-temporal dependencies. In
+addition, we will further point out the differences and con-
+duct performance comparisons in the experimental section.
+Set Matching. The objective of set matching is to accu-
+rately measure the similarity of two sets, which have re-
+ceived much attention over the years. Set matching tech-
+niques can be used to efficiently process complex data struc-
+tures [2, 72, 3] and has been applied in many computer vi-
+sion fields, including face recognition [63, 93, 92], object
+matching [73, 107], etc. Among them, Hausdorff distance
+is an important alternative to handle set matching problems.
+Hausdorff distance and its variants have been widely used
+in the field of image matching and achieved remarkable re-
+sults [34, 16, 35, 107, 82, 79]. Inspired by these great suc-
+cesses, we introduce set matching into the few-shot action
+recognition field for the first time.
+Temporal Coherence. Videos naturally involve temporal
+continuity, and there is much effort to effectively explore
+how to leverage this property [11, 22, 27, 58]. Inverse Dif-
+ference Moment (IDM) [11] is a commonly used measure of
+local homogeneity, which assumes that in a sequence, two
+elements are more similar if they are located next to each
+other. The idea of IDM has been widely applied to texture
+feature extraction [60], face recognition [59], and unsuper-
+vised representation learning [22, 27] and achieved remark-
+able performance. In this paper, we focus on constraining
+the few-shot matching process by exploiting temporal co-
+herence.
+Semi-supervised Few-shot Learning. In practical appli-
+cation scenarios, there are usually many unlabeled samples.
+Semi-supervised few-shot learning considers learning new
+concepts in the presence of extra unlabeled data. Ren et
+al. [71] first introduce the challenging semi-supervised few-
+shot learning paradigm and refine the prototypes by adopt-
+
+4
+Xiang Wang et al.
+ing a soft k-means on unlabeled data. LST [49] proposes a
+novel recursive-learning-based self-training strategy for ro-
+bust convergence of the inner loop. TransMatch [103] de-
+velops a new transfer learning framework by incorporat-
+ing MixMatch [4] and existing few-shot learning methods.
+PTN [31] employs the Poisson learning model to obtain in-
+formative presentations between the labeled and unlabeled
+data. PLCM [32] and iLPC [44] focus on cleaning predicted
+pseudo-labels and generating accurate confidence estima-
+tion. In the field of semi-supervised few-shot action recog-
+nition, LIM [113] utilizes a label-independent memory to
+preserve a feature bank and produces class prototypes for
+query classification.
+Unsupervised Few-shot Learning. The objective of un-
+supervised few-shot learning is to utilize unlabeled samples
+to construct meta-tasks for few-shot training. CACTUs [30]
+and UFLST [36] construct many tasks by clustering em-
+beddings and optimize the meta-learning process over the
+constructed tasks. UMTRA [38] generates artificial tasks
+by randomly sampling support examples from the training
+set and produces corresponding queries by augmentation.
+ULDA [69] and AAL [1] follow this paradigm to randomly
+group augmented images for meta-learning and point out the
+importance of data augmentation. More recently, MetaU-
+VFS [64] presents the first unsupervised meta-learning al-
+gorithm for few-shot action recognition and adopts a two-
+stream 2D and 3D CNN model to explore spatial and tem-
+poral features via contrastive learning.
+Few-shot Action Recognition.
+The difference between
+few-shot action recognition and the previous few-shot learn-
+ing approaches is that it deals with more complex higher
+dimensional video data instead of two-dimensional images.
+The existing methods mainly focus on metric-based learn-
+ing. OSS-Metric Learning [40] adopts OSS-Metric of video
+pairs to match videos. TARN [5] learns an attention-based
+deep-distance measure from an attribute to a class center
+for zero-shot and few-shot action recognition. CMN [112]
+utilizes a multi-saliency embedding algorithm to encode
+video representations. AMeFu-Net [20] uses depth infor-
+mation to assist learning. Xian et al. [95] propose to learn
+a generative adversarial network and produce video fea-
+tures of novel classes for generalization. Coskun et al. [12]
+leverage object-object interaction, hand grasp, optical flow,
+and hand trajectory to learn an egocentric few-shot classi-
+fier. OTAM [7] preserves the frame ordering in video data
+and estimates distances with ordered temporal alignment.
+ARN [105] introduces a self-supervised permutation invari-
+ant strategy for spatio-temporal modeling. ITANet [106]
+proposes a frame-wise implicit temporal alignment strategy
+to achieve accurate and robust video matching. TRX [68]
+matches actions by matching plentiful tuples of different
+sub-sequences. More recently, STRM [84] makes use of lo-
+cal and global enrichment mechanism for spatio-temporal
+modeling based on TRX [68] and enforces class-separability
+at different phase. Some works [33, 94, 108, 62] propose
+to design multi-level metrics for few-shot action recogni-
+tion. Note that most above methods focus on learning video
+embedding independently. Unlike these previous methods,
+our HyRSM++ improves the transferability of embedding
+by learning intra- and inter-relational patterns that can bet-
+ter generalize to unseen classes.
+3 Method
+In this section, we first formulate the definition of the
+few-shot action recognition task. Then we present our Hy-
+brid Relation guided temporal Set Matching (HyRSM++)
+method.
+3.1 Problem formulation
+Few-shot action recognition aims to obtain a model that can
+generalize well to new classes when limited labeled video
+data is available. To make training more faithful to the test
+environment, we adopt the episodic training manner [86] for
+few-shot adaptation as in previous work [86, 7, 68, 106]. In
+each episodic task, there are two sets, i.e., a support set S
+and a query set Q. The support set S contains N × K sam-
+ples from N different action classes, and each class contains
+K support videos, termed the N-way K-shot problem. The
+goal is to classify the query videos in Q into N classes with
+these support videos.
+3.2 HyRSM++
+Pipeline. The overall architecture of HyRSM++ is illus-
+trated in Figure 2. For each input video sequence, we first
+divide it into T segments and extract a snippet from each
+segment, as in previous methods [88, 7]. This way, in an
+episodic task, the support set can be denoted as S
+=
+{s1, s2, ..., sN×K}, where si = {s1
+i , s2
+i , ..., sT
+i }. For sim-
+plicity and convenience, we discuss the process of the N-
+way 1-shot problem, i.e., K = 1, and consider that the
+query set Q contains a single video q. Then we apply
+an embedding model to extract the feature representations
+for each video sequence and obtain the support features
+Fs = {fs1, fs2, ..., fsN } and the query feature fq, where
+fsi = {f 1
+i , f 2
+i , ..., f T
+i } and fq = {f 1
+q , f 2
+q , ..., f T
+q }. After
+that, we input Fs and fq to the hybrid relation module to
+learn task-specific features, resulting in ˜Fs and ˜fq. Finally,
+the enhanced representations ˜Fs and ˜fq are fed into the set
+matching metric to generate matching scores. Based on the
+output scores, we can train or test the total framework.
+
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+5
+Support set
+Query video
+Backbone
+Intra-relation
+Intra-relation
+Intra-relation
+Intra-relation
+A
+Inter-relation modeling
+Hybrid relation module
+0.1
+0.2
+0.7
+A
+Avg-pooling
+E
+Expend
+Concatenate
+Convolution
+Temporal set matching metric
+Backbone
+Backbone
+Backbone
+A
+A
+A
+E
+E
+E
+E
+Pull
+Push
+Fig. 2 Schematic illustration of the proposed Hybrid Relation guided temporal Set Matching (HyRSM++) approach on a 3-way 1-shot problem.
+Given an episode of video data, a feature embedding network is first employed to extract their feature vectors. Then, A hybrid relation module is
+followed to integrate rich information within each video and cross videos with intra-relation and inter-relation functions. Finally, the task-specific
+features are fed forward into a temporal set matching metric for matching score prediction. Best viewed in color.
+Hybrid relation module. Given the features Fs and fq
+output by the embedding network, current approaches, e.g.,
+OTAM [7], directly apply a classifier C in this feature space.
+They can be formulated as:
+yi = C(fsi, fq)
+(1)
+where yi is the matching score between fsi and fq. During
+training, yi = 1 if they belong to the same class, otherwise
+yi = 0. In the testing phase, yi can be adopted to predict
+the query label. From the perspective of probability theory,
+it makes decisions based on the priors fsi and fq:
+yi = P((fsi, fq)|fsi, fq)
+(2)
+which is a typical task-agnostic method. However, the task-
+agnostic embedding is often vulnerable to overfit irrelevant
+representations [29, 47] and may fail to transfer to unseen
+classes not yet observed in the training stage.
+Unlike the previous methods, we propose to learn task-
+specific features for each target task. To achieve this goal, we
+introduce a hybrid relation module to generate task-specific
+features by capturing rich information from different videos
+in an episode. Specifically, we elaborately design the hybrid
+relation module H in the following form:
+˜fi = H(fi, G); fi ∈ [Fs, fq], G = [Fs, fq]
+(3)
+That is, we improve the feature fi by aggregating seman-
+tic information cross video representations, i.e., G, in an
+episodic task, allowing the obtained task-specific feature ˜fi
+to be more discriminative than the isolated feature. For ef-
+ficiency, we further decompose hybrid relation module into
+two parts: intra-relation function Ha and inter-relation func-
+tion He.
+The intra-relation function aims to strengthen structural
+patterns within a video by capturing long-range temporal de-
+pendencies. We express this process as:
+f a
+i = Ha(fi)
+(4)
+here f a
+i
+∈ RT ×C is the output of fi through the intra-
+relation function and has the same shape as fi. Note that the
+intra-relation function has many alternative implements, in-
+cluding multi-head self-attention (MSA), Transformer [85],
+Bi-LSTM [25], Bi-GRU [10], etc., which is incredibly flex-
+ible and can be any one of them.
+Based on the features generated by the intra-relation
+function, an inter-relation function is deployed to semanti-
+cally enhance the features cross different videos:
+f e
+i = He
+i (f a
+i , Ga) =
+|Ga|
+�
+j
+(κ(ψ(f a
+i ), ψ(f a
+j )) ∗ ψ(f a
+j ))
+(5)
+where Ga = [F a
+s , f a
+q ], ψ(·) is a global average pooling layer,
+and κ(f a
+i , f a
+j ) is a learnable function that calculates the se-
+mantic correlation between f a
+i and f a
+j . The potential logic
+is that if the correlation score between f a
+i and f a
+j is high,
+i.e., κ(f a
+i , f a
+j ), it means they tend to have the same seman-
+tic content, hence we can borrow more information from f a
+j
+to elevate the representation f a
+i , and vice versa. In the same
+way, if the score κ(f a
+i , f a
+i ) is less than 1, it indicates that
+some irrelevant information in f a
+i should be suppressed.
+In this way, we can improve the feature discrimination
+by taking full advantage of the limited samples in each
+episodic task. The inter-relation function also has similar
+implements with the intra-relation function but with a dif-
+ferent target. After the inter-relation function, we employ
+an Expend-Concatenate-Convolution operation to aggregate
+
+6
+Xiang Wang et al.
+information, as shown in Figure 2, where the output feature
+˜fi has the same shape as f e
+i . In the form of prior, our method
+can be formulated as:
+yi = P(( ˜fsi, ˜fq)|H(fsi, G), H(fq, G)); G = [Fs, fq]
+(6)
+Intuitively, compared with Equation 2, it can be conducive
+to making better decisions because more priors are provided.
+In particular, the hybrid relation module is a plug-and-play
+unit. In the experiment, we will fully explore different con-
+figurations of the hybrid relation module and further inves-
+tigate its insertablility.
+Temporal set matching metric. Many prior few-shot
+action recognition algorithms usually impose a strict tempo-
+ral alignment strategy on generated video representations for
+few-shot classification. However, they suffer from causing
+some failed matches when encountering misaligned video
+instances. Instead, we develop a flexible metric based on set
+matching that explicitly discovers optimal frame matching
+pairs for the ability to be insensitive to misalignment. Con-
+cretely, the proposed temporal set matching metric contains
+two parts, bidirectional Mean Hausdorff Metric (Bi-MHM)
+and temporal coherence regularization, respectively. We will
+describe them in detail below.
+Given the relation-enhanced features ˜Fs and ˜fq, we
+present a novel metric to enable efficient and flexible match-
+ing. In this metric, we treat each video as a set of T frames
+and reformulate distance measurement between videos as
+a set matching problem, which is robust to complicated
+instances, whether they are aligned or not. Specifically,
+we achieve this goal by modifying the Hausdorff distance,
+which is a typical set matching approach. The standard
+Hausdorff distance D can be formulated as:
+d( ˜fi, ˜fq) = max
+˜
+f a
+i ∈ ˜fi
+( min
+˜
+f bq ∈ ˜
+fq
+��� ˜f a
+i − ˜f bq
+���)
+d( ˜fq, ˜fi) = max
+˜
+f b
+q ∈ ˜
+fq
+( min
+˜
+f a
+i ∈ ˜fi
+��� ˜f bq − ˜f a
+i
+���)
+D = max(d( ˜fi, ˜fq), d( ˜fq, ˜fi))
+(7)
+where ˜fi ∈ RT ×C contains T frame features, and
+��·
+�� is a
+distance measurement function, which is the cosine distance
+in our method.
+However, the previous methods [102, 21, 111, 16]
+pointed out that Hausdorff distance can be easily affected
+by noisy examples, resulting in inaccurate measurements.
+Hence they employ a directed modified Hausdorff distance
+that robust to noise as follows:
+dm( ˜fi, ˜fq) = 1
+Ni
+�
+˜
+f a
+i ∈ ˜fi
+( min
+˜
+f b
+q ∈ ˜
+fq
+��� ˜f a
+i − ˜f bq
+���)
+(8)
+where Ni is the length of ˜fi, and equal to T in this paper.
+Hausdorff distance and its variants achieve great success in
+image matching [82, 16, 34] and face recognition [21, 79].
+We thus propose to introduce the set matching strategy into
+the few-shot action recognition field and further design a
+novel bidirectional Mean Hausdorff Metric (Bi-MHM):
+Db = 1
+Ni
+�
+˜
+f a
+i ∈ ˜fi
+( min
+˜
+f bq ∈ ˜
+fq
+��� ˜f a
+i − ˜f bq
+���)+
+1
+Nq
+�
+˜
+f bq ∈ ˜
+fq
+( min
+˜
+f a
+i ∈ ˜fi
+��� ˜f bq − ˜f a
+i
+���)
+(9)
+where Ni and Nq are the lengths of the support feature ˜fi
+and the query feature ˜fq respectively.
+The proposed Bi-MHM is a symmetric function, and the
+two items are complementary to each other. From Equa-
+tion 9, we can find that Db can automatically find the best
+correspondencies between two videos, e.g., ˜fi and ˜fq. Note
+that our Bi-MHM is a non-parametric classifier and does not
+involve numerous non-parallel calculations, which helps to
+improve computing efficiency and transfer ability compared
+to the previous complex alignment classifiers [7, 68]. More-
+over, the hybrid relation module and Bi-MHM can mutually
+reinforce each other, consolidating the correlation between
+two videos collectively.
+The Bi-MHM approach described above assumes video
+sequence representations belonging to the same action have
+the same set structure in the feature space and does not
+explicitly utilize temporal order information. However, it
+would be much more general to take the inherent temporal
+information in videos into account. For this reason, we take
+advantage of the temporal coherence that naturally exists in
+sequential video data and construct a temporal coherence
+regularization to further constrain the matching process by
+incorporating temporal order information.
+IDM [11] is a commonly used means that can exploit
+temporal coherence within videos, which can be formulated
+as:
+I( ˜fi) =
+T
+�
+a=1
+T
+�
+b=1
+1
+(a − b)2 + 1 ·
+��� ˜f a
+i − ˜f b
+i
+���
+(10)
+where ˜fi is the input video feature, T is the temporal length
+of the video, and the above loss encourages frames that are
+close in time to be close in the feature space as well. In addi-
+tion, there is another way to use temporal order information
+in the literature [22, 59]:
+I( ˜fi; ˜f a
+i , ˜f b
+i ) =
+�
+�
+�
+��� ˜f a
+i − ˜f b
+i
+��� ,
+if |a − b| = 1
+max(0, m −
+��� ˜f a
+i − ˜f b
+i
+���)
+if |a − b| > 1
+(11)
+where m is the size of the margin. Equation 11 utilizes
+the video coherence property by pulling two frame features
+closer if they are adjacent, pushing farther apart by one mar-
+gin m if they are not adjacent. Through observation, we can
+
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+7
+see that in Equation 10, all frames are pulled close regardless
+of time distance. In Equation 11, all frame features are sep-
+arated by a margin m if they are not adjacent to the current
+frame, i.e., all pairs are treated equally. The above two man-
+ners do not fully exploit the smooth and continuous changes
+of the video. To this end, we propose a novel form to mine
+temporal coherence property:
+I( ˜fi; ˜f a
+i , ˜f b
+i ) =
+�
+�
+�
+1
+(a−b)2+1 ·
+��� ˜f a
+i − ˜f b
+i
+��� ,
+if |a − b| ≤ δ
+max(0, mab −
+��� ˜f a
+i − ˜f b
+i
+���)
+if |a − b| > δ
+(12)
+where δ is a window size and mab = 1 − e− (|a−b|−δ)2
+2σ2
+for smooth temporal coherence. Compared with the origi-
+nal forms, our proposed temporal coherence regularization
+can better reflect the continuous change of video and thus
+lead to better performance.
+In the training phase, we take the negative distance for
+each class as logit. Then we utilize the same cross-entropy
+loss as in [7, 68], the auxiliary semantic loss [46, 54] and the
+temporal coherence regularization to jointly train the model.
+The auxiliary semantic loss refers to the cross-entropy loss
+on the real action classes, which is widely used to improve
+training stability and generalization. During inference, we
+select the support class closest to the query for classification.
+3.3 Extended applications of HyRSM++
+3.3.1 Semi-supervised few-shot action recognition
+The objective of semi-supervised few-shot action recogni-
+tion [113] is to fully explore the auxiliary information from
+unlabeled video data to boost the few-shot classification.
+Compared with the standard supervised few-shot setting, in
+addition to the support set S and query set Q, an extra un-
+labeled set U is also included in a semi-supervised few-shot
+task to alleviate data scarcity. We demonstrate that the pro-
+posed HyRSM++ can build a bridge between labeled and
+unlabeled examples, leading to higher classification perfor-
+mance.
+Given an unlabeled set U, a common practice in semi-
+supervised learning literature [110, 104, 77] is to adopt the
+Pseudo Labeling technique [45], which assumes that the de-
+cision boundary usually lies in low-density areas and data
+samples in a high-density area have the same label. Sim-
+ilarly, traditional semi-supervised few-shot learning meth-
+ods [71, 49] usually produce pseudo labels for unlabeled
+data based on the known support set, and then the gener-
+ated high-confidence pseudo-label data is augmented into
+the support set. In this paper, we follow this paradigm and
+utilize HyRSM++ to leverage unlabeled examples. Since
+Algorithm 1 HyRSM++ for semi-supervised few-shot ac-
+tion recognition
+Require: A labeled support set S, an auxiliary unlabeled set U, and a
+query set Q
+Ensure: Optimized few-shot classifier HyRSM++
+1: Enter support set S and unlabeled set U into HyRSM++ and obtain
+the category prediction of U based on Equation 9;
+2: According to the prediction distribution, select the high-confidence
+samples to generate pseudo-labels and update S with the selected
+samples to get the augmented S
+′;
+3: Apply the augmented S
+′ and query set Q for supervised few-shot
+training as described in Section 3.2;
+noisy videos usually have higher losses in training, it is pos-
+sible to leverage the strong HyRSM++ to distinguish be-
+tween clean and noisy videos from the prediction scores.
+Based on this, we choose reliable pseudo-labeled samples
+in the unlabeled set by predictions and augment the support
+set with high-confidence pseudo-label data. Subsequently,
+we take advantage of the augmented support set to classify
+the query videos as in the supervised few-shot task. During
+the training stage, many semi-supervised few-shot tasks are
+sampled to optimize the whole model, as shown in Algo-
+rithm 1. For inference, the evaluation process is also con-
+ducted by sampling 10,000 episodic tasks.
+3.3.2 Unsupervised few-shot action recognition
+Unlike the previously described settings involving labelled
+data, unsupervised few-shot action recognition aims to use
+unlabeled data to construct few-shot tasks and learn adap-
+tations to different tasks. We further extend HyRSM++ to
+this unsupervised task and verify its capability of transfer-
+ring prior knowledge to learn to deal with unseen tasks effi-
+ciently.
+To perform unsupervised few-shot learning, construct-
+ing few-shot tasks is the first step. However, there are no
+label annotations that can be directly applied for few-shot
+learning in the challenging unsupervised setting. Following
+prior unsupervised few-shot algorithms [38, 36], we gener-
+ate few-shot tasks by first adopting existing unsupervised
+learning approaches to learn initialized feature embeddings
+of the input videos, and then leveraging deep clustering tech-
+niques to construct pseudo-classes of the videos. According
+to clustering results, we are able to produce few-shot tasks
+by sampling N-way K-shot episodes. We then use the con-
+structed few-shot tasks to train HyRSM++. During the test-
+ing phase, we sample 10,000 episodes from the test set to
+obtain the performance, and the label information is only
+used for evaluation.
+
+8
+Xiang Wang et al.
+Table 1 Comparison to recent few-shot action recognition methods on the meta-testing set of SSv2-Full, Kinetics, Epic-kitchens and HMDB51.
+The experiments are conducted under the 5-way setting, and results are reported as the shot increases from 1 to 5. ”-” means the result is not
+available in published works, and the underline indicates the second best result.
+Method
+Reference
+Dataset
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+CMN++ [112]
+ECCV’18
+SSv2-Full
+34.4
+-
+-
+-
+43.8
+TRN++ [109]
+ECCV’18
+38.6
+-
+-
+-
+48.9
+OTAM [7]
+CVPR’20
+42.8
+49.1
+51.5
+52.0
+52.3
+TTAN [48]
+ArXiv’21
+46.3
+52.5
+57.3
+59.3
+60.4
+ITANet [7]
+IJCAI’21
+49.2
+55.5
+59.1
+61.0
+62.3
+TRX (Ω={1}) [68]
+CVPR’21
+38.8
+49.7
+54.4
+58.0
+60.6
+TRX (Ω={2, 3})[68]
+CVPR’21
+42.0
+53.1
+57.6
+61.1
+64.6
+STRM [84]
+CVPR’22
+43.1
+53.3
+59.1
+61.7
+68.1
+MTFAN [94]
+CVPR’22
+45.7
+-
+-
+-
+60.4
+Nguyen et al. [62]
+ECCV’22
+43.8
+-
+-
+-
+61.1
+Huang et al. [33]
+ECCV’22
+49.3
+-
+-
+-
+66.7
+HCL [108]
+ECCV’22
+47.3
+54.5
+59.0
+62.4
+64.9
+HyRSM
+CVPR’22
+54.3 (+5.0)
+62.2 (+6.7)
+65.1 (+6.0)
+67.9 (+5.5)
+69.0 (+0.9)
+HyRSM++
+-
+55.0 (+5.7)
+63.5 (+8.0)
+66.0 (+6.9)
+68.8 (+6.4)
+69.8 (+1.7)
+MatchingNet [86]
+NeurIPS’16
+Kinetics
+53.3
+64.3
+69.2
+71.8
+74.6
+MAML [19]
+ICML’17
+54.2
+65.5
+70.0
+72.1
+75.3
+Plain CMN [112]
+ECCV’18
+57.3
+67.5
+72.5
+74.7
+76.0
+CMN-J [113]
+TPAMI’20
+60.5
+70.0
+75.6
+77.3
+78.9
+TARN [5]
+BMVC’19
+64.8
+-
+-
+-
+78.5
+ARN [105]
+ECCV’20
+63.7
+-
+-
+-
+82.4
+OTAM [7]
+CVPR’20
+73.0
+75.9
+78.7
+81.9
+85.8
+ITANet [106]
+IJCAI’21
+73.6
+-
+-
+-
+84.3
+TRX (Ω={1}) [68]
+CVPR’21
+63.6
+75.4
+80.1
+82.4
+85.2
+TRX (Ω={2, 3}) [68]
+CVPR’21
+63.6
+76.2
+81.8
+83.4
+85.9
+STRM [84]
+CVPR’22
+62.9
+76.4
+81.1
+83.8
+86.7
+MTFAN [94]
+CVPR’22
+74.6
+-
+-
+-
+87.4
+Nguyen et al. [62]
+ECCV’22
+74.3
+-
+-
+-
+87.4
+Huang et al. [33]
+ECCV’22
+73.3
+-
+-
+-
+86.4
+HCL [108]
+ECCV’22
+73.7
+79.1
+82.4
+84.0
+85.8
+HyRSM
+CVPR’22
+73.7 (-0.9)
+80.0 (+0.9)
+83.5 (+1.1)
+84.6 (+0.6)
+86.1 (-1.3)
+HyRSM++
+-
+74.0 (-0.6)
+80.8 (+1.7)
+83.9 (+1.5)
+85.3 (+1.3)
+86.4 (-1.0)
+OTAM [7]
+CVPR’20
+Epic-kitchens
+46.0
+50.3
+53.9
+54.9
+56.3
+TRX [68]
+CVPR’21
+43.4
+50.6
+53.5
+56.8
+58.9
+STRM [84]
+CVPR’22
+42.8
+50.4
+54.9
+58.0
+59.2
+HyRSM
+CVPR’22
+47.4 (+1.4)
+52.9 (+2.3)
+56.4 (+1.5)
+58.8 (+0.8)
+59.8 (+0.6)
+HyRSM++
+-
+48.0 (+2.0)
+54.9 (+4.3)
+57.5 (+2.6)
+59.6 (+1.6)
+60.8 (+1.6)
+ARN [105]
+ECCV’20
+HMDB51
+45.5
+-
+-
+-
+60.6
+OTAM [7]
+CVPR’20
+54.5
+63.5
+65.7
+67.2
+68.0
+TTAN [48]
+ArXiv’21
+57.1
+-
+-
+-
+74.0
+TRX [68]
+CVPR’21
+53.1
+62.5
+66.8
+70.2
+75.6
+STRM [84]
+CVPR’22
+52.3
+62.5
+67.4
+70.9
+77.3
+MTFAN [94]
+CVPR’22
+59.0
+-
+-
+-
+74.6
+Nguyen et al. [62]
+ECCV’22
+59.6
+-
+-
+-
+76.9
+Huang et al. [33]
+ECCV’22
+60.1
+-
+-
+-
+77.0
+HCL [108]
+ECCV’22
+59.1
+66.5
+71.2
+73.8
+76.3
+HyRSM
+CVPR’22
+60.3 (+0.2)
+68.2 (+1.7)
+71.7 (+0.5)
+75.3 (+1.5)
+76.0 (-1.3)
+HyRSM++
+-
+61.5 (+1.4)
+69.0 (+2.5)
+72.7 (+1.5)
+75.4 (+1.6)
+76.4 (-0.9)
+4 Experiments
+In this section, the following key questions will be answered
+in detail: (1) Is HyRSM++ competitive to other state-of-
+the-art methods on challenging few-shot benchmarks? (2)
+What components play an integral role in HyRSM++ so that
+HyRSM++ can work well? (3) Can the proposed hybrid re-
+lation module be viewed as a simple plug-and-play unit and
+have the same effect for other methods? (4) Does the pro-
+posed temporal set matching metric have an advantage over
+other measure competitors? (5) Can HyRSM++ have stable
+performance in a variety of different video scenarios?
+4.1 Datasets and experimental setups
+Datasets. We evaluate our HyRSM++ on six standard public
+few-shot benchmarks. For the Kinetics [8], SSv2-Full [23],
+and SSv2-Small [23] datasets, we adopt the existing splits
+proposed by [7, 112, 106, 68], and each dataset consists
+
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+9
+MSA
+Transformer
+Bi-LSTM
+Bi-GRU
+Inter-relation
+MSA
+Transformer
+Bi-LSTM
+Bi-GRU
+Intra-relation
+54.3
+53.0
+53.6
+53.3
+54.3
+53.8
+53.6
+54.1
+50.6
+50.4
+51.4
+50.9
+51.8
+50.6
+50.8
+51.8
+50.5
+51.0
+51.5
+52.0
+52.5
+53.0
+53.5
+54.0
+Fig. 3 Comparison between different components in hybrid relation
+module on 5-way 1-shot few-shot action classification without tempo-
+ral coherence regularization. Experiments are conducted on the SSv2-
+Full dataset.
+MSA
+Transformer
+Bi-LSTM
+Bi-GRU
+Inter-relation
+MSA
+Transformer
+Bi-LSTM
+Bi-GRU
+Intra-relation
+55.0
+55.0
+54.6
+54.4
+54.5
+54.6
+54.4
+54.5
+50.7
+50.6
+51.7
+51.9
+52.1
+51.7
+52.3
+52.0
+51.0
+51.5
+52.0
+52.5
+53.0
+53.5
+54.0
+54.5
+55.0
+Fig. 4 Comparison between different components in hybrid relation
+module on 5-way 1-shot few-shot action classification with temporal
+coherence regularization. Experiments are conducted on the SSv2-Full
+dataset.
+of 64 and 24 classes as the meta-training and meta-testing
+set, respectively. For UCF101 [78] and HMDB51 [42], we
+verify our proposed methods by leveraging existing splits
+from [105, 68]. In addition to the above, we also utilize
+the egocentric Epic-kitchens [14, 13] dataset to evaluate
+HyRSM++.
+Implementation details. Following previous works [112, 7,
+68, 106], ResNet-50 [28] initialized with ImageNet [15] pre-
+trained weights is utilized as the feature extractor in our ex-
+periments. We sparsely and uniformly sample 8 (i.e., T = 8)
+frames per video to construct input frame sequence, which is
+in line with previous methods [7, 106]. In the training phase,
+we also adopt basic data augmentation such as random crop-
+ping and color jitter, and use Adam [39] optimizer to train
+our model. During the inference stage, we conduct few-shot
+action recognition evaluation on 10,000 randomly sampled
+episodes from the meta-testing set and report the mean ac-
+curacy. For many shot classification, e.g., 5-shot, we follow
+ProtoNet [76] and calculate the mean features of support
+videos in each class as the prototypes, and classify the query
+videos according to their distances against the prototypes.
+4.2 Comparison with state-of-the-art
+In this section, we validate the effectiveness of the proposed
+HyRSM++ by comparing it with state-of-the-art methods
+under various settings. As indicated in Table 1 and Ta-
+ble 2, the proposed HyRSM++ surpasses other advanced
+approaches significantly and is able to achieve new state-
+of-the-art performance. For instance, HyRSM++ improves
+the state-of-the-art performance from 49.2% to 55.0% un-
+der the 1-shot setting on SSv2-Full and consistently outper-
+forms our original conference version [91]. Specially, ex-
+tensively compared with current strict temporal alignment
+techniques [7, 106] and complex fusion methods [48, 68],
+HyRSM++ produces results that are superior to them un-
+der most different shots, which implies that our approach
+is considerably flexible and efficient. Note that the SSv2-
+Full and SSv2-Small datasets tend to be motion-based and
+generally focus on temporal reasoning. While Kinetics and
+UCF101 are partly appearance-related datasets, and scene
+understanding is usually essential. Besides, Epic-kitchens
+and HMDB51 are relatively complicated and might involve
+diverse object interactions. Extensively evaluated on these
+benchmarks, HyRSM++ provides excellent performance. It
+reveals that our HyRSM++ has strong robustness and gen-
+eralization for different scenes. From Table 2, we observe
+that HyRSM++ outperforms current state-of-the-art meth-
+ods on UCF101 and SSv2-Small under the 1-shot and 3-
+shot settings, which suggests that our HyRSM++ can learn
+rich and effective representations with extremely limited
+samples. It’s worth noting that under the 5-shot evaluation,
+our HyRSM++ yields 95.9% and 58.0% 5-shot performance
+on UCF101 and SSv2-Small, respectively, which is slightly
+behind STRM and HCL. We attribute this to STRM and
+HCL are ensemble methods that weight each sample with
+attention or use multiple metrics for few-shot classification,
+which makes them more suitable for multi-shots, while our
+HyRSM++ is a simple and general method without involves
+complex ensemble operations. Moreover, we also observe
+that with the introduction of temporal coherence regulariza-
+tion, HyRSM++ has a significant improvement compared to
+HyRSM, which verifies the effectiveness of exploiting tem-
+poral order information during the set matching process.
+4.3 Ablation study
+For ease of comparison, we use a baseline method Pro-
+toNet [76] that applies global-average pooling to backbone
+representations to obtain a prototype for each class. We will
+explore the role and validity of our proposed modules in de-
+tail below.
+Design choices of relation modeling.
+To systematically
+investigate the effect of different relation modeling opera-
+
+10
+Xiang Wang et al.
+Table 2 Results on 1-shot, 3-shot, and 5-shot few-shot classification on the UCF101 and SSv2-Small datasets. ”-” means the result is not available
+in published works, and the underline indicates the second best result.
+UCF101
+SSv2-Small
+Method
+Reference
+1-shot
+3-shot
+5-shot
+1-shot
+3-shot
+5-shot
+MatchingNet [86]
+NeurIPS’16
+-
+-
+-
+31.3
+39.8
+45.5
+MAML [19]
+ICML’17
+-
+-
+-
+30.9
+38.6
+41.9
+Plain CMN [112]
+ECCV’18
+-
+-
+-
+33.4
+42.5
+46.5
+CMN-J [113]
+TPAMI’20
+-
+-
+-
+36.2
+44.6
+48.8
+ARN [105]
+ECCV’20
+66.3
+-
+83.1
+-
+-
+-
+OTAM [7]
+CVPR’20
+79.9
+87.0
+88.9
+36.4
+45.9
+48.0
+TTAN [48]
+ArXiv’21
+80.9
+-
+93.2
+-
+-
+-
+ITANet [106]
+IJCAI’21
+-
+-
+-
+39.8
+49.4
+53.7
+TRX [68]
+CVPR’21
+78.2
+92.4
+96.1
+36.0
+51.9
+59.1
+STRM [84]
+CVPR’22
+80.5
+92.7
+96.9
+37.1
+49.2
+55.3
+MTFAN [94]
+CVPR’22
+84.8
+-
+95.1
+-
+-
+-
+Nguyen et al. [62]
+ECCV’22
+84.9
+-
+95.9
+-
+-
+-
+Huang et al. [33]
+ECCV’22
+71.4
+-
+91.0
+38.9
+-
+61.6
+HCL [108]
+ECCV’22
+82.5
+91.0
+93.9
+38.7
+49.1
+55.4
+HyRSM
+CVPR’22
+83.9 (-1.0)
+93.0 (+0.3)
+94.7 (-2.2)
+40.6 (+0.8)
+52.3 (+0.4)
+56.1 (-5.5)
+HyRSM++
+-
+85.8 (+0.9)
+93.5 (+0.8)
+95.9 (-1.0)
+42.8 (+3.0)
+52.4 (+0.5)
+58.0 (-2.6)
+Table 3 Ablation study under 5-way 1-shot and 5-way 5-shot settings
+on the SSv2-Full dataset. “TCR” refers to temporal coherence regular-
+ization.
+Intra-relation
+Inter-relation
+Bi-MHM
+TCR
+1-shot
+5-shot
+35.2
+45.3
+�
+41.2
+55.0
+�
+43.7
+55.2
+�
+44.6
+56.0
+�
+�
+45.3
+57.1
+�
+�
+48.1
+60.5
+�
+�
+48.3
+61.2
+�
+�
+�
+49.2
+62.8
+�
+�
+51.4
+64.6
+�
+�
+�
+52.4
+65.8
+�
+�
+�
+54.3
+69.0
+�
+�
+�
+�
+55.0
+69.8
+Table 4 Generalization of hybrid relation module. We conduct exper-
+iments on SSv2-Full.
+Method
+1-shot
+5-shot
+OTAM [7]
+42.8
+52.3
+OTAM [7]+ Intra-relation
+48.9
+60.4
+OTAM [7]+ Inter-relation
+46.9
+57.8
+OTAM [7]+ Intra-relation + Inter-relation
+51.7
+63.9
+tions in hybrid relation module, we vary the components to
+construct some variants and report the results in Figure 3
+and Figure 4. The comparison experiments are conducted
+on the SSv2-Full dataset under the 5-way 1-shot setting. We
+can observe that different combinations have quite distinct
+properties, e.g., multi-head self-attention (MSA) and Trans-
+former are more effective to model intra-class relations than
+Bi-LSTM and Bi-GRU. For example, utilizing multi-head
+5-way
+6-way
+7-way
+8-way
+9-way
+10-way
+Accuracy (%)
+Kinetics
+OTAM
+TRX
+STRM
+HyRSM
+HyRSM++
+50
+66
+54
+58
+70
+62
+5-way
+6-way
+7-way
+8-way
+9-way
+10-way
+Accuracy (%)
+SSv2-Full
+OTAM
+TRX
+STRM
+HyRSM
+HyRSM++
+25
+30
+40
+35
+50
+45
+55
+74
+Fig. 5 N-way 1-shot performance trends of our HyRSM++ and other
+state-of-the-art methods with different N on SSv2-Full. The compari-
+son results prove the superiority of our HyRSM++.
+Accuracy (%)
+(a) Frames
+42
+46
+50
+54
+2
+3
+4
+5
+6
+7
+8
+9
+10
+1
+2
+4
+8
+16
+32
+Accuracy (%)
+1-shot
+5-shot
+45
+50
+60
+55
+70
+65
+(b) Head number
+Fig. 6 (a) Performance on SSv2-Full using a different number of
+frames under the 5-way 1-shot setting. (b) The effect of the number
+of heads on SSv2-Full.
+self-attention to learn intra-relation produces at least 2.5%
+improvements than with Bi-LSTM. Nevertheless, compared
+with other recent algorithms [68, 106], the performance of
+each combination can still be improved, which strongly sug-
+gests the necessity of structure design for learning task-
+specific features. For simplicity, we choose the same struc-
+ture to explore intra-relation and inter-relation, and the con-
+
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+11
+20
+30
+40
+50
+60
+70
+SSv2-Full (Resnet-18)
+OTAM
+TRX
+HyRSM
+HyRSM++
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+20
+30
+40
+50
+60
+70
+SSv2-Full (Resnet-34)
+OTAM
+TRX
+HyRSM
+HyRSM++
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+Accuracy (%)
+50
+55
+60
+65
+70
+75
+80
+85
+Kinetics (Resnet-18)
+OTAM
+TRX
+HyRSM
+HyRSM++
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+55
+60
+65
+70
+75
+80
+85
+Kinetics (Resnet-34)
+OTAM
+TRX
+HyRSM
+HyRSM++
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+Accuracy (%)
+Accuracy (%)
+Accuracy (%)
+Fig. 7 Comparison of the backbone with different depths on the SSv2-
+Full and Kinetics datasets.
+Table 5 Comparative experiments on SSv2-Full using the Inception-
+v3 [81] feature extractor.
+Method
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+OTAM [7]
+42.4
+46.6
+48.7
+49.2
+52.1
+TRX [68]
+37.7
+50.2
+55.5
+57.2
+60.1
+STRM [84]
+42.9
+53.9
+58.9
+62.3
+63.4
+HyRSM++
+53.3
+62.7
+65.3
+67.8
+69.3
+Table 6 Performance comparison on SSv2-Full with self-supervised
+initialization weights [97].
+Method
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+OTAM [7]
+41.2
+45.9
+48.8
+50.1
+51.0
+TRX [68]
+37.5
+43.8
+49.9
+51.6
+52.1
+STRM [84]
+38.0
+46.2
+49.9
+53.4
+54.4
+HyRSM++
+50.9
+59.1
+62.6
+65.5
+66.4
+Table 7 Performance comparison with different relation modeling
+paradigms on SSv2-Full and Kinetics.
+Setting
+Method
+Dataset
+1-shot
+5-shot
+Support-only
+HyRSM
+SSv2-Full
+52.1
+67.2
+Support-only
+HyRSM++
+53.7
+68.8
+Support&Query
+HyRSM
+54.3
+69.0
+Support&Query
+HyRSM++
+55.0
+69.8
+Support-only
+HyRSM
+Kinetics
+73.4
+85.5
+Support-only
+HyRSM++
+73.5
+85.7
+Support&Query
+HyRSM
+73.7
+86.1
+Support&Query
+HyRSM++
+74.0
+86.4
+figuration of multi-head self-attention is adopted in the ex-
+periments.
+Analysis of the proposed components. Table 3 summa-
+rizes the ablation study of each module in HyRSM++. To
+evaluate the function of the proposed components, Pro-
+toNet [76] is taken as our baseline. From the ablation results,
+we can conclude that each component is highly effective.
+In particular, compared to the baseline, intra-relation mod-
+eling can respectively bring 6.0% and 9.7% performance
+52.3 
+51.2 
+49.7 
+49.1 
+48.0 
+47.2 
+64.6 
+61.3 
+59.2 
+56.3 
+53.1 
+50.4 
+68.1 
+62.3 
+60.8 
+58.5 
+55.9 
+52.0 
+69.8 
+66.2 
+65.1 
+64.5 
+62.3 
+60.0 
+0%
+10%
+20%
+30%
+40%
+50%
+Accuracy (%)
+5-way 5-shot
+OTAM
+TRX
+STRM
+HyRSM++
+45
+61
+49
+53
+65
+57
+42.8 
+41.4 
+40.3 
+39.0 
+37.1 
+35.7 
+42.0 
+38.5 
+35.8 
+33.2 
+31.3 
+28.9 
+43.1
+40.4
+37.8
+34.5
+32.1
+30.0 
+55.0 
+49.8 
+48.1 
+46.4 
+43.2 
+41.3 
+0%
+10%
+20%
+30%
+40%
+50%
+Accuracy (%)
+5-way 1-shot
+OTAM
+TRX
+STRM
+HyRSM++
+25
+30
+40
+35
+50
+45
+55
+69
+Noisy ratio
+Noisy ratio
+Fig. 8 Robustness comparison experiments in the presence of noisy
+samples. X% represents the proportion of noisy labels included in the
+dataset.
+Table 8 Comparison with recent temporal alignment methods on the
+SSv2-Full dataset under the 5-way 1-shot and 5-way 5-shot settings.
+Diagonal means matching frame by frame.
+Metric
+Bi-direction
+1-shot
+5-shot
+Diagonal
+-
+38.3
+48.7
+Plain DTW [61]
+-
+39.6
+49.0
+OTAM [7]
+�
+39.3
+47.7
+OTAM [7]
+�
+42.8
+52.3
+Bi-MHM
+�
+44.6
+56.0
+Temporal set matching metric
+�
+45.3
+57.1
+Table 9 Comparison of different set matching strategies on the SSv2-
+Full dataset.
+Metric
+Bi-direction
+1-shot
+5-shot
+Hausdorff distance
+�
+32.4
+38.2
+Hausdorff distance
+�
+34.5
+39.1
+Modified Hausdorff distance
+�
+44.2
+50.0
+Bi-MHM
+�
+44.6
+56.0
+Temporal set matching metric
+�
+45.3
+57.1
+Table 10 Generalization of temporal coherence regularization. We
+conduct experiments on SSv2-Full. ”Hard margin” represents the
+method described in Equation 11.
+Method
+1-shot
+5-shot
+OTAM [7]
+42.8
+52.3
+OTAM [7] + IDM
+43.7
+55.0
+OTAM [7] + Hard margin
+43.2
+55.3
+OTAM [7] + Temporal coherence regularization
+44.1
+55.8
+Bi-MHM
+44.6
+56.0
+Bi-MHM + IDM
+44.7
+56.3
+Bi-MHM + Hard margin
+44.7
+56.5
+Bi-MHM + Temporal coherence regularization
+45.3
+57.1
+gains on 1-shot and 5-shot, and inter-relation function boosts
+the performance by 8.5% and 9.9% on 1-shot and 5-shot.
+In addition, the proposed set matching metric improves 1-
+shot and 5-shot classification by 9.4% and 10.7%, respec-
+tively, which indicates the ability to find better correspond-
+ing frames in the video pair. Adding temporal coherence
+regularization to the set matching metric also achieves sta-
+
+12
+Xiang Wang et al.
+ble performance improvements. Moreover, stacking the pro-
+posed modules can further improve performance, indicating
+the complementarity between components. When consider-
+ing all the proposed modules together to form HyRSM++,
+the performance of 1-shot and 5-shot is improved to 55.0%
+and 69.8%, respectively, which strongly supports the impor-
+tance of learning task-related features and flexible metrics.
+Pluggability of hybrid relation module. In Table 4, we
+experimentally show that the hybrid relation module gen-
+eralizes well to other methods by inserting it into the re-
+cent OTAM [7]. In this study, OTAM with our hybrid re-
+lation module benefits from relational information and fi-
+nally achieves 8.9% and 11.6% gains on 1-shot and 5-shot.
+This fully evidences that mining the rich information among
+videos to learn task-specific features is especially valuable.
+N-way few-shot classification.
+In the previous experi-
+ments, all of our comparative evaluation experiments were
+carried out under the 5-way setting. In order to further ex-
+plore the influence of different N, in Figure 5, we com-
+pare N-way (N ≥ 5) 1-shot results on SSv2-Full and Ki-
+netics. Results show that as N increases, the difficulty be-
+comes higher, and the performance decreases. Neverthe-
+less, the performance of our HyRSM++ is still consistently
+ahead of the recent state-of-the-art STRM [84], TRX [68]
+and OTAM [7], which shows the feasibility of our method
+to boost performance by introducing rich relations among
+videos and the power of the set matching metric.
+Varying the number of frames. To demonstrate the scal-
+ability of HyRSM++, we also explore the impact of differ-
+ent video frame numbers on performance. Of note, previous
+comparisons are performed under 8 frames of input. Results
+in Figure 6(a) show that as the number of frames increases,
+the performance improves. HyRSM++ gradually tends to be
+saturated when more than 7 frames.
+Influence of head number. Previous analyses have shown
+that multi-head self-attention can focus on different patterns
+and is critical to capturing diverse features [41]. We investi-
+gate the virtue of varying the number of heads in multi-head
+self-attention on performance in Figure 6(b). Experimental
+results indicate that the effect of multi-head is remarkable,
+and the performance starts to saturate beyond a particular
+point.
+Varying depth of the backbone. The proposed HyRSM++
+is general and compatible with feature extractors of various
+capacities. The previous methods all utilize ResNet-50 as
+backbone by default for a fair comparison, and the impact
+of backbone’s depth on performance is still under-explored.
+As presented in Figure 7, we attempt to answer this question
+by adopting ResNet-18 and ResNet-34 pre-trained on Ima-
+geNet as alternative backbones. Results demonstrate that the
+deeper network clearly benefits from greater learning capac-
+ity and results in better performance. In addition, we notice
+Acc = 100%
+Acc = 100% 
+(+ hybrid relation module)
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+1
+2
+3
+4
+5
+Acc = 40%
+Acc = 60%
+Acc = 60%
+Acc = 80%
+Acc = 60%
+Acc = 100% 
+(+ hybrid relation module)
+Acc = 100% 
+(+ hybrid relation module)
+Acc = 100% 
+(+ hybrid relation module)
+Acc = 100% 
+(+ hybrid relation module)
+Acc = 100% 
+(+ hybrid relation module)
+(a) Examples from SSv2-Full
+(b) Examples from Kinetics
+Fig. 9 Similarity visualization of how query videos (rows) match to
+support videos (columns). The boxes of different colors correspond to:
+correct match and incorrect match.
+Support
+Query
+(a) SSv2-Full: ”pretending to open something without actually opening it”
+(b) SSv2-Full: ”showing that something is empty”
+Support
+Query
+Support
+Query
+(c) Kinetics: ”cutting watermelon”
+Fig. 10 Visualization of matching results with the proposed set match-
+ing metric on SSv2-Full and Kinetics.
+that our proposed HyRSM++ consistently outperforms the
+competitors (i.e., OTAM and TRX), which indicates that our
+HyRSM++ is a generally effective framework.
+Influence of different backbones. To verify that our ap-
+proach is not limited to ResNet-like structures, we further
+perform experiments on Inception-v3 and report the results
+in Table 5. From the comparison, we note that HyRSM++ is
+significantly superior to other competitive algorithms. Com-
+pared with STRM [84], our proposed HyRSM++ leads to at
+least 5.5% performance gain under various settings.
+Impact of pretraining types. Supervised ImageNet initial-
+ization [15] is widely employed in many vision tasks [7,
+113, 90] and achieves impressive success. Recently, self-
+supervised techniques have also received widespread at-
+tention and revealed excellent application potential. In Ta-
+
+0.73
+0.063
+0.06
+0.082
+0.065
+0.16
+0.35
+0.21
+0.1 
+0.18
+0.067
+0.069
+0.67
+0.08
+0.11
+0.27
+0.038
+0.076
+0.42
+0.2 
+0.11
+0.27
+0.13
+0.09
+0.4 0.23
+0.25
+0.11
+0.19
+0.22
+0.18
+0.25
+0.24
+0.14
+0.19
+0.18
+0.18
+0.36
+0.091
+0.18
+0.23
+0.15
+0.11
+0.4
+0.1
+0.14
+0.32
+0.08
+0.11
+0.340.65
+0.029
+0.15
+0.088
+0.078
+0.092
+0.55
+0.04
+0.053
+0.27
+0.17
+0.076
+0.59
+0.094
+0.07
+0.11
+0.053
+0.042
+0.65
+0.14
+0.1
+0.039
+0.024
+0.087
+0.750.34
+0.16
+0.14
+0.23
+0.14
+0.17
+0.26
+0.19
+0.21
+0.18
+0.16
+0.2 
+0.27
+0.13
+0.24
+0.2 
+0.21
+0.16
+0.27
+0.16
+0.15
+0.23
+0.17
+0.18
+0.270.46
+0.064
+0.037
+0.38
+0.055
+0.035
+0.69
+0.021
+0.17
+0.082
+0.051
+0.042
+0.78
+0.06
+0.067
+0.09
+0.2
+0.18
+0.45
+0.087
+0.12
+0.18
+0.098
+0.13
+0.470.26
+0.13
+0.28
+0.21
+0.12
+0.13
+0.38
+0.23
+0.19
+0.069
+0.14
+0.16
+0.19
+0.2 
+0.32
+0.25
+0.13
+0.2 
+0.29
+0.12
+0.15
+0.09
+0.21
+0.19
+0.360.21
+0.25
+0.16
+0.25
+0.13
+0.24
+0.23
+0.16
+0.19
+0.18
+0.14
+0.16
+0.27
+0.12
+0.32
+0.22
+0.2 
+0.1 
+0.35
+0.13
+0.13
+0.19
+0.25
+0.15
+0.290.61
+0.074
+0.068
+0.14
+0.1
+0.076
+0.43
+0.084
+0.35
+0.063
+0.12
+0.061
+0.63
+0.077
+0.11
+0.21
+0.21
+0.06
+0.47
+0.053
+0.15
+0.055
+0.069
+0.054
+0.670.59
+0.043
+0.13
+0.13
+0.1
+0.11
+0.51
+0.14
+0.2
+0.036
+0.13
+0.27
+0.44
+0.091
+0.068
+0.27
+0.14
+0.17
+0.35
+0.066
+0.05
+0.057
+0.044
+0.038
+0.810.19
+0.24
+0.11
+0.2
+0.25
+0.27
+0.16
+0.21
+0.23
+0.13
+0.21
+0.22
+0.3
+0.14
+0.13
+0.15
+0.15
+0.11
+0.34
+0.26
+0.22
+0.16
+0.19
+0.18
+0.250.59
+0.18
+0.014
+0.095
+0.13
+0.21
+0.43
+0.025
+0.26
+0.076
+0.049
+0.042
+0.76
+0.071
+0.081
+0.31
+0.097
+0.019
+0.42
+0.16
+0.21
+0.063
+0.11
+0.13
+0.490.47
+0.1
+0.054
+0.21
+0.17
+0.19
+0.23
+0.08
+0.27
+0.24
+0.098
+0.15
+0.5 
+0.096
+0.16
+0.23
+0.082
+0.076
+0.4
+0.21
+0.12
+0.4
+0.13
+0.11
+0.25HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+13
+ble 6, we show the performance comparison with self-
+supervised pretraining weights [97]. Results demonstrate
+that our HyRSM++ is still powerful and not limited to the
+specific initialization weights.
+Other relation modeling forms. Previous few-shot image
+classification methods of learning task-specific features have
+also achieved promising results [101, 47]. However, many
+of them use some complex and fixed operations to learn the
+dependencies between images, while our method is straight-
+forward and flexible. Moreover, most previous works only
+use the information within the support set to learn task-
+specific features, ignoring the correlation with query sam-
+ples. In our hybrid relation module, we add the query video
+to the pool of inter-relation modeling to extract relevant in-
+formation suitable for query classification. As illustrated in
+Table 7, we try to remove the query video from the pool
+in HyRSM++, i.e., Support-only, but we can observe that
+after removing the query video, the performance of 1-shot
+and 5-shot on SSv2-Full reduces by 1.3% and 1.0%, respec-
+tively. There are similar conclusions on the Kinetics dataset.
+This evidences that the proposed hybrid relation module is
+reasonable and can effectively extract task-related features,
+thereby promoting query classification accuracy.
+Robustness to noise labels. To demonstrate the robustness
+of HyRSM++ to noise samples, we simulate the presence
+of noise labels in the dataset in Figure 8. From the results,
+we can observe that performance generally decreases as the
+proportion of noise rises. However, our HyRSM++ still ex-
+hibits higher performance than other methods, which illus-
+trates the robustness of our method and its adaptability to
+complex conditions.
+4.4 Comparison with other matching approaches
+Our proposed temporal set matching metric Bi-MHM aims
+to accurately find the corresponding video frames between
+video pairs by relaxing the strict temporal ordering con-
+straints. The following comparative experiments in Table 8
+are carried out under identical experimental setups, i.e., re-
+Table 11 Complexity analysis for 5-way 1-shot SSv2-Full evaluation.
+The experiments are carried out on one Nvidia V100 GPU.
+Method
+Backbone
+Param
+FLOPs
+Latency
+Acc
+HyRSM
+ResNet-18
+13.8M
+3.64G
+36.5ms
+46.6
+HyRSM++
+ResNet-18
+13.8M
+3.64G
+36.5ms
+47.7
+HyRSM
+ResNet-34
+23.9M
+7.34G
+67.5ms
+50.0
+HyRSM++
+ResNet-34
+23.9M
+7.34G
+67.5ms
+50.4
+OTAM [7]
+ResNet-50
+23.5M
+8.17G
+116.6ms
+42.8
+TRX [68]
+ResNet-50
+47.1M
+8.22G
+94.6ms
+42.0
+STRM [84]
+ResNet-50
+73.3M
+8.27G
+113.3ms
+43.1
+HyRSM
+ResNet-50
+65.6M
+8.36G
+83.5ms
+54.3
+HyRSM++
+ResNet-50
+65.6M
+8.36G
+83.5ms
+55.0
+place the OTAM directly with our Bi-MHM while keep-
+ing other settings unchanged. Results show that our Bi-
+MHM performs well and outperforms other temporal align-
+ment methods (e.g., OTAM). We further analyze different
+set matching approaches in Table 9, and the results indi-
+cate that Hausdorff distance is susceptible to noise interfer-
+ence, resulting in the mismatch and relatively poor perfor-
+mance. However, our Bi-MHM shows stability to noise and
+obtains better performance. Furthermore, compared with the
+single directional metric, our proposed bidirectional metric
+is more comprehensive in reflecting the actual distances be-
+tween videos and achieves better performance on few-shot
+tasks. In addition, we observe that the proposed temporal
+set matching metric achieves clear improvement over Bi-
+MHM after incorporating temporal coherence. For instance,
+the temporal set matching metric obtains 0.7%, 1.1% perfor-
+mance gains on 5-way 1-shot, and 5-way 5-shot SSv2-Full
+classification. It indicates the effectiveness of the proposed
+temporal set matching metric.
+4.5 Comparison of temporal coherence manners
+Pioneering work [11, 22, 59] also indicates the important
+role of temporal coherence and shows remarkable results in
+face recognition [59] and unsupervised representation learn-
+ing [22, 27]. However, they also have some limitations as
+noted in Section 3.2, and thus the temporal coherence reg-
+ularization is proposed for smooth video coherence. Ta-
+ble 10 compares the proposed temporal coherence regular-
+ization with existing temporal coherence schemes based on
+OTAM and Bi-MHM. Results show that exploiting tempo-
+ral coherence helps improve the classification accuracy of
+the metrics, which confirms our motivation for consider-
+ing temporal order information during the matching process.
+In addition, our proposed temporal coherence regularization
+achieves more significant improvements than other manners,
+and we attribute this to the smooth property of temporal co-
+herence regularization.
+4.6 Visualization results
+To qualitatively show the discriminative capability of the
+learned task-specific features in our proposed method, we
+visualize the similarities between query and support videos
+with and without the hybrid relation module. As depicted in
+Figure 9, by adding the hybrid relation module, the discrim-
+ination of features is significantly improved, contributing to
+predicting more accurately. Additionally, the matching re-
+sults of the set matching metric are visualized in Figure 10,
+and we can observe that our Bi-MHM is considerably flexi-
+ble in dealing with alignment and misalignment.
+
+14
+Xiang Wang et al.
+Support
+Query
+Support
+Query
+“tipping Sth over” from SSv2-Full
+OTAM
+HyRSM++
+“taking Sth out of Sth” from SSv2-Full
+OTAM
+HyRSM++
+Support
+Query
+“showing Sth next to Sth” from SSv2-Full
+OTAM
+HyRSM++
+Support
+Query
+“riding elephant” from Kinetics
+OTAM
+HyRSM++
+Support
+Query
+“playing trumpet” from Kinetics
+OTAM
+HyRSM++
+Support
+Query
+“filling eyebrows” from Kinetics
+OTAM
+HyRSM++
+Fig. 11 Visualization of activation maps with Grad-CAM [75]. Compared to OTAM [7], HyRSM++ focuses more precisely on classification-
+related regions.
+To further visually evaluate the proposed HyRSM++, we
+compare the activation visualization results of HyRSM++ to
+the competitive OTAM [7]. As shown in Figure 11, the fea-
+tures of OTAM usually contain non-target objects or ignore
+most discriminative parts since it lacks the mechanism of
+learning task-specific embeddings for feature adaptation. In
+contrast, our proposed HyRSM++ processes the query and
+support videos with an adaptive relation modeling operation,
+which allows it to focus on the different target objects. The
+above qualitative experiments illustrate the rationality of our
+model design and the necessity of learning task-related fea-
+tures.
+4.7 Limitations
+In order to further understand HyRSM++, Table 11 il-
+lustrates its differences with OTAM and TRX in terms
+of parameters, computation, and runtime. In the inference
+
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+15
+Table 12 Comparison to existing semi-supervised few-shot action recognition methods on the meta-testing set of Kinetics and SSv2-Small. The
+experiments are conducted under the 5-way setting, and results are reported as the shot increases from 1 to 5. ”w/o unlabeled data” indicates
+that there is no unlabeled set in a episode, i.e., the traditional few-shot action recognition setting, which can act as the lower bound of the semi-
+supervised counterpart.
+Dataset
+Method
+Backbone
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+Kinetics
+OTAM w/o unlabeled data [7]
+Inception-v3
+68.6
+72.7
+74.1
+75.7
+76.9
+DeepCluster CACTUs-MAML [30]
+Inception-v3
+65.1
+72.8
+76.5
+77.9
+79.5
+DeepCluster CACTUs-ProtoNets [30]
+Inception-v3
+66.9
+73.2
+77.0
+78.1
+79.9
+LIM [113]
+Inception-v3
+69.8
+75.9
+78.3
+80.4
+82.6
+HyRSM++ w/o unlabeled data
+Inception-v3
+69.1
+76.0
+78.6
+81.6
+81.9
+HyRSM++
+Inception-v3
+73.7
+79.4
+80.9
+81.8
+83.1
+CMN w/o unlabeled data [112]
+ResNet-50
+60.5
+70.0
+75.6
+77.3
+78.9
+OTAM w/o unlabeled data [7]
+ResNet-50
+73.0
+75.9
+78.7
+81.9
+85.8
+LIM (ensemble) [113]
+ResNet-50, Inception-v3, ResNet-18
+73.3
+78.3
+80.8
+82.4
+84.0
+HyRSM++ w/o unlabeled data
+ResNet-50
+74.0
+80.8
+83.6
+85.3
+86.4
+HyRSM++
+ResNet-50
+79.1
+84.3
+85.4
+86.4
+86.8
+SSv2-Small
+OTAM w/o unlabeled data [112]
+Inception-v3
+36.7
+41.0
+43.6
+44.1
+46.9
+DeepCluster CACTUs-MAML [30]
+Inception-v3
+37.9
+44.5
+45.9
+47.8
+49.9
+DeepCluster CACTUs-ProtoNets [30]
+Inception-v3
+38.4
+44.8
+46.1
+48.0
+50.1
+LIM [113]
+Inception-v3
+41.1
+46.9
+48.0
+51.5
+53.0
+HyRSM++ w/o unlabeled data
+Inception-v3
+41.5
+46.1
+49.5
+52.9
+55.1
+HyRSM++
+Inception-v3
+43.6
+49.5
+51.8
+52.4
+54.5
+CMN w/o unlabeled data [112]
+ResNet-50
+36.2
+42.1
+44.6
+47.0
+48.8
+OTAM w/o unlabeled data [112]
+ResNet-50
+36.4
+42.9
+45.9
+46.8
+48.0
+LIM (ensemble) [113]
+ResNet-50, Inception-v3, ResNet-18
+44.0
+49.8
+51.3
+53.9
+55.1
+HyRSM++ w/o unlabeled data
+ResNet-50
+42.8
+47.1
+52.4
+54.7
+58.0
+HyRSM++
+ResNet-50
+45.4
+51.1
+55.2
+57.4
+58.8
+74.0 
+77.7 
+79.1 
+79.7 
+79.9 
+80.8 
+83.0 
+84.3 
+84.3 
+84.6 
+83.6 
+84.8 
+85.4 
+85.5 
+85.9 
+85.3
+85.8
+86.4
+86.6
+86.8
+86.4
+86.5
+86.7
+86.8
+86.9
+0
+50
+100
+150
+200
+Accuracy (%)
+Kinetics
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+73
+75
+79
+77
+85
+81
+87
+83
+89
+Fig. 12 Performance comparison of different amounts of unlabeled
+data for testing in an episode on Kinetics.
+phase, HyRSM++ does not add additional computational
+burden compared to HyRSM because the temporal coher-
+ence regularization is not involved in the calculation. No-
+tably, HyRSM++ introduces extra parameters (i.e., hybrid
+relation module), resulting in increased GPU memory and
+computational consumption. Nevertheless, without complex
+non-parallel classifier heads, the whole inference speed of
+HyRSM++ is faster than OTAM and TRX. We will further
+investigate how to reduce complexity with no loss of perfor-
+mance in the future.
+42.8 
+45.0 
+45.4 
+45.8 
+46.2 
+47.1 
+50.1 
+51.1 
+51.2 
+51.6 
+52.4 
+54.1 
+55.2 
+55.5 
+55.8 
+54.7
+56.8
+57.4
+57.8
+57.9
+58.0 
+58.6 
+58.8 
+59.0 
+59.6 
+0
+50
+100
+150
+200
+Accuracy (%)
+SSv2-Small
+1-shot
+2-shot
+3-shot
+4-shot
+5-shot
+60
+38
+40
+44
+42
+50
+46
+58
+48
+62
+54
+52
+56
+Fig. 13 Performance comparison of different amounts of unlabeled
+data for testing in an episode on the SSv2-Small dataset.
+5 Extension to Semi-supervised Few-shot Action
+Recognition
+In this section, we demonstrate that the proposed HyRSM++
+can be extended to address the more challenging semi-
+supervised few-shot action recognition problem. Follow-
+ing LIM [113], we utilize two common datasets (Kinet-
+ics [8] and SSv2-Small [23]) to perform comparative exper-
+iments. These two datasets are subsets of Kinetics-400 [8]
+and Something-Something-v2 [23], respectively, and the un-
+labeled examples in our experiments are collected from the
+remaining videos of the same category as these subsets. To
+
+16
+Xiang Wang et al.
+Table 13 Comparison to state-of-the-art unsupervised few-shot action
+recognition approaches on UCF101, HMDB51, and Kinetics. ∗ indi-
+cates that the algorithm adopt the same 2D ResNet-50 backbone as
+HyRSM++.
+Method
+Supervision
+UCF101
+HMDB51 Kinetics
+MAML [19]
+Supervised
+-
+-
+54.2
+CMN [112]
+Supervised
+-
+-
+60.5
+TARN [5]
+Supervised
+-
+-
+66.6
+ProtoGAN [43]
+Supervised
+57.8
+34.7
+-
+ARN [105]
+Supervised
+66.3
+45.2
+63.7
+3DRotNet [37]
+Unsupervised
+39.4
+32.4
+27.5
+VCOP [96]
+Unsupervised
+32.9
+27.8
+26.5
+IIC [83]
+Unsupervised
+56.8
+34.7
+37.7
+Pace [87]
+Unsupervised
+25.6
+26.2
+22.4
+MemDPC [83]
+Unsupervised
+49.3
+30.3
+42.0
+CoCLR [26]
+Unsupervised
+52.0
+31.3
+37.6
+MetaUVFS∗ [64]
+Unsupervised
+66.1
+40.0
+50.9
+HyRSM++
+Unsupervised
+68.0
+41.0
+55.0
+64.0 
+64.7 
+66.3 
+66.5 
+68.0 
+66.5 
+66.4 
+50
+75
+100
+125
+150
+175
+200
+Accuracy (%)
+UCF101
+63
+36.1 
+36.1 
+36.9 
+39.2 
+41.0 
+38.9 
+39.7 
+50
+75
+100
+125
+150
+175
+200
+Accuracy (%)
+HMDB51
+51.2 
+52.0 
+52.0 
+54.7 
+55.0 
+54.9 
+54.3 
+50
+75
+100
+125
+150
+175
+200
+Accuracy (%)
+Kinetics
+49
+65
+67
+69
+55
+37
+39
+41
+43
+51
+53
+57
+35
+Fig. 14 Ablation study of different cluster numbers under 5-way 1-
+shot unsupervised few-shot settings.
+conduct the semi-supervised few-shot evaluation, we fol-
+low the mainstream distractor setting [30, 38, 113], where
+the unlabeled set contains other interference classes in each
+episodic task. This setting is more realistic and requires the
+model to be robust to the existence of noisy samples from
+other classes. In our experiments, we fixed the number of
+unlabeled videos in an episodic task to 100.
+Table 12 provides the comparison of our HyRSM++
+against state-of-the-art methods on the two standard semi-
+supervised few-shot benchmarks. We find that HyRSM++
+substantially surpasses the previous approaches, such as
+LIM [113]. Under the semi-supervised 5-way 1-shot sce-
+nario, HyRSM++ produces performance gains of 3.8% and
+2.5% on Kinetics and SSv2-Small than LIM with Inception-
+v3 backbone, respectively. In particular, when using the
+ResNet-50 backbone, our method is even superior to the
+multi-modal fusion method (i.e., LIM), which indicates that
+HyRSM++ enables more accurate pseudo-labels for unla-
+beled data and then can expand the support set to boost the
+classification accuracy of the query videos. In addition, com-
+pared to our supervised counterpart (i.e., HyRSM++ w/o un-
+labeled data), joining unlabeled data is beneficial to allevi-
+ating the data scarcity problem and promotes few-shot clas-
+sification accuracy. We can observe that when ResNet-50
+is adopted as the backbone, the performance of HyRSM++
+with unlabeled data is improved by 5.1% compared to that
+without unlabeled data under the 5-way 1-shot Kinetics
+evaluation.
+To further investigate the effect of unlabeled videos in
+an episode, we conduct comparative experiments with vary-
+ing numbers of unlabeled videos in Figure 12 and Figure 13.
+Experimental results show that as the number of unlabeled
+samples increases, the performance also increases gradually,
+indicating that the introduction of unlabeled data helps gen-
+eralize to unseen categories. Furthermore, we notice that the
+improvement in the 1-shot setting is more significant than
+that in the 5-shot, which shows that under the condition of
+low samples, unlabeled videos can improve the estimation
+of the distribution of new categories more effectively. Mean-
+while, as the amount of unlabeled data increases to a certain
+level, the performance starts to saturate slowly.
+6 Extension to Unsupervised Few-shot Action
+Recognition
+We also extend the proposed HyRSM++ to solve the
+challenging unsupervised few-shot action recognition task
+where labels for training videos are not available. Following
+previous work [38, 36], we adopt the idea of the ”cluster-
+ing first and then meta-learning” paradigm to construct few-
+shot tasks and exploit unlabeled data for training. Our ex-
+periments are based on unsupervised ResNet-50 initializa-
+tion [97], which is self-supervised pre-trained on Kinetics-
+400 [8] without accessing any label information. During the
+clustering process, we utilize the K-means clustering strat-
+egy for each dataset to obtain 150 clusters.
+As presented in Table 13, we compare HyRSM++ with
+current state-of-the-art methods on the UCF101, HMDB51
+and Kinetics datasets under the 5-way 1-shot setting. Note
+that HyRSM++ and MetaUVFS [64] use the same ResNet-
+50 structure as the feature extractor, and our HyRSM++
+shows better performance on each dataset. In particular, we
+observe that our method achieves 68.0% performance on
+the UCF101 dataset, a 1.9% improvement over MetaUVFS,
+and even surpasses the fully supervised ARN. The supe-
+rior performance of HyRSM++ reveals that our approach of
+leveraging relations within and cross videos and the flexible
+metric performs effectively in the low-shot regime. More-
+over, this phenomenon also demonstrates the potential of our
+method to learn a strongly robust few-shot model using only
+unlabeled videos, even though HyRSM++ is not specifically
+
+HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition
+17
+designed for the unsupervised few-shot action recognition
+task.
+In the experiments, one parameter involved in apply-
+ing HyRSM++ to the unsupervised few-shot setting is the
+number of clusters. In Figure 14, we display the perfor-
+mance comparison under different number of clusters. Re-
+sults show that when the number of clusters is 150, the per-
+formance reaches the peak value, which means that if the
+cluster number is too small, it may lead to under-clustering.
+If the number is too large, it may cause over-clustering, dam-
+aging the performance.
+7 Conclusion
+In this work, we have proposed a hybrid relation guided
+temporal set matching (HyRSM++) approach for few-shot
+action recognition. Firstly, we design a hybrid relation mod-
+ule to model the rich semantic relevance within one video
+and cross different videos in an episodic task to generate
+task-specific features. Secondly, built upon the representa-
+tive task-specific features, an efficient set matching metric is
+proposed to be resilient to misalignment and match videos
+accurately. During the matching process, a temporal coher-
+ence regularization is further imposed to exploit temporal
+order information. Furthermore, we extend HyRSM++ to
+solve the more challenging semi-supervised few-shot ac-
+tion recognition and unsupervised few-shot action recog-
+nition problems. Experimental results demonstrate that our
+HyRSM++ achieves the state-of-the-art performance on
+multiple standard benchmarks.
+Acknowledgements This work is supported by the National Natural
+Science Foundation of China under grant 61871435, Fundamental Re-
+search Funds for the Central Universities no.2019kfyXKJC024, 111
+Project on Computational Intelligence and Intelligent Control under
+Grant B18024, and Alibaba Group through Alibaba Research Intern
+Program.
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diff --git a/BdE1T4oBgHgl3EQfpQWt/content/tmp_files/load_file.txt b/BdE1T4oBgHgl3EQfpQWt/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf,len=2094
+page_content='Noname manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  (will be inserted by the editor)  HyRSM++: Hybrid Relation Guided Temporal Set Matching for  Few-shot Action Recognition  Xiang Wang · Shiwei Zhang · Zhiwu Qing · Zhengrong Zuo · Changxin Gao ·  Rong Jin · Nong Sang  Received: date / Accepted: date  Abstract Few-shot action recognition is a challenging but  practical problem aiming to learn a model that can be eas-  ily adapted to identify new action categories with only a few  labeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Recent attempts mainly focus on learning  deep representations for each video individually under the  episodic meta-learning regime and then performing tempo-  ral alignment to match query and support videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' However,  they still suffer from two drawbacks: (i) learning individ-  ual features without considering the entire task may result  in limited representation capability, and (ii) existing align-  ment strategies are sensitive to noises and misaligned in-  stances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' To handle the two limitations, we propose a novel  Hybrid Relation guided temporal Set Matching (HyRSM++)  approach for few-shot action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The core idea of  HyRSM++ is to integrate all videos within the task to learn  discriminative representations and involve a robust match-  ing technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' To be specific, HyRSM++ consists of two key  components, a hybrid relation module and a temporal set  matching metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Given the basic representations from the  feature extractor, the hybrid relation module is introduced  to fully exploit associated relations within and cross videos  in an episodic task and thus can learn task-specific embed-  dings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Subsequently, in the temporal set matching metric, we  carry out the distance measure between query and support  Xiang Wang · Zhiwu Qing · Zhengrong Zuo · Changxin Gao (Corre-  sponding author) · Nong Sang  Key Laboratory of Ministry of Education for Image Processing and  Intelligent Control, School of Artificial Intelligence and Automation,  Huazhong University of Science and Technology  E-mail: {wxiang, qzw, zhrzuo, cgao, nsang}@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='cn  Shiwei Zhang  Alibaba Group  E-mail: zhangjin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='zsw@alibaba-inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='com  Rong Jin  Twitter  E-mail: rongjinemail@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='com  videos from a set matching perspective and design a bidi-  rectional Mean Hausdorff Metric to improve the resilience  to misaligned instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In addition, we explicitly exploit  the temporal coherence in videos to regularize the matching  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In this way, HyRSM++ facilitates informative cor-  relation exchanged among videos and enables flexible pre-  dictions under the data-limited scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Furthermore, we  extend the proposed HyRSM++ to deal with the more chal-  lenging semi-supervised few-shot action recognition and un-  supervised few-shot action recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Experimental  results on multiple benchmarks demonstrate that our method  consistently outperforms existing methods and achieves  state-of-the-art performance under various few-shot set-  tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The source code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  com/alibaba-mmai-research/HyRSMPlusPlus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Keywords Few-shot Action Recognition · Set Match-  ing · Semi-supervised Few-shot Action Recognition ·  Unsupervised Few-shot Action Recognition  1 Introduction  Recently, the development of large-scale video bench-  marks [8, 23, 6, 13, 24] and deep networks [88, 51, 18,  89, 65, 52] have significantly boosted the progress of ac-  tion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' To achieve this success, we typically re-  quire large amounts of manually labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' However, ac-  quiring these labeled examples consumes a lot of manpower  and time, which actually limits further applications of this  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In this case, researchers look to alternatives to achieve  action classification without extensive costly labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Few-  shot action recognition is a promising direction to reduce  manual annotations and thus has attracted much attention  recently [112, 105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' It aims at learning to classify unseen  action classes with extremely few annotated examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='03330v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='CV]  9 Jan 2023  2  Xiang Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  CNN  CNN  CNN  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  Query video  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Support set  Hybrid relation module  Support: make coffee  Query: make coffee  Support: make coffee  Query: make coffee  \uf04f  \uf050  Metric space  (support)  Metric space  (quey)  “pour water”  “pour coffee powder”  Temporal alignment  Temporal set matching  (b)  Time line  Matching line  \uf04f  \uf050  (a)  Pull  Push  Pull  Push  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' 1 (a) Concept of the proposed hybrid relation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We adaptively produce task-specific video embeddings by extracting relevant discrim-  inative patterns cross videos in an episodic task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' (b) Example of make coffee, the current temporal alignment metrics tend to be strict, resulting in  an incorrect match on misaligned videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In contrast, the proposed temporal set matching metric involving set matching technique and temporal  coherence regularization is more flexible in finding the best correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  To solve the few-shot data-scarcity problem, popu-  lar attempts [112, 7, 68, 106] are mainly based on the  metric-based meta-learning technique [86], in which a com-  mon embedding space is first learnt via episodic training  and then an explicit or implicit alignment metric is em-  ployed to calculate the distances between the query (test)  videos and support (reference) videos for classification in  an episodic task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Typically, Ordered Temporal Alignment  Module (OTAM) [7] adopts a deep feature extractor to con-  vert an input video into a frame feature sequence indepen-  dently and explicitly explores the ordered temporal align-  ment path between support and query videos in this feature  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Temporal-Relational CrossTransformer (TRX) [68]  learns a deep embedding space and tries to exhaustively con-  struct temporally-corresponding sub-sequences of actions to  compare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Some recent works [33, 94, 108, 62] propose to  design multi-level metrics for few-shot action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Although these methods have achieved remarkable per-  formance, there are still two limitations: individual feature  learning and inflexible matching strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' First, discrimina-  tive interactive clues cross videos in an episode are ignored  when each video is considered independently during repre-  sentation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' As a result, these methods actually as-  sume the learned representations are equally effective on  different episodic tasks and maintain a fixed set of video fea-  tures for all test-time tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', task-agnostic, which hence  might overlook the most discriminative dimensions for the  current task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Existing work also shows that the task-agnostic  methods tend to suffer inferior generalization in other fields,  such as image recognition [47, 101], NLP [66, 57], and in-  formation retrieval [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Second, actions are usually com-  plicated and involve many subactions with different orders  and offsets, which may cause the failure of existing tempo-  ral alignment metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' For example, as shown in Figure 1(b),  to make coffee, you can pour water before pour coffee pow-  der, or in a reverse order, hence it is hard for recent temporal  alignment strategies to find the right correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Thus  a more flexible metric is required to cope with the misalign-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Inspired by the above observations, we thus solve the  few-shot action recognition problem by developing a novel  Hybrid Relation guided temporal Set Matching algorithm,  dubbed HyRSM++, which is architecturally composed of a  hybrid relation module and a temporal set matching metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  In the hybrid relation module, we argue that the considerable  relevant relations within and cross videos are beneficial to  generate a set of customized features that are discriminative  for a given task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' To this end, we first apply an intra-relation  function to strengthen structural patterns within a video via  modeling long-range temporal dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Then an inter-  relation function operates on different videos to extract rich  semantic information to reinforce the features which are  more relevant to query predictions, as shown in Figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  By this means, we can learn task-specific embeddings for  the few-shot task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' On top of the hybrid relation module,  we design a novel temporal set matching metric consist-  ing of a bidirectional Mean Hausdorff Metric and a tem-  poral coherence regularization to calculate the distances be-  tween query and support videos, as shown in Figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  The objective of the bidirectional Mean Hausdorff Metric  is to measure video distance from the set matching per-  spective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Concretely, we treat each video as a set of frames  and alleviate the strictly ordered constraints to acquire bet-  ter query-support correspondences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Furthermore, to exploit  long-range temporal order dependencies, we explicitly im-  pose temporal coherence regularization on the input videos  for more stable measurement without introducing extra net-  work parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In this way, by combining the hybrid re-  lation module and temporal set matching metric, the pro-  posed HyRSM++ can sufficiently integrate semantically re-  lational representations within the entire task and provide  flexible video matching in an end-to-end manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We evalu-  ate the proposed HyRSM++ on six challenging benchmarks  and achieve remarkable improvements again current state-  of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Although the intuition of HyRSM++ is straightforward,  it is elaborately designed for few-shot action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Can our HyRSM++ be applied to the more challenging  HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition  3  semi-supervised or unsupervised action recognition tasks  even if the settings are entirely different?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' To answer this  question, we extend HyRSM++ to the semi-supervised and  unsupervised objectives with minor task adaptation modi-  fications, and experimental results indicate that HyRSM++  can be well adapted to different scenarios well and achieves  impressive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  In summary, we make the following four contributions:  (1) We propose a novel hybrid relation module to cap-  ture the intra- and inter-relations inside the episodic task,  yielding task-specific representations for different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  (2) We reformulate the query-support video pair distance  metric as a set matching problem and develop a bidirectional  Mean Hausdorff Metric, which can be robust to complex ac-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' To utilize long-term temporal order cues, we further  design a new temporal coherence regularization on videos  without adding network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  (3) We conduct extensive experiments on six challeng-  ing datasets to verify that the proposed HyRSM++ achieves  superior performance over the state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  (4) We show that the proposed HyRSM++ can be di-  rectly extended to the more challenging semi-supervised  few-shot action recognition and unsupervised few-shot ac-  tion recognition task with minor modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  In this paper, we have extended our preliminary CVPR-  2022 conference version [91] in the following aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' i)  We integrate the temporal coherence regularization and set  matching strategy into a temporal set matching metric so  that the proposed metric can explicitly leverage temporal  order information in videos and match flexibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Note that  temporal coherence regularization does not introduce ad-  ditional parameters and will not increase the burden of in-  ference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' ii) We conduct more comprehensive ablation stud-  ies to verify the effectiveness and efficiency of the pro-  posed HyRSM++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' iii) We clearly improve the few-shot  action recognition performance over the previous version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Experimental results also manifest that HyRSM++ signifi-  cantly surpasses existing competitive methods and achieves  state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' iv) We show that the proposed  HyRSM++ can be easily extended to the more challeng-  ing semi-supervised few-shot recognition and unsupervised  few-shot action recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  2 Related Work  In the literature, there are some techniques related to this  paper, mainly including few-shot image classification, set  matching, temporal coherence, semi-supervised few-shot  learning, unsupervised few-shot learning, and few-shot ac-  tion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In this section, we will briefly review them  separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Few-shot Image Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Recently, the research  of few-shot learning [17, 55, 56] has proceeded roughly  along with the following directions: data augmentation,  optimization-based, and metric-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Data augmentation is  an intuitive method to increase the number of training sam-  ples and improve the diversity of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Mainstream strategies  include spatial deformation [70, 67] and semantic feature  augmentation [9, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Optimization-based methods learn  a meta-learner model that can quickly adopt to a new task  given a few training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' These algorithms include the  LSTM-based meta-learner [74], learning efficient model ini-  tialization [19], and learning stochastic gradient descent op-  timizer [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Metric-based methods attempt to address the  few-shot classification problem by ”learning to compare”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  This family of approaches aims to learn a feature space and  compare query and support images through Euclidean dis-  tance [76, 101, 99], cosine similarity [86, 98], or learnable  non-linear metric [80, 29, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Our work is more closely re-  lated to the metric-based methods [47, 101] that share the  same spirit of learning task-specific features, whereas we fo-  cus on solving the more challenging few-shot action recog-  nition task with diverse spatio-temporal dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In  addition, we will further point out the differences and con-  duct performance comparisons in the experimental section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Set Matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The objective of set matching is to accu-  rately measure the similarity of two sets, which have re-  ceived much attention over the years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Set matching tech-  niques can be used to efficiently process complex data struc-  tures [2, 72, 3] and has been applied in many computer vi-  sion fields, including face recognition [63, 93, 92], object  matching [73, 107], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Among them, Hausdorff distance  is an important alternative to handle set matching problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Hausdorff distance and its variants have been widely used  in the field of image matching and achieved remarkable re-  sults [34, 16, 35, 107, 82, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Inspired by these great suc-  cesses, we introduce set matching into the few-shot action  recognition field for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Temporal Coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Videos naturally involve temporal  continuity, and there is much effort to effectively explore  how to leverage this property [11, 22, 27, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Inverse Dif-  ference Moment (IDM) [11] is a commonly used measure of  local homogeneity, which assumes that in a sequence, two  elements are more similar if they are located next to each  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The idea of IDM has been widely applied to texture  feature extraction [60], face recognition [59], and unsuper-  vised representation learning [22, 27] and achieved remark-  able performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In this paper, we focus on constraining  the few-shot matching process by exploiting temporal co-  herence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Semi-supervised Few-shot Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In practical appli-  cation scenarios, there are usually many unlabeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Semi-supervised few-shot learning considers learning new  concepts in the presence of extra unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Ren et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' [71] first introduce the challenging semi-supervised few-  shot learning paradigm and refine the prototypes by adopt-  4  Xiang Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  ing a soft k-means on unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' LST [49] proposes a  novel recursive-learning-based self-training strategy for ro-  bust convergence of the inner loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' TransMatch [103] de-  velops a new transfer learning framework by incorporat-  ing MixMatch [4] and existing few-shot learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  PTN [31] employs the Poisson learning model to obtain in-  formative presentations between the labeled and unlabeled  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' PLCM [32] and iLPC [44] focus on cleaning predicted  pseudo-labels and generating accurate confidence estima-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In the field of semi-supervised few-shot action recog-  nition, LIM [113] utilizes a label-independent memory to  preserve a feature bank and produces class prototypes for  query classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Unsupervised Few-shot Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The objective of un-  supervised few-shot learning is to utilize unlabeled samples  to construct meta-tasks for few-shot training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' CACTUs [30]  and UFLST [36] construct many tasks by clustering em-  beddings and optimize the meta-learning process over the  constructed tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' UMTRA [38] generates artificial tasks  by randomly sampling support examples from the training  set and produces corresponding queries by augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  ULDA [69] and AAL [1] follow this paradigm to randomly  group augmented images for meta-learning and point out the  importance of data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' More recently, MetaU-  VFS [64] presents the first unsupervised meta-learning al-  gorithm for few-shot action recognition and adopts a two-  stream 2D and 3D CNN model to explore spatial and tem-  poral features via contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Few-shot Action Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  The difference between  few-shot action recognition and the previous few-shot learn-  ing approaches is that it deals with more complex higher  dimensional video data instead of two-dimensional images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  The existing methods mainly focus on metric-based learn-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' OSS-Metric Learning [40] adopts OSS-Metric of video  pairs to match videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' TARN [5] learns an attention-based  deep-distance measure from an attribute to a class center  for zero-shot and few-shot action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' CMN [112]  utilizes a multi-saliency embedding algorithm to encode  video representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' AMeFu-Net [20] uses depth infor-  mation to assist learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Xian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' [95] propose to learn  a generative adversarial network and produce video fea-  tures of novel classes for generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Coskun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' [12]  leverage object-object interaction, hand grasp, optical flow,  and hand trajectory to learn an egocentric few-shot classi-  fier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' OTAM [7] preserves the frame ordering in video data  and estimates distances with ordered temporal alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  ARN [105] introduces a self-supervised permutation invari-  ant strategy for spatio-temporal modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' ITANet [106]  proposes a frame-wise implicit temporal alignment strategy  to achieve accurate and robust video matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' TRX [68]  matches actions by matching plentiful tuples of different  sub-sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' More recently, STRM [84] makes use of lo-  cal and global enrichment mechanism for spatio-temporal  modeling based on TRX [68] and enforces class-separability  at different phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Some works [33, 94, 108, 62] propose  to design multi-level metrics for few-shot action recogni-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Note that most above methods focus on learning video  embedding independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Unlike these previous methods,  our HyRSM++ improves the transferability of embedding  by learning intra- and inter-relational patterns that can bet-  ter generalize to unseen classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  3 Method  In this section, we first formulate the definition of the  few-shot action recognition task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Then we present our Hy-  brid Relation guided temporal Set Matching (HyRSM++)  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1 Problem formulation  Few-shot action recognition aims to obtain a model that can  generalize well to new classes when limited labeled video  data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' To make training more faithful to the test  environment, we adopt the episodic training manner [86] for  few-shot adaptation as in previous work [86, 7, 68, 106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In  each episodic task, there are two sets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', a support set S  and a query set Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The support set S contains N × K sam-  ples from N different action classes, and each class contains  K support videos, termed the N-way K-shot problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The  goal is to classify the query videos in Q into N classes with  these support videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2 HyRSM++  Pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The overall architecture of HyRSM++ is illus-  trated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' For each input video sequence, we first  divide it into T segments and extract a snippet from each  segment, as in previous methods [88, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' This way, in an  episodic task, the support set can be denoted as S  =  {s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', sN×K}, where si = {s1  i , s2  i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', sT  i }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' For sim-  plicity and convenience, we discuss the process of the N-  way 1-shot problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', K = 1, and consider that the  query set Q contains a single video q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Then we apply  an embedding model to extract the feature representations  for each video sequence and obtain the support features  Fs = {fs1, fs2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', fsN } and the query feature fq, where  fsi = {f 1  i , f 2  i , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', f T  i } and fq = {f 1  q , f 2  q , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', f T  q }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' After  that, we input Fs and fq to the hybrid relation module to  learn task-specific features, resulting in ˜Fs and ˜fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Finally,  the enhanced representations ˜Fs and ˜fq are fed into the set  matching metric to generate matching scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Based on the  output scores, we can train or test the total framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition  5  Support set  Query video  Backbone  Intra-relation  Intra-relation  Intra-relation  Intra-relation  A  Inter-relation modeling  Hybrid relation module  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  A  Avg-pooling  E  Expend  Concatenate  Convolution  Temporal set matching metric  Backbone  Backbone  Backbone  A  A  A  E  E  E  E  Pull  Push  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' 2 Schematic illustration of the proposed Hybrid Relation guided temporal Set Matching (HyRSM++) approach on a 3-way 1-shot problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Given an episode of video data, a feature embedding network is first employed to extract their feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Then, A hybrid relation module is  followed to integrate rich information within each video and cross videos with intra-relation and inter-relation functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Finally, the task-specific  features are fed forward into a temporal set matching metric for matching score prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Best viewed in color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Hybrid relation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Given the features Fs and fq  output by the embedding network, current approaches, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=',  OTAM [7], directly apply a classifier C in this feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  They can be formulated as:  yi = C(fsi, fq)  (1)  where yi is the matching score between fsi and fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' During  training, yi = 1 if they belong to the same class, otherwise  yi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In the testing phase, yi can be adopted to predict  the query label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' From the perspective of probability theory,  it makes decisions based on the priors fsi and fq:  yi = P((fsi, fq)|fsi, fq)  (2)  which is a typical task-agnostic method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' However, the task-  agnostic embedding is often vulnerable to overfit irrelevant  representations [29, 47] and may fail to transfer to unseen  classes not yet observed in the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Unlike the previous methods, we propose to learn task-  specific features for each target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' To achieve this goal, we  introduce a hybrid relation module to generate task-specific  features by capturing rich information from different videos  in an episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Specifically, we elaborately design the hybrid  relation module H in the following form:  ˜fi = H(fi, G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' fi ∈ [Fs, fq], G = [Fs, fq]  (3)  That is, we improve the feature fi by aggregating seman-  tic information cross video representations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', G, in an  episodic task, allowing the obtained task-specific feature ˜fi  to be more discriminative than the isolated feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' For ef-  ficiency, we further decompose hybrid relation module into  two parts: intra-relation function Ha and inter-relation func-  tion He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  The intra-relation function aims to strengthen structural  patterns within a video by capturing long-range temporal de-  pendencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We express this process as:  f a  i = Ha(fi)  (4)  here f a  i  ∈ RT ×C is the output of fi through the intra-  relation function and has the same shape as fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Note that the  intra-relation function has many alternative implements, in-  cluding multi-head self-attention (MSA), Transformer [85],  Bi-LSTM [25], Bi-GRU [10], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', which is incredibly flex-  ible and can be any one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Based on the features generated by the intra-relation  function, an inter-relation function is deployed to semanti-  cally enhance the features cross different videos:  f e  i = He  i (f a  i , Ga) =  |Ga|  �  j  (κ(ψ(f a  i ), ψ(f a  j )) ∗ ψ(f a  j ))  (5)  where Ga = [F a  s , f a  q ], ψ(·) is a global average pooling layer,  and κ(f a  i , f a  j ) is a learnable function that calculates the se-  mantic correlation between f a  i and f a  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The potential logic  is that if the correlation score between f a  i and f a  j is high,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', κ(f a  i , f a  j ), it means they tend to have the same seman-  tic content, hence we can borrow more information from f a  j  to elevate the representation f a  i , and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In the same  way, if the score κ(f a  i , f a  i ) is less than 1, it indicates that  some irrelevant information in f a  i should be suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  In this way, we can improve the feature discrimination  by taking full advantage of the limited samples in each  episodic task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The inter-relation function also has similar  implements with the intra-relation function but with a dif-  ferent target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' After the inter-relation function, we employ  an Expend-Concatenate-Convolution operation to aggregate  6  Xiang Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  information, as shown in Figure 2, where the output feature  ˜fi has the same shape as f e  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In the form of prior, our method  can be formulated as:  yi = P(( ˜fsi, ˜fq)|H(fsi, G), H(fq, G));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' G = [Fs, fq]  (6)  Intuitively, compared with Equation 2, it can be conducive  to making better decisions because more priors are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  In particular, the hybrid relation module is a plug-and-play  unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In the experiment, we will fully explore different con-  figurations of the hybrid relation module and further inves-  tigate its insertablility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Temporal set matching metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Many prior few-shot  action recognition algorithms usually impose a strict tempo-  ral alignment strategy on generated video representations for  few-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' However, they suffer from causing  some failed matches when encountering misaligned video  instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Instead, we develop a flexible metric based on set  matching that explicitly discovers optimal frame matching  pairs for the ability to be insensitive to misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Con-  cretely, the proposed temporal set matching metric contains  two parts, bidirectional Mean Hausdorff Metric (Bi-MHM)  and temporal coherence regularization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We will  describe them in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Given the relation-enhanced features ˜Fs and ˜fq, we  present a novel metric to enable efficient and flexible match-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In this metric, we treat each video as a set of T frames  and reformulate distance measurement between videos as  a set matching problem, which is robust to complicated  instances, whether they are aligned or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Specifically,  we achieve this goal by modifying the Hausdorff distance,  which is a typical set matching approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The standard  Hausdorff distance D can be formulated as:  d( ˜fi, ˜fq) = max  ˜  f a  i ∈ ˜fi  ( min  ˜  f bq ∈ ˜  fq  ��� ˜f a  i − ˜f bq  ���)  d( ˜fq, ˜fi) = max  ˜  f b  q ∈ ˜  fq  ( min  ˜  f a  i ∈ ˜fi  ��� ˜f bq − ˜f a  i  ���)  D = max(d( ˜fi, ˜fq), d( ˜fq, ˜fi))  (7)  where ˜fi ∈ RT ×C contains T frame features, and  ��·  �� is a  distance measurement function, which is the cosine distance  in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  However, the previous methods [102, 21, 111, 16]  pointed out that Hausdorff distance can be easily affected  by noisy examples, resulting in inaccurate measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Hence they employ a directed modified Hausdorff distance  that robust to noise as follows:  dm( ˜fi, ˜fq) = 1  Ni  �  ˜  f a  i ∈ ˜fi  ( min  ˜  f b  q ∈ ˜  fq  ��� ˜f a  i − ˜f bq  ���)  (8)  where Ni is the length of ˜fi, and equal to T in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Hausdorff distance and its variants achieve great success in  image matching [82, 16, 34] and face recognition [21, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  We thus propose to introduce the set matching strategy into  the few-shot action recognition field and further design a  novel bidirectional Mean Hausdorff Metric (Bi-MHM):  Db = 1  Ni  �  ˜  f a  i ∈ ˜fi  ( min  ˜  f bq ∈ ˜  fq  ��� ˜f a  i − ˜f bq  ���)+  1  Nq  �  ˜  f bq ∈ ˜  fq  ( min  ˜  f a  i ∈ ˜fi  ��� ˜f bq − ˜f a  i  ���)  (9)  where Ni and Nq are the lengths of the support feature ˜fi  and the query feature ˜fq respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  The proposed Bi-MHM is a symmetric function, and the  two items are complementary to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' From Equa-  tion 9, we can find that Db can automatically find the best  correspondencies between two videos, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', ˜fi and ˜fq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Note  that our Bi-MHM is a non-parametric classifier and does not  involve numerous non-parallel calculations, which helps to  improve computing efficiency and transfer ability compared  to the previous complex alignment classifiers [7, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' More-  over, the hybrid relation module and Bi-MHM can mutually  reinforce each other, consolidating the correlation between  two videos collectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  The Bi-MHM approach described above assumes video  sequence representations belonging to the same action have  the same set structure in the feature space and does not  explicitly utilize temporal order information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' However, it  would be much more general to take the inherent temporal  information in videos into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' For this reason, we take  advantage of the temporal coherence that naturally exists in  sequential video data and construct a temporal coherence  regularization to further constrain the matching process by  incorporating temporal order information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  IDM [11] is a commonly used means that can exploit  temporal coherence within videos, which can be formulated  as:  I( ˜fi) =  T  �  a=1  T  �  b=1  1  (a − b)2 + 1 ·  ��� ˜f a  i − ˜f b  i  ���  (10)  where ˜fi is the input video feature, T is the temporal length  of the video, and the above loss encourages frames that are  close in time to be close in the feature space as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In addi-  tion, there is another way to use temporal order information  in the literature [22, 59]:  I( ˜fi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' ˜f a  i , ˜f b  i ) =  �  �  �  ��� ˜f a  i − ˜f b  i  ��� ,  if |a − b| = 1  max(0, m −  ��� ˜f a  i − ˜f b  i  ���)  if |a − b| > 1  (11)  where m is the size of the margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Equation 11 utilizes  the video coherence property by pulling two frame features  closer if they are adjacent, pushing farther apart by one mar-  gin m if they are not adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Through observation, we can  HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition  7  see that in Equation 10, all frames are pulled close regardless  of time distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In Equation 11, all frame features are sep-  arated by a margin m if they are not adjacent to the current  frame, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', all pairs are treated equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The above two man-  ners do not fully exploit the smooth and continuous changes  of the video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' To this end, we propose a novel form to mine  temporal coherence property:  I( ˜fi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' ˜f a  i , ˜f b  i ) =  �  �  �  1  (a−b)2+1 ·  ��� ˜f a  i − ˜f b  i  ��� ,  if |a − b| ≤ δ  max(0, mab −  ��� ˜f a  i − ˜f b  i  ���)  if |a − b| > δ  (12)  where δ is a window size and mab = 1 − e− (|a−b|−δ)2  2σ2  for smooth temporal coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Compared with the origi-  nal forms, our proposed temporal coherence regularization  can better reflect the continuous change of video and thus  lead to better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  In the training phase, we take the negative distance for  each class as logit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Then we utilize the same cross-entropy  loss as in [7, 68], the auxiliary semantic loss [46, 54] and the  temporal coherence regularization to jointly train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  The auxiliary semantic loss refers to the cross-entropy loss  on the real action classes, which is widely used to improve  training stability and generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' During inference, we  select the support class closest to the query for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3 Extended applications of HyRSM++  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1 Semi-supervised few-shot action recognition  The objective of semi-supervised few-shot action recogni-  tion [113] is to fully explore the auxiliary information from  unlabeled video data to boost the few-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Compared with the standard supervised few-shot setting, in  addition to the support set S and query set Q, an extra un-  labeled set U is also included in a semi-supervised few-shot  task to alleviate data scarcity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We demonstrate that the pro-  posed HyRSM++ can build a bridge between labeled and  unlabeled examples, leading to higher classification perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Given an unlabeled set U, a common practice in semi-  supervised learning literature [110, 104, 77] is to adopt the  Pseudo Labeling technique [45], which assumes that the de-  cision boundary usually lies in low-density areas and data  samples in a high-density area have the same label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Sim-  ilarly, traditional semi-supervised few-shot learning meth-  ods [71, 49] usually produce pseudo labels for unlabeled  data based on the known support set, and then the gener-  ated high-confidence pseudo-label data is augmented into  the support set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In this paper, we follow this paradigm and  utilize HyRSM++ to leverage unlabeled examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Since  Algorithm 1 HyRSM++ for semi-supervised few-shot ac-  tion recognition  Require: A labeled support set S, an auxiliary unlabeled set U, and a  query set Q  Ensure: Optimized few-shot classifier HyRSM++  1: Enter support set S and unlabeled set U into HyRSM++ and obtain  the category prediction of U based on Equation 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  2: According to the prediction distribution, select the high-confidence  samples to generate pseudo-labels and update S with the selected  samples to get the augmented S  ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  3: Apply the augmented S  ′ and query set Q for supervised few-shot  training as described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  noisy videos usually have higher losses in training, it is pos-  sible to leverage the strong HyRSM++ to distinguish be-  tween clean and noisy videos from the prediction scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Based on this, we choose reliable pseudo-labeled samples  in the unlabeled set by predictions and augment the support  set with high-confidence pseudo-label data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Subsequently,  we take advantage of the augmented support set to classify  the query videos as in the supervised few-shot task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' During  the training stage, many semi-supervised few-shot tasks are  sampled to optimize the whole model, as shown in Algo-  rithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' For inference, the evaluation process is also con-  ducted by sampling 10,000 episodic tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2 Unsupervised few-shot action recognition  Unlike the previously described settings involving labelled  data, unsupervised few-shot action recognition aims to use  unlabeled data to construct few-shot tasks and learn adap-  tations to different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We further extend HyRSM++ to  this unsupervised task and verify its capability of transfer-  ring prior knowledge to learn to deal with unseen tasks effi-  ciently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  To perform unsupervised few-shot learning, construct-  ing few-shot tasks is the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' However, there are no  label annotations that can be directly applied for few-shot  learning in the challenging unsupervised setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Following  prior unsupervised few-shot algorithms [38, 36], we gener-  ate few-shot tasks by first adopting existing unsupervised  learning approaches to learn initialized feature embeddings  of the input videos, and then leveraging deep clustering tech-  niques to construct pseudo-classes of the videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' According  to clustering results, we are able to produce few-shot tasks  by sampling N-way K-shot episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We then use the con-  structed few-shot tasks to train HyRSM++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' During the test-  ing phase, we sample 10,000 episodes from the test set to  obtain the performance, and the label information is only  used for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  8  Xiang Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Table 1 Comparison to recent few-shot action recognition methods on the meta-testing set of SSv2-Full, Kinetics, Epic-kitchens and HMDB51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  The experiments are conducted under the 5-way setting, and results are reported as the shot increases from 1 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' ”-” means the result is not  available in published works, and the underline indicates the second best result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Method  Reference  Dataset  1-shot  2-shot  3-shot  4-shot  5-shot  CMN++ [112]  ECCV’18  SSv2-Full  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  TRN++ [109]  ECCV’18  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  OTAM [7]  CVPR’20  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  TTAN [48]  ArXiv’21  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='4  ITANet [7]  IJCAI’21  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='3  TRX (Ω={1}) [68]  CVPR’21  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  TRX (Ω={2, 3})[68]  CVPR’21  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  STRM [84]  CVPR’22  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content=' [62]  ECCV’22  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='3  HyRSM  CVPR’22  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='3)  HyRSM++ 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='9)  4 Experiments  In this section, the following key questions will be answered  in detail: (1) Is HyRSM++ competitive to other state-of-  the-art methods on challenging few-shot benchmarks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' (2)  What components play an integral role in HyRSM++ so that  HyRSM++ can work well?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' (3) Can the proposed hybrid re-  lation module be viewed as a simple plug-and-play unit and  have the same effect for other methods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' (4) Does the pro-  posed temporal set matching metric have an advantage over  other measure competitors?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' (5) Can HyRSM++ have stable  performance in a variety of different video scenarios?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1 Datasets and experimental setups  Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We evaluate our HyRSM++ on six standard public  few-shot benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' For the Kinetics [8], SSv2-Full [23],  and SSv2-Small [23] datasets, we adopt the existing splits  proposed by [7, 112, 106, 68], and each dataset consists  HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition  9  MSA  Transformer  Bi-LSTM  Bi-GRU  Inter-relation  MSA  Transformer  Bi-LSTM  Bi-GRU  Intra-relation  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' 3 Comparison between different components in hybrid relation  module on 5-way 1-shot few-shot action classification without tempo-  ral coherence regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Experiments are conducted on the SSv2-  Full dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  MSA  Transformer  Bi-LSTM  Bi-GRU  Inter-relation  MSA  Transformer  Bi-LSTM  Bi-GRU  Intra-relation  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' 4 Comparison between different components in hybrid relation  module on 5-way 1-shot few-shot action classification with temporal  coherence regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Experiments are conducted on the SSv2-Full  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  of 64 and 24 classes as the meta-training and meta-testing  set, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' For UCF101 [78] and HMDB51 [42], we  verify our proposed methods by leveraging existing splits  from [105, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In addition to the above, we also utilize  the egocentric Epic-kitchens [14, 13] dataset to evaluate  HyRSM++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Following previous works [112, 7,  68, 106], ResNet-50 [28] initialized with ImageNet [15] pre-  trained weights is utilized as the feature extractor in our ex-  periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We sparsely and uniformly sample 8 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', T = 8)  frames per video to construct input frame sequence, which is  in line with previous methods [7, 106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In the training phase,  we also adopt basic data augmentation such as random crop-  ping and color jitter, and use Adam [39] optimizer to train  our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' During the inference stage, we conduct few-shot  action recognition evaluation on 10,000 randomly sampled  episodes from the meta-testing set and report the mean ac-  curacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' For many shot classification, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', 5-shot, we follow  ProtoNet [76] and calculate the mean features of support  videos in each class as the prototypes, and classify the query  videos according to their distances against the prototypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2 Comparison with state-of-the-art  In this section, we validate the effectiveness of the proposed  HyRSM++ by comparing it with state-of-the-art methods  under various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' As indicated in Table 1 and Ta-  ble 2, the proposed HyRSM++ surpasses other advanced  approaches significantly and is able to achieve new state-  of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' For instance, HyRSM++ improves  the state-of-the-art performance from 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2% to 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0% un-  der the 1-shot setting on SSv2-Full and consistently outper-  forms our original conference version [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Specially, ex-  tensively compared with current strict temporal alignment  techniques [7, 106] and complex fusion methods [48, 68],  HyRSM++ produces results that are superior to them un-  der most different shots, which implies that our approach  is considerably flexible and efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Note that the SSv2-  Full and SSv2-Small datasets tend to be motion-based and  generally focus on temporal reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' While Kinetics and  UCF101 are partly appearance-related datasets, and scene  understanding is usually essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Besides, Epic-kitchens  and HMDB51 are relatively complicated and might involve  diverse object interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Extensively evaluated on these  benchmarks, HyRSM++ provides excellent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' It  reveals that our HyRSM++ has strong robustness and gen-  eralization for different scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' From Table 2, we observe  that HyRSM++ outperforms current state-of-the-art meth-  ods on UCF101 and SSv2-Small under the 1-shot and 3-  shot settings, which suggests that our HyRSM++ can learn  rich and effective representations with extremely limited  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' It’s worth noting that under the 5-shot evaluation,  our HyRSM++ yields 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9% and 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0% 5-shot performance  on UCF101 and SSv2-Small, respectively, which is slightly  behind STRM and HCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We attribute this to STRM and  HCL are ensemble methods that weight each sample with  attention or use multiple metrics for few-shot classification,  which makes them more suitable for multi-shots, while our  HyRSM++ is a simple and general method without involves  complex ensemble operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Moreover, we also observe  that with the introduction of temporal coherence regulariza-  tion, HyRSM++ has a significant improvement compared to  HyRSM, which verifies the effectiveness of exploiting tem-  poral order information during the set matching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3 Ablation study  For ease of comparison, we use a baseline method Pro-  toNet [76] that applies global-average pooling to backbone  representations to obtain a prototype for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We will  explore the role and validity of our proposed modules in de-  tail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Design choices of relation modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  To systematically  investigate the effect of different relation modeling opera-  10  Xiang Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Table 2 Results on 1-shot, 3-shot, and 5-shot few-shot classification on the UCF101 and SSv2-Small datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' ”-” means the result is not available  in published works, and the underline indicates the second best result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  UCF101  SSv2-Small  Method  Reference  1-shot  3-shot  5-shot  1-shot  3-shot  5-shot  MatchingNet [86]  NeurIPS’16 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='1 Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' [62]  ECCV’22  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='9 Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' [33]  ECCV’22  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='5)  HyRSM++ 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='6)  Table 3 Ablation study under 5-way 1-shot and 5-way 5-shot settings  on the SSv2-Full dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' “TCR” refers to temporal coherence regular-  ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Intra-relation  Inter-relation  Bi-MHM  TCR  1-shot  5-shot  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='6  �  �  �  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  �  �  �  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  �  �  �  �  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='8  Table 4 Generalization of hybrid relation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We conduct exper-  iments on SSv2-Full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Method  1-shot  5-shot  OTAM [7]  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  OTAM [7]+ Intra-relation  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  OTAM [7]+ Inter-relation  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  OTAM [7]+ Intra-relation + Inter-relation  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  tions in hybrid relation module, we vary the components to  construct some variants and report the results in Figure 3  and Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The comparison experiments are conducted  on the SSv2-Full dataset under the 5-way 1-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We  can observe that different combinations have quite distinct  properties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', multi-head self-attention (MSA) and Trans-  former are more effective to model intra-class relations than  Bi-LSTM and Bi-GRU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' For example, utilizing multi-head  5-way  6-way  7-way  8-way  9-way  10-way  Accuracy (%)  Kinetics  OTAM  TRX  STRM  HyRSM  HyRSM++  50  66  54  58  70  62  5-way  6-way  7-way  8-way  9-way  10-way  Accuracy (%)  SSv2-Full  OTAM  TRX  STRM  HyRSM  HyRSM++  25  30  40  35  50  45  55  74  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' 5 N-way 1-shot performance trends of our HyRSM++ and other  state-of-the-art methods with different N on SSv2-Full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The compari-  son results prove the superiority of our HyRSM++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Accuracy (%)  (a) Frames  42  46  50  54  2  3  4  5  6  7  8  9  10  1  2  4  8  16  32  Accuracy (%)  1-shot  5-shot  45  50  60  55  70  65  (b) Head number  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' 6 (a) Performance on SSv2-Full using a different number of  frames under the 5-way 1-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' (b) The effect of the number  of heads on SSv2-Full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  self-attention to learn intra-relation produces at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5%  improvements than with Bi-LSTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Nevertheless, compared  with other recent algorithms [68, 106], the performance of  each combination can still be improved, which strongly sug-  gests the necessity of structure design for learning task-  specific features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' For simplicity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' we choose the same struc-  ture to explore intra-relation and inter-relation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' and the con-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='70  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='SSv2-Full (Resnet-18)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='OTAM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='TRX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='HyRSM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='HyRSM++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='70  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='SSv2-Full (Resnet-34)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='OTAM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='TRX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='HyRSM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='HyRSM++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='Accuracy (%)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='55  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='65  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='70  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='75  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='85  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='Kinetics (Resnet-18)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='OTAM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='TRX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='HyRSM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='HyRSM++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='55  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='65  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='70  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='75  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='85  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='Kinetics (Resnet-34)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='OTAM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='TRX  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='HyRSM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='HyRSM++  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5-shot  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='Accuracy (%)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='Accuracy (%)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='Accuracy (%)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' 7 Comparison of the backbone with different depths on the SSv2-  Full and Kinetics datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Table 5 Comparative experiments on SSv2-Full using the Inception-  v3 [81] feature extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Method  1-shot  2-shot  3-shot  4-shot  5-shot  OTAM [7]  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  TRX [68]  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  STRM [84]  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  HyRSM++  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  Table 6 Performance comparison on SSv2-Full with self-supervised  initialization weights [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Method  1-shot  2-shot  3-shot  4-shot  5-shot  OTAM [7]  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  TRX [68]  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  STRM [84]  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  HyRSM++  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  Table 7 Performance comparison with different relation modeling  paradigms on SSv2-Full and Kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Setting  Method  Dataset  1-shot  5-shot  Support-only  HyRSM  SSv2-Full  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  Support-only  HyRSM++  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  Support&Query  HyRSM  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  Support&Query  HyRSM++  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  Support-only  HyRSM  Kinetics  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  Support-only  HyRSM++  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  Support&Query  HyRSM  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  Support&Query  HyRSM++  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  figuration of multi-head self-attention is adopted in the ex-  periments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Analysis of the proposed components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Table 3 summa-  rizes the ablation study of each module in HyRSM++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' To  evaluate the function of the proposed components, Pro-  toNet [76] is taken as our baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' From the ablation results,  we can conclude that each component is highly effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  In particular, compared to the baseline, intra-relation mod-  eling can respectively bring 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0% and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7% performance  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='9  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  0%  10%  20%  30%  40%  50%  Accuracy (%)  5-way 5-shot  OTAM  TRX  STRM  HyRSM++  45  61  49  53  65  57  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='3  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='7  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  0%  10%  20%  30%  40%  50%  Accuracy (%)  5-way 1-shot  OTAM  TRX  STRM  HyRSM++  25  30  40  35  50  45  55  69  Noisy ratio  Noisy ratio  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' 8 Robustness comparison experiments in the presence of noisy  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' X% represents the proportion of noisy labels included in the  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Table 8 Comparison with recent temporal alignment methods on the  SSv2-Full dataset under the 5-way 1-shot and 5-way 5-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Diagonal means matching frame by frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Metric  Bi-direction  1-shot  5-shot  Diagonal 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  Plain DTW [61] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  OTAM [7]  �  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  OTAM [7]  �  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  Bi-MHM  �  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  Temporal set matching metric  �  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  Table 9 Comparison of different set matching strategies on the SSv2-  Full dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Metric  Bi-direction  1-shot  5-shot  Hausdorff distance  �  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  Hausdorff distance  �  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  Modified Hausdorff distance  �  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  Bi-MHM  �  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  Temporal set matching metric  �  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  Table 10 Generalization of temporal coherence regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We  conduct experiments on SSv2-Full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' ”Hard margin” represents the  method described in Equation 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Method  1-shot  5-shot  OTAM [7]  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  OTAM [7] + IDM  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  OTAM [7] + Hard margin  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  OTAM [7] + Temporal coherence regularization  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  Bi-MHM  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  Bi-MHM + IDM  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  Bi-MHM + Hard margin  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  Bi-MHM + Temporal coherence regularization  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  gains on 1-shot and 5-shot, and inter-relation function boosts  the performance by 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5% and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9% on 1-shot and 5-shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  In addition, the proposed set matching metric improves 1-  shot and 5-shot classification by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4% and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7%, respec-  tively, which indicates the ability to find better correspond-  ing frames in the video pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Adding temporal coherence  regularization to the set matching metric also achieves sta-  12  Xiang Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  ble performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Moreover, stacking the pro-  posed modules can further improve performance, indicating  the complementarity between components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' When consider-  ing all the proposed modules together to form HyRSM++,  the performance of 1-shot and 5-shot is improved to 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0%  and 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8%, respectively, which strongly supports the impor-  tance of learning task-related features and flexible metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Pluggability of hybrid relation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In Table 4, we  experimentally show that the hybrid relation module gen-  eralizes well to other methods by inserting it into the re-  cent OTAM [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In this study, OTAM with our hybrid re-  lation module benefits from relational information and fi-  nally achieves 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9% and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6% gains on 1-shot and 5-shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  This fully evidences that mining the rich information among  videos to learn task-specific features is especially valuable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  N-way few-shot classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  In the previous experi-  ments, all of our comparative evaluation experiments were  carried out under the 5-way setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In order to further ex-  plore the influence of different N, in Figure 5, we com-  pare N-way (N ≥ 5) 1-shot results on SSv2-Full and Ki-  netics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Results show that as N increases, the difficulty be-  comes higher, and the performance decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Neverthe-  less, the performance of our HyRSM++ is still consistently  ahead of the recent state-of-the-art STRM [84], TRX [68]  and OTAM [7], which shows the feasibility of our method  to boost performance by introducing rich relations among  videos and the power of the set matching metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Varying the number of frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' To demonstrate the scal-  ability of HyRSM++, we also explore the impact of differ-  ent video frame numbers on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Of note, previous  comparisons are performed under 8 frames of input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Results  in Figure 6(a) show that as the number of frames increases,  the performance improves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' HyRSM++ gradually tends to be  saturated when more than 7 frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Influence of head number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Previous analyses have shown  that multi-head self-attention can focus on different patterns  and is critical to capturing diverse features [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We investi-  gate the virtue of varying the number of heads in multi-head  self-attention on performance in Figure 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Experimental  results indicate that the effect of multi-head is remarkable,  and the performance starts to saturate beyond a particular  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Varying depth of the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The proposed HyRSM++  is general and compatible with feature extractors of various  capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The previous methods all utilize ResNet-50 as  backbone by default for a fair comparison, and the impact  of backbone’s depth on performance is still under-explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  As presented in Figure 7, we attempt to answer this question  by adopting ResNet-18 and ResNet-34 pre-trained on Ima-  geNet as alternative backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Results demonstrate that the  deeper network clearly benefits from greater learning capac-  ity and results in better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' we notice  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='Acc = 100%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='Acc = 100%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='(+ hybrid relation module)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='(a) Examples from SSv2-Full  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='(b) Examples from Kinetics  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' 9 Similarity visualization of how query videos (rows) match to  support videos (columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The boxes of different colors correspond to:  correct match and incorrect match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Support  Query  (a) SSv2-Full: ”pretending to open something without actually opening it”  (b) SSv2-Full: ”showing that something is empty”  Support  Query  Support  Query  (c) Kinetics: ”cutting watermelon”  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' 10 Visualization of matching results with the proposed set match-  ing metric on SSv2-Full and Kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  that our proposed HyRSM++ consistently outperforms the  competitors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', OTAM and TRX), which indicates that our  HyRSM++ is a generally effective framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Influence of different backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' To verify that our ap-  proach is not limited to ResNet-like structures, we further  perform experiments on Inception-v3 and report the results  in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' From the comparison, we note that HyRSM++ is  significantly superior to other competitive algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Com-  pared with STRM [84], our proposed HyRSM++ leads to at  least 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='25HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition  13  ble 6, we show the performance comparison with self-  supervised pretraining weights [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Results demonstrate  that our HyRSM++ is still powerful and not limited to the  specific initialization weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Other relation modeling forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Previous few-shot image  classification methods of learning task-specific features have  also achieved promising results [101, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' However, many  of them use some complex and fixed operations to learn the  dependencies between images, while our method is straight-  forward and flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Moreover, most previous works only  use the information within the support set to learn task-  specific features, ignoring the correlation with query sam-  ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In our hybrid relation module, we add the query video  to the pool of inter-relation modeling to extract relevant in-  formation suitable for query classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' As illustrated in  Table 7, we try to remove the query video from the pool  in HyRSM++, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', Support-only, but we can observe that  after removing the query video, the performance of 1-shot  and 5-shot on SSv2-Full reduces by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0%, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' There are similar conclusions on the Kinetics dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  This evidences that the proposed hybrid relation module is  reasonable and can effectively extract task-related features,  thereby promoting query classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Robustness to noise labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' To demonstrate the robustness  of HyRSM++ to noise samples, we simulate the presence  of noise labels in the dataset in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' From the results,  we can observe that performance generally decreases as the  proportion of noise rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' However, our HyRSM++ still ex-  hibits higher performance than other methods, which illus-  trates the robustness of our method and its adaptability to  complex conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4 Comparison with other matching approaches  Our proposed temporal set matching metric Bi-MHM aims  to accurately find the corresponding video frames between  video pairs by relaxing the strict temporal ordering con-  straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The following comparative experiments in Table 8  are carried out under identical experimental setups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', re-  Table 11 Complexity analysis for 5-way 1-shot SSv2-Full evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  The experiments are carried out on one Nvidia V100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Method  Backbone  Param  FLOPs  Latency  Acc  HyRSM  ResNet-18  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='8  TRX [68]  ResNet-50  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='0  place the OTAM directly with our Bi-MHM while keep-  ing other settings unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Results show that our Bi-  MHM performs well and outperforms other temporal align-  ment methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', OTAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We further analyze different  set matching approaches in Table 9, and the results indi-  cate that Hausdorff distance is susceptible to noise interfer-  ence, resulting in the mismatch and relatively poor perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' However, our Bi-MHM shows stability to noise and  obtains better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Furthermore, compared with the  single directional metric, our proposed bidirectional metric  is more comprehensive in reflecting the actual distances be-  tween videos and achieves better performance on few-shot  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In addition, we observe that the proposed temporal  set matching metric achieves clear improvement over Bi-  MHM after incorporating temporal coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' For instance,  the temporal set matching metric obtains 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1% perfor-  mance gains on 5-way 1-shot, and 5-way 5-shot SSv2-Full  classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' It indicates the effectiveness of the proposed  temporal set matching metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5 Comparison of temporal coherence manners  Pioneering work [11, 22, 59] also indicates the important  role of temporal coherence and shows remarkable results in  face recognition [59] and unsupervised representation learn-  ing [22, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' However, they also have some limitations as  noted in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2, and thus the temporal coherence reg-  ularization is proposed for smooth video coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Ta-  ble 10 compares the proposed temporal coherence regular-  ization with existing temporal coherence schemes based on  OTAM and Bi-MHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Results show that exploiting tempo-  ral coherence helps improve the classification accuracy of  the metrics, which confirms our motivation for consider-  ing temporal order information during the matching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  In addition, our proposed temporal coherence regularization  achieves more significant improvements than other manners,  and we attribute this to the smooth property of temporal co-  herence regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6 Visualization results  To qualitatively show the discriminative capability of the  learned task-specific features in our proposed method, we  visualize the similarities between query and support videos  with and without the hybrid relation module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' As depicted in  Figure 9, by adding the hybrid relation module, the discrim-  ination of features is significantly improved, contributing to  predicting more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Additionally, the matching re-  sults of the set matching metric are visualized in Figure 10,  and we can observe that our Bi-MHM is considerably flexi-  ble in dealing with alignment and misalignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  14  Xiang Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Support  Query  Support  Query  “tipping Sth over” from SSv2-Full  OTAM  HyRSM++  “taking Sth out of Sth” from SSv2-Full  OTAM  HyRSM++  Support  Query  “showing Sth next to Sth” from SSv2-Full  OTAM  HyRSM++  Support  Query  “riding elephant” from Kinetics  OTAM  HyRSM++  Support  Query  “playing trumpet” from Kinetics  OTAM  HyRSM++  Support  Query  “filling eyebrows” from Kinetics  OTAM  HyRSM++  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' 11 Visualization of activation maps with Grad-CAM [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Compared to OTAM [7], HyRSM++ focuses more precisely on classification-  related regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  To further visually evaluate the proposed HyRSM++, we  compare the activation visualization results of HyRSM++ to  the competitive OTAM [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' As shown in Figure 11, the fea-  tures of OTAM usually contain non-target objects or ignore  most discriminative parts since it lacks the mechanism of  learning task-specific embeddings for feature adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In  contrast, our proposed HyRSM++ processes the query and  support videos with an adaptive relation modeling operation,  which allows it to focus on the different target objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The  above qualitative experiments illustrate the rationality of our  model design and the necessity of learning task-related fea-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7 Limitations  In order to further understand HyRSM++, Table 11 il-  lustrates its differences with OTAM and TRX in terms  of parameters, computation, and runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In the inference  HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition  15  Table 12 Comparison to existing semi-supervised few-shot action recognition methods on the meta-testing set of Kinetics and SSv2-Small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The  experiments are conducted under the 5-way setting, and results are reported as the shot increases from 1 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' ”w/o unlabeled data” indicates  that there is no unlabeled set in a episode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', the traditional few-shot action recognition setting, which can act as the lower bound of the semi-  supervised counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Dataset  Method  Backbone  1-shot  2-shot  3-shot  4-shot  5-shot  Kinetics  OTAM w/o unlabeled data [7]  Inception-v3  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  DeepCluster CACTUs-MAML [30]  Inception-v3  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  DeepCluster CACTUs-ProtoNets [30]  Inception-v3  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  LIM [113]  Inception-v3  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  HyRSM++ w/o unlabeled data  Inception-v3  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  HyRSM++  Inception-v3  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='6  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  0  50  100  150  200  Accuracy (%)  SSv2-Small  1-shot  2-shot  3-shot  4-shot  5-shot  60  38  40  44  42  50  46  58  48  62  54  52  56  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' 13 Performance comparison of different amounts of unlabeled  data for testing in an episode on the SSv2-Small dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  5 Extension to Semi-supervised Few-shot Action  Recognition  In this section, we demonstrate that the proposed HyRSM++  can be extended to address the more challenging semi-  supervised few-shot action recognition problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Follow-  ing LIM [113], we utilize two common datasets (Kinet-  ics [8] and SSv2-Small [23]) to perform comparative exper-  iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' These two datasets are subsets of Kinetics-400 [8]  and Something-Something-v2 [23], respectively, and the un-  labeled examples in our experiments are collected from the  remaining videos of the same category as these subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' To  16  Xiang Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Table 13 Comparison to state-of-the-art unsupervised few-shot action  recognition approaches on UCF101, HMDB51, and Kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' ∗ indi-  cates that the algorithm adopt the same 2D ResNet-50 backbone as  HyRSM++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Method  Supervision  UCF101  HMDB51 Kinetics  MAML [19]  Supervised 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  CMN [112]  Supervised 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  TARN [5]  Supervised 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  ProtoGAN [43]  Supervised  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7 ARN [105]  Supervised  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  3DRotNet [37]  Unsupervised  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  VCOP [96]  Unsupervised  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  IIC [83]  Unsupervised  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  Pace [87]  Unsupervised  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  MemDPC [83]  Unsupervised  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  CoCLR [26]  Unsupervised  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='6  MetaUVFS∗ [64]  Unsupervised  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  HyRSM++  Unsupervised  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+page_content='0  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='4  50  75  100  125  150  175  200  Accuracy (%)  UCF101  63  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  50  75  100  125  150  175  200  Accuracy (%)  HMDB51  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='7  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='3  50  75  100  125  150  175  200  Accuracy (%)  Kinetics  49  65  67  69  55  37  39  41  43  51  53  57  35  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' 14 Ablation study of different cluster numbers under 5-way 1-  shot unsupervised few-shot settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  conduct the semi-supervised few-shot evaluation, we fol-  low the mainstream distractor setting [30, 38, 113], where  the unlabeled set contains other interference classes in each  episodic task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' This setting is more realistic and requires the  model to be robust to the existence of noisy samples from  other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In our experiments, we fixed the number of  unlabeled videos in an episodic task to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Table 12 provides the comparison of our HyRSM++  against state-of-the-art methods on the two standard semi-  supervised few-shot benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We find that HyRSM++  substantially surpasses the previous approaches, such as  LIM [113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Under the semi-supervised 5-way 1-shot sce-  nario, HyRSM++ produces performance gains of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='8% and  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='5% on Kinetics and SSv2-Small than LIM with Inception-  v3 backbone, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In particular, when using the  ResNet-50 backbone, our method is even superior to the  multi-modal fusion method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', LIM), which indicates that  HyRSM++ enables more accurate pseudo-labels for unla-  beled data and then can expand the support set to boost the  classification accuracy of the query videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In addition, com-  pared to our supervised counterpart (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=', HyRSM++ w/o un-  labeled data), joining unlabeled data is beneficial to allevi-  ating the data scarcity problem and promotes few-shot clas-  sification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' We can observe that when ResNet-50  is adopted as the backbone, the performance of HyRSM++  with unlabeled data is improved by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='1% compared to that  without unlabeled data under the 5-way 1-shot Kinetics  evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  To further investigate the effect of unlabeled videos in  an episode, we conduct comparative experiments with vary-  ing numbers of unlabeled videos in Figure 12 and Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Experimental results show that as the number of unlabeled  samples increases, the performance also increases gradually,  indicating that the introduction of unlabeled data helps gen-  eralize to unseen categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Furthermore, we notice that the  improvement in the 1-shot setting is more significant than  that in the 5-shot, which shows that under the condition of  low samples, unlabeled videos can improve the estimation  of the distribution of new categories more effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Mean-  while, as the amount of unlabeled data increases to a certain  level, the performance starts to saturate slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  6 Extension to Unsupervised Few-shot Action  Recognition  We also extend the proposed HyRSM++ to solve the  challenging unsupervised few-shot action recognition task  where labels for training videos are not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Following  previous work [38, 36], we adopt the idea of the ”cluster-  ing first and then meta-learning” paradigm to construct few-  shot tasks and exploit unlabeled data for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Our ex-  periments are based on unsupervised ResNet-50 initializa-  tion [97], which is self-supervised pre-trained on Kinetics-  400 [8] without accessing any label information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' During the  clustering process, we utilize the K-means clustering strat-  egy for each dataset to obtain 150 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  As presented in Table 13, we compare HyRSM++ with  current state-of-the-art methods on the UCF101, HMDB51  and Kinetics datasets under the 5-way 1-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Note  that HyRSM++ and MetaUVFS [64] use the same ResNet-  50 structure as the feature extractor, and our HyRSM++  shows better performance on each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In particular, we  observe that our method achieves 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='0% performance on  the UCF101 dataset, a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='9% improvement over MetaUVFS,  and even surpasses the fully supervised ARN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' The supe-  rior performance of HyRSM++ reveals that our approach of  leveraging relations within and cross videos and the flexible  metric performs effectively in the low-shot regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' More-  over, this phenomenon also demonstrates the potential of our  method to learn a strongly robust few-shot model using only  unlabeled videos, even though HyRSM++ is not specifically  HyRSM++: Hybrid Relation Guided Temporal Set Matching for Few-shot Action Recognition  17  designed for the unsupervised few-shot action recognition  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  In the experiments, one parameter involved in apply-  ing HyRSM++ to the unsupervised few-shot setting is the  number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' In Figure 14, we display the perfor-  mance comparison under different number of clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Re-  sults show that when the number of clusters is 150, the per-  formance reaches the peak value, which means that if the  cluster number is too small, it may lead to under-clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  If the number is too large, it may cause over-clustering, dam-  aging the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  7 Conclusion  In this work, we have proposed a hybrid relation guided  temporal set matching (HyRSM++) approach for few-shot  action recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Firstly, we design a hybrid relation mod-  ule to model the rich semantic relevance within one video  and cross different videos in an episodic task to generate  task-specific features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Secondly, built upon the representa-  tive task-specific features, an efficient set matching metric is  proposed to be resilient to misalignment and match videos  accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' During the matching process, a temporal coher-  ence regularization is further imposed to exploit temporal  order information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Furthermore, we extend HyRSM++ to  solve the more challenging semi-supervised few-shot ac-  tion recognition and unsupervised few-shot action recog-  nition problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content=' Experimental results demonstrate that our  HyRSM++ achieves the state-of-the-art performance on  multiple standard benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='  Acknowledgements This work is supported by the National Natural  Science Foundation of China under grant 61871435, Fundamental Re-  search Funds for the Central Universities no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
+page_content='2019kfyXKJC024, 111  Project on Computational Intelligence and Intelligent Control under  Grant B18024, and Alibaba Group through Alibaba Research Intern  Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BdE1T4oBgHgl3EQfpQWt/content/2301.03330v1.pdf'}
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+Safer Together: Machine Learning Models Trained on Shared Accident
+Datasets Predict Construction Injuries Better than Company-Specific
+Models
+Submitted to Automation in Construction
+Antoine J.-P. Tixier1a, Matthew R. Hallowella,b
+aSafetyAI R&D
+bUniversity of Colorado at Boulder
+Abstract
+Highlights
+• 9 companies from 3 domains (construction, electric T&D, oil & gas) shared their accident datasets.
+• Machine learning models were trained to predict safety outcomes from fundamental attributes.
+• Models trained on all datasets (full generic models) outperformed the company-specific models
+in 82% of the company-domain-outcome combinations, with large gains in F1 score (+4.4 on
+average and up to +15.3).
+• On average, generic models predicted 2.26 categories more than specific models (up to 7), making
+for more useful forecasts in practice.
+• Per-domain generic models were not always better than full generic models.
+• Combining generic and specific models (data quantity and relevance) was often very beneficial.
+• Generic models give companies devoid of accident datasets access to safety predictions.
+• Generic models address safety cross-organizational learning and dissemination in construction.
+In this study, we capitalized on a collective dataset repository of 57k accidents from 9 companies be-
+longing to 3 domains and tested whether models trained on multiple datasets (generic models) predicted
+safety outcomes better than the company-specific models. We experimented with full generic models
+(trained on all data), per-domain generic models (construction, electric T&D, oil & gas), and with en-
+sembles of generic and specific models. Results are very positive, with generic models outperforming the
+company-specific models in most cases while also generating finer-grained, hence more useful, forecasts.
+Successful generic models remove the needs for training company-specific models, saving a lot of time
+and resources, and give small companies, whose accident datasets are too limited to train their own mod-
+els, access to safety outcome predictions. It may still however be advantageous to train specific models
+to get an extra boost in performance through ensembling with the generic models. Overall, by learning
+lessons from a pool of datasets whose accumulated experience far exceeds that of any single company,
+and making these lessons easily accessible in the form of simple forecasts, generic models tackle the
+holy grail of safety cross-organizational learning and dissemination in the construction industry.
+Keywords: construction safety, artificial intelligence, supervised learning, injury prediction,
+transfer learning, data sharing, collective intelligence
+1antoine.tixier@safetyfunction.com
+arXiv:2301.03567v1  [cs.LG]  9 Jan 2023
+
+1. Introduction
+The SafetyAI council is a community of large organizations from the construction, oil &
+gas, and electric Transmission and Delivery (T&D) domains, that share their safety-related data
+with the SafetyAI Research and Development (R&D) team.
+Before exploiting the data, the R&D team is in charge of standardizing the datasets received
+by each company, which is crucial, as each one features different variables and different category
+names for each variable. Standardization makes sure that all datasets are based on the same
+taxonomy, i.e., speak the same language.
+The SafetyAI community dataset, comprising close to a million events including near misses,
+observations, good catches, etc., is only accessible to the R&D team, a neutral party, which guar-
+antees that it is impossible for companies to see each other’s data, and that the output of all the
+R&D conducted on the collective dataset is made available to the entire community. This is of
+paramount importance, in a very competitive environment.
+In this study, we started by extracting attributes from accident reports. We briefly introduce
+the attribute framework in what follows.
+1.1. Attribute-based framework
+Attributes are basic descriptors of construction work that are observable before accident
+occurrence, and cover means, methods, and environmental conditions [1, 2]. One advantage of
+the attribute-based framework over modeling at the task or work package level is that attributes
+are fundamental and universal. That is, any situation from any site around the world, in any
+industry sector, can be characterized by a set of attributes. Attributes can be recorded on-the-fly
+on site, or can be extracted offline from various mediums such as photos and text reports. For
+instance, four attributes can be extracted from the narrative worker tripped on a cable
+when carrying a 2x4 to his truck: (1) cable, (2) object on the floor, (3) lumber, and
+(4) light vehicle.
+Narratives are particularly well-suited if the goal is to use attributes for predictive modeling.
+Indeed, in incident report databases, narratives are often paired with outcomes such as accident
+type, injury severity, body part impacted, etc. Attributes also completely anonymize narratives,
+which is especially desirable when considering a pool of datasets aggregated from different
+companies. For any given event, everything that remains is a set of attributes and a set of
+standardized safety outcomes.
+However, manually extracting attributes from large amounts of text reports is very costly in
+terms of human resources and pose inter-annotator agreement issues. To solve this problem, we
+developed and validated a Natural Language Processing (NLP) tool based on rules and lexicons
+[3]. We later proved that using the attributes extracted by the tool to predict safety outcomes
+was effective and valid [4, 5]. We also used the attributes extracted by the tool for unsupervised
+learning applications, such as clustering and visualization [6], and risk modeling and simulation
+[7].
+1.2. Differences with our previous research and objective of the current study
+In our original study [4], we provided a proof for the concept of predicting safety outcomes
+from attributes, both extracted with the NLP tool. Then, in [5], we showed that attributes were
+still highly predictive when the safety outcomes were given by independent human annotations,
+which definitely validated the approach. We also used a much larger dataset than in the orig-
+inal study, two new supervised learning algorithms, model stacking, a healthier experimental
+setup with more appropriate performance metrics, and we analyzed per-category attribute im-
+portance scores. We also showed that unlike what we had concluded in [4], injury severity was
+predictable from attributes.
+2
+
+In the present research, we interested ourselves with a new, completely different problem.
+We had access to a pool of accident datasets coming from 9 companies, and our goal was to:
+“Test whether predictive models trained on a generic dataset (i.e., aggregated from the datasets
+of multiple companies) outperformed the models trained on the specific dataset of each com-
+pany.”
+More precisely, we experimented with two types of generic models:
+• Full generic model: one model trained on the datasets of all companies.
+• Per-domain generic models: one model per industry sector, trained only on the datasets
+of the companies involved in that sector (or the parts thereof, as some companies belong
+to multiple domains).
+The potential advantages of generic models are numerous:
+1. Usually with machine learning, the more data, the better, so generic models are expected
+to bring improvements in predictive skill compared to the company-specific models. This
+is not guaranteed however, as one important question is whether (1) more data (generic
+datasets) or (2) more relevant data (specific datasets) is better.
+2. By being trained on larger datasets, the generic models learn to predict a greater variety
+of outcome categories than the specific models, making for more useful forecasts.
+3. Successful generic models would remove the needs for training specific models for each
+company, saving a lot of time and resources.
+4. Alternatively, if company-specific models are already available, combining them with the
+generic models may provide an extra boost in performance.
+5. Last but not least, successful generic models would give small companies -whose accident
+datasets are too limited to train their own specific models- access to high quality safety
+outcome forecasts.
+From a high level, generic models tackle the holy grail of safety cross-organizational learn-
+ing and dissemination in the construction industry. Indeed, generic models (1) learn lessons
+from a pool of datasets whose quantity and diversity2 of accumulated experience far exceeds
+that of any single company, and (2) disseminate these lessons as forecasts, which are clear, di-
+rect, and easily accessible information, via, e.g., a user interface (desktop or mobile) or API
+taking attributes as input and returning probabilities for each category of each outcome.
+Moreover, one should note that in the pool, the individual biases of each dataset, due to
+specific annotators, reporting practices and policies, etc., tend to average out. Consequently, the
+lessons learned by the supervised learning algorithms on the generic datasets are more objective
+and broadly applicable than that learned on the specific datasets.
+2. Background
+The needs to share standardized incident data at the industry level to enable collaborative
+learning have long been recognized in aviation and transportation [8]. Some examples include
+2Diversity of situations, means and methods, environmental conditions, geographical areas...
+3
+
+the NASA-managed Aviation Safety Reporting System (ASRS) database, created in 1976 and
+featuring over a million incidents, or the European Coordination Center for Accident and In-
+cident Reporting Systems (ECCAIRS) database, started in 2004. Such collective repositories
+also exist in the chemical industry, with the Major Accident Reporting System (eMARS) of the
+European Commission, launched in 1982, and the Process Safety Incident Database (PSID) of
+the Center for Chemical Process Safety [9].
+However, the construction industry still lacks comparable initiatives. The needs for data
+storage and access infrastructures for construction safety did start to receive some attention
+recently [10, 11], but most efforts placed themselves at the company or project level. Cross-
+organizational safety data collection is still rare in practice [12, 13]. This is a major issue,
+as collaborative machine learning at the industry level is not possible until a common pool of
+standardized datasets has been put together. This provided the motivation for us to create the
+SafetyAI council in 2020.
+One should note that some consortiums already exist, such as the INGAA Foundation, the
+Edison Electric Institute (EEI), the Construction Safety Research Alliance (CSRA), or the Na-
+tional Safety Council (NSC), but their activities do not revolve around systematic large-scale
+accident data collection and analysis. These initiatives rather involve working groups, com-
+munities of practice, qualitative analyses, and conferences, towards building communications,
+policies, best practices, business intelligence, safety culture and leadership, training material,
+etc. In other words, they are based on “soft” methods for knowledge sharing and collabo-
+rative learning at the human level. They do not primarily conduct “hard” scientific research
+and software development, and do not pool accident datasets for AI applications and automatic
+large-scale learning and dissemination.
+3. Data Description
+As already explained, as part of the SafetyAI initiative, we had access to a pool of safety
+datasets coming from nine large companies from the construction, oil & gas, and electric Trans-
+mission and Delivery (T&D) domains. One company, Company73, also had about 600 corporate
+services (office) events for the severity outcome. We kept these cases as training data for the
+full generic model but did not train a specific model on them.
+Member companies conduct work mostly in North America, and rely on their own teams as
+well as contractors. The collective dataset covers the period 2000 to 2022, with a distribution
+biased towards the last decade and especially more recent years.
+While the entire pool comprises almost a million events including near misses and observa-
+tions, we focused on accident cases only in this effort. As can be seen in Table 1, the sizes of
+the individual datasets ranged from 2k to 20k cases, with an average of 6k per company. There
+were 57262 accident cases in total, recorded over tens of millions of work hours.
+We considered the same outcomes as in [5]: injury severity, body part impacted, injury
+type, and accident type. The columns corresponding to each outcome were selected from the
+company datasets and normalized to use a common, standard set of categories, shown in Table
+2. Not all outcomes were available for every event of every company. From the narrative of
+each report, we extracted with the NLP tool [3] the original set of 80 attributes [3, 5], plus 11
+new items (see Table A.7). We also used the tool to extract a fifth outcome, energy source, that
+was not available in the company datasets.
+3Company names have been anonymized.
+4
+
+Comp.1
+Comp.2
+Comp.3
+Comp.4
+Comp.5
+Comp.6
+Comp.7
+Comp.8
+Comp.9
+Domains
+Constr.,
+elec.
+Oilgas
+Constr.,
+oilgas
+Elec.
+Constr.
+Constr.,
+elec.
+Elec.,
+oilgas,
+corp.
+Oilgas
+Elec.
+Regions
+Canada
+California
+NAM
+NAM
+NAM
+NAM
+NAM,
+Mexico
+World⋆
+Southeast
+USA
+n
+4481
+1965
+4072
+5321
+7245
+4310
+8345
+19298
+2225
+Table 1: Company overview. NAM: North America (Canada + USA). Constr.: construction. Elec: electric T&D.
+Oilgas: oil & gas. ⋆Including ships and rigs. Corp: corporate.
+Injury Severity
+Body Part
+Injury Type
+Accident Type
+Energy Source
+first aid
+38994
+hand
+15782
+cut
+14086
+handling
+6379
+motion
+33958
+report-only
+6993
+head
+10296
+strain
+10069
+fall
+5374
+gravity
+15904
+lost time
+5319
+leg
+6550
+contusion
+8558
+exposure
+3986
+chemical
+2411
+medical
+4913
+arm
+5943
+foreign body
+3348
+struck
+3834
+biological
+2044
+recordable
+1043
+trunk
+5375
+pinch
+1756
+contact
+2269
+thermal
+1691
+foot
+4632
+fracture
+1681
+caught
+1758
+mechanical
+611
+multiple/entire
+942
+burn
+1454
+overexertion
+1523
+pressure
+296
+irritation
+1222
+equipment
+1449
+electricity
+181
+pain
+1194
+PPE
+949
+radiation
+166
+exhaustion
+1054
+transitioning
+578
+bite
+710
+error
+425
+Table 2: Outcome category counts, across all companies and domains. PPE: personal protective equipment.
+4. Experimental Setup
+4.1. Splits
+Train, validation and test splits were created for each of the 51 company-domain-outcome
+combinations for which at least 2 categories with more than 100 observations each were avail-
+able (shown in Table 4), by randomly sampling without replacement 64%, 16%, and 20% of
+cases, respectively. The counts summed over companies are shown in Table 3. Note that the
+proportions we used in our previous work [5] were 81%, 9% and 10%, but in the present re-
+search, we decided to reserve more observations for the validation and test sets to make them
+more representative of the training sets, in order to increase the stability and validity of hyper-
+parameter tuning and evaluation4.
+A specific model was trained on each of the 51 company-domain-outcome combinations for
+which sufficient data were available, except for that one combination involving the corporate
+cases, making for a total of 50 specific models.
+For a given domain and a given outcome, the splits of the per-domain generic model were
+obtained by combining, across all companies, the splits corresponding to that domain and that
+outcome. In total, there was one per-domain generic model for each domain and for each
+outcome, hence a total of 3 × 5 = 15 per-domain generic models.
+For a given outcome, the splits of the full generic model were obtained by combining, across
+all companies and across all domains, the splits corresponding to that outcome. In total, there
+was one full generic model for each outcome, hence a total of 5 full generic models.
+For each of the aforementioned cases, we tried 3 different algorithms, as will be explained
+in subsection 4.3. Hence, a total of (15 + 5) × 3 = 60 generic models were trained.
+4Increasing the sizes of the validation and test sets was a good alternative to k-fold cross-validation, which would
+have taken too much time.
+5
+
+# Companies
+Train
+Val
+Test
+Severity
+Construction
+4
+9980
+2494
+3119
+Electric T&D
+4
+6672
+1669
+2085
+Oil & Gas
+4
+18381
+4595
+5744
+Corporate
+1
+418
+105
+131
+Full
+9
+35451
+8863
+11079
+Body Part
+Construction
+4
+8209
+2052
+2565
+Electric T&D
+4
+6036
+1508
+1885
+Oil & Gas
+3
+15788
+3947
+4933
+Full
+9
+30033
+7507
+9383
+Injury Type
+Construction
+4
+6267
+1566
+1958
+Electric T&D
+4
+4764
+1191
+1489
+Oil & Gas
+3
+14960
+3740
+4675
+Full
+9
+25991
+6497
+8122
+Acc. Type
+Construction
+2
+2740
+685
+856
+Electric T&D
+2
+1600
+400
+500
+Oil & Gas
+3
+2910
+728
+910
+Full
+6
+7250
+1813
+2266
+En. Source
+Construction
+4
+4875
+1218
+1524
+Electric T&D
+3
+2637
+660
+825
+Oil & Gas
+2
+2600
+650
+813
+Full
+8
+10112
+2528
+3162
+Table 3: Split counts for each domain-outcome combination, summed over companies. For # Companies, full ̸=
+total as some companies belong to multiple domains (see Tables 1 and 4).
+Construction
+Electric T&D
+Oil & Gas
+Corp.
+Comp.
+S
+B
+IT
+AT
+E
+S
+B
+IT
+AT
+E
+S
+B
+IT
+AT
+E
+S
+1
+x
+x
+x
+x
+2
+x
+x
+x
+3
+x
+x
+x
+x
+x
+x
+x
+4
+x
+x
+x
+x
+x
+5
+x
+x
+x
+x
+x
+6
+x
+x
+x
+x
+x
+x
+x
+x
+7
+x
+x
+x
+x
+x
+x
+x
+x
+x
+8
+x
+x
+x
+x
+x
+9
+x
+x
+x
+x
+x
+Table 4: The 51 company-domain-outcome combinations associated with at least 2 categories with more than 100
+observations each. S: severity, B: body part, IT: injury type, AT: accident type, E: energy source. Corp.: corportate.
+4.2. Class imbalance
+To address the problem of class imbalance, weights inversely proportional to category
+counts in the training set were computed with the formula max(counts)/counts, like in
+[5]. During training, these weights forced the models to pay more attention to the cases from
+the minority categories. Per-category counts with training weights can be found in Tables B.8
+and B.9 for the 15 domain-outcome combinations.
+4.3. Supervised learning algorithms
+Like in [5], we relied on three popular machine learning models: Random Forest (RF) [14],
+eXtreme Gradient Boosting (XGBoost or XGB) [15], and linear Support Vector Machine (SVM)
+[16]. More precisely, we used the Python’s scikit-learn implementations of Random
+Forest5 and linear SVM6, while, for XGBoost, we used the original Python library7 and in
+5https://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestClassifier.html
+6https://scikit-learn.org/stable/modules/generated/sklearn.svm.LinearSVC.html
+7https://xgboost.readthedocs.io/en/latest/python/python api.html#module-xgboost.sklearn
+6
+
+particular the GPU-accelerated implementation of the “fast histogram” algorithm (gpu hist)
+as the tree method8.
+For theoretical details about each algorithm, we refer the reader to our paper [5], publicly
+available9.
+4.4. Hyperparameter optimization
+We tuned the models by performing grid searches on the validation sets. Details about the
+parameters searched are available in Appendix C. The final models were trained on the union
+of the training and validation sets with the best parameter values. Both the specific models and
+the generic models were tested on the test sets of the specific models, to ensure fair comparison.
+As already explained, there were 50 such test sets, one for each company-domain-outcome
+combination.
+4.5. Transfer learning by stacking generic and specific models
+As was mentioned in the introduction, one important question is the extent to which (1) more
+data (generic datasets) or (2) more relevant data (specific datasets) is better. In what follows,
+we explore a way to move past this binary choice and have a tradeoff between quantity and
+relevance.
+Inspired by transfer learning, which is very successful in computer vision [17] and NLP
+[18, 19, 20, 21], we experimented with combining the predictions of the generic and specific
+models via an ensemble model.
+Very briefly, in AI, transfer learning refers to a two-step process. First, a model is trained
+at solving a general task on large amounts of data. This phase is called the pretraining phase,
+as it allows the model to acquire generic knowledge (e.g., in NLP, reading and writing), that
+is applicable to a great variety of situations downstream. Second, the pretrained model is fine-
+tuned on a specific task of interest, often associated with a much smaller dataset (e.g. in NLP,
+summarization, classification, question answering, paraphrase detection, etc.).
+In our case, the generic and the specific models have to perform the same task, i.e., predict-
+ing a given safety outcome10, and there is no pretraining phase per se, in that the generic and the
+specific models are two different models. However, our approach is similar in spirit to transfer
+learning, as our goal is to capitalize on generic knowledge gained from large amounts of data to
+improve performance on a specific task associated with a smaller dataset.
+More precisely, for each company-domain-outcome combination, we trained a meta-model
+taking as input the weighted elementwise sum of the probabilistic forecasts of the best generic
+and specific models11. We used a simple logistic regression12 as our meta-model, with the C
+parameter fixed and equal to 0.2, like in [5]. We grid searched the validation set to find the best
+values of coefficients a and b where:
+inputensemble = a × outputgeneric + b × outputspecific
+(1)
+Besides performance considerations, using tunable weights improves interpretability, by
+providing information regarding which of the generic model or the specific model makes the
+most important contribution to predictive skill.
+8https://xgboost.readthedocs.io/en/latest/gpu/
+9https://arxiv.org/pdf/1908.05972.pdf
+10The generic model has to perform a more difficult version of the task, though (more categories to predict).
+11The entries of the specific model vector for the categories that it did not predict were set to zero.
+12https://scikit-learn.org/stable/modules/generated/sklearn.linear model.LogisticRegression.html
+7
+
+We tried values from 0.1 to 1 with 0.1 steps, holding the other parameter equal to 1, and
+conversely. That is, the following 19 pairs: (0.1, 1), (0.2, 1), ... , (1, 1), (1, 0.1), (1, 0.2), ... ,
+(1, 0.9).
+SVM issue. By design, the implementation of the linear SVM model we used, linearSVC,
+only returns discrete predictions, that is, a single label corresponding to the most likely category,
+rather than a probability distribution over all categories. To address this issue, in [5], we tried
+retraining the best SVM using the SVC implementation13 with linear Kernel. However, results
+were not convincing. Therefore, in the present study, we decided simply not to use model
+stacking when one of the two models involved (e.g., best generic or specific model) was a
+SVM.
+4.6. Performance metrics
+Due to the large class imbalance for all outcomes, measuring classification performance
+with accuracy was inadequate. Rather, we computed precision, recall, and F1-score.
+Precision, respectively recall, for category i, is equal to the number of correct predictions
+for category i (number of hits), divided by the number of predictions made for category i (hits
+and false alarms), respectively by the number of observations in category i (hits and misses).
+precision =
+Ci,i
+�K
+j=1 Cj,i
+recall =
+Ci,i
+�K
+j=1 Ci,j
+(2)
+Where the confusion matrix C is a square matrix of dimension K ×K (K being the number
+of categories) and whose (i, j)th element Ci,j indicates how many of the observations known
+to be in category i were predicted to be in category j. Finally, we computed the F1-score, the
+harmonic mean of precision and recall:
+F1 = 2 × precision × recall
+precision + recall
+(3)
+4.7. Configuration
+We relied on a single Ubuntu 20.04.4 machine featuring a 4.9 MHz 12-thread CPU, a 12
+GB Nvidia Titan V GPU, 64 GB of RAM, R version 4.1.3 [22], and Python version 3.8.13 with
+scikit-learn version 1.1.1 [23]. Running all experiments took approximately ten days.
+5. Results
+Each generic model (full and per-domain), as well as ensembles thereof (stacking approach
+described in section 4.5) was tested on the test set of each company-domain-outcome combina-
+tion and compared against the best performing specific model for this combination.
+Results are very positive. As can be seen in Table 5, across all companies, the generic mod-
+els (full or per-domain) outperform the specific models 82% of the time, i.e., for 41 company-
+domain-outcome combinations out of 50. Detailed per-company results can be found in Ap-
+pendix E for the full generic models and Appendix F for the per-domain generic models. At
+the company level, improvements are brought on average for 80.6% of outcomes (across all
+domains), ranging from 33.3% for Company2 to 100% for Company1, Company4, Company6,
+and Company9.
+13https://scikit-learn.org/stable/modules/generated/sklearn.svm.SVC.html
+8
+
+Construction
+Electric T&D
+Oil & Gas
+C
+S
+B
+IT
+AT
+E
+S
+B
+IT
+AT
+E
+S
+B
+IT
+AT
+E
+1
++1.26
++0.2
++2.55
++3.07
+2
+x
++3.15
+x
+3
+x
++6.56
++0.49
+x
++12.47
++0.99
+x
+4
++3.49
++0.47
++3.06
++0.98
++0.63
+5
+x
++2.64
++1.29
++3.14
++2.79
+6
++12.86
++4.39
++1.63
++7.09
++12.87
++5
++11.19
++0.59
+7
+x
++6.69
++15.3
+x
++1.04
++9.54
++2.12
++2.2
+8
++1.11
++0.47
++2.78
+x
++3.13
+9
++1.56
++5.38
++12.16
++5.01
++6.16
+Table 5: Company-level max gains. x: no improvement. S: severity, B: body part, IT: injury type, AT: accident type,
+E: energy source. C: company
+Furthermore, as shown in Fig. 1, gains are high on average (+4.4 in F1 score) and reach im-
+pressive values, e.g., +15.3 for Company7 on electric T&D-injury type, +12.87 for Company6
+on electric T&D-severity, +12.86 for Company6 on construction-severity, +6.56 for Company3
+on construction-body part, etc. And all of that, while predicting more categories.
+Distribution of Company−level Max Gains
+Gain in F1 score over specific models
+0
+5
+10
+15
+0
+2
+4
+6
+8
+10
+12
+14
+Counts
++
++
++++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
+++
++
++
+++
++
++
++
++
++
+Figure 1: Company-level max gains, across all domains and outcomes. n=41, min=0.2, max=15.3, mean=4.4.
+There are only 9 domain-outcome combinations over 50, across 5 companies, on which
+the generic models do not bring any quantitative improvement. However, since their forecasts
+are more informative (more categories predicted), it may still make sense in practice to use the
+generic models in lieu of the specific models, even on these combinations. For instance, for
+Company3-oil & gas-accident type, the specific model only predicts exposure and struck, but
+the generic model also predicts the categories caught, fall, and overexertion.
+The F1 scores averaged over all companies are shown in Table 6. Overall, the generic mod-
+els bring improvement over the specific models for 73.3 % of the domain-outcome combinations
+(11 out of 15). As shown on the right of Fig. 2, maximum gains range from 0.95 (for electric
+T&D-energy source) to 9.98 (for electric T&D-injury type) with an average of 3.37. Also, not
+only do the 11 best generic models outperform their specific counterparts with a comfortable
+margin, but they also generate finer-grained forecasts, which are much more useful in practice.
+More specifically, generic models predict 2.26 additional categories on average, even up to 7
+for construction-injury type (while still providing a gain of 3.48 in F1 score). This is remarkable,
+considering that the more categories to be predicted, the more difficult the task (see Appendix
+D).
+9
+
+Distribution of All Gains
+Gain in F1 score over specific models
+Counts
+0
+2
+4
+6
+8
+10
+0
+5
+10
+15
+20
+25
+30
+Counts
+++
++
++
++
++
++ + +
++
++
++
++
++
++
++
++
++
++
+++
++
++
++
++
++
++
++
++
++
++
++
++
++
++
++++
++
++++
+++
++
++
++
++
+Distribution of Max Gains
+Gain in F1 score over specific models
+Frequency
+0
+2
+4
+6
+8
+10
+0
+1
+2
+3
+4
+5
+6
+Counts
++++
++
++
++
++
++
++
++
++
+Figure 2: Gains averaged over companies. Left: n=48, min=0.16, max=9.98, mean=2.22. Right: n=11, min=0.95,
+max=9.98, mean=3.37.
+The construction and oil & gas domains see gains for 3 outcomes out of 5, while on the
+electric T&D domain, we observe improvement for every outcome. Further, for the body part,
+injury type, and energy source outcomes, there is at least one generic model that outperforms
+its specific counterpart, on every domain, while the severity and accident type outcomes see
+improvements only on the electric T&D domain.
+However, it is important to note that even on those 4 domain-outcome combinations on
+which the generic models do not offer gains in predictive performance, it can still be desirable
+to use them in practice over the specific models, as they generate more informative forecasts,
+with 2 additional categories predicted, on average.
+Overall, more than half of all F1 scores recorded for the generic models (79 out of 150,
+or 53%) are greater or within two points of that of the specific models, while predicting 1.83
+more categories on average. And, as shown on the left of Fig. 2, the 48 generic models that
+outperform their specific counterparts bring on average an improvement of 2.22 in F1 score.
+5.1. Body part, injury type, and energy source
+Some of the greatest improvements are observed for injury type, where the best generic
+models provide large average gains of 3.48, 9.98, and 3.07, respectively on the construction,
+electric T&D, and oil & gas domains, while predicting on average 4.25 more categories than
+the company-specific models. This large boost in performance is remarkable considering the
+significant increase in task difficulty.
+Similarly, for energy source, the best generic models provide 4.48, 0.95, and 1.83 improve-
+ments in F1 score, while predicting 1.53 more categories on average; and for body part, the
+gains are 3.38, 3.42, and 1.37, with 0.17 more categories predicted.
+5.2. Severity and accident type
+For severity and accident type, the generic models outperform the company-specific ones
+on the electric T&D domain, with 3.09 and 2.03 gains in F1 scores, while predicting 2 and 0.5
+more categories on average.
+On the construction and oil & gas domains, the best generic models are between 2.4 and 6
+points below the company-specific ones. However, they still offer the benefit of predicting more
+categories (+1.6 on average).
+10
+
+Construction
+Electric T&D
+Oil & Gas
+Full
+Dom.
+Full
+Dom.
+Full
+Dom.
+Severity
+F1
+SVM
+gen
+31.66
+30.29
+35.52
+35.79
+30.92
+28.97
+RF
+gen
+27.26
+31.04
+33.37
+41.28
+23.48
+26.08
+ens
+30.33
+31.82
+41.17†
+43.88†
+28.98
+30.76
+XGB
+gen
+26.98
+28.74
+36.4
+39.69⋆
+24.25
+24.19
+ens
+29.67
+31.81
+40.01⋆
+40.95
+29.14
+31.44
+spec
+35.34†
+40.79
+33.86†
+Count
+# categories
+spec
+3.5
+3
+3.5
+gen
+5
+5
+5
+5
+5
+5
+# datasets
+9
+4
+9
+4
+9
+4
+Body Part
+F1
+SVM
+gen
+25.51
+31.32
+26.68
+25.24
+23.65
+24.66
+RF
+gen
+34.41†
+31.81
+34.94†
+33.05
+30.22†
+28.85†
+ens
+30.33⋆
+30.39⋆
+34.08
+29.25
+26.43
+27.54⋆
+XGB
+gen
+32.86
+32.23†
+33.76
+35.21†
+29.22
+28.59⋆
+ens
+29.05⋆
+29.2⋆
+32.31
+34.71
+26.37
+27.74⋆
+spec
+31.03
+31.79
+28.85
+Count
+# categories
+spec
+6
+5.5
+6
+gen
+6
+6
+6
+6
+6
+6
+# datasets
+9
+4
+9
+4
+9
+3
+Injury Type
+F1
+SVM
+gen
+38.73
+42.78⋆
+53.7†
+42.91⋆
+35.11⋆
+33.89
+RF
+gen
+40.03
+42.11⋆
+42.44⋆
+41.68⋆
+32.15
+32.82
+ens
+41.75
+45.46†
+49.42
+45.10
+38.65†
+38.97
+XGB
+gen
+36.76
+42.55⋆
+41.03
+41.48
+31.33
+32.34
+ens
+47.4†
+45.45
+51.44
+49.28†
+38.05
+39.79†
+spec
+43.92
+43.72
+36.72
+Count
+# categories
+spec
+4
+5.25
+7
+gen
+11
+6
+11
+8
+11
+11
+# datasets
+9
+4
+9
+4
+9
+3
+Accident Type
+F1
+SVM
+gen
+42.39
+42.20
+44.58
+44.84
+60.69
+64.85
+RF
+gen
+42.46
+42.42
+48.58†
+50.2†
+63.14
+64.12
+ens
+44.35
+40.8
+41.29
+39.72
+66.40
+65.74
+XGB
+gen
+48.27
+49.21
+47.80⋆
+49.58
+58.58
+63.04
+ens
+42.02
+43.40
+41.08
+43.53
+64.56
+67.13
+spec
+54.98†
+48.17
+73.16†
+Count
+# categories
+spec
+3.5
+4.5
+2.67
+gen
+5
+5
+5
+5
+5
+4
+# datasets
+6
+2
+6
+2
+6
+3
+Energy Source
+F1
+SVM
+gen
+74.12†
+73.78
+77.64⋆
+78.87†
+69.54⋆
+55.63
+RF
+gen
+72.71
+73.91†
+77.12⋆
+78.61
+71.12
+70.72†
+ens
+69.06⋆
+69.64⋆
+75.83
+76.48⋆
+70.58
+69.12⋆
+XGB
+gen
+73.77
+71.04
+78.83†
+75.61
+72.22†
+70.35⋆
+ens
+69.05⋆
+70.23
+76.47⋆
+75.03
+70.98
+70.38⋆
+spec
+69.64
+77.92
+70.39
+Count
+# categories
+spec
+2.25
+2.67
+3
+gen
+5
+3
+5
+3
+5
+4
+# datasets
+8
+4
+8
+3
+8
+2
+Table 6: Results averaged over companies. †: best of their sub-column. Bold/⋆: better than/within 2 pts of spec. Full: full generic
+model (one per outcome, same across domains). Dom.: per-domain generic model (one per outcome per domain). Gen/spec:
+generic/specific. Ens: ensemble thereof. # datasets: number of company datasets forming the generic dataset. Note: for the same
+outcome, # categories and # datasets are the same for Full across domains, we repeat them only to ease comparison.
+5.3. Full vs. per-domain
+In what follows, we refer to the full and per-domain models and their ensemble versions.
+When considering full generic models, the average improvement in F1-score over the specific
+models is 2.85 and there are 2.44 additional categories predicted (min=0, max=7), while when
+11
+
+considering per-domain generic models, the average improvement is 2.57 and 1.38 additional
+categories are predicted (min=0, max=4). The per-domain models reach a higher max score than
+the full models on 9 combinations out of 15 (60%), and in 5 out of 11 (45%) when the specific
+models are outperformed. The full and per-domain models outperform the specific models on
+the same 11 domain-outcome combinations.
+So, in terms of performance, there is no clear winner. However, since the full generic models
+predict more categories, and are also simpler conceptually (just one model per outcome), full
+models seem like the way to go. This conclusion however will need to be validated when more
+datasets are available for each domain. One thing to note, however, is that specific models may
+still be desirable in the context of model stacking, as covered next.
+5.4. Generic vs. ensemble (generic + specific)
+The transfer learning-like stacking approach, i.e., combining the predictions of the generic
+and specific models, boosts performance over the generic models (both full and per-domain)
+on all domains for the severity and injury type outcomes, in some cases for accident type, and
+nowhere for body part and energy source.
+For severity, the average gains are of 3.93, and range from 0.78 to an impressive 7.8 (for
+electric T&D-full-RF). Results are even more impressive for injury type. Gains range from 1.72
+to 10.64 (for construction-full-XGB), with a high average of 6.17.
+It is interesting to note that for severity and injury type, very few of the generic models
+outperform the specific models in the first place, and it is only by combining their predictions
+with that of the specific models that absolute best performance can be reached, on the electrical
+domain for severity, and on all domains for injury type.
+We also observe that conversely, for body part and energy source, where model stacking
+does not bring additional skill, the generic models are stronger than the specific models in the
+first place.
+All in all, these results may suggest that ensembling only works when the generic models are
+not already better than the specific models. However, this rule does not hold everywhere (e.g.,
+construction-accident type-XGB), so additional data, experiments and results will be necessary
+to draw any general conclusion here.
+5.5. Quantity vs. relevance
+As far as whether more data or more relevant data is best, Fig. 3 shows the distributions of
+the best a and b coefficients as determined on the validation sets. It tends to indicate that, on
+average, the best tradeoff involves anywhere from a little bit to a lot of generic model (anywhere
+in the [0.1,1] range, with peaks towards [0.1,0.2] and [0.9,1]), but almost always a lot of specific
+model (between 0.9 and 1). In other words, data relevance always seems important, while
+the contribution of data quantity fluctuates. However, this is only a general trend. As can be
+seen in the detailed results per company (Appendix E and Appendix F), in some cases, the
+contribution of the generic model is more important than that of the specific model, e.g., (1,0.6)
+for Company6-XGB in the first table of Appendix F.
+5.6. Best model type
+For the full generic models, the best algorithm is RF (6 domain-outcome combinations over
+15), followed by SVM (5/15) and XGB (4/15). When stacked with the specific model, RF
+reaches best performance in 10 out of 15 combinations.
+12
+
+Coefficient Distributions
+Coefficient Value
+Count
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+10
+20
+30
+40
+50
+60
+generic
+specific
+Coefficient Distributions
+Coefficient Value
+Count
+0.2
+0.4
+0.6
+0.8
+1.0
+0
+10
+20
+30
+40
+50
+generic
+specific
+Figure 3: Distributions of the best coefficient values a (generic) and b (specific). Left: full. Right: per-domain.
+When considering the per-domain generic models, SVM obtains the best score 7 times out
+of 15, followed by RF (5/15) and XGB (3/5). However, when used in the ensemble, XGB is the
+best (10/15).
+RF and XGB are better choices than SVM as they consistently top the scores and can be
+used in ensembles. In terms of performance though, there is no clear winner between the two.
+One or the other could be used interchangeably. However, XGBoost is superior in practice as
+far as deployment is concerned, as the Random Forest models take a lot of disk space, even after
+applying some compression tricks.
+6. Conclusion
+We showed that generic models provide consistent and large improvements over company-
+specific models. Moreover, generic models issue finer-grained forecasts that are more useful in
+practice, as they predict more categories of each safety outcome.
+Generic models remove the needs for training company-specific models, saving a lot of time
+and resources, and give small companies, whose accident datasets are too limited to train their
+own models, access to safety outcome predictions.
+Per-domain generic models (trained on data from a specific industry sector) are not always
+better than full generic models (trained on all data). Ensembling generic and specific models is
+often very beneficial. Therefore, it might still be worth training specific models to combine their
+predictions with that of the generic models. If specific models are already in use, combining
+them with the generic models may provide a boost in performance.
+The forecasts are in essence clear and direct information that can be accessed via a user
+interface (as a desktop or mobile webpage or application), or via an API for integration into any
+existing ecosystem. In each case, the only input required is a set of attributes, and the output are
+probabilities for each category of each outcome.
+By learning lessons from a pool of datasets whose accumulated experience far exceeds that
+of any single company, and making these lessons easily accessible, generic models tackle the
+holy grail of safety cross-organizational learning and dissemination in the construction industry.
+7. Acknowledgements
+We thank the Nvidia corporation for donating the Titan V GPU that was used in this re-
+search, as part of their GPU grant program.
+13
+
+8. References
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+Appendices
+Appendix A. Attribute List
+adverse low temps
+fuses⋆
+machinery
+spark
+bolt
+grinding
+manlift
+splinter/sliver
+breaker⋆
+grout
+mud
+spool
+cable
+guardrail/handrail
+nail
+stairs
+cable tray
+hammer
+no/improper PPE
+steel/steel sections
+chipping
+hand size pieces
+object at height
+stripping
+cleaning
+hazardous substance
+object on the floor
+stud
+clearance⋆
+heat source/high temps
+piping
+switch/switching⋆
+concrete
+heater⋆
+pole⋆
+tank
+concrete liquid
+heavy material/tool
+pontoon
+transformer⋆
+conduit
+heavy vehicle
+poor housekeeping
+uneven surface
+confined work space
+hose
+poor visibility
+unpowered tool
+congested work space
+improper body position
+powered tool
+unpowered transporter
+crane
+improper procedure/inattention
+rebar
+unstable support/surface
+door
+improper security of materials
+relay⋆
+valve
+drill
+improper security of tools
+repetitive motion
+vault⋆
+dunnage
+insect/animal
+scaffold
+welding
+electricity
+job trailer
+screw
+wind
+exiting
+ladder
+sharp edge
+wire
+fan⋆
+lifting/pulling/manipulating
+slag
+working at height
+fatigued dizzy
+light vehicle
+slippery surface
+working below elev wksp/mat
+forklift
+LOTO/labeling⋆
+small particle
+working overhead
+formwork
+lumber
+soffit
+wrench
+Table A.7: 92 attributes used in this study. LOTO: lockout-tagout. PPE: personal protective equipment. ⋆: eleven
+new attributes added since [4, 5].
+15
+
+Appendix B. Detailed split counts
+Severity
+Train
+w
+Val
+Test
+Construction
+report-only
+917
+8.2
+226
+283
+1st aid
+7486
+1.0
+1876
+2369
+medical
+470
+15.9
+114
+140
+recordable
+147
+50.9
+28
+42
+lost time
+960
+7.8
+250
+285
+total
+9980
+2494
+3119
+Electric T&D
+report-only
+2392
+1.2
+576
+712
+1st aid
+2809
+1.0
+736
+905
+medical
+554
+5.1
+140
+162
+recordable
+310
+9.1
+74
+101
+lost time
+607
+4.6
+143
+205
+total
+6672
+1669
+2085
+Oil & Gas
+report-only
+929
+14.8
+244
+279
+1st aid
+13766
+1.0
+3405
+4279
+medical
+1919
+7.2
+489
+618
+recordable
+152
+90.6
+42
+52
+lost time
+1615
+8.5
+415
+516
+total
+18381
+4595
+5744
+Corporate
+report-only
+97
+3.3
+31
+22
+1st aid
+321
+1.0
+74
+109
+total
+418
+105
+131
+Full
+report-only
+4335
+5.6
+1077
+1296
+1st aid
+24382
+1.0
+6091
+7662
+medical
+2943
+8.3
+743
+920
+recordable
+609
+40.0
+144
+195
+lost time
+3182
+7.7
+808
+1006
+total
+35451
+8863
+11079
+Body Part
+Train
+w
+Val
+Test
+Construction
+arm
+1059
+2.6
+285
+338
+foot
+694
+3.9
+167
+232
+hand
+2732
+1.0
+701
+864
+head
+1682
+1.6
+394
+494
+leg
+958
+2.9
+262
+307
+trunk
+1084
+2.5
+243
+330
+total
+8209
+2052
+2565
+Electric T&D
+arm
+1061
+1.4
+274
+319
+foot
+372
+4.0
+89
+135
+hand
+1473
+1.0
+368
+452
+head
+1246
+1.2
+318
+403
+leg
+1084
+1.4
+251
+307
+trunk
+800
+1.8
+208
+269
+total
+6036
+1508
+1885
+Oil & Gas
+arm
+1445
+3.9
+386
+477
+foot
+1741
+3.2
+421
+568
+hand
+5586
+1.0
+1385
+1740
+head
+3514
+1.6
+887
+1088
+leg
+2053
+2.7
+498
+596
+trunk
+1449
+3.9
+370
+464
+total
+15788
+3947
+4933
+Full
+arm
+3565
+2.7
+945
+1134
+foot
+2807
+3.5
+677
+935
+hand
+9791
+1.0
+2454
+3056
+head
+6442
+1.5
+1599
+1985
+leg
+4095
+2.4
+1011
+1210
+trunk
+3333
+2.9
+821
+1063
+total
+30033
+7507
+9383
+Accident Type
+Train
+w
+Val
+Test
+Construction
+caught
+396
+2.3
+105
+137
+exposure
+119
+7.8
+38
+40
+fall
+803
+1.2
+200
+243
+overexertion
+492
+1.9
+128
+160
+struck
+930
+1.0
+214
+276
+total
+2740
+685
+856
+Electric T&D
+caught
+207
+2.2
+55
+62
+exposure
+454
+1.0
+123
+142
+fall
+403
+1.1
+102
+143
+overexertion
+288
+1.6
+51
+65
+struck
+248
+1.8
+69
+88
+total
+1600
+400
+500
+Oil & Gas
+caught
+198
+7.7
+43
+53
+exposure
+526
+2.9
+127
+184
+fall
+1527
+1.0
+393
+463
+struck
+659
+2.3
+165
+210
+total
+2910
+728
+910
+Full
+caught
+801
+3.4
+203
+252
+exposure
+1099
+2.5
+288
+366
+fall
+2733
+1.0
+695
+849
+overexertion
+780
+3.5
+179
+225
+struck
+1837
+1.5
+448
+574
+total
+7250
+1813
+2266
+Energy Source
+Train
+w
+Val
+Test
+Construction
+chemical
+76
+42.7
+21
+14
+gravity
+1551
+2.1
+405
+479
+motion
+3248
+1.0
+792
+1031
+total
+4875
+1218
+1524
+Electric
+biological
+221
+7.6
+52
+88
+gravity
+733
+2.3
+179
+230
+motion
+1683
+1.0
+429
+507
+total
+2637
+660
+825
+Oil & Gas
+chemical
+70
+21.2
+13
+21
+gravity
+1485
+1.0
+361
+448
+motion
+914
+1.6
+246
+300
+thermal
+131
+11.3
+30
+44
+total
+2600
+650
+813
+Full
+biological
+221
+26.4
+52
+88
+chemical
+146
+40.0
+34
+35
+gravity
+3769
+1.6
+945
+1157
+motion
+5845
+1.0
+1467
+1838
+thermal
+131
+44.6
+30
+44
+total
+10112
+2528
+3162
+Table B.8: Split counts (1/2). w: training weights.
+16
+
+Injury Type
+Train
+w
+Val
+Test
+Construction
+contusion
+728
+3.6
+185
+229
+cut
+2644
+1.0
+682
+795
+fob
+399
+6.6
+84
+118
+fracture
+100
+26.4
+24
+39
+pinch
+267
+9.9
+90
+97
+strain
+2129
+1.2
+501
+680
+total
+6267
+1566
+1958
+Electric T&D
+bite
+129
+12.3
+35
+42
+burn
+75
+21.2
+14
+21
+contusion
+861
+1.8
+216
+277
+cut
+1305
+1.2
+330
+400
+fob
+209
+7.6
+46
+69
+fracture
+176
+9.0
+39
+53
+irritation
+420
+3.8
+101
+141
+strain
+1589
+1.0
+410
+486
+total
+4764
+1191
+1489
+Oil & Gas
+bite
+168
+27.6
+39
+52
+burn
+572
+8.1
+150
+179
+contusion
+3587
+1.30
+848
+1091
+cut
+4638
+1.0
+1160
+1509
+exhaustion
+75
+61.8
+24
+25
+fob
+1440
+3.2
+381
+455
+fracture
+622
+7.5
+160
+199
+irritation
+127
+36.5
+37
+42
+pain
+704
+6.6
+176
+215
+pinch
+720
+6.4
+181
+231
+strain
+2307
+2.0
+584
+677
+total
+14960
+3740
+4675
+Full
+bite
+297
+28.9
+74
+94
+burn
+647
+13.3
+164
+200
+contusion
+5176
+1.7
+1249
+1597
+cut
+8587
+1.0
+2172
+2704
+exhaustion
+75
+114.5
+24
+25
+fob
+2048
+4.2
+511
+642
+fracture
+898
+9.6
+223
+291
+irritation
+547
+15.7
+138
+183
+pain
+704
+12.2
+176
+215
+pinch
+987
+8.7
+271
+328
+strain
+6025
+1.4
+1495
+1843
+total
+25991
+6497
+8122
+Table B.9: Split counts (2/2). w: training weights.
+Appendix C. Hyperparameter Optimization Details
+For Random Forest14, we searched the number of trees (ntree parameter, from 100 to
+1600 with steps of 100), the number of variables to try when making each split (mtry, from 5
+to 45 with steps of 5), and the leaf size (nodesize, 1, 2, 5, 10, 25, and 50).
+For XGBoost15, we searched the maximum depth of a tree in the sequence (max depth,
+from 3 to 6 with steps of 1), the learning rate (learning rate, 0.01, 0.05, and 0.1), the
+minimum leaf size (min child weight, 1, 3, 5, and 10), the percentage of training instances
+to be used in building each tree (subsample, 0.3, 0.5, 0.7, and 1) , and the percentage of
+predictors to be considered in making each split of a given tree (colsample bylevel, 0.3,
+0.5, 0.7, and 1). The number of trees in the sequence (ntrees) was set to 2000. The loss was
+the multinomial one. Finally, for the SVM model, we optimized the C parameter (C, 10x with
+x taking 3000 evenly spaced values in [−9, 9]).
+14https://scikit-learn.org/stable/modules/generated/sklearn.ensemble.RandomForestClassifier.html
+15https://xgboost.readthedocs.io/en/latest/parameter.html
+17
+
+Appendix D. Illustration of Task Difficulty vs. Number of Categories
+To illustrate how the prediction task gets more and more difficult as the number of cate-
+gories increases, we designed a synthetic example in which 105 observations were drawn from
+an increasing number of categories (2 to 12). Class imbalance was simulated by drawing from
+the categories with probabilities following the lognormal distribution (mean=0, sd=2). We con-
+sidered two baselines: a random baseline, that predicts categories uniformly at random, and a
+most frequent baseline, which always returns the most frequent category. Our proxy for diffi-
+culty was one minus the F1 score of the baselines. In other words, the less well the baselines
+are doing, the more difficult the task. We can see on Fig. D.4 that the task difficulty rapidly
+increases with the number of categories, and that going from 2 to 6 categories almost makes the
+task twice as hard.
+2
+4
+6
+8
+10
+12
+0.5
+0.6
+0.7
+0.8
+0.9
+Prediction Task Difficulty vs Number of Categories
+Number of Categories
+1 − F1 score
+Random Baseline
+Most Frequent Baseline
+Figure D.4
+Appendix E. Per-Company Results for the Full Generic Models
+Note: the ensemble (“ens”) rows are left blank whenever the specific model is a SVM, as
+we could not use ensembling in this case (the forecast of the SVM is not probabilistic).
+Appendix E.1. Severity
+Comp.1
+Comp.3
+Comp.5
+Comp.6
+Avg
+spec
+29.51
+32.62
+45.35
+33.9
+35.34†
+SVM
+gen
+20.23
+25.64
+34.01
+46.76
+31.66
+gen
+25.75
+21.54
+29.75
+31.99
+27.26
+RF
+ens
+28.68
+31.62
+30.69
+30.33
+coef.
+(0.4,1)
+(0.8,1)
+(0.4,1)
+gen
+27.58
+23.26
+27.58
+29.48
+26.98
+XGB
+ens
+28.85
+28.34
+31.82
+29.67
+coef.
+(0.1,1)
+(0.3,1)
+(0.5,1)
+#lev. spec
+4
+4
+3
+3
+3.5
+#lev. gen
+5
+5
+5
+5
+5
+Table E.10: Severity, construction. †: best model on average.
+18
+
+Comp.4
+Comp.6
+Comp.7
+Comp.9
+Avg
+spec
+29.48
+45.66
+57.67
+30.34
+40.79
+SVM
+gen
+20.93
+42.56
+46.67
+31.9
+35.52
+gen
+27.46
+38.61
+39.97
+27.42
+33.37
+RF
+ens
+28.73
+53.62
+41.17†
+coef.
+(1,0.9)
+(0.5,1)
+gen
+27.39
+53.02
+39.24
+25.95
+36.4
+XGB
+ens
+28.74
+51.27
+40.01⋆
+coef.
+(0.8,1)
+(0.2,1)
+#lev. spec
+4
+2
+2
+4
+3
+#lev. gen
+5
+5
+5
+5
+5
+Table E.11: Severity, electric T&D. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Comp.2
+Comp.3
+Comp.8
+Comp.7
+Avg
+spec
+42.53
+24.74
+39.72
+28.44
+33.86†
+SVM
+gen
+37.91
+22.53
+38.85
+24.41
+30.92
+gen
+17.96
+17.12
+35.69
+23.14
+23.48
+RF
+ens
+27.87
+24.05
+39.81
+24.2
+28.98
+coef.
+(0.2,1)
+(0.7,1)
+(0.7,1)
+(0.1,1)
+gen
+16.75
+23.25
+35.27
+21.72
+24.25
+XGB
+ens
+27.89
+25.36
+39.61
+23.7
+29.14
+coef.
+(0.2,1)
+(1,0.8)
+(0.3,1)
+(0.1,1)
+#lev. spec
+3
+4
+3
+4
+3.5
+#lev. gen
+5
+5
+5
+5
+5
+Table E.12: Severity, oil & gas. †: best model on average.
+Appendix E.2. Body Part
+Comp.1
+Comp.3
+Comp.5
+Comp.6
+Avg
+spec
+34.14
+26.48
+32.09
+31.39
+31.03
+SVM
+gen
+23.26
+25.09
+27.02
+26.66
+25.51
+gen
+34.14
+33.04
+34.68
+35.78
+34.41†
+RF
+ens
+33.49
+22.43
+32.7
+32.7
+30.33⋆
+coef.
+(0.4,1)
+(0.1,1)
+(0.7,1)
+(0.6,1)
+gen
+31.92
+30.57
+34.73
+34.22
+32.86
+XGB
+ens
+32.44
+20.38
+32.62
+30.77
+29.05⋆
+coef.
+(0.1,1)
+(0.2,1)
+(0.2,1)
+(0.5,1)
+#lev. spec
+6
+6
+6
+6
+6
+#lev. gen
+6
+6
+6
+6
+6
+Table E.13: Body part, construction. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+19
+
+Comp.4
+Comp.6
+Comp.7
+Comp.9
+Avg
+spec
+29.25
+27.7
+46.34
+23.86
+31.79
+SVM
+gen
+19.21
+28.86
+38.26
+20.4
+26.68
+gen
+27.96
+32
+51.02
+28.76
+34.94†
+RF
+ens
+27.94
+50.75
+23.56
+34.08
+coef.
+(0.4,1)
+(0.1,1)
+(0.4,1)
+gen
+28.24
+31.12
+46.44
+29.24
+33.76
+XGB
+ens
+27.96
+41.17
+27.81
+32.31
+coef.
+(0.2,1)
+(0.1,1)
+(0.5,1)
+#lev. spec
+6
+6
+4
+6
+5.5
+#lev. gen
+6
+6
+6
+6
+6
+Table E.14: Body part, electric T&D. †: best model on average. Bold: better the company-specific model.
+Comp.2
+Comp.8
+Comp.7
+Avg
+spec
+22.96
+32.41
+31.17
+28.85
+SVM
+gen
+22.66
+26.23
+22.06
+23.65
+gen
+26.11
+32.34
+32.21
+30.22†
+RF
+ens
+20.31
+32.88
+26.09
+26.43
+coef.
+(0.1,1)
+(1,0.1)
+(0.1,1)
+gen
+25.5
+32.36
+29.81
+29.22
+XGB
+ens
+16.26
+32.28
+30.56
+26.37
+coef.
+(0.1,1)
+(1,0.3)
+(0.2,1)
+#lev. spec
+6
+6
+6
+6
+#lev. gen
+6
+6
+6
+6
+Table E.15: Body part, oil & gas. †: best model on average. Bold: better the company-specific model.
+Appendix E.3. Injury Type
+Comp.1
+Comp.3
+Comp.5
+Comp.6
+Avg
+spec
+54
+37.7
+33.91
+50.07
+43.92
+SVM
+gen
+34.67
+36.66
+34.78
+48.81
+38.73
+gen
+47.84
+33.86
+33.11
+45.3
+40.03
+RF
+ens
+47.6
+31.98
+45.67
+41.75
+coef.
+(0.2,1)
+(0.1,1)
+(0.4,1)
+gen
+46.46
+23.99
+31.9
+44.7
+36.76
+XGB
+ens
+56.55
+35.2
+50.46
+47.4†
+coef.
+(0.6,1)
+(0.4,1)
+(0.2,1)
+#lev. spec
+3
+3
+6
+4
+4
+#lev. gen
+11
+11
+11
+11
+11
+Table E.16: Injury type, construction. †: best model on average. Bold: better the company-specific model.
+Comp.4
+Comp.6
+Comp.7
+Comp.9
+Avg
+spec
+39.21
+43.4
+47.28
+44.98
+43.72
+SVM
+gen
+42.27
+54.59
+60.78
+57.14
+53.7†
+gen
+26.44
+42.41
+56.74
+44.16
+42.44⋆
+RF
+ens
+39.33
+59.52
+49.42
+coef.
+(1,0.5)
+(1,0.2)
+gen
+28.57
+41.6
+51.45
+42.49
+41.03
+XGB
+ens
+40.31
+62.58
+51.44
+coef.
+(1,0.8)
+(1,0.1)
+#lev. spec
+5
+6
+4
+6
+5.25
+#lev. gen
+11
+11
+11
+11
+11
+Table E.17: Injury type, electric T&D. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+20
+
+Comp.2
+Comp.8
+Comp.7
+Avg
+spec
+35.39
+34.04
+40.72
+36.72
+SVM
+gen
+27.97
+30.67
+46.69
+35.11⋆
+gen
+23.72
+32.22
+40.52
+32.15
+RF
+ens
+36.82
+40.48
+38.65†
+coef.
+(0.5,1)
+(0.7,1)
+gen
+23.69
+31.01
+39.28
+31.33
+XGB
+ens
+35.09
+41
+38.05
+coef.
+(1,0.7)
+(1,0.6)
+#lev. spec
+3
+10
+8
+7
+#lev. gen
+11
+11
+11
+11
+Table E.18: Injury type, oil & gas. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Appendix E.4. Accident Type
+Comp.3
+Comp.5
+Avg
+spec
+68.63
+41.34
+54.98†
+SVM
+gen
+41.87
+42.91
+42.39
+gen
+40.44
+44.48
+42.46
+RF
+ens
+44.35
+44.35
+coef.
+(1,0.7)
+gen
+54.04
+42.51
+48.27
+XGB
+ens
+42.02
+42.02
+coef.
+(1,1)
+#lev. spec
+2
+5
+3.5
+#lev. gen
+5
+5
+5
+Table E.19: Accident type, construction. †: best model on average.
+Comp.4
+Comp.9
+Avg
+spec
+43.15
+53.2
+48.17
+SVM
+gen
+36.46
+52.71
+44.58
+gen
+40.05
+57.11
+48.58†
+RF
+ens
+41.29
+41.29
+coef.
+(0.4,1)
+gen
+38.13
+57.46
+47.8⋆
+XGB
+ens
+41.08
+41.08
+coef.
+(0.4,1)
+#lev. spec
+5
+4
+4.5
+#lev. gen
+5
+5
+5
+Table E.20: Accident type, electric T&D. †: best model on average. Bold/⋆: better/within 2pts of the company-
+specific model.
+21
+
+Comp.3
+Comp.8
+Comp.7
+Avg
+spec
+80.91
+85
+53.58
+73.16†
+SVM
+gen
+58.06
+78.09
+45.92
+60.69
+gen
+61.67
+78.03
+49.71
+63.14
+RF
+ens
+78.46
+54.35
+66.4
+coef.
+(0.1,1)
+(1,0.1)
+gen
+46.65
+76.93
+52.16
+58.58
+XGB
+ens
+73.8
+55.31
+64.56
+coef.
+(1,0.7)
+(1,0.7)
+#lev. spec
+2
+2
+4
+2.67
+#lev. gen
+5
+5
+5
+5
+Table E.21: Accident type, oil & gas. †: best model on average.
+Appendix E.5. Energy Source
+Comp.1
+Comp.3
+Comp.5
+Comp.6
+Avg
+spec
+71.69
+70.97
+68.07
+67.82
+69.64
+SVM
+gen
+74.76
+78.16
+70.86
+72.69
+74.12†
+gen
+70.36
+76.03
+70.14
+74.31
+72.71
+RF
+ens
+71.05
+68.02
+68.1
+69.06⋆
+coef.
+(0.9,1)
+(0.2,1)
+(0.4,1)
+gen
+74.33
+83.44
+64.62
+72.7
+73.77
+XGB
+ens
+71.88
+66.81
+68.47
+69.05⋆
+coef.
+(0.4,1)
+(0.1,1)
+(0.4,1)
+#lev. spec
+2
+2
+3
+2
+2.25
+#lev. gen
+5
+5
+5
+5
+5
+Table E.22: Energy source, construction.
+†: best model on average. Bold/⋆: better/within 2pts of the company-
+specific model.
+Comp.4
+Comp.6
+Comp.9
+Avg
+spec
+79.5
+73.22
+81.05
+77.92
+SVM
+gen
+76.59
+70.61
+85.73
+77.64⋆
+gen
+74.99
+73.06
+83.32
+77.12⋆
+RF
+ens
+77.85
+73.81
+75.83
+coef.
+(0.9,1)
+(0.2,1)
+gen
+76.43
+72.85
+87.21
+78.83†
+XGB
+ens
+79.41
+73.52
+76.47⋆
+coef.
+(0.2,1)
+(0.3,1)
+#lev. spec
+3
+2
+3
+2.67
+#lev. gen
+5
+5
+5
+5
+Table E.23: Energy source, electric T&D. †: best model on average. Bold/⋆: better/within 2pts of the company-
+specific model.
+22
+
+Comp.8
+Comp.7
+Avg
+spec
+68.98
+71.8
+70.39
+SVM
+gen
+68.73
+70.36
+69.54⋆
+gen
+70.43
+71.81
+71.12
+RF
+ens
+68.27
+72.89
+70.58
+coef.
+(0.4,1)
+(1,0.2)
+gen
+70.44
+74
+72.22†
+XGB
+ens
+68.72
+73.25
+70.98
+coef.
+(0.1,1)
+(0.3,1)
+#lev. spec
+4
+2
+3
+#lev. gen
+5
+5
+5
+Table E.24: Energy source, oil & gas. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Appendix F. Per-Company Results for the Per-Domain Generic Models
+Note: the ensemble (‘ens’) rows are left blank whenever the specific model is a SVM, as we
+could not use ensembling in this case (the forecast of the SVM is not probabilistic).
+Appendix F.1. Severity
+Comp.5
+Comp.3
+Comp.6
+Comp.1
+Avg
+spec
+45.35
+32.62
+33.9
+29.51
+35.34†
+SVM
+gen
+39.86
+26.03
+32.61
+22.66
+30.29
+gen
+34.1
+27.7
+32.62
+29.74
+31.04
+RF
+ens
+31.2
+33.5
+30.77
+31.82
+Coeffs
+(0.8,1)
+(0.6,1)
+(1,0.3)
+gen
+30.84
+26.84
+28.33
+28.95
+28.74
+XGB
+ens
+31.3
+34.14
+30
+31.81
+Coeffs
+(0.3,1)
+(1,0.6)
+(0.5,1)
+#categories spec
+3
+4
+3
+4
+3.5
+#categories gen
+5
+5
+5
+5
+5
+Table F.25: Severity, construction. †: best model on average.
+Comp.7
+Comp.4
+Comp.9
+Comp.6
+Avg
+spec
+57.67
+29.48
+30.34
+45.66
+40.79
+SVM
+gen
+36
+30.47
+24.62
+52.06
+35.79
+gen
+47.19
+30.91
+28.5
+58.53
+41.28
+RF
+ens
+54.8
+32.97
+43.88†
+Coeffs
+(1,0.6)
+(1,0.9)
+gen
+44.23
+30.59
+26.49
+57.44
+39.69⋆
+XGB
+ens
+54.95
+26.96
+40.95
+Coeffs
+(0.4,1)
+(0.1,1)
+#categories spec
+2
+4
+4
+2
+3
+#categories gen
+5
+5
+5
+5
+5
+Table F.26: Severity, elec. †: best model on average. Bold/⋆: better/within 2pts of the company-specific model.
+23
+
+Comp.7
+Comp.3
+Comp.8
+Comp.2
+Avg
+spec
+28.44
+24.74
+39.72
+42.53
+33.86†
+SVM
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+26.7
+21.05
+40.83
+27.31
+28.97
+gen
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+19.82
+35.7
+22.59
+26.08
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+ens
+27.59
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+40.22
+32.86
+30.76
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+(1,0.9)
+(1,0.8)
+(0.7,1)
+gen
+23.82
+19.7
+33.09
+20.15
+24.19
+XGB
+ens
+27.97
+25.73
+40.1
+31.96
+31.44
+Coeffs
+(0.4,1)
+(1,0.7)
+(1,0.2)
+(0.7,1)
+#categories spec
+4
+4
+3
+3
+3.5
+#categories gen
+5
+5
+5
+5
+5
+Table F.27: Severity, oil & gas. †: best model on average.
+Appendix F.2. Body part
+Comp.5
+Comp.3
+Comp.6
+Comp.1
+Avg
+spec
+32.09
+26.48
+31.39
+34.14
+31.03
+SVM
+gen
+31.08
+28.14
+31.92
+34.13
+31.32
+gen
+32.19
+27.06
+33.64
+34.34
+31.81
+RF
+ens
+31.23
+25.77
+35.14
+29.41
+30.39⋆
+Coeffs
+(0.1,1)
+(0.2,1)
+(0.6,1)
+(0.1,1)
+gen
+33.6
+29.91
+32.48
+32.92
+32.23†
+XGB
+ens
+32.34
+20.72
+32.33
+31.41
+29.2⋆
+Coeffs
+(0.1,1)
+(0.2,1)
+(0.5,1)
+(0.1,1)
+#categories spec
+6
+6
+6
+6
+6
+#categories gen
+6
+6
+6
+6
+6
+Table F.28: Body part, construction. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Comp.7
+Comp.4
+Comp.9
+Comp.6
+Avg
+spec
+46.34
+29.25
+23.86
+27.7
+31.79
+SVM
+gen
+34.02
+18.16
+19.6
+29.17
+25.24
+gen
+48.21
+26.31
+25.71
+31.97
+33.05
+RF
+ens
+39.52
+28.63
+19.59
+29.25
+Coeffs
+(0.1,1)
+(0.1,1)
+(0.1,1)
+gen
+53.03
+28.55
+26.56
+32.7
+35.21†
+XGB
+ens
+49.41
+29.72
+25.01
+34.71
+Coeffs
+(1,1)
+(0.6,1)
+(0.2,1)
+#categories spec
+4
+6
+6
+6
+5.5
+#categories gen
+6
+6
+6
+6
+6
+Table F.29: Body part, elec. †: best model on average. Bold/⋆: better/within 2pts of the company-specific model.
+24
+
+Comp.7
+Comp.8
+Comp.2
+Avg
+spec
+31.17
+32.41
+22.96
+28.85†
+SVM
+gen
+27.22
+25.91
+20.84
+24.66
+gen
+29.64
+31.8
+25.12
+28.85†
+RF
+ens
+30.15
+31.66
+20.81
+27.54⋆
+Coeffs
+(0.1,1)
+(0.1,1)
+(0.4,1)
+gen
+29.69
+32.36
+23.72
+28.59⋆
+XGB
+ens
+31.84
+32.1
+19.28
+27.74⋆
+Coeffs
+(1,0.5)
+(1,0.1)
+(0.2,1)
+#categories spec
+6
+6
+6
+6
+#categories gen
+6
+6
+6
+6
+Table F.30: Body part, oil & gas.
+†: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Appendix F.3. Injury type
+Comp.5
+Comp.3
+Comp.6
+Comp.1
+Avg
+spec
+33.91
+37.7
+50.07
+54
+43.92
+SVM
+gen
+34.16
+36.31
+51.7
+48.97
+42.78⋆
+gen
+33.56
+33.91
+49.34
+51.64
+42.11⋆
+RF
+ens
+33.38
+48.57
+54.42
+45.46†
+Coeffs
+(0.1,1)
+(0.1,1)
+(1,0.2)
+gen
+33.3
+38.19
+48.17
+50.54
+42.55⋆
+XGB
+ens
+34.74
+47.08
+54.53
+45.45
+Coeffs
+(0.3,1)
+(0.1,1)
+(0.5,1)
+#categories spec
+6
+3
+4
+3
+4
+#categories gen
+6
+6
+6
+6
+6
+Table F.31: Injury type, construction. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Comp.7
+Comp.4
+Comp.9
+Comp.6
+Avg
+spec
+47.28
+39.21
+44.98
+43.4
+43.72
+SVM
+gen
+57.28
+28.12
+44.54
+41.7
+42.91⋆
+gen
+53.99
+29.2
+43.07
+40.47
+41.68⋆
+RF
+ens
+51.33
+38.87
+45.1
+Coeffs
+(0.8,1)
+(0.4,1)
+gen
+49.62
+29.63
+40.26
+46.42
+41.48
+XGB
+ens
+59.48
+39.09
+49.28†
+Coeffs
+(1,0.3)
+(1,0.3)
+#categories spec
+4
+5
+6
+6
+5.25
+#categories gen
+8
+8
+8
+8
+8
+Table F.32: Injury type, elec. †: best model on average. Bold/⋆: better/within 2pts of the company-specific model.
+25
+
+Comp.7
+Comp.8
+Comp.2
+Avg
+spec
+40.72
+34.04
+35.39
+36.72
+SVM
+gen
+50.26
+33.24
+18.18
+33.89
+gen
+39.57
+33.87
+25.02
+32.82
+RF
+ens
+41.69
+36.25
+38.97
+Coeffs
+(1,0.7)
+(0.8,1)
+gen
+38.32
+32.64
+26.07
+32.34
+XGB
+ens
+42.88
+36.7
+39.79†
+Coeffs
+(1,0.7)
+(1,0.1)
+#categories spec
+8
+10
+3
+7
+#categories gen
+11
+11
+11
+11
+Table F.33: Injury type, oil & gas. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Appendix F.4. Accident type
+Comp.5
+Comp.3
+Avg
+spec
+41.34
+68.63
+54.98†
+SVM
+gen
+44.25
+40.15
+42.2
+gen
+41.37
+43.48
+42.42
+RF
+ens
+40.8
+40.8
+Coeffs
+(0.1,1)
+gen
+43.21
+55.21
+49.21
+XGB
+ens
+43.4
+43.4
+Coeffs
+(1,0.1)
+#categories spec
+5
+2
+3.5
+#categories gen
+5
+5
+5
+Table F.34: Accident type, construction. †: best model on average.
+Comp.4
+Comp.9
+Avg
+spec
+43.15
+53.2
+48.17
+SVM
+gen
+39.45
+50.22
+44.84
+gen
+44.13
+56.28
+50.2†
+RF
+ens
+39.72
+39.72
+Coeffs
+(0.3,1)
+gen
+40.96
+58.21
+49.58
+XGB
+ens
+43.53
+43.53
+Coeffs
+(0.4,1)
+#categories spec
+5
+4
+4.5
+#categories gen
+5
+5
+5
+Table F.35: Accident type, elec. †: best model on average. Bold/⋆: better/within 2pts of the company-specific model.
+Comp.7
+Comp.3
+Comp.8
+Avg
+spec
+53.58
+80.91
+85
+73.16†
+SVM
+gen
+55.04
+59.76
+79.75
+64.85
+gen
+51.77
+58.06
+82.53
+64.12
+RF
+ens
+53.02
+78.46
+65.74
+Coeffs
+(1,0.1)
+(0.1,1)
+gen
+49.03
+62.16
+77.93
+63.04
+XGB
+ens
+55.7
+78.56
+67.13
+Coeffs
+(1,0.9)
+(0.1,1)
+#categories spec
+4
+2
+2
+2.67
+#categories gen
+4
+4
+4
+4
+Table F.36: Accident type, oil & gas. †: best model on average.
+26
+
+Appendix F.5. Energy source
+Comp.5
+Comp.3
+Comp.6
+Comp.1
+Avg
+spec
+68.07
+70.97
+67.82
+71.69
+69.64
+SVM
+gen
+70.3
+76.99
+73.32
+74.5
+73.78
+gen
+68.17
+79.98
+74.28
+73.21
+73.91†
+RF
+ens
+67.85
+69.62
+71.45
+69.64⋆
+Coeffs
+(0.1,1)
+(0.9,1)
+(0.4,1)
+gen
+62.88
+73.28
+74.91
+73.09
+71.04
+XGB
+ens
+68.05
+70.75
+71.9
+70.23
+Coeffs
+(0.1,1)
+(0.7,1)
+(0.5,1)
+#categories spec
+3
+2
+2
+2
+2.25
+#categories gen
+3
+3
+3
+3
+3
+Table F.37: Energy source, construction.
+†: best model on average. Bold/⋆: better/within 2pts of the company-
+specific model.
+Comp.4
+Comp.9
+Comp.6
+Avg
+spec
+79.5
+81.05
+73.22
+77.92
+SVM
+gen
+78.31
+85.83
+72.46
+78.87†
+gen
+80.13
+83.73
+71.96
+78.61
+RF
+ens
+79.75
+73.22
+76.48⋆
+Coeffs
+(0.5,1)
+(0.1,1)
+gen
+75.63
+82.34
+68.86
+75.61
+XGB
+ens
+77.15
+72.91
+75.03
+Coeffs
+(1,0.8)
+(0.1,1)
+#categories spec
+3
+3
+2
+2.67
+#categories gen
+3
+3
+3
+3
+Table F.38: Energy source, elec.
+†: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+Comp.7
+Comp.8
+Avg
+spec
+71.8
+68.98
+70.39
+SVM
+gen
+49.94
+61.33
+55.63
+gen
+69.34
+72.11
+70.72†
+RF
+ens
+70.44
+67.8
+69.12⋆
+Coeffs
+(1,0.5)
+(0.4,1)
+gen
+72.58
+68.12
+70.35⋆
+XGB
+ens
+72.06
+68.69
+70.38⋆
+Coeffs
+(0.2,1)
+(0.1,1)
+#categories spec
+2
+4
+3
+#categories gen
+4
+4
+4
+Table F.39: Energy source, oil & gas. †: best model on average. Bold/⋆: better/within 2pts of the company-specific
+model.
+27
+
diff --git a/ENE1T4oBgHgl3EQf-QbO/content/tmp_files/load_file.txt b/ENE1T4oBgHgl3EQf-QbO/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..2feca50a89e29a0e981031323d4d581a025ce6d7
--- /dev/null
+++ b/ENE1T4oBgHgl3EQf-QbO/content/tmp_files/load_file.txt
@@ -0,0 +1,2384 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf,len=2383
+page_content='Safer Together: Machine Learning Models Trained on Shared Accident  Datasets Predict Construction Injuries Better than Company-Specific  Models  Submitted to Automation in Construction  Antoine J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Tixier1a, Matthew R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Hallowella,b  aSafetyAI R&D  bUniversity of Colorado at Boulder  Abstract  Highlights  9 companies from 3 domains (construction, electric T&D, oil & gas) shared their accident datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Machine learning models were trained to predict safety outcomes from fundamental attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Models trained on all datasets (full generic models) outperformed the company-specific models  in 82% of the company-domain-outcome combinations, with large gains in F1 score (+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4 on  average and up to +15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  On average, generic models predicted 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='26 categories more than specific models (up to 7), making  for more useful forecasts in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Per-domain generic models were not always better than full generic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Combining generic and specific models (data quantity and relevance) was often very beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Generic models give companies devoid of accident datasets access to safety predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Generic models address safety cross-organizational learning and dissemination in construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  In this study, we capitalized on a collective dataset repository of 57k accidents from 9 companies be-  longing to 3 domains and tested whether models trained on multiple datasets (generic models) predicted  safety outcomes better than the company-specific models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' We experimented with full generic models  (trained on all data), per-domain generic models (construction, electric T&D, oil & gas), and with en-  sembles of generic and specific models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Results are very positive, with generic models outperforming the  company-specific models in most cases while also generating finer-grained, hence more useful, forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Successful generic models remove the needs for training company-specific models, saving a lot of time  and resources, and give small companies, whose accident datasets are too limited to train their own mod-  els, access to safety outcome predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' It may still however be advantageous to train specific models  to get an extra boost in performance through ensembling with the generic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Overall, by learning  lessons from a pool of datasets whose accumulated experience far exceeds that of any single company,  and making these lessons easily accessible in the form of simple forecasts, generic models tackle the  holy grail of safety cross-organizational learning and dissemination in the construction industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Keywords: construction safety, artificial intelligence, supervised learning, injury prediction,  transfer learning, data sharing, collective intelligence  1antoine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='tixier@safetyfunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='com  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='03567v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='LG]  9 Jan 2023  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Introduction  The SafetyAI council is a community of large organizations from the construction, oil &  gas, and electric Transmission and Delivery (T&D) domains, that share their safety-related data  with the SafetyAI Research and Development (R&D) team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Before exploiting the data, the R&D team is in charge of standardizing the datasets received  by each company, which is crucial, as each one features different variables and different category  names for each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Standardization makes sure that all datasets are based on the same  taxonomy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=', speak the same language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  The SafetyAI community dataset, comprising close to a million events including near misses,  observations, good catches, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=', is only accessible to the R&D team, a neutral party, which guar-  antees that it is impossible for companies to see each other’s data, and that the output of all the  R&D conducted on the collective dataset is made available to the entire community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' This is of  paramount importance, in a very competitive environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  In this study, we started by extracting attributes from accident reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' We briefly introduce  the attribute framework in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Attribute-based framework  Attributes are basic descriptors of construction work that are observable before accident  occurrence, and cover means, methods, and environmental conditions [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' One advantage of  the attribute-based framework over modeling at the task or work package level is that attributes  are fundamental and universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' That is, any situation from any site around the world, in any  industry sector, can be characterized by a set of attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Attributes can be recorded on-the-fly  on site, or can be extracted offline from various mediums such as photos and text reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' For  instance, four attributes can be extracted from the narrative worker tripped on a cable  when carrying a 2x4 to his truck: (1) cable, (2) object on the floor, (3) lumber, and  (4) light vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Narratives are particularly well-suited if the goal is to use attributes for predictive modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Indeed, in incident report databases, narratives are often paired with outcomes such as accident  type, injury severity, body part impacted, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Attributes also completely anonymize narratives,  which is especially desirable when considering a pool of datasets aggregated from different  companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' For any given event, everything that remains is a set of attributes and a set of  standardized safety outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  However, manually extracting attributes from large amounts of text reports is very costly in  terms of human resources and pose inter-annotator agreement issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' To solve this problem, we  developed and validated a Natural Language Processing (NLP) tool based on rules and lexicons  [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' We later proved that using the attributes extracted by the tool to predict safety outcomes  was effective and valid [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' We also used the attributes extracted by the tool for unsupervised  learning applications, such as clustering and visualization [6], and risk modeling and simulation  [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Differences with our previous research and objective of the current study  In our original study [4], we provided a proof for the concept of predicting safety outcomes  from attributes, both extracted with the NLP tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Then, in [5], we showed that attributes were  still highly predictive when the safety outcomes were given by independent human annotations,  which definitely validated the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' We also used a much larger dataset than in the orig-  inal study, two new supervised learning algorithms, model stacking, a healthier experimental  setup with more appropriate performance metrics, and we analyzed per-category attribute im-  portance scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' We also showed that unlike what we had concluded in [4], injury severity was  predictable from attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  2  In the present research, we interested ourselves with a new, completely different problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  We had access to a pool of accident datasets coming from 9 companies, and our goal was to:  “Test whether predictive models trained on a generic dataset (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=', aggregated from the datasets  of multiple companies) outperformed the models trained on the specific dataset of each com-  pany.”  More precisely, we experimented with two types of generic models:  Full generic model: one model trained on the datasets of all companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Per-domain generic models: one model per industry sector, trained only on the datasets  of the companies involved in that sector (or the parts thereof, as some companies belong  to multiple domains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  The potential advantages of generic models are numerous:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Usually with machine learning, the more data, the better, so generic models are expected  to bring improvements in predictive skill compared to the company-specific models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' This  is not guaranteed however, as one important question is whether (1) more data (generic  datasets) or (2) more relevant data (specific datasets) is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' By being trained on larger datasets, the generic models learn to predict a greater variety  of outcome categories than the specific models, making for more useful forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Successful generic models would remove the needs for training specific models for each  company, saving a lot of time and resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Alternatively, if company-specific models are already available, combining them with the  generic models may provide an extra boost in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Last but not least, successful generic models would give small companies -whose accident  datasets are too limited to train their own specific models- access to high quality safety  outcome forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  From a high level, generic models tackle the holy grail of safety cross-organizational learn-  ing and dissemination in the construction industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Indeed, generic models (1) learn lessons  from a pool of datasets whose quantity and diversity2 of accumulated experience far exceeds  that of any single company, and (2) disseminate these lessons as forecasts, which are clear, di-  rect, and easily accessible information, via, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=', a user interface (desktop or mobile) or API  taking attributes as input and returning probabilities for each category of each outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Moreover, one should note that in the pool, the individual biases of each dataset, due to  specific annotators, reporting practices and policies, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=', tend to average out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Consequently, the  lessons learned by the supervised learning algorithms on the generic datasets are more objective  and broadly applicable than that learned on the specific datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Background  The needs to share standardized incident data at the industry level to enable collaborative  learning have long been recognized in aviation and transportation [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Some examples include  2Diversity of situations, means and methods, environmental conditions, geographical areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  3  the NASA-managed Aviation Safety Reporting System (ASRS) database, created in 1976 and  featuring over a million incidents, or the European Coordination Center for Accident and In-  cident Reporting Systems (ECCAIRS) database, started in 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Such collective repositories  also exist in the chemical industry, with the Major Accident Reporting System (eMARS) of the  European Commission, launched in 1982, and the Process Safety Incident Database (PSID) of  the Center for Chemical Process Safety [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  However, the construction industry still lacks comparable initiatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' The needs for data  storage and access infrastructures for construction safety did start to receive some attention  recently [10, 11], but most efforts placed themselves at the company or project level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Cross-  organizational safety data collection is still rare in practice [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' This is a major issue,  as collaborative machine learning at the industry level is not possible until a common pool of  standardized datasets has been put together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' This provided the motivation for us to create the  SafetyAI council in 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  One should note that some consortiums already exist, such as the INGAA Foundation, the  Edison Electric Institute (EEI), the Construction Safety Research Alliance (CSRA), or the Na-  tional Safety Council (NSC), but their activities do not revolve around systematic large-scale  accident data collection and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' These initiatives rather involve working groups, com-  munities of practice, qualitative analyses, and conferences, towards building communications,  policies, best practices, business intelligence, safety culture and leadership, training material,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' In other words, they are based on “soft” methods for knowledge sharing and collabo-  rative learning at the human level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' They do not primarily conduct “hard” scientific research  and software development, and do not pool accident datasets for AI applications and automatic  large-scale learning and dissemination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Data Description  As already explained, as part of the SafetyAI initiative, we had access to a pool of safety  datasets coming from nine large companies from the construction, oil & gas, and electric Trans-  mission and Delivery (T&D) domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' One company, Company73, also had about 600 corporate  services (office) events for the severity outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' We kept these cases as training data for the  full generic model but did not train a specific model on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Member companies conduct work mostly in North America, and rely on their own teams as  well as contractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' The collective dataset covers the period 2000 to 2022, with a distribution  biased towards the last decade and especially more recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  While the entire pool comprises almost a million events including near misses and observa-  tions, we focused on accident cases only in this effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' As can be seen in Table 1, the sizes of  the individual datasets ranged from 2k to 20k cases, with an average of 6k per company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' There  were 57262 accident cases in total, recorded over tens of millions of work hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  We considered the same outcomes as in [5]: injury severity, body part impacted, injury  type, and accident type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' The columns corresponding to each outcome were selected from the  company datasets and normalized to use a common, standard set of categories, shown in Table  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Not all outcomes were available for every event of every company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' From the narrative of  each report, we extracted with the NLP tool [3] the original set of 80 attributes [3, 5], plus 11  new items (see Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' We also used the tool to extract a fifth outcome, energy source, that  was not available in the company datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  3Company names have been anonymized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  4  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  Domains  Constr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=',  elec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Oilgas  Constr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=',  oilgas  Elec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Constr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Constr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=',  elec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Elec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=',  oilgas,  corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Oilgas  Elec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Regions  Canada  California  NAM  NAM  NAM  NAM  NAM,  Mexico  World⋆  Southeast  USA  n  4481  1965  4072  5321  7245  4310  8345  19298  2225  Table 1: Company overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' NAM: North America (Canada + USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='  Oilgas: oil & gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='578  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='bite  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='710  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='error  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='425  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='Table 2: Outcome category counts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' across all companies and domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' PPE: personal protective equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Experimental Setup  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Splits  Train, validation and test splits were created for each of the 51 company-domain-outcome  combinations for which at least 2 categories with more than 100 observations each were avail-  able (shown in Table 4), by randomly sampling without replacement 64%, 16%, and 20% of  cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' The counts summed over companies are shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Note that the  proportions we used in our previous work [5] were 81%, 9% and 10%, but in the present re-  search, we decided to reserve more observations for the validation and test sets to make them  more representative of the training sets, in order to increase the stability and validity of hyper-  parameter tuning and evaluation4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  A specific model was trained on each of the 51 company-domain-outcome combinations for  which sufficient data were available, except for that one combination involving the corporate  cases, making for a total of 50 specific models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  For a given domain and a given outcome, the splits of the per-domain generic model were  obtained by combining, across all companies, the splits corresponding to that domain and that  outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' In total, there was one per-domain generic model for each domain and for each  outcome, hence a total of 3 × 5 = 15 per-domain generic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  For a given outcome, the splits of the full generic model were obtained by combining, across  all companies and across all domains, the splits corresponding to that outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' In total, there  was one full generic model for each outcome, hence a total of 5 full generic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  For each of the aforementioned cases, we tried 3 different algorithms, as will be explained  in subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Hence, a total of (15 + 5) × 3 = 60 generic models were trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  4Increasing the sizes of the validation and test sets was a good alternative to k-fold cross-validation, which would  have taken too much time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  5  # Companies  Train  Val  Test  Severity  Construction  4  9980  2494  3119  Electric T&D  4  6672  1669  2085  Oil & Gas  4  18381  4595  5744  Corporate  1  418  105  131  Full  9  35451  8863  11079  Body Part  Construction  4  8209  2052  2565  Electric T&D  4  6036  1508  1885  Oil & Gas  3  15788  3947  4933  Full  9  30033  7507  9383  Injury Type  Construction  4  6267  1566  1958  Electric T&D  4  4764  1191  1489  Oil & Gas  3  14960  3740  4675  Full  9  25991  6497  8122  Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Type  Construction  2  2740  685  856  Electric T&D  2  1600  400  500  Oil & Gas  3  2910  728  910  Full  6  7250  1813  2266  En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Source  Construction  4  4875  1218  1524  Electric T&D  3  2637  660  825  Oil & Gas  2  2600  650  813  Full  8  10112  2528  3162  Table 3: Split counts for each domain-outcome combination, summed over companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' For # Companies, full ̸=  total as some companies belong to multiple domains (see Tables 1 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Construction  Electric T&D  Oil & Gas  Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  S  B  IT  AT  E  S  B  IT  AT  E  S  B  IT  AT  E  S  1  x  x  x  x  2  x  x  x  3  x  x  x  x  x  x  x  4  x  x  x  x  x  5  x  x  x  x  x  6  x  x  x  x  x  x  x  x  7  x  x  x  x  x  x  x  x  x  8  x  x  x  x  x  9  x  x  x  x  x  Table 4: The 51 company-domain-outcome combinations associated with at least 2 categories with more than 100  observations each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' S: severity, B: body part, IT: injury type, AT: accident type, E: energy source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Corp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=': corportate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Class imbalance  To address the problem of class imbalance, weights inversely proportional to category  counts in the training set were computed with the formula max(counts)/counts, like in  [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' During training, these weights forced the models to pay more attention to the cases from  the minority categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Per-category counts with training weights can be found in Tables B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9 for the 15 domain-outcome combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Supervised learning algorithms  Like in [5], we relied on three popular machine learning models: Random Forest (RF) [14],  eXtreme Gradient Boosting (XGBoost or XGB) [15], and linear Support Vector Machine (SVM)  [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' More precisely, we used the Python’s scikit-learn implementations of Random  Forest5 and linear SVM6, while, for XGBoost, we used the original Python library7 and in  5https://scikit-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='RandomForestClassifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='html  6https://scikit-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='svm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='LinearSVC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='html  7https://xgboost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='io/en/latest/python/python api.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='html#module-xgboost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='sklearn  6  particular the GPU-accelerated implementation of the “fast histogram” algorithm (gpu hist)  as the tree method8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  For theoretical details about each algorithm, we refer the reader to our paper [5], publicly  available9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Hyperparameter optimization  We tuned the models by performing grid searches on the validation sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Details about the  parameters searched are available in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' The final models were trained on the union  of the training and validation sets with the best parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Both the specific models and  the generic models were tested on the test sets of the specific models, to ensure fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  As already explained, there were 50 such test sets, one for each company-domain-outcome  combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Transfer learning by stacking generic and specific models  As was mentioned in the introduction, one important question is the extent to which (1) more  data (generic datasets) or (2) more relevant data (specific datasets) is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' In what follows,  we explore a way to move past this binary choice and have a tradeoff between quantity and  relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Inspired by transfer learning, which is very successful in computer vision [17] and NLP  [18, 19, 20, 21], we experimented with combining the predictions of the generic and specific  models via an ensemble model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Very briefly, in AI, transfer learning refers to a two-step process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' First, a model is trained  at solving a general task on large amounts of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' This phase is called the pretraining phase,  as it allows the model to acquire generic knowledge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=', in NLP, reading and writing), that  is applicable to a great variety of situations downstream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Second, the pretrained model is fine-  tuned on a specific task of interest, often associated with a much smaller dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' in NLP,  summarization, classification, question answering, paraphrase detection, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  In our case, the generic and the specific models have to perform the same task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=', predict-  ing a given safety outcome10, and there is no pretraining phase per se, in that the generic and the  specific models are two different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' However, our approach is similar in spirit to transfer  learning, as our goal is to capitalize on generic knowledge gained from large amounts of data to  improve performance on a specific task associated with a smaller dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  More precisely, for each company-domain-outcome combination, we trained a meta-model  taking as input the weighted elementwise sum of the probabilistic forecasts of the best generic  and specific models11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' We used a simple logistic regression12 as our meta-model, with the C  parameter fixed and equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2, like in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' We grid searched the validation set to find the best  values of coefficients a and b where:  inputensemble = a × outputgeneric + b × outputspecific  (1)  Besides performance considerations, using tunable weights improves interpretability, by  providing information regarding which of the generic model or the specific model makes the  most important contribution to predictive skill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  8https://xgboost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='org/pdf/1908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='pdf  10The generic model has to perform a more difficult version of the task, though (more categories to predict).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  11The entries of the specific model vector for the categories that it did not predict were set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  12https://scikit-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='org/stable/modules/generated/sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='linear model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='LogisticRegression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='html  7  We tried values from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1 to 1 with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1 steps, holding the other parameter equal to 1, and  conversely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' That is, the following 19 pairs: (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1, 1), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' ,  (1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  SVM issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' By design, the implementation of the linear SVM model we used, linearSVC,  only returns discrete predictions, that is, a single label corresponding to the most likely category,  rather than a probability distribution over all categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' To address this issue, in [5], we tried  retraining the best SVM using the SVC implementation13 with linear Kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' However, results  were not convincing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Therefore, in the present study, we decided simply not to use model  stacking when one of the two models involved (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=', best generic or specific model) was a  SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Performance metrics  Due to the large class imbalance for all outcomes, measuring classification performance  with accuracy was inadequate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Rather, we computed precision, recall, and F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Precision, respectively recall, for category i, is equal to the number of correct predictions  for category i (number of hits), divided by the number of predictions made for category i (hits  and false alarms), respectively by the number of observations in category i (hits and misses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  precision =  Ci,i  �K  j=1 Cj,i  recall =  Ci,i  �K  j=1 Ci,j  (2)  Where the confusion matrix C is a square matrix of dimension K ×K (K being the number  of categories) and whose (i, j)th element Ci,j indicates how many of the observations known  to be in category i were predicted to be in category j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Finally, we computed the F1-score, the  harmonic mean of precision and recall:  F1 = 2 × precision × recall  precision + recall  (3)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Configuration  We relied on a single Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4 machine featuring a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9 MHz 12-thread CPU, a 12  GB Nvidia Titan V GPU, 64 GB of RAM, R version 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3 [22], and Python version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='13 with  scikit-learn version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1 [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Running all experiments took approximately ten days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Results  Each generic model (full and per-domain), as well as ensembles thereof (stacking approach  described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5) was tested on the test set of each company-domain-outcome combina-  tion and compared against the best performing specific model for this combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Results are very positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' As can be seen in Table 5, across all companies, the generic mod-  els (full or per-domain) outperform the specific models 82% of the time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=', for 41 company-  domain-outcome combinations out of 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Detailed per-company results can be found in Ap-  pendix E for the full generic models and Appendix F for the per-domain generic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' At  the company level, improvements are brought on average for 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6% of outcomes (across all  domains), ranging from 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3% for Company2 to 100% for Company1, Company4, Company6,  and Company9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='13  9  +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='56  +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='38  +12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='16  +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='01  +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='16  Table 5: Company-level max gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' x: no improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' S: severity, B: body part, IT: injury type, AT: accident type,  E: energy source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' C: company  Furthermore, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' 1, gains are high on average (+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4 in F1 score) and reach im-  pressive values, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=', +15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3 for Company7 on electric T&D-injury type, +12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='87 for Company6  on electric T&D-severity, +12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='86 for Company6 on construction-severity, +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='56 for Company3  on construction-body part, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' And all of that, while predicting more categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Distribution of Company−level Max Gains  Gain in F1 score over specific models  0  5  10  15  0  2  4  6  8  10  12  14  Counts  +  +  +++  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  ++  +  +  ++  +  +  +  +  +  Figure 1: Company-level max gains, across all domains and outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' n=41, min=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2, max=15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3, mean=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  There are only 9 domain-outcome combinations over 50, across 5 companies, on which  the generic models do not bring any quantitative improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' However, since their forecasts  are more informative (more categories predicted), it may still make sense in practice to use the  generic models in lieu of the specific models, even on these combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' For instance, for  Company3-oil & gas-accident type, the specific model only predicts exposure and struck, but  the generic model also predicts the categories caught, fall, and overexertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  The F1 scores averaged over all companies are shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Overall, the generic mod-  els bring improvement over the specific models for 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3 % of the domain-outcome combinations  (11 out of 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' As shown on the right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' 2, maximum gains range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='95 (for electric  T&D-energy source) to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98 (for electric T&D-injury type) with an average of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Also, not  only do the 11 best generic models outperform their specific counterparts with a comfortable  margin, but they also generate finer-grained forecasts, which are much more useful in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  More specifically, generic models predict 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='26 additional categories on average, even up to 7  for construction-injury type (while still providing a gain of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48 in F1 score).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' This is remarkable,  considering that the more categories to be predicted, the more difficult the task (see Appendix  D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  9  Distribution of All Gains  Gain in F1 score over specific models  Counts  0  2  4  6  8  10  0  5  10  15  20  25  30  Counts  ++  +  +  +  +  + + +  +  +  +  +  +  +  +  +  +  +  ++  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +++  +  +++  ++  +  +  +  +  Distribution of Max Gains  Gain in F1 score over specific models  Frequency  0  2  4  6  8  10  0  1  2  3  4  5  6  Counts  +++  +  +  +  +  +  +  +  +  Figure 2: Gains averaged over companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Left: n=48, min=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='16, max=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98, mean=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Right: n=11, min=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='95,  max=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98, mean=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  The construction and oil & gas domains see gains for 3 outcomes out of 5, while on the  electric T&D domain, we observe improvement for every outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Further, for the body part,  injury type, and energy source outcomes, there is at least one generic model that outperforms  its specific counterpart, on every domain, while the severity and accident type outcomes see  improvements only on the electric T&D domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  However, it is important to note that even on those 4 domain-outcome combinations on  which the generic models do not offer gains in predictive performance, it can still be desirable  to use them in practice over the specific models, as they generate more informative forecasts,  with 2 additional categories predicted, on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Overall, more than half of all F1 scores recorded for the generic models (79 out of 150,  or 53%) are greater or within two points of that of the specific models, while predicting 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='83  more categories on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' And, as shown on the left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' 2, the 48 generic models that  outperform their specific counterparts bring on average an improvement of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22 in F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Body part, injury type, and energy source  Some of the greatest improvements are observed for injury type, where the best generic  models provide large average gains of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='07, respectively on the construction,  electric T&D, and oil & gas domains, while predicting on average 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25 more categories than  the company-specific models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' This large boost in performance is remarkable considering the  significant increase in task difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Similarly, for energy source, the best generic models provide 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='95, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='83 improve-  ments in F1 score, while predicting 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='53 more categories on average;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' and for body part, the  gains are 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='38, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='42, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='37, with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='17 more categories predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Severity and accident type  For severity and accident type, the generic models outperform the company-specific ones  on the electric T&D domain, with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='09 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='03 gains in F1 scores, while predicting 2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  more categories on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  On the construction and oil & gas domains, the best generic models are between 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4 and 6  points below the company-specific ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' However, they still offer the benefit of predicting more  categories (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6 on average).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  10  Construction  Electric T&D  Oil & Gas  Full  Dom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Full  Dom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Full  Dom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Severity  F1  SVM  gen  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='66  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='29  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='52  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='79  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='92  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='97  RF  gen  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='26  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='04  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='37  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='28  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='08  ens  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='33  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='82  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='17†  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='88†  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='76  XGB  gen  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='74  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='69⋆  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='19  ens  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='67  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='81  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='01⋆  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='95  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='14  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='44  spec  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34†  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='79  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='86†  Count  # categories  spec  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  gen  5  5  5  5  5  5  # datasets  9  4  9  4  9  4  Body Part  F1  SVM  gen  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='51  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='32  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='68  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='24  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='65  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='66  RF  gen  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='41†  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='81  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='94†  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22†  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='85†  ens  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='33⋆  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='39⋆  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='08  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='43  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='54⋆  XGB  gen  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='86  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='23†  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='76  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='21†  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='59⋆  ens  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05⋆  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2⋆  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='31  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='71  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='37  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='74⋆  spec  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='03  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='79  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='85  Count  # categories  spec  6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='76  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='79†  spec  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='92  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='20  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='58  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='84  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='85  RF  gen  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='46  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='35  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='74  XGB  gen  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='27  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='21  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='80⋆  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='58  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='58  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='04  ens  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='02  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='40  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='08  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='53  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='56  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='13  spec  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98†  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='17  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='16†  Count  # categories  spec  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='67  gen  5  5  5  5  5  4  # datasets  6  2  6  2  6  3  Energy Source  F1  SVM  gen  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12†  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='78  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='64⋆  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='87†  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='54⋆  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='63  RF  gen  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='71  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='91†  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12⋆  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='61  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72†  ens  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='06⋆  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='64⋆  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='83  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48⋆  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='58  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12⋆  XGB  gen  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='77  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='04  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='83†  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='61  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22†  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='35⋆  ens  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05⋆  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='23  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='47⋆  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='03  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='38⋆  spec  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='64  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='92  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='39  Count  # categories  spec  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='67  3  gen  5  3  5  3  5  4  # datasets  8  4  8  3  8  2  Table 6: Results averaged over companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best of their sub-column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better than/within 2 pts of spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Full: full generic  model (one per outcome, same across domains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Dom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' : per-domain generic model (one per outcome per domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Gen/spec:  generic/specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Ens: ensemble thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' # datasets: number of company datasets forming the generic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Note: for the same  outcome, # categories and # datasets are the same for Full across domains, we repeat them only to ease comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Full vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' per-domain  In what follows, we refer to the full and per-domain models and their ensemble versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  When considering full generic models, the average improvement in F1-score over the specific  models is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='85 and there are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='44 additional categories predicted (min=0, max=7), while when  11  considering per-domain generic models, the average improvement is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='57 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='38 additional  categories are predicted (min=0, max=4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' The per-domain models reach a higher max score than  the full models on 9 combinations out of 15 (60%), and in 5 out of 11 (45%) when the specific  models are outperformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' The full and per-domain models outperform the specific models on  the same 11 domain-outcome combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  So, in terms of performance, there is no clear winner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' However, since the full generic models  predict more categories, and are also simpler conceptually (just one model per outcome), full  models seem like the way to go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' This conclusion however will need to be validated when more  datasets are available for each domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' One thing to note, however, is that specific models may  still be desirable in the context of model stacking, as covered next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Generic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' ensemble (generic + specific)  The transfer learning-like stacking approach, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=', combining the predictions of the generic  and specific models, boosts performance over the generic models (both full and per-domain)  on all domains for the severity and injury type outcomes, in some cases for accident type, and  nowhere for body part and energy source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  For severity, the average gains are of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='93, and range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='78 to an impressive 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8 (for  electric T&D-full-RF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Results are even more impressive for injury type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Gains range from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  to 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='64 (for construction-full-XGB), with a high average of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  It is interesting to note that for severity and injury type, very few of the generic models  outperform the specific models in the first place, and it is only by combining their predictions  with that of the specific models that absolute best performance can be reached, on the electrical  domain for severity, and on all domains for injury type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  We also observe that conversely, for body part and energy source, where model stacking  does not bring additional skill, the generic models are stronger than the specific models in the  first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  All in all, these results may suggest that ensembling only works when the generic models are  not already better than the specific models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' However, this rule does not hold everywhere (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=',  construction-accident type-XGB), so additional data, experiments and results will be necessary  to draw any general conclusion here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Quantity vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' relevance  As far as whether more data or more relevant data is best, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' 3 shows the distributions of  the best a and b coefficients as determined on the validation sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' It tends to indicate that, on  average, the best tradeoff involves anywhere from a little bit to a lot of generic model (anywhere  in the [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1] range, with peaks towards [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2] and [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9,1]), but almost always a lot of specific  model (between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9 and 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' In other words, data relevance always seems important, while  the contribution of data quantity fluctuates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' However, this is only a general trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' As can be  seen in the detailed results per company (Appendix E and Appendix F), in some cases, the  contribution of the generic model is more important than that of the specific model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=', (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6)  for Company6-XGB in the first table of Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Best model type  For the full generic models, the best algorithm is RF (6 domain-outcome combinations over  15), followed by SVM (5/15) and XGB (4/15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' When stacked with the specific model, RF  reaches best performance in 10 out of 15 combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  12  Coefficient Distributions  Coefficient Value  Count  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='0  0  10  20  30  40  50  60  generic  specific  Coefficient Distributions  Coefficient Value  Count  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='0  0  10  20  30  40  50  generic  specific  Figure 3: Distributions of the best coefficient values a (generic) and b (specific).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Left: full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Right: per-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  When considering the per-domain generic models, SVM obtains the best score 7 times out  of 15, followed by RF (5/15) and XGB (3/5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' However, when used in the ensemble, XGB is the  best (10/15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  RF and XGB are better choices than SVM as they consistently top the scores and can be  used in ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' In terms of performance though, there is no clear winner between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  One or the other could be used interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' However, XGBoost is superior in practice as  far as deployment is concerned, as the Random Forest models take a lot of disk space, even after  applying some compression tricks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Conclusion  We showed that generic models provide consistent and large improvements over company-  specific models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Moreover, generic models issue finer-grained forecasts that are more useful in  practice, as they predict more categories of each safety outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Generic models remove the needs for training company-specific models, saving a lot of time  and resources, and give small companies, whose accident datasets are too limited to train their  own models, access to safety outcome predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Per-domain generic models (trained on data from a specific industry sector) are not always  better than full generic models (trained on all data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Ensembling generic and specific models is  often very beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Therefore, it might still be worth training specific models to combine their  predictions with that of the generic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' If specific models are already in use, combining  them with the generic models may provide a boost in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  The forecasts are in essence clear and direct information that can be accessed via a user  interface (as a desktop or mobile webpage or application), or via an API for integration into any  existing ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' In each case, the only input required is a set of attributes, and the output are  probabilities for each category of each outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  By learning lessons from a pool of datasets whose accumulated experience far exceeds that  of any single company, and making these lessons easily accessible, generic models tackle the  holy grail of safety cross-organizational learning and dissemination in the construction industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Acknowledgements  We thank the Nvidia corporation for donating the Titan V GPU that was used in this re-  search, as part of their GPU grant program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  13  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content=' Liu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Goyal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Ghazvininejad, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Mohamed, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Levy, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Stoyanov, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Zettlemoyer, Bart:  Denoising sequence-to-sequence pre-training for natural language generation, translation, and comprehension,  arXiv preprint arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='13461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Eddine, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Tixier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Vazirgiannis, Barthez: a skilled pretrained french sequence-to-sequence  model, arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  [22] R Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Com-  puting, Vienna, Austria (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='org/  [23] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Pedregosa, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Varoquaux, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Gramfort, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Michel, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Thirion, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Grisel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Blondel, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Prettenhofer,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Weiss, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Dubourg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Vanderplas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Passos, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Cournapeau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Brucher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Perrot, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Duchesnay, Scikit-  learn: Machine learning in Python, Journal of Machine Learning Research 12 (2011) 2825–2830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Appendices  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Attribute List  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='adverse low temps  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='fuses⋆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='machinery  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='spark  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='bolt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='manlift  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='splinter/sliver  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='breaker⋆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='grout  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='guardrail/handrail  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='steel/steel sections  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='clearance⋆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='heavy material/tool  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='uneven surface  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='confined work space  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='working below elev wksp/mat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='forklift  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='LOTO/labeling⋆  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='small particle  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='working overhead  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='formwork  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='lumber  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='soffit  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='wrench  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7: 92 attributes used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' LOTO: lockout-tagout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' PPE: personal protective equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' ⋆: eleven  new attributes added since [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  15  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Detailed split counts  Severity  Train  w  Val  Test  Construction  report-only  917  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  226  283  1st aid  7486  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='0  1876  2369  medical  470  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  114  140  recordable  147  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  28  42  lost time  960  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  250  285  total  9980  2494  3119  Electric T&D  report-only  2392  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  576  712  1st aid  2809  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='0  736  905  medical  554  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  140  162  recordable  310  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  74  101  lost time  607  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  143  205  total  6672  1669  2085  Oil & Gas  report-only  929  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  244  279  1st aid  13766  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='0  3405  4279  medical  1919  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  489  618  recordable  152  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  42  52  lost time  1615  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  415  516  total  18381  4595  5744  Corporate  report-only  97  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  31  22  1st aid  321  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='0  74  109  total  418  105  131  Full  report-only  4335  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  1077  1296  1st aid  24382  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='0  6091  7662  medical  2943  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  743  920  recordable  609  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='0  144  195  lost time  3182  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  808  1006  total  35451  8863  11079  Body Part  Train  w  Val  Test  Construction  arm  1059  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='9  167  232  hand  2732  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='0  701  864  head  1682  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  394  494  leg  958  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  262  307  trunk  1084  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  243  330  total  8209  2052  2565  Electric T&D  arm  1061  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  274  319  foot  372  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='0  89  135  hand  1473  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='0  368  452  head  1246  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  318  403  leg  1084  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  251  307  trunk  800  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  208  269  total  6036  1508  1885  Oil & Gas  arm  1445  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  386  477  foot  1741  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  421  568  hand  5586  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='6  887  1088  leg  2053  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='9  370  464  total  15788  3947  4933  Full  arm  3565  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  945  1134  foot  2807  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  677  935  hand  9791  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='5  1599  1985  leg  4095  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  1011  1210  trunk  3333  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  821  1063  total  30033  7507  9383  Accident Type  Train  w  Val  Test  Construction  caught  396  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  105  137  exposure  119  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  38  40  fall  803  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  200  243  overexertion  492  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  128  160  struck  930  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='0  214  276  total  2740  685  856  Electric T&D  caught  207  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  55  62  exposure  454  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='1  102  143  overexertion  288  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='8  69  88  total  1600  400  500  Oil & Gas  caught  198  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  43  53  exposure  526  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  127  184  fall  1527  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='3  165  210  total  2910  728  910  Full  caught  801  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  203  252  exposure  1099  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  288  366  fall  2733  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='0  695  849  overexertion  780  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  179  225  struck  1837  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  448  574  total  7250  1813  2266  Energy Source  Train  w  Val  Test  Construction  chemical  76  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  21  14  gravity  1551  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  405  479  motion  3248  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='0  792  1031  total  4875  1218  1524  Electric  biological  221  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  52  88  gravity  733  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='0  429  507  total  2637  660  825  Oil & Gas  chemical  70  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  13  21  gravity  1485  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='6  246  300  thermal  131  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  30  44  total  2600  650  813  Full  biological  221  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  52  88  chemical  146  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='6  30  44  total  10112  2528  3162  Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8: Split counts (1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' w: training weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='0  584  677  total  14960  3740  4675  Full  bite  297  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  74  94  burn  647  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='6  223  291  irritation  547  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  138  183  pain  704  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  176  215  pinch  987  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  271  328  strain  6025  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  1495  1843  total  25991  6497  8122  Table B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9: Split counts (2/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' w: training weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Hyperparameter Optimization Details  For Random Forest14, we searched the number of trees (ntree parameter, from 100 to  1600 with steps of 100), the number of variables to try when making each split (mtry, from 5  to 45 with steps of 5), and the leaf size (nodesize, 1, 2, 5, 10, 25, and 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  For XGBoost15, we searched the maximum depth of a tree in the sequence (max depth,  from 3 to 6 with steps of 1), the learning rate (learning rate, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1), the  minimum leaf size (min child weight, 1, 3, 5, and 10), the percentage of training instances  to be used in building each tree (subsample, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7, and 1) , and the percentage of  predictors to be considered in making each split of a given tree (colsample bylevel, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7, and 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' The number of trees in the sequence (ntrees) was set to 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' The loss was  the multinomial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Finally, for the SVM model, we optimized the C parameter (C, 10x with  x taking 3000 evenly spaced values in [−9, 9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  14https://scikit-learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='org/stable/modules/generated/sklearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='RandomForestClassifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='html  15https://xgboost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='io/en/latest/parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='html  17  Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Illustration of Task Difficulty vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Number of Categories  To illustrate how the prediction task gets more and more difficult as the number of cate-  gories increases, we designed a synthetic example in which 105 observations were drawn from  an increasing number of categories (2 to 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Class imbalance was simulated by drawing from  the categories with probabilities following the lognormal distribution (mean=0, sd=2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' We con-  sidered two baselines: a random baseline, that predicts categories uniformly at random, and a  most frequent baseline, which always returns the most frequent category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Our proxy for diffi-  culty was one minus the F1 score of the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' In other words, the less well the baselines  are doing, the more difficult the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' We can see on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4 that the task difficulty rapidly  increases with the number of categories, and that going from 2 to 6 categories almost makes the  task twice as hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  2  4  6  8  10  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  Prediction Task Difficulty vs Number of Categories  Number of Categories  1 − F1 score  Random Baseline  Most Frequent Baseline  Figure D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Per-Company Results for the Full Generic Models  Note: the ensemble (“ens”) rows are left blank whenever the specific model is a SVM, as  we could not use ensembling in this case (the forecast of the SVM is not probabilistic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Severity  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Avg  spec  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='51  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='62  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='35  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34†  SVM  gen  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='23  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='64  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='01  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='76  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='66  gen  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='75  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='54  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='99  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='26  RF  ens  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='68  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='58  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='98  XGB  ens  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content=' spec  4  4  3  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' gen  5  5  5  5  5  Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='10: Severity, construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  18  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  Avg  spec  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='66  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='67  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='79  SVM  gen  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='93  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='56  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='67  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='52  gen  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='46  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='61  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='97  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='42  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='37  RF  ens  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='73  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='62  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='17†  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5,1)  gen  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='39  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='02  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='24  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='95  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  XGB  ens  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='74  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='27  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='01⋆  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2,1)  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' spec  4  2  2  4  3  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' gen  5  5  5  5  5  Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='11: Severity, electric T&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-specific  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Avg  spec  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='53  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='74  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='44  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='86†  SVM  gen  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='91  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='53  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='85  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='41  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='92  gen  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='96  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='69  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='14  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48  RF  ens  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='87  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='81  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  gen  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='75  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='27  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25  XGB  ens  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='89  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='36  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='61  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='14  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2,1)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' spec  3  4  3  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' gen  5  5  5  5  5  Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12: Severity, oil & gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Body Part  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Avg  spec  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='14  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='09  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='39  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='03  SVM  gen  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='26  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='09  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='02  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='66  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='51  gen  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='14  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='04  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='68  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='78  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='41†  RF  ens  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='49  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='43  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='33⋆  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='6,1)  gen  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='92  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='57  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='73  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='86  XGB  ens  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='44  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='38  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='62  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='77  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05⋆  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='2,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5,1)  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' spec  6  6  6  6  6  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' gen  6  6  6  6  6  Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='13: Body part, construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-specific  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  19  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  Avg  spec  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='86  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='79  SVM  gen  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='21  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='86  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='26  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='68  gen  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='96  32  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='02  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='76  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='94†  RF  ens  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='94  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='75  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='56  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='08  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  gen  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='24  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='44  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='24  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='76  XGB  ens  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='96  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='17  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='81  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='31  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5,1)  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' spec  6  6  4  6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' gen  6  6  6  6  6  Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='14: Body part, electric T&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold: better the company-specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Avg  spec  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='96  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='41  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='17  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='85  SVM  gen  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='66  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='23  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='06  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='65  gen  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='11  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='21  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22†  RF  ens  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='31  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='88  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='09  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='43  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  gen  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='36  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='81  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22  XGB  ens  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='26  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='28  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='56  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='37  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2,1)  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' spec  6  6  6  6  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' gen  6  6  6  6  Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='15: Body part, oil & gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold: better the company-specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Injury Type  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Avg  spec  54  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='91  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='07  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='92  SVM  gen  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='67  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='66  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='78  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='81  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='73  gen  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='84  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='86  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='11  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='03  RF  ens  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='67  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='75  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  gen  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='46  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='99  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='76  XGB  ens  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='55  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='46  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4†  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2,1)  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' spec  3  3  6  4  4  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' gen  11  11  11  11  11  Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='16: Injury type, construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold: better the company-specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  Avg  spec  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='21  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='28  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  SVM  gen  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='27  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='59  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='78  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='14  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7†  gen  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='44  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='41  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='74  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='16  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='44⋆  RF  ens  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='33  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='52  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='42  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2)  gen  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='57  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='45  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='49  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='03  XGB  ens  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='31  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='58  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='44  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1)  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' spec  5  6  4  6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' gen  11  11  11  11  11  Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='17: Injury type, electric T&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-specific  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  20  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Avg  spec  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='39  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='04  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  SVM  gen  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='97  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='67  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='69  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='11⋆  gen  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='52  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='15  RF  ens  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='82  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='65†  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7,1)  gen  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='69  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='01  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='28  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='33  XGB  ens  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='09  41  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6)  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' spec  3  10  8  7  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' gen  11  11  11  11  Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='18: Injury type, oil & gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-specific  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Accident Type  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  Avg  spec  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='63  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98†  SVM  gen  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='87  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='91  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='39  gen  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='44  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='46  RF  ens  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='35  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='35  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7)  gen  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='04  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='51  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='27  XGB  ens  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='02  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='02  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (1,1)  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' spec  2  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' gen  5  5  5  Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='19: Accident type, construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  Avg  spec  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='15  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='17  SVM  gen  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='46  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='71  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='58  gen  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='11  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='58†  RF  ens  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='29  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='29  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  gen  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='13  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='46  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8⋆  XGB  ens  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='08  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='08  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' spec  5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' gen  5  5  5  Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='20: Accident type, electric T&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-  specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  21  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Avg  spec  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='91  85  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='58  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='16†  SVM  gen  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='06  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='09  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='92  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='69  gen  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='67  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='03  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='71  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='14  RF  ens  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='46  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='35  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1)  gen  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='65  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='93  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='16  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='58  XGB  ens  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='31  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='56  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7)  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' spec  2  2  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='67  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' gen  5  5  5  5  Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='21: Accident type, oil & gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Energy Source  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Avg  spec  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='69  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='97  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='07  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='82  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='64  SVM  gen  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='76  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='16  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='86  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='69  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12†  gen  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='36  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='03  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='14  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='31  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='71  RF  ens  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='02  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='06⋆  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  gen  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='33  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='44  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='62  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='77  XGB  ens  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='88  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='81  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='47  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05⋆  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' spec  2  2  3  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' gen  5  5  5  5  5  Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22: Energy source, construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-  specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  Avg  spec  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='92  SVM  gen  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='59  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='61  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='73  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='64⋆  gen  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='99  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='06  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='32  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12⋆  RF  ens  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='85  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='81  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='83  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2,1)  gen  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='43  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='85  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='21  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='83†  XGB  ens  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='41  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='52  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='47⋆  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3,1)  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' spec  3  2  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='67  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' gen  5  5  5  5  Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='23: Energy source, electric T&D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-  specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  22  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Avg  spec  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='39  SVM  gen  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='73  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='36  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='54⋆  gen  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='43  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='81  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12  RF  ens  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='27  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='89  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='58  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2)  gen  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='44  74  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22†  XGB  ens  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98  coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3,1)  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' spec  4  2  3  #lev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' gen  5  5  5  Table E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='24: Energy source, oil & gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-specific  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Per-Company Results for the Per-Domain Generic Models  Note: the ensemble (‘ens’) rows are left blank whenever the specific model is a SVM, as we  could not use ensembling in this case (the forecast of the SVM is not probabilistic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Severity  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  Avg  spec  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='35  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='62  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='51  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34†  SVM  gen  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='86  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='03  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='61  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='66  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='29  gen  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='62  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='74  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='04  RF  ens  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='77  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='82  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6,1)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3)  gen  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='84  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='84  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='74  XGB  ens  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='14  30  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='81  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3,1)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5,1)  #categories spec  3  4  3  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  #categories gen  5  5  5  5  5  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25: Severity, construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Avg  spec  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='67  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='66  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='79  SVM  gen  36  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='47  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='62  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='06  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='79  gen  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='19  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='91  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='53  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='28  RF  ens  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='97  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='88†  Coeffs  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9)  gen  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='23  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='59  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='49  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='44  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='69⋆  XGB  ens  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='95  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='96  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='95  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  #categories spec  2  4  4  2  3  #categories gen  5  5  5  5  5  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='26: Severity, elec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  23  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  Avg  spec  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='44  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='74  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='53  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='86†  SVM  gen  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='83  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='31  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='97  gen  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='82  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='59  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='08  RF  ens  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='59  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='38  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='86  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='76  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8,1)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7,1)  gen  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='82  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='09  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='15  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='19  XGB  ens  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='97  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='73  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='96  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='44  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7,1)  #categories spec  4  4  3  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  #categories gen  5  5  5  5  5  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='27: Severity, oil & gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Body part  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  Avg  spec  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='09  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='39  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='14  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='03  SVM  gen  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='08  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='14  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='92  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='13  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='32  gen  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='19  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='06  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='64  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='81  RF  ens  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='23  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='77  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='14  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='6  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='23†  XGB  ens  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='1,1)  #categories spec  6  6  6  6  6  #categories gen  6  6  6  6  6  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-specific  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Avg  spec  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='86  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+page_content='79  SVM  gen  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='02  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='16  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='17  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='24  gen  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='21  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='31  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='71  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='97  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05  RF  ens  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='52  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='63  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='59  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  gen  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='03  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='55  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='56  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='21†  XGB  ens  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='41  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='01  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='71  Coeffs  (1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2,1)  #categories spec  4  6  6  6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  #categories gen  6  6  6  6  6  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='29: Body part, elec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  24  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  Avg  spec  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='17  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='41  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='96  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='85†  SVM  gen  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='91  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='84  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='66  gen  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='64  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='85†  RF  ens  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='15  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='66  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='81  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='54⋆  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  gen  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='69  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='36  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='59⋆  XGB  ens  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='84  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='28  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='74⋆  Coeffs  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2,1)  #categories spec  6  6  6  6  #categories gen  6  6  6  6  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='30: Body part, oil & gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-specific  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Injury type  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  Avg  spec  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='91  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='07  54  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='92  SVM  gen  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='16  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='31  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='97  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='78⋆  gen  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='56  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='91  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='64  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='11⋆  RF  ens  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='38  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='57  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='42  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='46†  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2)  gen  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='19  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='17  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='54  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='55⋆  XGB  ens  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='74  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='08  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='53  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='45  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5,1)  #categories spec  6  3  4  3  4  #categories gen  6  6  6  6  6  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='31: Injury type, construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-specific  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Avg  spec  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='28  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='21  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  SVM  gen  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='28  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='54  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='91⋆  gen  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='99  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='07  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='47  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='68⋆  RF  ens  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='33  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='87  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  gen  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='62  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='63  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='26  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='42  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48  XGB  ens  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='09  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='28†  Coeffs  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3)  #categories spec  4  5  6  6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25  #categories gen  8  8  8  8  8  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='32: Injury type, elec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  25  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  Avg  spec  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='04  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='39  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  SVM  gen  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='26  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='24  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='18  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='89  gen  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='57  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='87  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='02  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='82  RF  ens  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='69  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='97  Coeffs  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8,1)  gen  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='32  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='64  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='07  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34  XGB  ens  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='88  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='79†  Coeffs  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7)  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1)  #categories spec  8  10  3  7  #categories gen  11  11  11  11  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='33: Injury type, oil & gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-specific  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Accident type  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  Avg  spec  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='63  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98†  SVM  gen  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='15  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  gen  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='37  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='42  RF  ens  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  gen  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='21  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='21  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='21  XGB  ens  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  Coeffs  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1)  #categories spec  5  2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  #categories gen  5  5  5  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34: Accident type, construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  Avg  spec  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='15  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='17  SVM  gen  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='45  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='84  gen  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='13  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='28  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2†  RF  ens  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3,1)  gen  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='96  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='21  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='58  XGB  ens  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='53  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='53  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  #categories spec  5  4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  #categories gen  5  5  5  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='35: Accident type, elec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  Avg  spec  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='58  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='91  85  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='16†  SVM  gen  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='04  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='76  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='75  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='85  gen  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='77  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='06  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='53  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12  RF  ens  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='02  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='46  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='74  Coeffs  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  gen  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='03  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='16  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='93  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='04  XGB  ens  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='56  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='13  Coeffs  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  #categories spec  4  2  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='67  #categories gen  4  4  4  4  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='36: Accident type, oil & gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  26  Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Energy source  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1  Avg  spec  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='07  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='97  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='82  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='69  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='64  SVM  gen  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='3  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='99  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='32  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='78  gen  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='17  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='28  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='21  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='91†  RF  ens  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='85  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='62  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='45  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='64⋆  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  gen  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='88  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='28  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='91  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='09  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='04  XGB  ens  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='75  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='23  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5,1)  #categories spec  3  2  2  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='25  #categories gen  3  3  3  3  3  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='37: Energy source, construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-  specific model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='9  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='6  Avg  spec  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='05  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='92  SVM  gen  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='31  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='83  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='46  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='87†  gen  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='13  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='73  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='96  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='61  RF  ens  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='75  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='22  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='48⋆  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  gen  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='63  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='86  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='61  XGB  ens  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='15  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='91  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='03  Coeffs  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  #categories spec  3  3  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='67  #categories gen  3  3  3  3  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='38: Energy source, elec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-specific  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='7  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  Avg  spec  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='98  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='39  SVM  gen  49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='94  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='33  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='63  gen  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='34  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='11  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='72†  RF  ens  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='44  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='8  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12⋆  Coeffs  (1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='5)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='4,1)  gen  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='58  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='12  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='35⋆  XGB  ens  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='06  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='69  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='38⋆  Coeffs  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='2,1)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='1,1)  #categories spec  2  4  3  #categories gen  4  4  4  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='39: Energy source, oil & gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' †: best model on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content=' Bold/⋆: better/within 2pts of the company-specific  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
+page_content='  27' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE1T4oBgHgl3EQf-QbO/content/2301.03567v1.pdf'}
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+arXiv:2301.01730v1  [quant-ph]  4 Jan 2023
+Multitime Quantum Communication: Interesting But
+Not Counterfactual
+Robert B. Griffiths∗
+Department of Physics
+Carnegie Mellon University
+Pittsburgh, PA 15213
+Version of 4 January 2023
+Abstract
+A protocol for transmission of information between two parties introduced by Salih
+et al., Phys. Rev. Lett. 110 (2013) 170502 (hereafter SLAZ), involves sending quan-
+tum amplitude back and forth through a quantum channel in a series of steps, rather
+than simply sending a signal in one direction. The authors claimed that their protocol
+was “counterfactual” in the sense that while a quantum channel is needed to connect
+the parties, its actual usage becomes vanishingly small in the asymptotic limit as the
+number of steps tends to infinity. Here we show that this claim is incorrect because it
+uses probabilistic reasoning that is not valid at intermediate times in the presence of
+quantum interference. When ill-defined probabilities are replaced with a well-defined
+measure of channel usage here called “Cost”, equal to the absolute square of the am-
+plitude sent through the channel, the total Cost does not go to zero in the asymptotic
+limit of a large number of steps, but is bounded below by a rigorous inequality. A
+detailed analysis shows that this bound is satisfied in the SLAZ protocol. The analysis
+leading to the bound uses the fact that the Gram matrix formed by inner products of
+a collection of pure quantum states is additive over Hilbert subspaces and invariant
+under unitary time transformations. Its off-diagonal elements, which in general are not
+positive, play a significant role in the formal argument as well as providing a somewhat
+strange way of visualizing the transfer of information.
+Contents
+I
+Introduction
+2
+II
+One-Way Protocols
+3
+II.1
+Multiple Channels in Parallel . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+II.2
+One Channel Used Multiple Times . . . . . . . . . . . . . . . . . . . . . . .
+5
+∗Electronic address: rgrif@cmu.edu
+1
+
+III Two-way Protocols
+6
+III.1 Gram Matrices
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+III.2 Basic Two-Way Protocol
+. . . . . . . . . . . . . . . . . . . . . . . . . . . .
+6
+III.3 Sending One Classical Bit
+. . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+III.4 Lower Bound on Costs
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+10
+IV The SLAZ Protocol
+11
+IV.1 Description of the Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . .
+11
+IV.2 Calculation of Costs and Overlap
+. . . . . . . . . . . . . . . . . . . . . . .
+13
+IV.3 Discussion of Costs and Probabilities . . . . . . . . . . . . . . . . . . . . . .
+14
+V
+Conclusion
+15
+I
+Introduction
+The motivation for this paper is a scheme for the transmission of quantum informatiom
+introduced by Salih et. al [1] with the title “Protocol for direct counterfactual quantum com-
+munication”, and referred to hereafter as SLAZ, the initials of the authors. One ordinarily
+thinks of the transmission of information as sending a signal through a channel from sender
+to receiver. However the idea in SLAZ is that information can be sent from Bob to Alice if
+the quantum particle used to carry the information starts off in Alice’s domain, and a part
+of its quantum amplitude is sent to Bob through a quantum channel. Bob modifies this is
+some way before sending (or possibly not sending) it back to Alice, depending on the signal
+he wants to send. Alice then employs what Bob has returned to begin a second round of
+sending amplitude to Bob, who again modifies it before returning it, and so forth. This
+back-and-forth motion can continue for a large number of rounds until the information that
+Bob is sending has arrived in Alice’s domain, where she can carry out a measurement or
+perhaps perform additional processing. A key feature of protocols of this type is that all the
+intermediate steps can be represented by purely unitary time evolution, with intermediate
+time measurements, if any replaced by unitaries—a process of purification.
+The use of amplitude rather than particle in the previous paragraph is intentional, be-
+cause the state of the photon or other particle is in general a coherent superposition of parts
+associated with different spatial locations: Alice’s domain, Bob’s domain, and the channel
+connecting them. One generally thinks of a particle as something with a spatial location,
+but in quantum mechanics one cannot simultaneously ascribe particle and wave properties
+to the same entity at the same time because of wave-particle duality. In Hilbert-space quan-
+tum mechanics physical properties, such as location in space, are represented by projectors
+(Sec. III.5 of [2]), and when a projector representing a wave, think of |ψ⟩⟨ψ|, does not com-
+mute with a projector specifying a spatial location, ignoring this fact can rapidly lead to
+paradoxes. The double-slit paradox is an example: when a coherent wave passes through
+the slit system one cannot say through which slit the particle passed.
+The term “counterfactual” in the original SLAZ paper has the following significance. A
+quantum channel connecting the communicating parties is essential: this is not a case of
+mysterious nonlocal influences of the sort which are sometimes invoked to explain quantum
+violations of Bell inequalities.
+However, if the number of steps in an SLAZ protocol is
+2
+
+sufficiently large, the magnitude of the amplitude sent through the channel in each step can
+be made very small, and vanishes in the limit as the number of steps tends to infinity.
+A similar claim of counterfactuality has been made in much of the rather substantial
+literature motivated by the original SLAZ publication, which contains various modifications
+and extensions of the original protocol. There have also been criticisms of these counter-
+factual claims, and (of course) replies to criticisms. The Conclusion, Sec. V, of the present
+paper contains a few remarks about how its results apply to some of these publications, but
+a review, much less a detailed discussion, lies far outside its scope. The interested reader is
+referred to the extensive bibliographies found in [3,4].
+The aim of the present paper is to study the use of quantum channels in protocols of the
+SLAZ type, in particular the sense in which this usage is or is not counterfactual. To this end
+the concept of the Cost of using a quantum channel, the absolute square of the amplitude
+passing through it in a particular step in the protocol, is suggested, for reasons discussed
+in Sec. II, as a useful substitute for “probabiilities”, which in a quantum context are often
+ill-defined. The example of multiple channels in parallel, which few would want to claim are
+“counterfactual”, serves as an introduction to how information can be sent through a single
+channel in a single direction at multiple times, in a process in which all of the intermediate
+steps are represented by unitary maps.
+The main mathematical results of this paper are in Sec. III: Gram matrices and some of
+their properties are discssed in Sec. III.1, while Sec. III.2 gives the basic structure of simple
+two-way multiple time protocols. Section III.3 considers simple schemes for transmitting one
+classical bit, while the rigorous lower bound that undermines various counterfactual claims
+is the topic of Sec. III.4.
+The original SLAZ protocol is studied in detail in Sec. IV. In particular the total Cost
+of transmitting a classical bit λ = 0, in which Bob reflects the amplitude back to Alice, and
+for transmitting λ = 1, in which he absorbs rather than returns it, are evaluated explicitly.
+It turns out that in the asymptotic limit the λ = 1 Cost is miniscule, but that for λ = 0
+is enormous, while the product of the two remains finite and satisfies the rigourous bound
+in Sec. III.4. The mistaken claim that the SLAZ protocol is counterfactual results from two
+errors: a concept of channel use which would be questionable even for a classical stochastic
+process, and an improper use of probabilities in a way that violates quantum principles.
+The concluding Sec. V has a summary of the main results of this paper, a few comments
+on some parts of the literature related to SLAZ, and some suggestions for future directions
+of research.
+This author believes that protocols of the SLAZ type are quite interesting,
+deserve further exploration, and might contribute to useful ways of studying multipartite
+and multitime transmission of quantum information, as in quantum networks. And that such
+studies would prove more fruitful in the absence of misleading claims of counterfactuality.
+II
+One-Way Protocols
+II.1
+Multiple Channels in Parallel
+Think of quantum information as the information carried by a photon as it passes through
+a quantum channel, such as an optical fiber.
+The information could be encoded in its
+polarization. Rather than using a single channel, one could imagine sending the photon as a
+3
+
+coherent state through a set of N channels in parallel, using a collection of beamsplitters to
+divide up the initial amplitude among the different channels, and a corresponding collection
+to later recombine them. Let us suppose that the normalized |Φ⟩ that represents the photon
+at some intermediate time is a superposition of amplitudes
+|Φ⟩ =
+N
+�
+n=1
+cn|φn⟩
+(1)
+associated with the individual channels, labeled by n. Define the Cost qn associated with
+the use of channel n, and the total Cost Q for the channel system as:
+qn := |cn|2,
+Q :=
+N
+�
+n=1
+qn.
+(2)
+If the |φn⟩ and |Φ⟩ are normalized, Q is equal to 1, so one might identify qn with the prob-
+ability that the photon is in channel n. But what does that mean? In standard (textbook)
+quantum mechanics probability refers to the outcome of a measurement, but a measurement
+carried out at an intermediate time, when the quantum state is a coherent superposition
+over various locations, can alter what occurs later, and hence it is dangerous to associate
+such a probability with a situation in which a measurement does not take place.
+Another way of viewing this difficulty is to recall that von Neumann (Sec. III.5 of [2])
+identified quantum physical properties—which in classical physics are asociated with sets
+of points in the classical phase space—with projectors, self-adjoint idempotent operators,
+P = P † = P 2, on the quantum Hilbert space. For example, in the case of a spin-half particle
+the projectors
+P = (I − σz)/2
+R = (I + σx)/2,
+(3)
+where I is the identity and σz and σx are Pauli operators, represent the properties Sz =
+−¯h/2 and Sx = +¯h/2, respectively. In general, if two projectors P and R commute their
+product PR = RP represents the property P AND R. But if they do not commute, neither
+PR nor RP is a projector, and so neither represents a quantum property. In some sense
+noncommutation is the very essence of quantum mechanics; it is what distinguishes it from
+classical physics. The use of standard (Kolmogorov) probabilities requires a sample space of
+mutually-exclusive possibilities, one and only one of which occurs in a particular run of an
+experiment. In quantum theory such a sample space is a collection of mutually orthogonal
+projectors that sum to the identity, a projective decomposition of the identity. For example, R
+and I −R in (3) in the case of spin half; see (7) below for the general definition. In quantum
+mechanics there are often many possible sample spaces that one might be interested in,
+and carelessly combining incompatible spaces—some projectors in one do not commute with
+projectors in the other—inevitably leads to paradoxes rather than physical understanding.
+In the present context the dyad |Φ⟩⟨Φ| is a projector that does not commute with any
+of the projectos |φn⟩⟨φn| for which cn is nonzero, and thus it is meaningless to assign a
+probability to the latter in a situation where the coherent state |Φ⟩ will later be transformed
+by the final beamsplitters into the original state that entered the channel system.
+4
+
+II.2
+One Channel Used Multiple Times
+The possible advantanges, if any, of using many channels in parallel can also be realized
+by employing a single channel and sending quantum amplitude through it at a succession of
+times; this is what makes protocols of the SLAZ type of some interest. Let us suppose that
+information is being sent from Bob to Alice. One can think of the photon at a particular time
+as being in a coherent superposition of amplitudes in three different physical locations: Alice’s
+domain A, Bob’s domain B, and the channel C connecting them. The same symbols can be
+used for the Hilbert-space projectors associated with these locations, thus operators which are
+self-adjoint and idempotent, A = A† = A2, and mutually orthogonal, AB = BC = AC = 0.
+They sum to the identity
+A + C + B = I
+(4)
+and hence form a projective decomposition of the identity—see the general definition in
+(7) below. A projective decomposition of the identity is the quantum counterpart of the
+sample space of mutually exclusive possibilities essential for using standard (Kolmogorov)
+probability theory in the case of a quantum system. Note that A, B, and C are subspaces of a
+single Hilbert space, not subsystems represented by a tensor product. If the quantum particle
+possesses other degrees of freedom, these projectors are to be understood using the usual
+convention as including the identity operator on these additional degrees of freedom. Thus
+for a photon, A means that it is located in Alice’s domain, whatever may be its polarization.
+Bob can send a particular type of information λ to Alice by starting with a normalized
+reference state |ψ0⟩ = B|ψ0⟩, the particle is somewhere in his domain B, and using a unitary
+transformation Bλ acting on the subspace B + C to place it in a state
+|ψλ
+1⟩ = C|ψλ
+1⟩ = Bλ|ψ0⟩,
+(5)
+in the channel, at which point it travels through the channel to Alice. As the channel has no
+effect except to transmit the particle from one end to the other, we simplify the discussion
+(here and later) by using the same symbol for the ket that arrives at Alice’s end. She then
+applies a unitary A that does not depend on λ, for she does not know what Bob is sending,
+to empty the channel and arrive at a final state
+|ψλ
+2⟩ = A|ψλ
+2⟩ = A|ψλ
+1⟩,
+(6)
+which she can then measure or subject to further processing.
+This single-round transmission process can be carried out in a number of rounds in which
+during the n’th round Bob employs a unitary Bλ
+n acting on the B + C subspace to map an
+amplitude cn|ψ0⟩ into C, which is initially empty, and which travels to Alice, who uses a
+unitary An acting on A + C to remove it from the channel, which is then empty and ready
+for the next round. One way to visualize this is that Bob has a domain B of high dimension,
+and at the outset splits up the initial amplitude |ψ0⟩ into pieces placed in different subspaces
+of B with the help of a suitable set of beamsplitters. At round n the unitary Bλ
+n interchanges
+the appropriate subspace of B with the empty C. Alice’s A is also large, and her An maps
+whatever Bob has sent into an empty subspace reserved for this purpose. When the run is
+completed Alice can then combine the amplitudes in these different subspaces into a smaller
+space—e.g., using beamsplitters—or she can do a similar combination at the end of each
+5
+
+round. Of course Alice’s and Bob’s unitaries cannot be chosen independently; the two must
+work together to design the protocol. What is unknown to Alice is Bob’s choice of λ for a
+particular run; this is the information that she can extract at the end.
+Some multiple-time protocols employ measurements by Alice at intermediate times. In
+cases such as the original SLAZ scheme, discussed below in Sec. IV, it is possible to store the
+amplitude that could have triggered the measuring device in an empty subspace in Alice’s
+domain and put off the measurement until the protocol is finished. Of course, amplitudes
+that correspond to several measurements in succession can be combined, just as in the case
+of simultaneous transmission through several channels in parallel, as discussed earlier.
+III
+Two-way Protocols
+III.1
+Gram Matrices
+Let {Pj} be a projective decomposition of the Hilbert space identity I:
+I =
+�
+j
+Pj,
+Pj = P †
+j ,
+PjPk = δjkPj,
+(7)
+and let {|ψµ⟩}, µ = 0, 1, . . ., be a collection of kets on the same Hilbert space. The Gram
+matrix
+Gµν = ⟨ψµ|ψν⟩ =
+�
+j
+Gµν(Pj) =
+�
+j
+⟨ψµ|Pj|ψν⟩
+(8)
+is additive in that it is a sum over contributions from the different subspaces. In addition,
+Gµν is invariant (or conserved) under a unitary operation U that acts on every ket in the
+collection {|ψµ⟩}. Also, if this unitary acts on only some of the subspaces, say P1 and P2, and
+is the identity operator on the others, then while both Gµν(P1) and Gµν(P2) may change,
+their sum Gµν(P1) + Gµν(P2) remains unchanged. That Gram matrices are additive and
+conserved plays an important role in what follows.
+We shall refer to the diagonal elements Gµµ(Pj), which are non-negative, as weights.
+As these are rather like probabilities, their additivity and conservation is not surprising.
+However, that the same is true of the nondiagonal elements Gµν(Pj) with µ ̸= ν, hereafter
+referred to as overlaps, comes as something of a surprise, especially since |ψµ⟩ and |ψν⟩ may
+refer to two different runs of an experiment, one on Friday and one on Monday. Nonetheless,
+overlaps play a key role in the following analysis, not only as part of the mathematics but
+also in a surprising but useful “intuitive” way of thinking about what is going on. The
+absolute value of an overlap corresponds to a notion of fidelity in quantum information, but
+in general an overlap is a complex number, and the fact that it can be negative as well as
+positive is a key element in what follows.
+III.2
+Basic Two-Way Protocol
+In the following discussion the projective decomposition of the identity (7) that will
+concern us is {A, C, B}, where A means that the photon or other quantum particle is in
+Alice’s domain, B that it is Bob’s domain, and C in the channel connecting them. At the
+6
+
+beginning of a two-way protocol of the SLAZ type in which Bob is sending information to
+Alice of the photon amplitude is in Alice’s domain A. She initiates the run by sending some
+amplitude to Bob through the channel. He then modifies it and returns some or all of it to
+Alice, in a manner that depends on the information λ he wishes to transmit. Alice processes
+what Bob has returned, and begins the second round by again sending amplitude to Bob,
+who again returns it, etc. This can go on for N rounds, following which Alice makes a
+measurement to determine the value of λ.
+In further detail: At the beginning of round n, Alice uses a unitary An1 acting on A + C
+to map some of the amplitude in A into an empty channel C. This amplitude then flows
+through the channel to Bob, where he empties the channel into B, does some processing,
+and then maps some amplitude back into C. This flows to Alice, who empties C into A
+using a unitary An2. We assume that “flow through the channel” does not change anything,
+and hence it is convenient not to think of C as divided into close-to-Alice, close-to-Bob, and
+in-between subspaces, but simply imagine that Alice and then Bob and then Alice are acting
+on a single C subspace. Alice uses unitaries that act on A + C and are independent of λ,
+while Bob uses unitaries Bλ
+n, that depend on the information λ he wants to transmit, which
+act on C + B. Both the Alice and Bob unitaries will in general depend upon the round n,
+but Alice’s do not depend upon λ. In addition we impose the restriction that Bob’s actions
+are passive in the sense that that the magnitude of the amplitude he sends back to Alice
+in round n cannot be greater than what he has just received. This last condition clearly
+differentiates these two-way protocols from the one-way protocols of Sec. II.2.
+The requirement that Alice and Bob only employ unitary operations simplifies the anal-
+ysis. It is true that various published protocols of this type, including the original SLAZ
+version to be discussed in Sec. IV, employ nonunitary measurements at intermediate times.
+In the cases of interest to us these measurements can be replaced by unitary operations
+which allow the measurements to be put off until the end of the run, in a manner indicated
+in Sec. II.2 and employed in the discussion in Sec. IV.
+To quantify the channel usage for these protocols we use the notions of Cost, equal to the
+absolute square of the amplitude for a single use of the channel, and total Cost for the sum of
+the Costs involved in a single experimental run, as in Sec. II.1, see (2). An important issue
+connected to claims that these protocols are counterfactual has to do with the difference
+between Cost and probability, as will be discussed later for the SLAZ protocol in Sec. IV—
+the importance of this has already been noted in Sec. II.1. In particular we will be interested
+in identifying protocols that minimize the overall Cost, as in the example discussed next.
+III.3
+Sending One Classical Bit
+In the simplest SLAZ protocol Bob wants to send a single classical bit, λ = 0 or 1, to
+Alice. At the start all of the amplitude is in A for both a λ = 0 and a λ = 1 run, so all four
+of the initial Gram matrix elements Gµν
+0 (A), µ and ν equal to 0 or 1, are equal to 1. The
+goal is that after N rounds the result will be
+Gµν
+N (A) = δµν,
+(9)
+so that Alice can determine the value of λ Bob has sent by making a measurement in an
+appropriate basis. Thus the desired change is that during the course of the run the overlaps,
+7
+
+the off diagonal elements G01(A) and G10(A), decrease from 1 to 0, while the weights G00(A)
+and G11(A), remain equal to 1.
+At this point it is worth noting that if both weights are not maintained—for example if
+at the end G00(A) = 1 while G11(A) = G01(A) = 0, Alice can still extract the value of λ by
+measuring whether or not the photon is in the state |ψ0⟩. Let us call this, for want of a better
+term, a partial protocol in contrast to a full protocol that results in (9). A partial protocol
+can be used for one-way transmission, and the obvious advantage is that it costs nothing to
+transmit λ = 1. A possible disadvantage is that when Alice’s measurement reveals nothing
+it could be because of some failure in the channel or in the measuring device. In the present
+discussion we focus on full protocols.
+A very simple way to implement such a protocol is that on the very first step Alice sends
+the entire amplitude to Bob, with a Cost of 1 for this use of the channel. Bob then simply
+modifies this using the unitary Bλ and sends it back to Alice, either in one round or several
+rounds, with Alice sending nothing back. The Cost for using the channel in the Bob-to-Alice
+direction is also 1, see the discussion in Sec. II.2. Hence a total Cost of 2 for the protocol
+as a whole. Notice that since there is no restriction on Bλ this rather trivial protocol can be
+used to send “quantum” information. From the perspective of Cost, two-way protocols of
+the kind under discussion are interesting because a classical bit, λ = 0 or 1, can be sent at
+a total Cost of 1 rather than 2. And as shown below in Sec. III.4, the product of the Costs
+for λ = 0 and 1 cannot be less than 1.
+To discuss the successive steps in protocols that optimize the Cost, we need an appropriate
+notation. We will represent kets as row vectors as in the following example
+|ψ⟩ = |a; c; b⟩ = |a1, a2, a3; c1, c2; b1, b2⟩
+(10)
+where the dimensions of the A, B, and C subspaces are d(A) = 3, d(B) = 2 and d(C) = 2.
+Note that we are dealing with a direct sum of subspaces, A ⊕ B ⊕ C, not a tensor product
+of subsystems. In much of what follows, B is empty or can be ignored, so |a; c⟩ will suffice;
+this and other minor variants in notation should be self-explanatory.
+Let us start with an extremely simple one-round full protocol with d(A) = 2, d(C) = 1.
+It consists of the following steps:
+|a1, a2; c⟩ = |1, 0; 0⟩ → |1/
+√
+2, 0; 1/
+√
+2⟩
+⇒ |1/
+√
+2, 0; (−1)λ/
+√
+2⟩ → |1/
+√
+2, (−1)λ/
+√
+2; 0⟩,
+(11)
+where 0 means this amplitude is equal to zero; do not confuse it with the label 0 for one of
+the two orthogonal states of a qubit. Here → indicates the action of a unitary on A + C
+carried out by Alice, and ⇒ a λ-dependent unitary on C carried out by Bob. The action by
+Bob could involve intermediate steps requiring the B subspace, but its net effect is only to
+change the contents of C, so there is no need to include B in the discussion.
+In words: At the outset all of the amplitude is in Alice’s A, a1 = 1. She maps half (in
+the sense of the absolute square) of it into C and sends it to Bob, who either sends it back
+unchanged in order to transmit λ = 0, or with the opposite phase to send λ = 1. Alice then
+empties the channel into the a2 position, using a unitary on A + C that is independent of λ,
+as it simply requires interchanging two subspaces. A final measurement by Alice determines
+which of the two orthogonal states is present in A, and thus which bit Bob was sending.
+8
+
+Next consider what is happening to the Gram matrices Gµν(A) and Gµν(C) during the
+successive steps. In particular, the overlap G01(A) is equal to 1 at the outset, and the first
+step reduces it to 1/2 by placing 1/2 in C. Bob’s action changes G01(C) from +1/2 to −1/2,
+and this negative contribution to the overlap moves back into A when Alice empties the
+channel, leading to the desired G01(A) = 0. On the other hand, whereas the weight G00(A)
+is reduced to 1/2 during the first step, Bob’s action does not change the sign of G00(C), so in
+the final step Alice moves this weight back to its initial value of 1, and similarly for G11(A).
+Thus the goals of a full protocol have been achieved.
+The Costs of using the channel are easily evaluated: 1/2 for the Alice-to-Bob step and
+the same for Bob-to-Alice, for a total Cost of Qλ = 1, the same for λ = 0 and 1. These
+satisfy the rigorous lower bound worked out below in Sec. III.4, so this protocol is optimal
+if one uses total Cost as an appropriate measure of channel usage.
+This protocol is easily extended to an equally efficient version involving N rounds, N any
+positive integer. Let
+ǫ = 1/2N,
+(12)
+and for the first, n = 1, round replace (11) with
+|1, 0; 0⟩ → |
+√
+1 − ǫ, 0; √ǫ ⟩ ⇒ |
+√
+1 − ǫ, 0; (−1)λ√ǫ ⟩ → |
+√
+1 − ǫ, (−1)λ√ǫ; 0⟩,
+(13)
+while for round n + 1,
+|
+√
+1 − nǫ, (−1)λ√nǫ; 0⟩ → |
+�
+1 − (n + 1)ǫ, (−1)λ√nǫ; √ǫ⟩
+⇒ |
+�
+1 − (n + 1)ǫ, (−1)λ√nǫ; (−1)λ√ǫ ⟩ → |
+�
+1 − (n + 1)ǫ, (−1)λ�
+(n + 1)ǫ; 0⟩,
+(14)
+where it is straightforward to show that there exists a λ-independent unitary for the last
+step. The final result at the end of round N is the same as in (11), the case in which N = 1,
+and again the total Cost is Q0 = Q1 = 1, independent of λ. One can also let ǫ depend on n,
+thus ǫn > 0 for round n, subject to the condition
+�
+n
+ǫn = 1/2,
+(15)
+and the Cost is again equal to 1.
+There are other protocols with larger Costs which may have some practical advantage.
+Thus rather than a scalar amplitude, Alice might use photon polarization, say horizontal
+H, which Bob could return as H to send λ = 0 or rotate to vertical V to send λ = 1.
+In this case the Costs are Q0 = Q1 = 2, so twice that for an optimal one-way protocol.
+However, there is now no need to maintain a particular phase relation between what is in
+Alice’s domain and what is available to Bob during each round. If polarization is easier to
+maintain than phase—one leaves that up to the experts—one could imagine the added Cost
+being worthwhile if Alice has a large apparatus capable of generating single photons, while
+Bob, off on a trip to spy on Eve, needs only something easily carried in a suitcase.
+The protocol used in SLAZ, in which Bob returns the amplitude for λ = 0, but absorbs
+it or feeds it to a measuring apparatus for λ = 1, looks less promising. Because the λ = 1
+weight only moves from Alice to Bob it is difficult to have G11(A) = 1 at the end of the
+protocol. In fact SLAZ, discussed in Sec. IV, employs a clever trick (“Zeno effect”) to get
+around this problem, albeit at the cost of a large number of rounds to keep the probability
+of failure small, and a large channel usage Cost for one of the bits.
+9
+
+III.4
+Lower Bound on Costs
+The additivity and conservation properties of the Gram matrix Gµν introduced in Sec. III.1
+will now be used to obtain lower bounds on the total Cost of two-way protocols of the sort
+exemplified by, but not limited to, the case of 1 classical bit discussed above in Sec. III.3.
+Using the |a; c⟩ notation of (10)—the b entry is not needed in the following discussion—round
+n of an N round protocol consists of the following steps carried out on A + C:
+|aµ; 0⟩n → |¯aµ; cµ⟩n ⇒ |¯aµ; ˆcµ⟩n → |aµ; 0⟩n+1.
+(16)
+Here µ labels the bit which Bob is transmitting during this run. Thus after Alice uses a
+unitary An1 on A + C to move some amplitude, |cµ⟩n into an initially empty channel. Bob
+applies a unitary Bµ
+n to C + B, leading to an amplitude |ˆcµ⟩n—note the hat added to c—in
+the channel. If Bob’s action is passive, as assumed in Sec. III.3 (and in the later discussion
+of SLAZ in Sec. IV), one would have
+∥ˆcµ∥n ≤ ∥cµ∥n,
+(17)
+but this conditions is actually not needed to obtain the general results and inequalities given
+below, which thus apply equally to one-way multi-time transmission. As a final step Alice
+employs a unitary An2 on A + C to empty the channel by placing its amplitude into A. It is
+important that Alice’s unitaries An1 and An2, unlike Bob’s Bµ
+n, do not depend upon µ, which
+can be different in different runs of the experiment.
+The change in the Gram matrix associated with A during round n is given by
+Gµν
+n+1(A) − Gµν
+n (A) = ⟨aµ|aν⟩n+1 − ⟨aµ|aν⟩n = ⟨ˆcµ|ˆcν⟩n − ⟨cµ|cν⟩n,
+(18)
+where ⟨aµ|aν⟩n is the inner product of |aµ⟩n and |aν⟩n. The equality follows from the fact that
+Gµν(A + C) is invariant under An1 and An2, and additive: Gµν(A + C) = Gµν(A) + Gµν(C).
+To discuss the total change during N rounds, n = 1, 2, . . . N, it is convenient to define
+|Cµ⟩ := {|cµ⟩1, |cµ⟩2, . . . |cµ⟩N},
+| ˆCµ⟩ := {|ˆcµ⟩1, |ˆcµ⟩2, . . . |ˆcµ⟩N}
+(19)
+with inner products
+⟨Cµ|Cν⟩ =
+N
+�
+n=1
+⟨cµ|cν⟩n,
+⟨ ˆCµ| ˆCν⟩ =
+N
+�
+n=1
+⟨ˆcµ|ˆcν⟩n.
+(20)
+Summing (18) over N rounds yields the following formula
+∆Gµν(A) = Gµν
+N (A) − Gµν
+0 (A) = ⟨ ˆCµ| ˆCν⟩ − ⟨Cµ|Cν⟩,
+(21)
+for the total change in the A portion of the Gram matrix during the full protocol. This
+quantity is bounded by
+|∆Gµν(A)| ≤ |⟨ ˆCµ| ˆCν⟩| + |⟨Cµ|Cν⟩| ≤ ∥ ˆCµ∥ · ∥ ˆCν∥ + ∥Cµ∥ · ∥Cν∥
+(22)
+using the norm ⟨Cµ|Cµ⟩ = ∥Cµ∥2.
+10
+
+Next define the total Cost Kµ for Alice-to-Bob and ˆKµ for Bob-to-Alice uses of the
+channel, with Qµ their sum:
+Kµ = ⟨Cµ|Cµ⟩ = ∥Cµ∥2,
+ˆKµ = ⟨Cµ|Cµ⟩ = ∥ ˆCµ∥2,
+Qµ = Kµ + ˆKµ.
+(23)
+Combining (22) and (23) gives
+|∆Gµν(A)| ≤
+√
+KµKν +
+�
+ˆKµ ˆKν ≤
+�
+QµQν.
+(24)
+This yields an upper bound
+∆Gµµ(A) ≤ Qµ
+(25)
+for a non-negative diagonal weight, and for the off-diagonal overlap:
+|∆Gµν(A)| ≤
+�
+QµQν.
+(26)
+In the particular case of the 1-bit two-way protocol, Sec. III.3, the aim is to reduce G01(A)
+from its initial value of 1 to 0 after N rounds. Setting µ = 0 and ν = 1 in (26), we see that
+to achieve this result it is necessarily the case that the Costs Q0 and Q1 for sending bits
+λ = 0 and λ = 1 must satisfy the condition
+Q0Q1 ≥ 1.
+(27)
+This is satisfied as an equality with Q0 = Q1 = 1 for the specific protocols discussed in
+Sec. III.3, which shows that they are optimal if total Cost is used as a measure. For more
+general protocols there is no reason to expect that the two Costs will be equal, and in that
+case if, say, the Cost for λ = 1 is made very small, that for λ = 0 must be very large. This
+is in fact the case for the original SLAZ protocol, as discussed below in Sec. IV, which thus
+provides an interesting illustration of such a tradeoff.
+IV
+The SLAZ Protocol
+IV.1
+Description of the Protocol
+The original SLAZ protocol differs from the simpler situation discussed in Sec. III.3 in
+two respects. First, it has a hierarchical structure: there are a large number M of outer
+rounds or cycles, each of which consists of a large number N of inner rounds or cycles, and
+the protocol will succeed with high probability provided
+1 ≪ M ≪ N.
+(28)
+Second, while Bob sends a bit λ = 0 by reflecting the amplitude sent by Alice back into
+the channel, for λ = 1 he simply empties the channel, which can be described as a unitary
+operation in which the C amplitude is placed in Bob’s subspace B. In addition, the original
+SLAZ protocol and some of its modifications involve measurements at intermediate times,
+and these will be replaced in the discussion below by unitary operations in the manner
+suggested at the end of Sec. II.2.
+11
+
+We use a notation
+|ψ⟩ = |a1, a2, a3, a4; c; b⟩
+(29)
+of the form introduced in (10), where the aj are scalar amplitudes in Alice’s domain A =
+A1+A2+A3+A4, c is the amplitude the channel C, and b is in Bob’s domain B. Here capital
+letters are used to denote subspaces and the corresponding projectors, while lower case letters
+indicate (in general complex) scalar amplitudes. While A4 and B are one-dimensional, one
+can also make these larger spaces for reasons that will appear during the discussion. An
+abbreviated notation is often convenient: |a2, a3⟩ in the case of a unitary acting on A2 + A3
+while all the other amplitudes remain unchanged.
+Central to the discussion are unitary operators that represent a rotation by an angle θ
+on a 2-dimensional space:
+R(θ)|α, β⟩ = |α cos θ − β sin θ, α sin θ + β cos θ⟩.
+(30)
+In particular, RM and RN, defined in terms of small angles, play a central role:
+RM := R(θM),
+θM := π/(2M),
+RN := R(θN),
+θN := π/(2N).
+(31)
+Note in particular that
+(RM)M = (RN)N = R(π/2);
+R(π/2) |α, β⟩ = | − β, α⟩.
+(32)
+In view of the fact that θN is a small angle, the following approximations turn out to be
+userful:
+cos θN ≈ exp[−θ2
+N/2] = exp[−π2/(8N2)] ≈ 1 − π2/(8N2),
+(cos θN)N ≈ exp[−π2/(8N)] ≈ 1 − π2/(8N) ≈ 1,
+(33)
+and similarly if N is replaced by M.
+These approximations are useful for understanding the overall structure of the protocol,
+which is the following. At the beginning of outer round m, 1 ≤ m ≤ M, RM is applied to
+A1 + A2 to yield,
+|a1, a2⟩λ = RM|¯a1, ¯a2⟩λ,
+(34)
+where ¯a1 and ¯a2 are the values of these amplitudes at the end of the previous outer round. In
+general they depend upon which bit λ = 0 or 1 is being transmitted, whence the superscript
+label, even though Alice’s operations do not depend upon λ. The very first outer round
+m = 1 begins by applying (34) to the starting state (29) with a1 = 1 and all the other
+amplitudes equal to zero.
+The initial step (34) of outer round m is followed by a sequance of N inner rounds, each
+involving the following steps, here displayed using the type of notation employed in Sec. III.3,
+but now with reference to the subspace A2 + A3 + C.
+|a2, a3; c = 0⟩ → |a′
+2, a′
+3; c = 0⟩ → |a′
+2, 0; a′
+3⟩
+⇒ |a′
+2, 0; (1 − λ)a′
+3⟩ → |a′
+2, (1 − λ)a′
+3; 0⟩,
+(35)
+where
+|a′
+2, a′
+3⟩ = RN|a2, a3⟩.
+(36)
+12
+
+In words, Alice applies the unitary rotation RN, (31), to A2 +A3, and then maps A3 into the
+empty channel. Next comes Bob’s action, indicated by ⇒, to either reflect the amplitude a′
+3
+back into C if he is sending λ = 0, or shift it into his domain B, leaving the channel empty
+if sending λ = 1. Alice, who does not know the value of λ, maps whatever is in the channel
+back into A3 by a unitary that simply exchanges the contents of A3 and C, and then begins
+the next inner round. The result of N inner rounds in succession is
+|a2, a3⟩ →
+�
+|0, a2⟩ for λ = 0,
+|(cos θN)Na2, 0⟩ ≈ |a2, 0⟩ for λ = 1.
+(37)
+where the λ = 1 approximation is justified when N is very large, see (33).
+Following the N inner rounds Alice completes this outer round by applying a unitary to
+A3 + A4 that empties the contents of A3 into A4. For λ = 1, a3 = 0, (37), so this emptying
+step is trivial, while for λ = 0 it is nontrivial, and plays a signficant role in understanding
+the true Costs of the protocol. In the original SLAZ protocol this emptying step is replaced
+by a measurement, but instead of a measurement one can just as well let the amplitudes
+accumulate in A4, which is the perspective used here.
+At the end of the protocol after
+completing M outer rounds the final result is
+λ = 0 : |a1 = 1 − r1, a2 = 0, a3 = 0, a4 = r4, c = 0, b = 0⟩
+λ = 1 : |a1 = s1, a2 = 1 − s2, a3 = 0, a4 = 0, c = 0, b = sb⟩,
+(38)
+where the quantities denoted by rj and sk are small corrections, of order 1/M or M/N
+If these are ignored, all the amplitude is in A1 for λ = 0 or A2 for λ = 1, and a simple
+measurement allows Alice to determine which bit Bob sent.
+IV.2
+Calculation of Costs and Overlap
+It is fairly straightforward to work out the Costs for the SLAZ protocol using approxi-
+mations justified by 1 ≪ M ≪ N, and the results are summarized in Sec. IV.3 below. We
+begin with the case λ = 1. If one ignores small quantities, the nonzero components of |ψ⟩m
+at the beginning and at the end of outer round m are
+a1 = cos(mθM),
+a2 = sin(mθM),
+(39)
+and since MθM = π/2, at the end of outer round M the result is the λ = 1 line in (38).
+The probability that the photon arrives in B during outer round m—the probability that
+Bob will detect it if he uses a measuring device—is the sum of the absolute squares of the
+amplitudes in the channel C in the N inner rounds, as this is an incoherent process:
+N(sin(mθM))2(sin(θN))2 ≈ (π2/4)(sin(mθ/M))2/N.
+(40)
+Summing over m gives the total probability
+K1 = Q1 = (π2/8)(M/N).
+(41)
+that the photon will end up in Bob’s domain by the end of the protocol, which is the same
+as the total Cost for λ = 1.
+13
+
+In the case λ = 0, any amplitude placed by Alice in C is immediately returned by Bob,
+and at the end of each outer round is emptied into a4, so that at the end of outer round m
+the state is
+|ψ⟩m = |a1 = (cos θM)m, a2 = 0, a3 = 0, a4, c = 0, b = 0⟩.
+(42)
+For m = M this is (38) with r1 = (π2/8M). Thus at the end of the protocol a2, a3, c and
+b are strictly zero. The Cost associated with inner round n—note that the channel is used
+twice—is
+2[sin θM · sin(nπ/2N)]2.
+(43)
+Summing over n gives a total of (π2/4)(N/M2) for each outer round, and hence for M outer
+rounds a total Cost of
+Q0 = (π2/4)(N/M2).
+(44)
+To compute the total change in overlap ∆G01(A), note that since for λ = 1 Bob does not
+return an amplitude, only the ⟨Cµ|Cν⟩ term in (21) contributes. The contribution for inner
+round n of outer round m is the product of the factors
+[sin θM sin(nθN)] · [sin(mθM) sin θN]
+(45)
+corresponding to λ = 0 and 1. Summing them yields
+(sin θM sin θN)
+M,N
+�
+m,n
+sin(mθM) sin(nθN) = (π2/4MN)(4MN/π2) = 1,
+(46)
+and hence
+∆G01(A) = −1,
+(47)
+as expected.
+IV.3
+Discussion of Costs and Probabilities
+To summarize the results of Sec. IV.2: The total Costs Q0 and Q1 for λ = 0 and 1 are:
+Q0 = (π2/4)(N/M),
+Q1 = (π2/8)(M/N),
+Q0Q1 ≈ 3.044.
+Q0/Q1 = 2N2/M2.
+(48)
+Given that M ≪ N, Q1 is miniscule, Q0 is enormous, while their product is of order 1, and
+satisfies the rigorous bound (27). The case λ = 1 is the easiest to understand. Since Bob
+does not return the amplitude put into the channel by Alice, the Bob-to-Alice Cost ˆK1 is
+zero. The Alice-to-Bob Cost is |sb|2 in (38), i.e., the probability that at the very end the
+photon is in Bob’s domain. The physical reason for this is that the process by which the
+amplitude gets there is incoherent, no quantum interference, since no amplitude goes back
+through the channel. Bob could either accumulate these amplitudes until the end of the
+protocol and then measure to see if the photon is in B, or carry out a measurement at the
+end of each inner round; in either case the probabilility of his detecting the photon is |sb|2 in
+(38). The situation is analogous to the use of intermediate time measurements in a one-way
+protocol as discussed at the end of Sec. II.2.
+The enormous Cost Q0 for λ = 0 comes about because Bob repeatedly returns the
+amplitude sent by Alice in a coherent process. While the amplitude bouncing back and
+14
+
+forth through the channel is relatively small, of order 1/M, multiplying its absolute square
+by 2N, the number of times this amplitude is is in the channel during each outer round,
+leads to a Cost of order N/M2 for each outer round, and hence a total of order N/M for the
+complete process.
+Clearly the large value of Q0 means the claim that protocol is counterfactual cannot
+be maintained if Cost is used as a criterion for channel use, so it is worth discussing how
+the authors of SLAZ reached a different conclusion. In essence their reasoning was based
+on the small value of the amplitude in A3 at the end of an outer round just before it is
+transferred to A4, as per the discussion in Sec. IV.1. The absolute square of this amplitude
+is the probability that the corresponding detector D3 in Fig. 2(b) in the SLAZ paper will
+be triggered. This amplitude was earlier oscillating back and forth inside the subspace with
+projector S = A2+A3+C, and hence it is reasonable to assume that if this detector triggers,
+the photon was earlier in S during all N inner rounds making up this particular outer round1.
+As this probability is of order 1/M2, the probability that one of the D3 detectors triggers
+during the M outer rounds that make up a given run is of order 1/M, and hence small.
+There are two serious objections to using this small probability to justify the claim that
+the protocol is counterfactual: one classical and the other quantum. Let us start with the
+former. During a particular outer round the photon amplitude in a λ = 0 run rattles back
+and forth inside S a total of N times, and in particular it is in C a total of 2N times.
+Consider a stochastic classical protocol for transmitting information in which most of the
+time Alice and Bob exchange no information at all. However, with a small probability ǫ
+Alice sends a little white ball into the channel leading to Bob, who colors it green or red
+and sends it back to Alice to convey one bit of information. She records the color, paints
+the ball white, and returns it to Bob who again colors it to send a second bit, and so forth,
+for a total of N rounds. The average rate of transmitting information is Nǫ bits, and one
+cannot simply throw away the factor of N and claim that this protocol is in some sense
+‘counterfactual’.
+The quantum difficulty has to do with what can be inferred from the probability that
+the photon was in S = A2 + A3 + C during the inner rounds that make up a particular outer
+round. One may be tempted to use classical reasoning and assume that the probabilities of
+being in each of the mutually exclusive regions, A2, A3, and C, that combine to make up S
+are well-defined and sum to the probability of being in S. But in the presence of quantum
+interference this sort of reasoning is invalid and leads to paradoxes. See the discussion of
+parallel channels in Sec. II.1.
+V
+Conclusion
+The original SLAZ proposal has motivated a large number of papers; see the extensive
+bibliographies in [3, 4]. Merely trying to summarize them, much less provide a detailed re-
+view, lies outside the scope of the present paper. Broadly speaking, this literature consists
+of modifications, extensions, or improvements of the original SLAZ scheme; along with crit-
+icisms of the claim that these protocols are counterfactual and replies to such criticisms. It
+is hoped that the following rather brief comments will provide some orientation.
+1This assumption can be justified using Consistent Histories; see the discussion of measurements in [5,6]
+15
+
+Significant extensions of the original SLAZ scheme by the last three members of the
+original collaboration include: the use of a phase change rather than absorption to transmit
+the λ = 1 bit [7]; a scheme to transmit quantum states by multiple iterations of the original
+SLAZ scheme [8]; using many photons in place of a single photon to transmit a classical
+bit [4]. These and others are certainly interesting ideas from the perspective of transmitting
+quantum information, and worth further exploration.
+On the other hand, in these and all other extensions or modifications of SLAZ this author
+has examined, the claim that the protocol is “counterfactual,” in the sense that the total use
+of a quantum channel is negligible in the asymptotic limit, is subject to the same objections
+discussed in Sec. IV.3: An improper use of probabilistic reasoning in a situation where
+quantum interference means probabilities cannot be defined, and where even in a classical
+situation Cost would be better than probability as a measure of channel usage. The total
+Cost remains finite in the asymptotic limit of a very large number of steps, which means that
+counterfactual claims should be dropped. Doing so will aid, not hinder, the serious study of
+these interesting quantum schemes for transmitting information.
+Shortly after the original SLAZ publication, Vaidman published a Comment [9] claiming
+that in the λ = 0 case in which Bob reflects the amplitude rather than absorbing it, the
+photon which was later (with high probability) detected by Alice must at an earlier time have
+been in the channel C. In their Reply [10] the SLAZ authors pointed out this way of reasoning
+about events at an intermediate time in the presence of quantum interference was invalid,
+and leads to paradoxes, a position supported by the analysis in Sec. IV.2 above. However,
+they then repeated their original counterfactual claim which itself is based on a defective
+understanding of probabilities at an intermediate time. A later and much more extended
+criticism of counterfactuality claims by Vaidman [11] suffers from the same difficulty as his
+earlier Comment.
+Some years later Aharonov and Vaidman [12] claimed to have found a scheme of the
+general SLAZ type which is genuinely counterfactual.
+However, when measurements or
+absortion of a photon at intermediate times are replaced by unitary processes—mapping
+amplitude into an empty subspace reserved for this purpose, as discussed in Sec. IV.1—the
+inequality in Sec. III.4 applies to this case and undermines the counterfactual claim. The
+fundamental difficulty with such claims is that the Hilbert space projector which identifies the
+position of a particle at some intermediate time does not commute with the one representing
+the quantum state evolving unitarily in time.
+The most significant contributions of the present paper to the analysis of SLAZ-type
+protocols is the use of Cost as a measure of channel usage, and the use of Gram matrices for
+discussing information transfer at intermediate times in the presence of quantum interference.
+In particular, the fact that these Gram matrices are additive over subspaces and invariant
+(“conserved”) under unitary time transformations, plays a key part in the discussions in
+Sec. III. A rather surprising feature is the role of off-diagonal elements, “overlaps”, as a type
+of information measure which, unlike most such measures, is not in general positive. That it
+can be negative plays a very signficant part in understanding its intuitive role in information
+transfer. That its total change on Alice’s side must be −1 during the course of a successful
+protocol is confirmed for the SLAZ protocol in Sec. IV.2.
+This use of Gram matrices requires that the intermediate time steps be unitary. In the
+case of SLAZ, measurements at intermediate times can be eliminated by mapping photon
+16
+
+amplitude into empty subspaces, and this can be achieved in certain other cases, e.g., the
+Aharonov and Vaidman protocol [12]. However, it is less clear whether something similar
+could be done in a case in which, for example, Alice uses measurements at intermediate times
+to change later steps in the protocol in hopes of reducing the total Cost. This author believes
+that such an improvement is impossible, because measurements themselves are quantum
+processes whose description simply requires a large enough Hilbert space in Alice’s domain
+[13]. But this has not yet been demonstrated.
+And what is special about classical information? Sending an arbitrary one-qubit quantum
+state from Alice to Bob using the 2-way protocol of Sec. III.3 could be done with a Cost of 2,
+which is to say twice that of simply using a 1-way protocol from Bob to Alice. That this is
+the minimum seems likely, but has not been demonstrated. What about a two-way protocol
+with all the amplitude starting on Alice’s side, with the aim of a perfect transmission of each
+of two specified nonorthogonal states from Bob to Alice—what would be the minimum total
+Cost?
+An interesting feature of the original SLAZ protocol is the enormous ratio 2N2/M2, see
+(48), of the Costs to transmit λ = 0 and 1, in contrast to the relatively simple protocols
+discussed in Sec. III.3 for which the ratio is 1. Because the success of SLAZ depends upon N
+being much larger than M, this large ratio presumably has something to do with Bob’s not
+sending anything back through the channel when λ = 1. Might there be some interesting
+physical principles, in addition to the Zeno effect, hiding here and waiting to be explored?
+In conclusion it is hoped that the thinking and tools employed in this paper will be useful
+for studying other problems of quantum information at intermediate times in situations
+where the careless use of ill-defined probabilities generates paradoxes rather than physical
+understanding. In particular, information transfer among three or more parties, of current
+interest in the study of quantum networks, might benefit from the sort of analysis used here.
+Acknowledgements
+The author expresses his appreciation to Carnegie-Mellon University and its Physics
+Department for continuing support of his activities as an emeritus faculty member.
+References
+[1] Hatim Salih, Zheng-Hong Li, M. Al-Amri, and M. Suhail Zubairy. Protocol for di-
+rect counterfactual quantum communication.
+Phys. Rev. Lett., 110:170502, 2013.
+arXiv:1206.2042.
+[2] Johann von Neumann. Mathematische Grundlagen der Quantenmechanik. Springer-
+Verlag, Berlin, 1932. English translation by R. T. Beyer: Mathematical Foundations of
+Quantum Mechanics, Princeton University Press, Princeton, New Jersey (1955).
+[3] Jonte R. Hance, James Ladyman, and John Rarity. How quantum is quantum counter-
+factual communication? Found. Phys., 51:12, 2021. arXiv:1909.07530.
+17
+
+[4] Zheng-Hong Li, Shang-Yue Feng, M. Al-Amri, and M. Suhail Zubairy. Direct counter-
+factual quantum communication protocol beyond a single photon source. Phys. Rev. A,
+106:032610, 2022. arXiv:2202.03935.
+[5] Robert B. Griffiths. What quantum measurements measure. Phys. Rev. A, 96:032110,
+2017. arXiv:1704.08725.
+[6] Robert B. Griffiths.
+The Consistent Histories Approach to Quantum Mechanics.
+Stanford Encyclopedia of Philosophy, 2019.
+https://plato.stanford.edu/entries/qm-
+consistent-histories/.
+[7] Zheng-Hong Li, M. Al-Amri, and M. Suhail Zubairy. Direct quantum communication
+with almost invisible photons. Phys. Rev. A, 89:052334, 2014.
+[8] Zheng-Hong Li, M. Al-Amri, and M. Suhail Zubairy. Direct counterfactual transmission
+of a quantum state. Phys. Rev. A, 92:052315, 2015.
+[9] Lev Vaidman. Tracing the past of a quantum particle. Phys. Rev. A, 89:024102, 2014.
+arXiv:1312.7566.
+[10] Hatim Salih, Zheng-Hong Li, M. Al-Amri, and M. Suhail Zubairy. Salih et al. reply.
+Phys. Rev. Lett., 112:208902, 2014. arXiv:1404.5392.
+[11] L. Vaidman.
+Counterfactuality of ‘counterfactual’ communication.
+J. Phys. A,
+48:465303, 2015. arXiv:1410.2723.
+[12] Yakir Aharonov and Lev Vaidman.
+Modification of counterfactual communication
+protocols that eliminates weak particle traces.
+Phys. Rev. A, 99:010103, 2019.
+arXiv:1805.10634.
+[13] For a consistent quantum-mechanical description of the measuring process, see [5], the
+relevant sections of [6], and Chs. 17 and 18 of [14].
+[14] Robert B. Griffiths. Consistent Quantum Theory. Cambridge University Press, Cam-
+bridge, U.K., 2002. http://quantum.phys.cmu.edu/CQT/.
+18
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf,len=795
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='01730v1  [quant-ph]  4 Jan 2023  Multitime Quantum Communication: Interesting But  Not Counterfactual  Robert B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Griffiths∗  Department of Physics  Carnegie Mellon University  Pittsburgh, PA 15213  Version of 4 January 2023  Abstract  A protocol for transmission of information between two parties introduced by Salih  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' 110 (2013) 170502 (hereafter SLAZ), involves sending quan-  tum amplitude back and forth through a quantum channel in a series of steps, rather  than simply sending a signal in one direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The authors claimed that their protocol  was “counterfactual” in the sense that while a quantum channel is needed to connect  the parties, its actual usage becomes vanishingly small in the asymptotic limit as the  number of steps tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Here we show that this claim is incorrect because it  uses probabilistic reasoning that is not valid at intermediate times in the presence of  quantum interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' When ill-defined probabilities are replaced with a well-defined  measure of channel usage here called “Cost”, equal to the absolute square of the am-  plitude sent through the channel, the total Cost does not go to zero in the asymptotic  limit of a large number of steps, but is bounded below by a rigorous inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' A  detailed analysis shows that this bound is satisfied in the SLAZ protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The analysis  leading to the bound uses the fact that the Gram matrix formed by inner products of  a collection of pure quantum states is additive over Hilbert subspaces and invariant  under unitary time transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Its off-diagonal elements, which in general are not  positive, play a significant role in the formal argument as well as providing a somewhat  strange way of visualizing the transfer of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Contents  I  Introduction  2  II  One-Way Protocols  3  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='1  Multiple Channels in Parallel .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
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+page_content='  3  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='2  One Channel Used Multiple Times .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
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+page_content='  5  ∗Electronic address: rgrif@cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='edu  1  III Two-way Protocols  6  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='1 Gram Matrices  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  11  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='2 Calculation of Costs and Overlap  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  13  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='3 Discussion of Costs and Probabilities .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
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+page_content='  14  V  Conclusion  15  I  Introduction  The motivation for this paper is a scheme for the transmission of quantum informatiom  introduced by Salih et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' al [1] with the title “Protocol for direct counterfactual quantum com-  munication”, and referred to hereafter as SLAZ, the initials of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' One ordinarily  thinks of the transmission of information as sending a signal through a channel from sender  to receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' However the idea in SLAZ is that information can be sent from Bob to Alice if  the quantum particle used to carry the information starts off in Alice’s domain, and a part  of its quantum amplitude is sent to Bob through a quantum channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Bob modifies this is  some way before sending (or possibly not sending) it back to Alice, depending on the signal  he wants to send.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Alice then employs what Bob has returned to begin a second round of  sending amplitude to Bob, who again modifies it before returning it, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' This  back-and-forth motion can continue for a large number of rounds until the information that  Bob is sending has arrived in Alice’s domain, where she can carry out a measurement or  perhaps perform additional processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' A key feature of protocols of this type is that all the  intermediate steps can be represented by purely unitary time evolution, with intermediate  time measurements, if any replaced by unitaries—a process of purification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The use of amplitude rather than particle in the previous paragraph is intentional, be-  cause the state of the photon or other particle is in general a coherent superposition of parts  associated with different spatial locations: Alice’s domain, Bob’s domain, and the channel  connecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' One generally thinks of a particle as something with a spatial location,  but in quantum mechanics one cannot simultaneously ascribe particle and wave properties  to the same entity at the same time because of wave-particle duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In Hilbert-space quan-  tum mechanics physical properties, such as location in space, are represented by projectors  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='5 of [2]), and when a projector representing a wave, think of |ψ⟩⟨ψ|, does not com-  mute with a projector specifying a spatial location, ignoring this fact can rapidly lead to  paradoxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The double-slit paradox is an example: when a coherent wave passes through  the slit system one cannot say through which slit the particle passed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The term “counterfactual” in the original SLAZ paper has the following significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' A  quantum channel connecting the communicating parties is essential: this is not a case of  mysterious nonlocal influences of the sort which are sometimes invoked to explain quantum  violations of Bell inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  However, if the number of steps in an SLAZ protocol is  2  sufficiently large, the magnitude of the amplitude sent through the channel in each step can  be made very small, and vanishes in the limit as the number of steps tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  A similar claim of counterfactuality has been made in much of the rather substantial  literature motivated by the original SLAZ publication, which contains various modifications  and extensions of the original protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' There have also been criticisms of these counter-  factual claims, and (of course) replies to criticisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The Conclusion, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' V, of the present  paper contains a few remarks about how its results apply to some of these publications, but  a review, much less a detailed discussion, lies far outside its scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The interested reader is  referred to the extensive bibliographies found in [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The aim of the present paper is to study the use of quantum channels in protocols of the  SLAZ type, in particular the sense in which this usage is or is not counterfactual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' To this end  the concept of the Cost of using a quantum channel, the absolute square of the amplitude  passing through it in a particular step in the protocol, is suggested, for reasons discussed  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' II, as a useful substitute for “probabiilities”, which in a quantum context are often  ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The example of multiple channels in parallel, which few would want to claim are  “counterfactual”, serves as an introduction to how information can be sent through a single  channel in a single direction at multiple times, in a process in which all of the intermediate  steps are represented by unitary maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The main mathematical results of this paper are in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III: Gram matrices and some of  their properties are discssed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='1, while Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='2 gives the basic structure of simple  two-way multiple time protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='3 considers simple schemes for transmitting one  classical bit, while the rigorous lower bound that undermines various counterfactual claims  is the topic of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The original SLAZ protocol is studied in detail in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In particular the total Cost  of transmitting a classical bit λ = 0, in which Bob reflects the amplitude back to Alice, and  for transmitting λ = 1, in which he absorbs rather than returns it, are evaluated explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  It turns out that in the asymptotic limit the λ = 1 Cost is miniscule, but that for λ = 0  is enormous, while the product of the two remains finite and satisfies the rigourous bound  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The mistaken claim that the SLAZ protocol is counterfactual results from two  errors: a concept of channel use which would be questionable even for a classical stochastic  process, and an improper use of probabilities in a way that violates quantum principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The concluding Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' V has a summary of the main results of this paper, a few comments  on some parts of the literature related to SLAZ, and some suggestions for future directions  of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  This author believes that protocols of the SLAZ type are quite interesting,  deserve further exploration, and might contribute to useful ways of studying multipartite  and multitime transmission of quantum information, as in quantum networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' And that such  studies would prove more fruitful in the absence of misleading claims of counterfactuality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  II  One-Way Protocols  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='1  Multiple Channels in Parallel  Think of quantum information as the information carried by a photon as it passes through  a quantum channel, such as an optical fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The information could be encoded in its  polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Rather than using a single channel, one could imagine sending the photon as a  3  coherent state through a set of N channels in parallel, using a collection of beamsplitters to  divide up the initial amplitude among the different channels, and a corresponding collection  to later recombine them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Let us suppose that the normalized |Φ⟩ that represents the photon  at some intermediate time is a superposition of amplitudes  |Φ⟩ =  N  �  n=1  cn|φn⟩  (1)  associated with the individual channels, labeled by n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Define the Cost qn associated with  the use of channel n, and the total Cost Q for the channel system as:  qn := |cn|2,  Q :=  N  �  n=1  qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (2)  If the |φn⟩ and |Φ⟩ are normalized, Q is equal to 1, so one might identify qn with the prob-  ability that the photon is in channel n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' But what does that mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In standard (textbook)  quantum mechanics probability refers to the outcome of a measurement, but a measurement  carried out at an intermediate time, when the quantum state is a coherent superposition  over various locations, can alter what occurs later, and hence it is dangerous to associate  such a probability with a situation in which a measurement does not take place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Another way of viewing this difficulty is to recall that von Neumann (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='5 of [2])  identified quantum physical properties—which in classical physics are asociated with sets  of points in the classical phase space—with projectors, self-adjoint idempotent operators,  P = P † = P 2, on the quantum Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' For example, in the case of a spin-half particle  the projectors  P = (I − σz)/2  R = (I + σx)/2,  (3)  where I is the identity and σz and σx are Pauli operators, represent the properties Sz =  −¯h/2 and Sx = +¯h/2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In general, if two projectors P and R commute their  product PR = RP represents the property P AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' But if they do not commute, neither  PR nor RP is a projector, and so neither represents a quantum property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In some sense  noncommutation is the very essence of quantum mechanics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' it is what distinguishes it from  classical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The use of standard (Kolmogorov) probabilities requires a sample space of  mutually-exclusive possibilities, one and only one of which occurs in a particular run of an  experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In quantum theory such a sample space is a collection of mutually orthogonal  projectors that sum to the identity, a projective decomposition of the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' For example, R  and I −R in (3) in the case of spin half;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' see (7) below for the general definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In quantum  mechanics there are often many possible sample spaces that one might be interested in,  and carelessly combining incompatible spaces—some projectors in one do not commute with  projectors in the other—inevitably leads to paradoxes rather than physical understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  In the present context the dyad |Φ⟩⟨Φ| is a projector that does not commute with any  of the projectos |φn⟩⟨φn| for which cn is nonzero, and thus it is meaningless to assign a  probability to the latter in a situation where the coherent state |Φ⟩ will later be transformed  by the final beamsplitters into the original state that entered the channel system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  4  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='2  One Channel Used Multiple Times  The possible advantanges, if any, of using many channels in parallel can also be realized  by employing a single channel and sending quantum amplitude through it at a succession of  times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' this is what makes protocols of the SLAZ type of some interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Let us suppose that  information is being sent from Bob to Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' One can think of the photon at a particular time  as being in a coherent superposition of amplitudes in three different physical locations: Alice’s  domain A, Bob’s domain B, and the channel C connecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The same symbols can be  used for the Hilbert-space projectors associated with these locations, thus operators which are  self-adjoint and idempotent, A = A† = A2, and mutually orthogonal, AB = BC = AC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  They sum to the identity  A + C + B = I  (4)  and hence form a projective decomposition of the identity—see the general definition in  (7) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' A projective decomposition of the identity is the quantum counterpart of the  sample space of mutually exclusive possibilities essential for using standard (Kolmogorov)  probability theory in the case of a quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Note that A, B, and C are subspaces of a  single Hilbert space, not subsystems represented by a tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' If the quantum particle  possesses other degrees of freedom, these projectors are to be understood using the usual  convention as including the identity operator on these additional degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Thus  for a photon, A means that it is located in Alice’s domain, whatever may be its polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Bob can send a particular type of information λ to Alice by starting with a normalized  reference state |ψ0⟩ = B|ψ0⟩, the particle is somewhere in his domain B, and using a unitary  transformation Bλ acting on the subspace B + C to place it in a state  |ψλ  1⟩ = C|ψλ  1⟩ = Bλ|ψ0⟩,  (5)  in the channel, at which point it travels through the channel to Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' As the channel has no  effect except to transmit the particle from one end to the other, we simplify the discussion  (here and later) by using the same symbol for the ket that arrives at Alice’s end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' She then  applies a unitary A that does not depend on λ, for she does not know what Bob is sending,  to empty the channel and arrive at a final state  |ψλ  2⟩ = A|ψλ  2⟩ = A|ψλ  1⟩,  (6)  which she can then measure or subject to further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  This single-round transmission process can be carried out in a number of rounds in which  during the n’th round Bob employs a unitary Bλ  n acting on the B + C subspace to map an  amplitude cn|ψ0⟩ into C, which is initially empty, and which travels to Alice, who uses a  unitary An acting on A + C to remove it from the channel, which is then empty and ready  for the next round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' One way to visualize this is that Bob has a domain B of high dimension,  and at the outset splits up the initial amplitude |ψ0⟩ into pieces placed in different subspaces  of B with the help of a suitable set of beamsplitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' At round n the unitary Bλ  n interchanges  the appropriate subspace of B with the empty C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Alice’s A is also large, and her An maps  whatever Bob has sent into an empty subspace reserved for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' When the run is  completed Alice can then combine the amplitudes in these different subspaces into a smaller  space—e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=', using beamsplitters—or she can do a similar combination at the end of each  5  round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Of course Alice’s and Bob’s unitaries cannot be chosen independently;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' the two must  work together to design the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' What is unknown to Alice is Bob’s choice of λ for a  particular run;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' this is the information that she can extract at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Some multiple-time protocols employ measurements by Alice at intermediate times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In  cases such as the original SLAZ scheme, discussed below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' IV, it is possible to store the  amplitude that could have triggered the measuring device in an empty subspace in Alice’s  domain and put off the measurement until the protocol is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Of course, amplitudes  that correspond to several measurements in succession can be combined, just as in the case  of simultaneous transmission through several channels in parallel, as discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  III  Two-way Protocols  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='1  Gram Matrices  Let {Pj} be a projective decomposition of the Hilbert space identity I:  I =  �  j  Pj,  Pj = P †  j ,  PjPk = δjkPj,  (7)  and let {|ψµ⟩}, µ = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=', be a collection of kets on the same Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The Gram  matrix  Gµν = ⟨ψµ|ψν⟩ =  �  j  Gµν(Pj) =  �  j  ⟨ψµ|Pj|ψν⟩  (8)  is additive in that it is a sum over contributions from the different subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In addition,  Gµν is invariant (or conserved) under a unitary operation U that acts on every ket in the  collection {|ψµ⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Also, if this unitary acts on only some of the subspaces, say P1 and P2, and  is the identity operator on the others, then while both Gµν(P1) and Gµν(P2) may change,  their sum Gµν(P1) + Gµν(P2) remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' That Gram matrices are additive and  conserved plays an important role in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  We shall refer to the diagonal elements Gµµ(Pj), which are non-negative, as weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  As these are rather like probabilities, their additivity and conservation is not surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  However, that the same is true of the nondiagonal elements Gµν(Pj) with µ ̸= ν, hereafter  referred to as overlaps, comes as something of a surprise, especially since |ψµ⟩ and |ψν⟩ may  refer to two different runs of an experiment, one on Friday and one on Monday.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Nonetheless,  overlaps play a key role in the following analysis, not only as part of the mathematics but  also in a surprising but useful “intuitive” way of thinking about what is going on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The  absolute value of an overlap corresponds to a notion of fidelity in quantum information, but  in general an overlap is a complex number, and the fact that it can be negative as well as  positive is a key element in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='2  Basic Two-Way Protocol  In the following discussion the projective decomposition of the identity (7) that will  concern us is {A, C, B}, where A means that the photon or other quantum particle is in  Alice’s domain, B that it is Bob’s domain, and C in the channel connecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' At the  6  beginning of a two-way protocol of the SLAZ type in which Bob is sending information to  Alice of the photon amplitude is in Alice’s domain A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' She initiates the run by sending some  amplitude to Bob through the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' He then modifies it and returns some or all of it to  Alice, in a manner that depends on the information λ he wishes to transmit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Alice processes  what Bob has returned, and begins the second round by again sending amplitude to Bob,  who again returns it, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' This can go on for N rounds, following which Alice makes a  measurement to determine the value of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  In further detail: At the beginning of round n, Alice uses a unitary An1 acting on A + C  to map some of the amplitude in A into an empty channel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' This amplitude then flows  through the channel to Bob, where he empties the channel into B, does some processing,  and then maps some amplitude back into C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' This flows to Alice, who empties C into A  using a unitary An2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' We assume that “flow through the channel” does not change anything,  and hence it is convenient not to think of C as divided into close-to-Alice, close-to-Bob, and  in-between subspaces, but simply imagine that Alice and then Bob and then Alice are acting  on a single C subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Alice uses unitaries that act on A + C and are independent of λ,  while Bob uses unitaries Bλ  n, that depend on the information λ he wants to transmit, which  act on C + B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Both the Alice and Bob unitaries will in general depend upon the round n,  but Alice’s do not depend upon λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In addition we impose the restriction that Bob’s actions  are passive in the sense that that the magnitude of the amplitude he sends back to Alice  in round n cannot be greater than what he has just received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' This last condition clearly  differentiates these two-way protocols from the one-way protocols of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The requirement that Alice and Bob only employ unitary operations simplifies the anal-  ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' It is true that various published protocols of this type, including the original SLAZ  version to be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' IV, employ nonunitary measurements at intermediate times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  In the cases of interest to us these measurements can be replaced by unitary operations  which allow the measurements to be put off until the end of the run, in a manner indicated  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='2 and employed in the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  To quantify the channel usage for these protocols we use the notions of Cost, equal to the  absolute square of the amplitude for a single use of the channel, and total Cost for the sum of  the Costs involved in a single experimental run, as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='1, see (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' An important issue  connected to claims that these protocols are counterfactual has to do with the difference  between Cost and probability, as will be discussed later for the SLAZ protocol in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' IV—  the importance of this has already been noted in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In particular we will be interested  in identifying protocols that minimize the overall Cost, as in the example discussed next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='3  Sending One Classical Bit  In the simplest SLAZ protocol Bob wants to send a single classical bit, λ = 0 or 1, to  Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' At the start all of the amplitude is in A for both a λ = 0 and a λ = 1 run, so all four  of the initial Gram matrix elements Gµν  0 (A), µ and ν equal to 0 or 1, are equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The  goal is that after N rounds the result will be  Gµν  N (A) = δµν,  (9)  so that Alice can determine the value of λ Bob has sent by making a measurement in an  appropriate basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Thus the desired change is that during the course of the run the overlaps,  7  the off diagonal elements G01(A) and G10(A), decrease from 1 to 0, while the weights G00(A)  and G11(A), remain equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  At this point it is worth noting that if both weights are not maintained—for example if  at the end G00(A) = 1 while G11(A) = G01(A) = 0, Alice can still extract the value of λ by  measuring whether or not the photon is in the state |ψ0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Let us call this, for want of a better  term, a partial protocol in contrast to a full protocol that results in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' A partial protocol  can be used for one-way transmission, and the obvious advantage is that it costs nothing to  transmit λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' A possible disadvantage is that when Alice’s measurement reveals nothing  it could be because of some failure in the channel or in the measuring device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In the present  discussion we focus on full protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  A very simple way to implement such a protocol is that on the very first step Alice sends  the entire amplitude to Bob, with a Cost of 1 for this use of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Bob then simply  modifies this using the unitary Bλ and sends it back to Alice, either in one round or several  rounds, with Alice sending nothing back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The Cost for using the channel in the Bob-to-Alice  direction is also 1, see the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Hence a total Cost of 2 for the protocol  as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Notice that since there is no restriction on Bλ this rather trivial protocol can be  used to send “quantum” information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' From the perspective of Cost, two-way protocols of  the kind under discussion are interesting because a classical bit, λ = 0 or 1, can be sent at  a total Cost of 1 rather than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' And as shown below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='4, the product of the Costs  for λ = 0 and 1 cannot be less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  To discuss the successive steps in protocols that optimize the Cost, we need an appropriate  notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' We will represent kets as row vectors as in the following example  |ψ⟩ = |a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' b⟩ = |a1, a2, a3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' c1, c2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' b1, b2⟩  (10)  where the dimensions of the A, B, and C subspaces are d(A) = 3, d(B) = 2 and d(C) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Note that we are dealing with a direct sum of subspaces, A ⊕ B ⊕ C, not a tensor product  of subsystems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In much of what follows, B is empty or can be ignored, so |a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' c⟩ will suffice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  this and other minor variants in notation should be self-explanatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Let us start with an extremely simple one-round full protocol with d(A) = 2, d(C) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  It consists of the following steps:  |a1, a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' c⟩ = |1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' 0⟩ → |1/  √  2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' 1/  √  2⟩  ⇒ |1/  √  2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' (−1)λ/  √  2⟩ → |1/  √  2, (−1)λ/  √  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' 0⟩,  (11)  where 0 means this amplitude is equal to zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' do not confuse it with the label 0 for one of  the two orthogonal states of a qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Here → indicates the action of a unitary on A + C  carried out by Alice, and ⇒ a λ-dependent unitary on C carried out by Bob.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The action by  Bob could involve intermediate steps requiring the B subspace, but its net effect is only to  change the contents of C, so there is no need to include B in the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  In words: At the outset all of the amplitude is in Alice’s A, a1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' She maps half (in  the sense of the absolute square) of it into C and sends it to Bob, who either sends it back  unchanged in order to transmit λ = 0, or with the opposite phase to send λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Alice then  empties the channel into the a2 position, using a unitary on A + C that is independent of λ,  as it simply requires interchanging two subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' A final measurement by Alice determines  which of the two orthogonal states is present in A, and thus which bit Bob was sending.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  8  Next consider what is happening to the Gram matrices Gµν(A) and Gµν(C) during the  successive steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In particular, the overlap G01(A) is equal to 1 at the outset, and the first  step reduces it to 1/2 by placing 1/2 in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Bob’s action changes G01(C) from +1/2 to −1/2,  and this negative contribution to the overlap moves back into A when Alice empties the  channel, leading to the desired G01(A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' On the other hand, whereas the weight G00(A)  is reduced to 1/2 during the first step, Bob’s action does not change the sign of G00(C), so in  the final step Alice moves this weight back to its initial value of 1, and similarly for G11(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Thus the goals of a full protocol have been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The Costs of using the channel are easily evaluated: 1/2 for the Alice-to-Bob step and  the same for Bob-to-Alice, for a total Cost of Qλ = 1, the same for λ = 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' These  satisfy the rigorous lower bound worked out below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='4, so this protocol is optimal  if one uses total Cost as an appropriate measure of channel usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  This protocol is easily extended to an equally efficient version involving N rounds, N any  positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Let  ǫ = 1/2N,  (12)  and for the first, n = 1, round replace (11) with  |1, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' 0⟩ → |  √  1 − ǫ, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' √ǫ ⟩ ⇒ |  √  1 − ǫ, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' (−1)λ√ǫ ⟩ → |  √  1 − ǫ, (−1)λ√ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' 0⟩,  (13)  while for round n + 1,  |  √  1 − nǫ, (−1)λ√nǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' 0⟩ → |  �  1 − (n + 1)ǫ, (−1)λ√nǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' √ǫ⟩  ⇒ |  �  1 − (n + 1)ǫ, (−1)λ√nǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' (−1)λ√ǫ ⟩ → |  �  1 − (n + 1)ǫ, (−1)λ�  (n + 1)ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' 0⟩,  (14)  where it is straightforward to show that there exists a λ-independent unitary for the last  step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The final result at the end of round N is the same as in (11), the case in which N = 1,  and again the total Cost is Q0 = Q1 = 1, independent of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' One can also let ǫ depend on n,  thus ǫn > 0 for round n, subject to the condition  �  n  ǫn = 1/2,  (15)  and the Cost is again equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  There are other protocols with larger Costs which may have some practical advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Thus rather than a scalar amplitude, Alice might use photon polarization, say horizontal  H, which Bob could return as H to send λ = 0 or rotate to vertical V to send λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  In this case the Costs are Q0 = Q1 = 2, so twice that for an optimal one-way protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  However, there is now no need to maintain a particular phase relation between what is in  Alice’s domain and what is available to Bob during each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' If polarization is easier to  maintain than phase—one leaves that up to the experts—one could imagine the added Cost  being worthwhile if Alice has a large apparatus capable of generating single photons, while  Bob, off on a trip to spy on Eve, needs only something easily carried in a suitcase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The protocol used in SLAZ, in which Bob returns the amplitude for λ = 0, but absorbs  it or feeds it to a measuring apparatus for λ = 1, looks less promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Because the λ = 1  weight only moves from Alice to Bob it is difficult to have G11(A) = 1 at the end of the  protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In fact SLAZ, discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' IV, employs a clever trick (“Zeno effect”) to get  around this problem, albeit at the cost of a large number of rounds to keep the probability  of failure small, and a large channel usage Cost for one of the bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  9  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='4  Lower Bound on Costs  The additivity and conservation properties of the Gram matrix Gµν introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='1  will now be used to obtain lower bounds on the total Cost of two-way protocols of the sort  exemplified by, but not limited to, the case of 1 classical bit discussed above in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Using the |a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' c⟩ notation of (10)—the b entry is not needed in the following discussion—round  n of an N round protocol consists of the following steps carried out on A + C:  |aµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' 0⟩n → |¯aµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' cµ⟩n ⇒ |¯aµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' ˆcµ⟩n → |aµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' 0⟩n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (16)  Here µ labels the bit which Bob is transmitting during this run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Thus after Alice uses a  unitary An1 on A + C to move some amplitude, |cµ⟩n into an initially empty channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Bob  applies a unitary Bµ  n to C + B, leading to an amplitude |ˆcµ⟩n—note the hat added to c—in  the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' If Bob’s action is passive, as assumed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='3 (and in the later discussion  of SLAZ in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' IV), one would have  ∥ˆcµ∥n ≤ ∥cµ∥n,  (17)  but this conditions is actually not needed to obtain the general results and inequalities given  below, which thus apply equally to one-way multi-time transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' As a final step Alice  employs a unitary An2 on A + C to empty the channel by placing its amplitude into A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' It is  important that Alice’s unitaries An1 and An2, unlike Bob’s Bµ  n, do not depend upon µ, which  can be different in different runs of the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The change in the Gram matrix associated with A during round n is given by  Gµν  n+1(A) − Gµν  n (A) = ⟨aµ|aν⟩n+1 − ⟨aµ|aν⟩n = ⟨ˆcµ|ˆcν⟩n − ⟨cµ|cν⟩n,  (18)  where ⟨aµ|aν⟩n is the inner product of |aµ⟩n and |aν⟩n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The equality follows from the fact that  Gµν(A + C) is invariant under An1 and An2, and additive: Gµν(A + C) = Gµν(A) + Gµν(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  To discuss the total change during N rounds, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' N, it is convenient to define  |Cµ⟩ := {|cµ⟩1, |cµ⟩2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' |cµ⟩N},  | ˆCµ⟩ := {|ˆcµ⟩1, |ˆcµ⟩2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' |ˆcµ⟩N}  (19)  with inner products  ⟨Cµ|Cν⟩ =  N  �  n=1  ⟨cµ|cν⟩n,  ⟨ ˆCµ| ˆCν⟩ =  N  �  n=1  ⟨ˆcµ|ˆcν⟩n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (20)  Summing (18) over N rounds yields the following formula  ∆Gµν(A) = Gµν  N (A) − Gµν  0 (A) = ⟨ ˆCµ| ˆCν⟩ − ⟨Cµ|Cν⟩,  (21)  for the total change in the A portion of the Gram matrix during the full protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' This  quantity is bounded by  |∆Gµν(A)| ≤ |⟨ ˆCµ| ˆCν⟩| + |⟨Cµ|Cν⟩| ≤ ∥ ˆCµ∥ · ∥ ˆCν∥ + ∥Cµ∥ · ∥Cν∥  (22)  using the norm ⟨Cµ|Cµ⟩ = ∥Cµ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  10  Next define the total Cost Kµ for Alice-to-Bob and ˆKµ for Bob-to-Alice uses of the  channel, with Qµ their sum:  Kµ = ⟨Cµ|Cµ⟩ = ∥Cµ∥2,  ˆKµ = ⟨Cµ|Cµ⟩ = ∥ ˆCµ∥2,  Qµ = Kµ + ˆKµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (23)  Combining (22) and (23) gives  |∆Gµν(A)| ≤  √  KµKν +  �  ˆKµ ˆKν ≤  �  QµQν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (24)  This yields an upper bound  ∆Gµµ(A) ≤ Qµ  (25)  for a non-negative diagonal weight, and for the off-diagonal overlap:  |∆Gµν(A)| ≤  �  QµQν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (26)  In the particular case of the 1-bit two-way protocol, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='3, the aim is to reduce G01(A)  from its initial value of 1 to 0 after N rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Setting µ = 0 and ν = 1 in (26), we see that  to achieve this result it is necessarily the case that the Costs Q0 and Q1 for sending bits  λ = 0 and λ = 1 must satisfy the condition  Q0Q1 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (27)  This is satisfied as an equality with Q0 = Q1 = 1 for the specific protocols discussed in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='3, which shows that they are optimal if total Cost is used as a measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' For more  general protocols there is no reason to expect that the two Costs will be equal, and in that  case if, say, the Cost for λ = 1 is made very small, that for λ = 0 must be very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' This  is in fact the case for the original SLAZ protocol, as discussed below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' IV, which thus  provides an interesting illustration of such a tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  IV  The SLAZ Protocol  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='1  Description of the Protocol  The original SLAZ protocol differs from the simpler situation discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='3 in  two respects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' First, it has a hierarchical structure: there are a large number M of outer  rounds or cycles, each of which consists of a large number N of inner rounds or cycles, and  the protocol will succeed with high probability provided  1 ≪ M ≪ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (28)  Second, while Bob sends a bit λ = 0 by reflecting the amplitude sent by Alice back into  the channel, for λ = 1 he simply empties the channel, which can be described as a unitary  operation in which the C amplitude is placed in Bob’s subspace B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In addition, the original  SLAZ protocol and some of its modifications involve measurements at intermediate times,  and these will be replaced in the discussion below by unitary operations in the manner  suggested at the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  11  We use a notation  |ψ⟩ = |a1, a2, a3, a4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' b⟩  (29)  of the form introduced in (10), where the aj are scalar amplitudes in Alice’s domain A =  A1+A2+A3+A4, c is the amplitude the channel C, and b is in Bob’s domain B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Here capital  letters are used to denote subspaces and the corresponding projectors, while lower case letters  indicate (in general complex) scalar amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' While A4 and B are one-dimensional, one  can also make these larger spaces for reasons that will appear during the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' An  abbreviated notation is often convenient: |a2, a3⟩ in the case of a unitary acting on A2 + A3  while all the other amplitudes remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Central to the discussion are unitary operators that represent a rotation by an angle θ  on a 2-dimensional space:  R(θ)|α, β⟩ = |α cos θ − β sin θ, α sin θ + β cos θ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (30)  In particular, RM and RN, defined in terms of small angles, play a central role:  RM := R(θM),  θM := π/(2M),  RN := R(θN),  θN := π/(2N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (31)  Note in particular that  (RM)M = (RN)N = R(π/2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  R(π/2) |α, β⟩ = | − β, α⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (32)  In view of the fact that θN is a small angle, the following approximations turn out to be  userful:  cos θN ≈ exp[−θ2  N/2] = exp[−π2/(8N2)] ≈ 1 − π2/(8N2),  (cos θN)N ≈ exp[−π2/(8N)] ≈ 1 − π2/(8N) ≈ 1,  (33)  and similarly if N is replaced by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  These approximations are useful for understanding the overall structure of the protocol,  which is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' At the beginning of outer round m, 1 ≤ m ≤ M, RM is applied to  A1 + A2 to yield,  |a1, a2⟩λ = RM|¯a1, ¯a2⟩λ,  (34)  where ¯a1 and ¯a2 are the values of these amplitudes at the end of the previous outer round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In  general they depend upon which bit λ = 0 or 1 is being transmitted, whence the superscript  label, even though Alice’s operations do not depend upon λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The very first outer round  m = 1 begins by applying (34) to the starting state (29) with a1 = 1 and all the other  amplitudes equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The initial step (34) of outer round m is followed by a sequance of N inner rounds, each  involving the following steps, here displayed using the type of notation employed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='3,  but now with reference to the subspace A2 + A3 + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  |a2, a3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' c = 0⟩ → |a′  2, a′  3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' c = 0⟩ → |a′  2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' a′  3⟩  ⇒ |a′  2, 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' (1 − λ)a′  3⟩ → |a′  2, (1 − λ)a′  3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' 0⟩,  (35)  where  |a′  2, a′  3⟩ = RN|a2, a3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (36)  12  In words, Alice applies the unitary rotation RN, (31), to A2 +A3, and then maps A3 into the  empty channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Next comes Bob’s action, indicated by ⇒, to either reflect the amplitude a′  3  back into C if he is sending λ = 0, or shift it into his domain B, leaving the channel empty  if sending λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Alice, who does not know the value of λ, maps whatever is in the channel  back into A3 by a unitary that simply exchanges the contents of A3 and C, and then begins  the next inner round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The result of N inner rounds in succession is  |a2, a3⟩ →  �  |0, a2⟩ for λ = 0,  |(cos θN)Na2, 0⟩ ≈ |a2, 0⟩ for λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (37)  where the λ = 1 approximation is justified when N is very large, see (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Following the N inner rounds Alice completes this outer round by applying a unitary to  A3 + A4 that empties the contents of A3 into A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' For λ = 1, a3 = 0, (37), so this emptying  step is trivial, while for λ = 0 it is nontrivial, and plays a signficant role in understanding  the true Costs of the protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In the original SLAZ protocol this emptying step is replaced  by a measurement, but instead of a measurement one can just as well let the amplitudes  accumulate in A4, which is the perspective used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  At the end of the protocol after  completing M outer rounds the final result is  λ = 0 : |a1 = 1 − r1, a2 = 0, a3 = 0, a4 = r4, c = 0, b = 0⟩  λ = 1 : |a1 = s1, a2 = 1 − s2, a3 = 0, a4 = 0, c = 0, b = sb⟩,  (38)  where the quantities denoted by rj and sk are small corrections, of order 1/M or M/N  If these are ignored, all the amplitude is in A1 for λ = 0 or A2 for λ = 1, and a simple  measurement allows Alice to determine which bit Bob sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='2  Calculation of Costs and Overlap  It is fairly straightforward to work out the Costs for the SLAZ protocol using approxi-  mations justified by 1 ≪ M ≪ N, and the results are summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' We  begin with the case λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' If one ignores small quantities, the nonzero components of |ψ⟩m  at the beginning and at the end of outer round m are  a1 = cos(mθM),  a2 = sin(mθM),  (39)  and since MθM = π/2, at the end of outer round M the result is the λ = 1 line in (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The probability that the photon arrives in B during outer round m—the probability that  Bob will detect it if he uses a measuring device—is the sum of the absolute squares of the  amplitudes in the channel C in the N inner rounds, as this is an incoherent process:  N(sin(mθM))2(sin(θN))2 ≈ (π2/4)(sin(mθ/M))2/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (40)  Summing over m gives the total probability  K1 = Q1 = (π2/8)(M/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (41)  that the photon will end up in Bob’s domain by the end of the protocol, which is the same  as the total Cost for λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  13  In the case λ = 0, any amplitude placed by Alice in C is immediately returned by Bob,  and at the end of each outer round is emptied into a4, so that at the end of outer round m  the state is  |ψ⟩m = |a1 = (cos θM)m, a2 = 0, a3 = 0, a4, c = 0, b = 0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (42)  For m = M this is (38) with r1 = (π2/8M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Thus at the end of the protocol a2, a3, c and  b are strictly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The Cost associated with inner round n—note that the channel is used  twice—is  2[sin θM · sin(nπ/2N)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (43)  Summing over n gives a total of (π2/4)(N/M2) for each outer round, and hence for M outer  rounds a total Cost of  Q0 = (π2/4)(N/M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (44)  To compute the total change in overlap ∆G01(A), note that since for λ = 1 Bob does not  return an amplitude, only the ⟨Cµ|Cν⟩ term in (21) contributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The contribution for inner  round n of outer round m is the product of the factors  [sin θM sin(nθN)] · [sin(mθM) sin θN]  (45)  corresponding to λ = 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Summing them yields  (sin θM sin θN)  M,N  �  m,n  sin(mθM) sin(nθN) = (π2/4MN)(4MN/π2) = 1,  (46)  and hence  ∆G01(A) = −1,  (47)  as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='3  Discussion of Costs and Probabilities  To summarize the results of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='2: The total Costs Q0 and Q1 for λ = 0 and 1 are:  Q0 = (π2/4)(N/M),  Q1 = (π2/8)(M/N),  Q0Q1 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Q0/Q1 = 2N2/M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  (48)  Given that M ≪ N, Q1 is miniscule, Q0 is enormous, while their product is of order 1, and  satisfies the rigorous bound (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The case λ = 1 is the easiest to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Since Bob  does not return the amplitude put into the channel by Alice, the Bob-to-Alice Cost ˆK1 is  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The Alice-to-Bob Cost is |sb|2 in (38), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=', the probability that at the very end the  photon is in Bob’s domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The physical reason for this is that the process by which the  amplitude gets there is incoherent, no quantum interference, since no amplitude goes back  through the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Bob could either accumulate these amplitudes until the end of the  protocol and then measure to see if the photon is in B, or carry out a measurement at the  end of each inner round;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' in either case the probabilility of his detecting the photon is |sb|2 in  (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The situation is analogous to the use of intermediate time measurements in a one-way  protocol as discussed at the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The enormous Cost Q0 for λ = 0 comes about because Bob repeatedly returns the  amplitude sent by Alice in a coherent process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' While the amplitude bouncing back and  14  forth through the channel is relatively small, of order 1/M, multiplying its absolute square  by 2N, the number of times this amplitude is is in the channel during each outer round,  leads to a Cost of order N/M2 for each outer round, and hence a total of order N/M for the  complete process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Clearly the large value of Q0 means the claim that protocol is counterfactual cannot  be maintained if Cost is used as a criterion for channel use, so it is worth discussing how  the authors of SLAZ reached a different conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In essence their reasoning was based  on the small value of the amplitude in A3 at the end of an outer round just before it is  transferred to A4, as per the discussion in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The absolute square of this amplitude  is the probability that the corresponding detector D3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' 2(b) in the SLAZ paper will  be triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' This amplitude was earlier oscillating back and forth inside the subspace with  projector S = A2+A3+C, and hence it is reasonable to assume that if this detector triggers,  the photon was earlier in S during all N inner rounds making up this particular outer round1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  As this probability is of order 1/M2, the probability that one of the D3 detectors triggers  during the M outer rounds that make up a given run is of order 1/M, and hence small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  There are two serious objections to using this small probability to justify the claim that  the protocol is counterfactual: one classical and the other quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Let us start with the  former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' During a particular outer round the photon amplitude in a λ = 0 run rattles back  and forth inside S a total of N times, and in particular it is in C a total of 2N times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Consider a stochastic classical protocol for transmitting information in which most of the  time Alice and Bob exchange no information at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' However, with a small probability ǫ  Alice sends a little white ball into the channel leading to Bob, who colors it green or red  and sends it back to Alice to convey one bit of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' She records the color, paints  the ball white, and returns it to Bob who again colors it to send a second bit, and so forth,  for a total of N rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The average rate of transmitting information is Nǫ bits, and one  cannot simply throw away the factor of N and claim that this protocol is in some sense  ‘counterfactual’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The quantum difficulty has to do with what can be inferred from the probability that  the photon was in S = A2 + A3 + C during the inner rounds that make up a particular outer  round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' One may be tempted to use classical reasoning and assume that the probabilities of  being in each of the mutually exclusive regions, A2, A3, and C, that combine to make up S  are well-defined and sum to the probability of being in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' But in the presence of quantum  interference this sort of reasoning is invalid and leads to paradoxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' See the discussion of  parallel channels in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  V  Conclusion  The original SLAZ proposal has motivated a large number of papers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' see the extensive  bibliographies in [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Merely trying to summarize them, much less provide a detailed re-  view, lies outside the scope of the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Broadly speaking, this literature consists  of modifications, extensions, or improvements of the original SLAZ scheme;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' along with crit-  icisms of the claim that these protocols are counterfactual and replies to such criticisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' It  is hoped that the following rather brief comments will provide some orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  1This assumption can be justified using Consistent Histories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' see the discussion of measurements in [5,6]  15  Significant extensions of the original SLAZ scheme by the last three members of the  original collaboration include: the use of a phase change rather than absorption to transmit  the λ = 1 bit [7];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' a scheme to transmit quantum states by multiple iterations of the original  SLAZ scheme [8];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' using many photons in place of a single photon to transmit a classical  bit [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' These and others are certainly interesting ideas from the perspective of transmitting  quantum information, and worth further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  On the other hand, in these and all other extensions or modifications of SLAZ this author  has examined, the claim that the protocol is “counterfactual,” in the sense that the total use  of a quantum channel is negligible in the asymptotic limit, is subject to the same objections  discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='3: An improper use of probabilistic reasoning in a situation where  quantum interference means probabilities cannot be defined, and where even in a classical  situation Cost would be better than probability as a measure of channel usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The total  Cost remains finite in the asymptotic limit of a very large number of steps, which means that  counterfactual claims should be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Doing so will aid, not hinder, the serious study of  these interesting quantum schemes for transmitting information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Shortly after the original SLAZ publication, Vaidman published a Comment [9] claiming  that in the λ = 0 case in which Bob reflects the amplitude rather than absorbing it, the  photon which was later (with high probability) detected by Alice must at an earlier time have  been in the channel C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In their Reply [10] the SLAZ authors pointed out this way of reasoning  about events at an intermediate time in the presence of quantum interference was invalid,  and leads to paradoxes, a position supported by the analysis in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='2 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' However,  they then repeated their original counterfactual claim which itself is based on a defective  understanding of probabilities at an intermediate time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' A later and much more extended  criticism of counterfactuality claims by Vaidman [11] suffers from the same difficulty as his  earlier Comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Some years later Aharonov and Vaidman [12] claimed to have found a scheme of the  general SLAZ type which is genuinely counterfactual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  However, when measurements or  absortion of a photon at intermediate times are replaced by unitary processes—mapping  amplitude into an empty subspace reserved for this purpose, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='1—the  inequality in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='4 applies to this case and undermines the counterfactual claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' The  fundamental difficulty with such claims is that the Hilbert space projector which identifies the  position of a particle at some intermediate time does not commute with the one representing  the quantum state evolving unitarily in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  The most significant contributions of the present paper to the analysis of SLAZ-type  protocols is the use of Cost as a measure of channel usage, and the use of Gram matrices for  discussing information transfer at intermediate times in the presence of quantum interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  In particular, the fact that these Gram matrices are additive over subspaces and invariant  (“conserved”) under unitary time transformations, plays a key part in the discussions in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' A rather surprising feature is the role of off-diagonal elements, “overlaps”, as a type  of information measure which, unlike most such measures, is not in general positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' That it  can be negative plays a very signficant part in understanding its intuitive role in information  transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' That its total change on Alice’s side must be −1 during the course of a successful  protocol is confirmed for the SLAZ protocol in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  This use of Gram matrices requires that the intermediate time steps be unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In the  case of SLAZ, measurements at intermediate times can be eliminated by mapping photon  16  amplitude into empty subspaces, and this can be achieved in certain other cases, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=', the  Aharonov and Vaidman protocol [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' However, it is less clear whether something similar  could be done in a case in which, for example, Alice uses measurements at intermediate times  to change later steps in the protocol in hopes of reducing the total Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' This author believes  that such an improvement is impossible, because measurements themselves are quantum  processes whose description simply requires a large enough Hilbert space in Alice’s domain  [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' But this has not yet been demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  And what is special about classical information?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Sending an arbitrary one-qubit quantum  state from Alice to Bob using the 2-way protocol of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='3 could be done with a Cost of 2,  which is to say twice that of simply using a 1-way protocol from Bob to Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' That this is  the minimum seems likely, but has not been demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' What about a two-way protocol  with all the amplitude starting on Alice’s side, with the aim of a perfect transmission of each  of two specified nonorthogonal states from Bob to Alice—what would be the minimum total  Cost?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  An interesting feature of the original SLAZ protocol is the enormous ratio 2N2/M2, see  (48), of the Costs to transmit λ = 0 and 1, in contrast to the relatively simple protocols  discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='3 for which the ratio is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Because the success of SLAZ depends upon N  being much larger than M, this large ratio presumably has something to do with Bob’s not  sending anything back through the channel when λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Might there be some interesting  physical principles, in addition to the Zeno effect, hiding here and waiting to be explored?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  In conclusion it is hoped that the thinking and tools employed in this paper will be useful  for studying other problems of quantum information at intermediate times in situations  where the careless use of ill-defined probabilities generates paradoxes rather than physical  understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' In particular, information transfer among three or more parties, of current  interest in the study of quantum networks, might benefit from the sort of analysis used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  Acknowledgements  The author expresses his appreciation to Carnegie-Mellon University and its Physics  Department for continuing support of his activities as an emeritus faculty member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
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+page_content=' Suhail Zubairy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
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+page_content='  arXiv:1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='10634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  [13] For a consistent quantum-mechanical description of the measuring process, see [5], the  relevant sections of [6], and Chs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' 17 and 18 of [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  [14] Robert B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Griffiths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Consistent Quantum Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' Cambridge University Press, Cam-  bridge, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=', 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content=' http://quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='edu/CQT/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
+page_content='  18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtAzT4oBgHgl3EQfw_5J/content/2301.01730v1.pdf'}
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+arXiv:2301.05112v1  [gr-qc]  12 Jan 2023
+GWitchHunters: Machine Learning and citizen science
+to improve the performance of Gravitational Wave
+detector
+Massimiliano Razzanoa,b,1,∗, Francesco Di Renzoa,b, Francesco Fidecaroa,b,
+Gary Hemmingc, Stavros Katsanevasc
+aDepartment of Physics, University of Pisa, Largo B. Pontecorvo 3, Pisa, I-56127
+bINFN Section of Pisa, Largo B. Pontecorvo 3, Pisa, I-56127
+cEuropean Gravitational Observatory (EGO),Via E. Amaldi, 5, Cascina,I-56021
+Abstract
+The Gravitational waves have opened a new window on the Universe and
+paved the way to a new era of multimessenger observations of cosmic sources.
+Second-generation ground-based detectors such as Advanced LIGO and Ad-
+vanced Virgo have been extremely successful in detecting gravitational wave
+signals from coalescence of black holes and/or neutron stars. However, in
+order to reach the required sensitivities, the background noise must be inves-
+tigated and removed. In particular, transient noise events called “glitches”
+can affect data quality and mimic real astrophysical signals, and it is there-
+fore of paramount importance to characterize them and find their origin,
+a task that will support the activities of detector characterization of Virgo
+and other interferometers. Machine learning is one of the most promising
+approaches to characterize and remove noise glitches in real time, thus im-
+proving the sensitivity of interferometers. A key input to the preparation of
+a training dataset for these machine learning algorithms can originate from
+citizen science initiatives, where volunteers contribute to classify and analyze
+signals collected by detectors. We will present GWitchHunters, a new citi-
+zen science project focused on the study of gravitational wave noise, that has
+been developed within the REINFORCE project (a ”Science With And For
+Society” project funded under the EU’s H2020 program). We will present
+∗Corresponding author
+Email address: massimiliano.razzano@unipi.it (Massimiliano Razzano)
+1on behalf of the REINFORCE Consortium
+Preprint submitted to Nuclear Instruments and Methods in Physics Research AJanuary 13, 2023
+
+the project, its development and the key tasks that citizens are participating
+in, as well as its impact on the study of noise in the Advanced Virgo detector.
+Keywords:
+gravitational waves, machine learning, citizen science
+PACS: 04.20.–q, 04.30.Tv,
+2000 MSC: 83C35,
+1. Introduction
+Gravitational wave physics is opening an entire new window on the Uni-
+verse. Since their discovery in 2015 [1], the Advanced LIGO [2] and Advanced
+Virgo [3] detectors have carried on three observing runs (O1, O2, O3) and
+unveiled 90 signals produced by the coalescence of compact objects, mostly
+binary black hole with a small fraction of neutron star and/or black hole
+binaries [4].
+Advanced LIGO and Virgo are second-generation laser interferometers with
+Fabry-Perot cavities hosted in km perpendicular arms, that are capable of
+detecting the tiny deformations induced in the fabric of spacetime by the
+passage of gravitational waves. In order to improve the sensitivity of the
+detectors, there is a continuous effort to reduce the background noise due to
+local disturbances. In particular, at low frequencies the noise is dominated by
+seismic and Newtonian noise, while at mid frequencies the main component
+is related to the thermal noise and at high frequencies the noise is mostly
+related to quantum effects.
+The activity of detector characterization and
+noise hunting in gravitational wave detectors is focused on the investigation
+of stationary and non stationary noise sources. In particular, non station-
+ary transient noise events called glitches are of particular importance in the
+noise studies. In fact, glitches can affect data quality and stability and mimic
+real astrophysical signals, thus reducing the effective duty cycle of interfer-
+ometers. The classification and characterization of glitches is therefore key
+to understand the origin of noise in detector. However, glitches have com-
+plex temporal signatures, that make difficult to classify them using standard
+methods. Various works have shown that Machine Learning methods can be
+promising for the classification of glitches [5, 7]. In particular, images built
+from the time-frequency spectrograms of glitches are very effective in show-
+ing the complex morphology of glitches and can be easily given in input to
+machine learning algorithms, including deep convolutional neural networks
+[6]. A possible approach to this problem is based on supervised learning,
+2
+
+that requires large number of labeled glitch samples, that could be produced
+by dedicated citizen science initiatives, where volunteers look at images and
+clssify them.
+A successful example of this method is provided by Gravi-
+tySpy2, a citizen science project focused on the classification of glitches in
+LIGO and Virgo[8]. Here we present GWitchHunters3, a new citizen science
+project complementary to GravitySpy and aimed at improving sensitivity of
+gravitational wave detectors combining citizen science and machine learning.
+2. The REINFORCE Project
+GwitchHunters has been developed within the Research Infrastructures
+FOR Citizens in Europe (REINFORCE) project4. REINFORCE is a Re-
+search & Innovation Project, supported by the EU H2020 SWAFS “Science
+with and for Society” work programme and aimed at creating a series of
+cutting-edge citizen science projects on Frontier Physics research, with the
+goal of engaging >100,000 citizens. REINFORCE is based on four citizen
+science demonstrators focused Gravitational Waves (GWitchHunters), Astro-
+physical neutrinos (Deep Sea Explorers), High Energy Physics (New Particle
+Search at CERN) and muon-based tomography (Cosmic Muon Images). All
+demonstrators are hosted on Zooniverse [9], the world leading platform for
+citizen science projects.
+3. Overview of GWitchHunters
+GwitchHunters has been officially launched on Zooniverse in November
+2021 after a dedicated review phase and offers to citizens a set of different
+tasks of increasing difficulty. Data are presented as spectrograms and come
+from the Virgo O3 run. A Playground task is specifically devoted to learning
+the basics of glitch morphology and its classification. Three other levels offer
+(1) the possibility to classify glitches among a larger set of classes, (2) localize
+the glitches in the time-frequency space, and (3) compare the spectrogram
+in the main channel of Virgo with that produced by auxiliary sensors. This
+last task is particularly innovative, since it offer the possibility of linking the
+glitches observed in the main channel to local disturbancies in the detector,
+2http://https://gravityspy.org/
+3https://www.zooniverse.org/projects/reinforce/gwitchhunters
+4https://www.reinforceeu.eu/
+3
+
+Figure 1: Example of a GWitchHunters spectrogram showing two glitches, as well as the
+rectagle drawn by citizens to locate them
+thus suggesting a possible hint to the origin of each glitch. These tasks can
+be carried both on a personal computer and on mobile devices. The project
+also features a set of tutorials and examples to teach the volunteers how to
+perform the different tasks, as well as a ”Field Guide” containing information
+on the Advanced Virgo detector, the various glitch classes and the auxiliary
+channels used in the project.
+4. First Results and Conclusions
+Since its official launch in November 2021, ∼2800 volunteers have sub-
+scribed to the project, although another significant amount have contributed
+without officially registering. This collective effort has produced more than
+∼400000 classifications of ∼ 40000 data samples so far. In order to pro-
+mote the project and engage citizens, the REINFORCE consortium has or-
+ganized many initiatives, including workshops, press activities, online chal-
+4
+
+Virgo strain channel
+Frequency [Hz]
+Normalizedenergy
+100
+0.8
+0.6
+-0.4
+-0.2
+0:2
+0.4
+0.6
+0.8
+Time [s]lenges5 and training school6. A monitoring of the project website has been
+carried on, showing that these initiatives successfully attracted more volun-
+teers to GWitchHunters, leading to peaks of ∼5000 classifications per day.
+The results of the volunteers analysis are used for training a machine learning
+algorithm that automatically analyze the glitch data. In particular, we fo-
+cused on a 2D convolutional neural network architecture, that has been also
+tested on simulations [6, 10] reaching an accuracy greater than 99%. These
+first tests show how the GWitchHunters project could be successfully used
+to join citizen science and machine learning with the goal of contributing to
+increase the sensitivity of gravitational wave detectors.
+Acknowledgements
+REINFORCE has received funding from the European Union’s Horizon
+2020 research and innovation program, under Grant Agreement no. 872859.
+References
+[1] Abbott, B. P. et al. 2016, Physical Review Letters, 116, 061102.
+doi:10.1103/PhysRevLett.116.061102
+[2] Aasi, J. et al. 2015, Classical and Quantum Gravity, 32, 074001.
+doi:10.1088/0264-9381/32/7/074001
+[3] Acernese, F. et al. 2015, Classical and Quantum Gravity, 32, 024001.
+doi:10.1088/0264-9381/32/2/024001
+[4] Abbott, B. et al. 2021, arXiv:2111.03606
+[5] George, D., Shen, H., & Huerta, E. A. 2017, arXiv:1711.07468
+[6] Razzano, M. & Cuoco, E. 2018, Classical and Quantum Gravity, 35,
+095016. doi:10.1088/1361-6382/aab793
+[7] Powell, J. et al. 2015, Classical and Quantum Gravity, 32, 215012.
+doi:10.1088/0264-9381/32/21/215012
+5e.g. https://www.reinforceeu.eu/winter-challenge-2022
+6e.g. https://reinforce.ea.gr/international-training-course/
+5
+
+[8] Zevin, M. et al. 2017, Classical and Quantum Gravity, 34, 064003.
+doi:10.1088/1361-6382/aa5cea
+[9] Lintott, C. J. et al. 2008, MNRAS, 389, 1179. doi:10.1111/j.1365-
+2966.2008.13689.x
+[10] Cuoco E., et al 2021 Mach. Learn.: Sci. Technol. 2 011002
+6
+
diff --git a/F9E4T4oBgHgl3EQfgA1N/content/tmp_files/load_file.txt b/F9E4T4oBgHgl3EQfgA1N/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf,len=127
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='05112v1  [gr-qc]  12 Jan 2023  GWitchHunters: Machine Learning and citizen science  to improve the performance of Gravitational Wave  detector  Massimiliano Razzanoa,b,1,∗, Francesco Di Renzoa,b, Francesco Fidecaroa,b,  Gary Hemmingc, Stavros Katsanevasc  aDepartment of Physics, University of Pisa, Largo B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Pontecorvo 3, Pisa, I-56127  bINFN Section of Pisa, Largo B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Pontecorvo 3, Pisa, I-56127  cEuropean Gravitational Observatory (EGO),Via E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Amaldi, 5, Cascina,I-56021  Abstract  The Gravitational waves have opened a new window on the Universe and  paved the way to a new era of multimessenger observations of cosmic sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  Second-generation ground-based detectors such as Advanced LIGO and Ad-  vanced Virgo have been extremely successful in detecting gravitational wave  signals from coalescence of black holes and/or neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' However, in  order to reach the required sensitivities, the background noise must be inves-  tigated and removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In particular, transient noise events called “glitches”  can affect data quality and mimic real astrophysical signals, and it is there-  fore of paramount importance to characterize them and find their origin,  a task that will support the activities of detector characterization of Virgo  and other interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Machine learning is one of the most promising  approaches to characterize and remove noise glitches in real time, thus im-  proving the sensitivity of interferometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' A key input to the preparation of  a training dataset for these machine learning algorithms can originate from  citizen science initiatives, where volunteers contribute to classify and analyze  signals collected by detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' We will present GWitchHunters, a new citi-  zen science project focused on the study of gravitational wave noise, that has  been developed within the REINFORCE project (a ”Science With And For  Society” project funded under the EU’s H2020 program).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' We will present  ∗Corresponding author  Email address: massimiliano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='razzano@unipi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='it (Massimiliano Razzano)  1on behalf of the REINFORCE Consortium  Preprint submitted to Nuclear Instruments and Methods in Physics Research AJanuary 13, 2023  the project, its development and the key tasks that citizens are participating  in, as well as its impact on the study of noise in the Advanced Virgo detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  Keywords:  gravitational waves, machine learning, citizen science  PACS: 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='–q, 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='Tv,  2000 MSC: 83C35,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Introduction  Gravitational wave physics is opening an entire new window on the Uni-  verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Since their discovery in 2015 [1], the Advanced LIGO [2] and Advanced  Virgo [3] detectors have carried on three observing runs (O1, O2, O3) and  unveiled 90 signals produced by the coalescence of compact objects, mostly  binary black hole with a small fraction of neutron star and/or black hole  binaries [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  Advanced LIGO and Virgo are second-generation laser interferometers with  Fabry-Perot cavities hosted in km perpendicular arms, that are capable of  detecting the tiny deformations induced in the fabric of spacetime by the  passage of gravitational waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In order to improve the sensitivity of the  detectors, there is a continuous effort to reduce the background noise due to  local disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In particular, at low frequencies the noise is dominated by  seismic and Newtonian noise, while at mid frequencies the main component  is related to the thermal noise and at high frequencies the noise is mostly  related to quantum effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  The activity of detector characterization and  noise hunting in gravitational wave detectors is focused on the investigation  of stationary and non stationary noise sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In particular, non station-  ary transient noise events called glitches are of particular importance in the  noise studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In fact, glitches can affect data quality and stability and mimic  real astrophysical signals, thus reducing the effective duty cycle of interfer-  ometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' The classification and characterization of glitches is therefore key  to understand the origin of noise in detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' However, glitches have com-  plex temporal signatures, that make difficult to classify them using standard  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Various works have shown that Machine Learning methods can be  promising for the classification of glitches [5, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In particular, images built  from the time-frequency spectrograms of glitches are very effective in show-  ing the complex morphology of glitches and can be easily given in input to  machine learning algorithms, including deep convolutional neural networks  [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' A possible approach to this problem is based on supervised learning,  2  that requires large number of labeled glitch samples, that could be produced  by dedicated citizen science initiatives, where volunteers look at images and  clssify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  A successful example of this method is provided by Gravi-  tySpy2, a citizen science project focused on the classification of glitches in  LIGO and Virgo[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Here we present GWitchHunters3, a new citizen science  project complementary to GravitySpy and aimed at improving sensitivity of  gravitational wave detectors combining citizen science and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' The REINFORCE Project  GwitchHunters has been developed within the Research Infrastructures  FOR Citizens in Europe (REINFORCE) project4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' REINFORCE is a Re-  search & Innovation Project, supported by the EU H2020 SWAFS “Science  with and for Society” work programme and aimed at creating a series of  cutting-edge citizen science projects on Frontier Physics research, with the  goal of engaging >100,000 citizens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' REINFORCE is based on four citizen  science demonstrators focused Gravitational Waves (GWitchHunters), Astro-  physical neutrinos (Deep Sea Explorers), High Energy Physics (New Particle  Search at CERN) and muon-based tomography (Cosmic Muon Images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' All  demonstrators are hosted on Zooniverse [9], the world leading platform for  citizen science projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Overview of GWitchHunters  GwitchHunters has been officially launched on Zooniverse in November  2021 after a dedicated review phase and offers to citizens a set of different  tasks of increasing difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Data are presented as spectrograms and come  from the Virgo O3 run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' A Playground task is specifically devoted to learning  the basics of glitch morphology and its classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' Three other levels offer  (1) the possibility to classify glitches among a larger set of classes, (2) localize  the glitches in the time-frequency space, and (3) compare the spectrogram  in the main channel of Virgo with that produced by auxiliary sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' This  last task is particularly innovative, since it offer the possibility of linking the  glitches observed in the main channel to local disturbancies in the detector,  2http://https://gravityspy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
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+page_content='eu/  3  Figure 1: Example of a GWitchHunters spectrogram showing two glitches, as well as the  rectagle drawn by citizens to locate them  thus suggesting a possible hint to the origin of each glitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' These tasks can  be carried both on a personal computer and on mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' The project  also features a set of tutorials and examples to teach the volunteers how to  perform the different tasks, as well as a ”Field Guide” containing information  on the Advanced Virgo detector, the various glitch classes and the auxiliary  channels used in the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' First Results and Conclusions  Since its official launch in November 2021, ∼2800 volunteers have sub-  scribed to the project, although another significant amount have contributed  without officially registering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' This collective effort has produced more than  ∼400000 classifications of ∼ 40000 data samples so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In order to pro-  mote the project and engage citizens, the REINFORCE consortium has or-  ganized many initiatives, including workshops, press activities, online chal-  4  Virgo strain channel  Frequency [Hz]  Normalizedenergy  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='2  0:2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='8  Time [s]lenges5 and training school6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' A monitoring of the project website has been  carried on, showing that these initiatives successfully attracted more volun-  teers to GWitchHunters, leading to peaks of ∼5000 classifications per day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  The results of the volunteers analysis are used for training a machine learning  algorithm that automatically analyze the glitch data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' In particular, we fo-  cused on a 2D convolutional neural network architecture, that has been also  tested on simulations [6, 10] reaching an accuracy greater than 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' These  first tests show how the GWitchHunters project could be successfully used  to join citizen science and machine learning with the goal of contributing to  increase the sensitivity of gravitational wave detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  Acknowledgements  REINFORCE has received funding from the European Union’s Horizon  2020 research and innovation program, under Grant Agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' 872859.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='  References  [1] Abbott, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
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+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='1088/0264-9381/32/21/215012  5e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
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+page_content='reinforceeu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='eu/winter-challenge-2022  6e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content=' https://reinforce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
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+page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
+page_content='1365-  2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
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+page_content='13689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/F9E4T4oBgHgl3EQfgA1N/content/2301.05112v1.pdf'}
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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
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+page_content='CO]  6 Jan 2023  Vertex-Critical (P5, chair)-Free Graphs  Shenwei Huang*†  Zeyu Li‡§  January 4, 2022  Abstract  Given two graphs H1 and H2, a graph G is (H1, H2)-free if it contains no  induced subgraph isomorphic to H1 or H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A Pt is the path on t vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A chair  is a P4 with an additional vertex adjacent to one of the middle vertices of the P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A  graph G is k-vertex-critical if G has chromatic number k but every proper induced  subgraph of G has chromatic number less than k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' In this paper, we prove that there  are finitely many 5-vertex-critical (P5, chair)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Graph coloring;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' k-vertex-critical graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' forbidden induced subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  1  Introduction  All graphs in this paper are finite and simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We say that a graph G contains a graph  H if H is isomorphic to an induced subgraph of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A graph G is H-free if it does  not contain H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For a family of graphs H, G is H-free if G is H-free for every H ∈  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' When H consists of two graphs, we write (H1, H2)-free instead of {H1, H2}-  free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' As usual, Pt and Cs denote the path on t vertices and the cycle on s vertices,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A clique (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' independent set) in a graph is a set of pairwise adjacent  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' nonadjacent) vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The complete graph on n vertices is denoted by Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The  graph K3 is also referred to as the triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The clique number of G, denoted by ω(G),  is the size of a largest clique in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For two graphs G and H, we use G + H to denote  the disjoint union of G and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If a graph G can be partitioned into k independent sets  S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' , Sk such that there is an edge between every vertex in Si and every vertex in Sj  for all 1 ≤ i < j ≤ k, G is called a complete k-partite graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' each Si is called a part  of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If we do not specify the number of parts in G, we simply say that G is a complete  multipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We denote by Kn1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=',nk the complete k-partite graph such that the  ith part Si has size ni, for each 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  A q-coloring of a graph G is a function φ : V (G) −→ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' , q} such that φ(u) ̸=  φ(v) whenever u and v are adjacent in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' And a q-coloring of G is also a partition of  V (G) into q independent sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A graph is q-colorable if it admits a q-coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The  College  of  Computer  Science,  Nankai  University,  Tianjin  300350,  China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Email:  shenweihuang@nankai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Supported by Natural Science Foundation of Tianjin (20JCY-  BJC01190).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  †Tianjin Key Laboratory of Network and Data Security Technology, Nankai University, Tianjin 300071,  China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  ‡College of Computer Science, Nankai University, Tianjin 300350, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  §Tianjin Key Laboratory of Network and Data Security Technology, Nankai University, Tianjin 300071,  China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  1  Figure 1: The graph chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  chromatic number of a graph G, denoted by χ(G), is the minimum number q for which  G is q-colorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We call a graph G is k-chromatic when χ(G) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  A graph G is k-critical if it is k-chromatic and χ(G − e) < χ(G) for any edge  e ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We call a graph is critical if it is k-critical for some integer k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A graph  G is k-vertex-critical if χ(G) = k and χ(G−v) < k for any v ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For a set H of  graphs and a graph G, we say that G is k-vertex-critical H-free if it is k-vertex-critical  and H-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Our research is mainly motivated by the following theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Theorem 1 ([7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For any fixed k ≥ 5, there are infinitely many k-vertex-critical P5-  free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Thus, it is natural to consider which subclasses of P5-free graphs have finitely many  k-vertex-critical graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The reason for finiteness is that if we know there are only  finitely many k-vertex-critical graphs, then there is a polynomial-time algorithm for  (k − 1)-coloring graphs in that class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' In 2021, Kameron, Goedgebeur, Huang and Shi  [4] obtained the following dichotomy result for k-vertex-critical (P5, H)-free graphs  when |H| = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Theorem 2 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let H be a graph of order 4 and k ≥ 5 be a fixed integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then  there are infinitely many k-vertex-critical (P5, H)-free graphs if and only if H is 2P2  or P1 + K3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  In [4], it was also asked which five-vertex graphs H can lead to finitely many  k-vertex-critical (P5, H)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' It is known that there are finitely many 5-vertex-  critical (P5,banner)-free graphs [3, 9], and finitely many k-vertex-critical (P5, P5)-  free graphs for every fixed k [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Hell and Huang proved that there are finitely many  k-vertex-critical (P6, C4)-free graphs [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This was later generalized to (Pt, Kr,s)-  free graphs in the context of H-coloring [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This gives an affirmative answer for  H = K2,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Recently, it was also shown that the answer to the above question is  positive if H is gem or P2 + P3 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Moreover, it was proved that there are finitely  many 5-vertex-critical (P5, bull)-free graphs [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  In this article, we continue such a study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' A chair is a P4 with an additional vertex  adjacent to one of the middle vertices of the P4 (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' In particular, we prove  the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' There are finitely many 5-vertex-critical (P5, chair)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  2  Preliminaries  For general graph theory notation we follow [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let G = (V, E) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If uv ∈  E, we say that u and v are neighbors or adjacent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' otherwise u and v are nonneighbors  2  or nonadjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We use u ∼ v to mean that u and v are neighbors and u ≁ v to mean  that u and v are nonneighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The neighborhood of a vertex v, denoted by NG(v), is  the set of neighbors of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For a set X ⊆ V (G), let NG(X) = �  v∈X NG(v) \\ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We  shall omit the subscript whenever the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For X, Y ⊆ V , we say that X is  complete (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' anticomplete) to Y if every vertex in X is adjacent (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' nonadjacent)  to every vertex in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If X = {x}, we write “x is complete (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' anticomplete) to  Y ” instead of “{x} is complete (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' anticomplete) to Y ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If a vertex v is neither  complete nor anticomplete to a set S, we say that v is mixed on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If a vertex v is  neither complete nor anticomplete to two ends of an edge, we say that v is distinguish  the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We say that H is a homogeneous set if no vertex in V − H is mixed on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  More generally, we say that H is homogeneous with respect to a subset S ⊆ V if no  vertex in S can be mixed on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For S ⊆ V , the subgraph induced by S, is denoted by  G[S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  A pair of comparable vertices of G is pairwise nonadjacent vertices u, v such that  N(v) ⊆ N(u) or N(u) ⊆ N(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' It is well-known that k-vertex-critical graphs cannot  contain comparable vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We shall use the following generalization in later proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Lemma 1 ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let G be a k-vertex-critical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then G has no two nonempty  disjoint subsets X and Y of V (G) that satisfy all the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  X and Y are anticomplete to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  χ(G[X]) ≤ χ(G[Y ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Y is complete to N(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  3  New Results  In this section, we prove our new results: there are finitely many 5-vertex-critical  (P5, chair)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' To prove Theorem 3, we prove the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let G be a 5-vertex-critical (P5, chair)-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If G contains a C5,  then G has finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof of Theorem 3 assuming Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let G be a 5-vertex-critical(P5, chair)-free  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If G contains C5, then G has finite order by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If G is C5-free, then G  has finite order by a result in [7] that there are only thirteen 5-vertex-critical (P5, C5)-  free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' In either case, G has finite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Next we prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='1  Structure Around C5  In this subsection, we discuss some structural properties of (P5, chair)-free graphs  containing a C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let G be a connected (P5, chair)-free graph containing an induced  C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let C = v1, v2, v3, v4, v5 be an induced C5 with vivi+1 being an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We divide  V \\V (C) as follows, where all indices are modulo 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  S0 = {v ∈ V \\V (C) : NC(v) = ∅},  S1(i) = {v ∈ V \\V (C) : NC(v) = {vi}},  S1  2(i) = {v ∈ V \\V (C) : NC(v) = {vi, vi+1}},  3  S2  2(i) = {v ∈ V \\V (C) : NC(v) = {vi, vi+2}},  S1  3(i) = {v ∈ V \\V (C) : NC(v) = {vi−1, vi, vi+1}},  S2  3(i) = {v ∈ V \\V (C) : NC(v) = {vi−2, vi, vi+2}},  S4(i) = {v ∈ V \\V (C) : NC(v) = {vi−2, vi−1, vi+1, vi+2}},  S5 = {v ∈ V \\V (C) : NC(v) = V (C)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  We use Sm  3 (i ± 1) to denote Sm  3 (i + 1) ∪ Sm  3 (i − 1) for m = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The notations  Sm  3 (i±2), S4(i±1) and S4(i±2) are defined similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We now prove some properties  about these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' S1(i) ∪ S1  2(i) ∪ S2  2(i) = ∅, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let u, v be arbitrary two vertices such that v ∈ S1(i) ∪ S1  2(i),  u ∈ S2  2(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then {v, vi, vi−1, vi−2, vi−3} induces a P5, and {u, vi, vi−1, vi−2} and  {vi+1} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' S0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We will first show that N(S0) ⊆ S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Since G is connected,  there is a pair of vertices u and v such that u ∈ S0, v ∈ V (G)\\S0 and u ∼ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If  v ∈ S1  3(i) for any i, then {u, v, vi+1, vi+2, vi−2} induces a P5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If  v ∈ S2  3(i) ∪ S4(i + 1) for any i, then {vi+1, vi, v, vi−2} and {u} induce a chair, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Thus, v can only belong to S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then, two nonempty disjoint subsets S0  and C of V (G) satisfy the three conditions of Lemma 1, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Therefore,  S0 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' S1  3(i) is clique, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We assume that there are two vertices u, v ∈ S1  3(i) with u ≁ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Then {v, vi+1, vi+2, vi−2} and {u} induce a chair in G, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Each vertex in S4(i) ∪ S5 is either complete or anticomplete to a component  of S2  3(i), for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We assume that there is an edge uv of S2  3(i) can be distinguished by vertex  s ∈ S4(i) ∪ S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Without loss of generality, let s ∼ u, s ≁ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then {vi−1, s, u, v} and  {vi+1} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Each vertex in V (G)−(S2  3(i)∪S4(i)∪S5) is either complete or anticomplete  to S2  3(i), for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By symmetry, it suffices to prove the claim for i, i+ 1 and i+ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let v ∈ S2  3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  If v is adjacent to s1 ∈ S1  3(i + 1), then {vi−1, vi−2, v, s1, vi+1} is an induced P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If v  is not adjacent to s2 ∈ S1  3(i) ∪ S2  3(i + 1) ∪ S4(i + 2), then {vi−1, s2, vi+1, vi+2, v} is  an induced P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If v is not adjacent to s3 ∈ S2  3(i + 2) ∪ S4(i + 1), then {vi−1, s3, vi+2, v}  and {vi+1} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If v is not adjacent to s4 ∈ S1  3(i + 2), then {vi−1, vi, vi+1, v}  and {s4} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Every component of S2  3(i) is a homogeneous set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By Claim 4 and Claim 5, there is no vertex of G\\S2  3(i) that can distinguish an  edge of S2  3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  4  Let Ti = S1  3(i ± 2) ∪ S2  3(i ± 1) ∪ S2  3(i ± 2) for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' S4(i) is complete to Ti, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By the symmetry, it suffers to prove the claim for S1  3(i+2)∪S2  3(i+1)∪S2  3(i+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Let v ∈ S4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If v is not adjacent to s1 ∈ S1  3(i + 2), then {vi, vi−1, v, vi+2, s1} in-  duces a P5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If v is not adjacent to s2 ∈ S2  3(i + 1), then {vi, vi−1, v, vi+2}  and {s2} induce a chair, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If v is not adjacent to s3 ∈ S2  3(i + 2), then  {s3, vi, vi+1, v, vi−2} induces a P5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For each s ∈ S1  3(i) ∪ S4(i ± 2), u, v ∈ S4(i) with uv /∈ E, s cannot mix on  {u, v}, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By the symmetry, it suffers to prove the claim for S1  3(i) ∪ S4(i + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let  s ∈ S1  3(i) ∪ S4(i + 2) with s ∼ u, s ≁ v , then {vi, s, u, vi+2, v} induces a P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Let Ri = S1  3(i ± 1) ∪ S2  3(i) ∪ S4(i ± 1) ∪ S5, for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For each s ∈ Ri, u, v ∈ S4(i) with uv /∈ E, s is adjacent to at least one of  {u, v}, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By the symmetry, it suffers to prove the claim for S1  3(i + 1) ∪ S2  3(i) ∪ S4(i +  1) ∪ S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let s1 ∈ S1  3(i + 1) ∪ S2  3(i) ∪ S4(i − 1), if s1 is nonadjacent to both {u, v},  then {v, vi−1, vi, s1} and {u} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let s2 ∈ S5, if s2 is nonadjacent to both  {u, v}, then {vi, s2, vi−2, v} and {u} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Every vertex in S4(i ± 2) is complete to x, y ∈ S4(i) with xy /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By symmetry, let v ∈ S4(i + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' v can not mix on x, y by Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If v ≁ x  and v ≁ y, {vi, v, vi−2, x} and {y} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then v is complete to {x, y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='2  Proof of Theorem 4  Let graph family F = {K5, W, P, Q1, Q2, Q3} (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' The adjacency lists of  F are given in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' It is routine to verify that every graph in F is a 5-vertex-  critical (P5, chair)-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let G be a 5-vertex-critical (P5, chair)-free graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If G contains  a induced F ∈ F, then G is isomorphic to F since G is 5-vertex-critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Therefore,  we may assume that G is F-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  By Claim 1 and Claim 2, G has a finite order if and only if S3 ∪ S4 ∪ S5 has finite  size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' |S1  3(i)| ≤ 2, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If |S1  3(i)| ≥ 3, then S1  3(i) ∪ {vi, vi+1} contains a K5 by Claim 3, a contradic-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' χ(S2  3(i) ∪ S4(i) ∪ S5) ≤ 2, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If χ(S2  3(i) ∪ S4(i) ∪ S5) ≥ 3, then the proper subgraph S2  3(i) ∪ S4(i) ∪ S5 ∪  {vi−2, vi+2} has chromatic number at least 5, contradicting that G is 5-vertex-critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' S5 is an independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  5  0  1  2  3  4  K5  0  1  2  3  4  5  6  W  0  1  2  3  4  5  6  7  8  P  0  1  2  3  4  5  6  7  8  Q1  0  1  2  3  4  5  6  7  8  Q2  0  1  2  3  4  5  6  7  8  Q3  Figure 2: Graph Family F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If there are two adjacent vertices u, v ∈ S5, then G contains a W ∈ F, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Every homogeneous component of S2  3(i) or S4(i) is isomorphic to K1 or  K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let K be a component of S2  3(i) or S4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Since G has no K5 or W, K has  no triangles or C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Since G is P5-free, G is bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So χ(K) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Clearly, if  χ(K) = 1, then K is isomorphic to K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Now assume that χ(K) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let X and  Y be the bipartition of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let x ∈ X and y ∈ Y with xy ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Suppose that  (X ∪ Y ) \\ {x, y} ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Since G is 5-vertex-critical, G − ((X ∪ Y ) \\ {x, y}) has a  4-coloring φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Without loss of generality, we may assume that φ(x) = 1 and φ(y) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Now if we color every vertex in X with color 1 and color every vertex in Y with color  2, the resulting coloring is a 4-coloring of G by Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This contradicts that G is  5-vertex-critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So K is isomorphic to K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' |S2  3(i)| ≤ 3, for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let K be a component of S2  3(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We say that K is of type i if χ(K) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We  show that there is at most one component of type i for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Take two components  K, K′ of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let k ∈ K and k′ ∈ K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By Lemma 1, there are vertices u, v  such that u ∈ N(K) \\ N(K′) and v ∈ N(K′) \\ N(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By Claim 6, uk ∈ E, vk′ ∈ E  and uk′, vk /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Any vertex in V (G)−(S2  3(i)∪S4(i)∪S5) can’t mix on two vertices  of S2  3(i) by Claim 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So u, v ∈ S4(i) ∪ S5 by our assumption about k, k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If u ≁ v,  {k, u, vi+1, v, k′} induces a P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Therefore, u ∼ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By Claim 13, u, v cannot be in  S5 at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' It is easy to see that C ∪ {k, k′, u, v} contains an induced P, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  As a result, |S2  3(i)| ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' S4(i) is a star, or S4(i) is complete to S4(i + 2) ∪ S4(i − 2), for all 1 ≤  i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If S4(i) is disconnected, S4(i) is complete to S4(i+2)∪S4(i − 2) by Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  If S4(i) is connected, then S4(i) is a bipartite graph by Claim 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If χ(S4(i)) = 1,  S4(i) is isomorphic to K1 and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Now assume that |S4(i)| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let X, Y  be the bipartition of S4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If |X| ≥ 2 and |Y | ≥ 2, then every vertex in S4(i ± 2) is  6  complete to X ∪ Y by Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Thus, S4(i) is complete to S4(i ± 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Therefore, we  may assume that |X| = 1 and so S4(i) is a star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Recall that Ri = S1  3(i ± 1) ∪ S2  3(i) ∪ S4(i ± 1) ∪ S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If S4(i) is a star, then |S4(i)| ≤ 2 for all 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Suppose that S4(i) = X ∪ Y with Y  = {y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  We show that |X| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let x1, x2 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By Lemma 1, there exist a ∈ N(x1)\\N(x2) and  b ∈ N(x2)\\N(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Note that any vertex of G − Ri can’t mix on two nonadjacent  vertices of X by Claim 7 - Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So a, b ∈ Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If a ≁ b, {x1, a, vi, b, x2} induces  a P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So a ∼ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' It is not hard to check that G contains one of Q1, Q2 and Q3, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Thus, there are at most two vertices in X, and so |S4(i)| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For each i, when S4(i) is complete to S4(i ± 2) and Ri is not empty, then  |S4(i)| ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' When S4(i) is (P1 + P2)-free, S4(i) is a complete bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let (X, Y )  be a partition of S4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We show that |X|, |Y | ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Suppose not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let x1, x2, x3, x4 be  vertices in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By Lemma 1, there vertices a1 ∈ N(x1)\\N(x2), a2 ∈ N(x2)\\N(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Notice that a1, a2 ∈ Ri by Claim 7 - Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If a1 ≁ a2, G contains an induced  P5 = {x1, a1, vi, a2, x2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So a1 ∼ a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then a1 ∈ S1  3(i − 1) ∪ S4(i + 1) and a2 ∈  S1  3(i + 1) ∪ S4(i − 1), otherwise, it is easy to check that G contains one of Q1 and  Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Similarly, there exists a3 ∈ N(x3)\\N(x4), a4 ∈ N(x4)\\N(x3) and a3, a4 ∈  Ri, a3 ∼ a4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Thus {x3, x4} is complete to {a1, a2}, and {x1, x2} is complete to  {a3, a4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This shows that a1, a2, a3, a4 are pairwise different vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then a3 ∈  S1  3(i − 1) ∪ S4(i + 1), a4 ∈ S1  3(i + 1) ∪ S4(i − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Recall that S1  3(i − 1) or S1  3(i + 1)  is a clique by Claim 3, and S1  3(i − 1) is complete to S4(i + 1), S1  3(i + 1) is complete  to S4(i − 1) by Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If a1 ≁ a3 and a2 ≁ a4, then a1, a3 ∈ S4(i + 1) and a2, a4 ∈  S4(i − 1), then {vi−2, vi+2, x3, a1, a2} is an induced K5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Otherwise, if a1 ∼ a3,  {vi−1, vi−2, x3, a1, a3} induces K5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So a2 ∼ a4, then {vi+1, vi+2, x3, a2, a4} induces  a K5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So |S4(i)| ≤ 6 if S4(i) is (P1 + P2)-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Now suppose that S4(i) contains a P1 + P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let P1 + P2 = {a, b, c : a ≁ b, a ≁  c, b ∼ c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We first prove some useful facts about P1 + P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  S1  3(i) is anticomplete to P1 + P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  (1)  Every x ∈ S1  3(i) is either complete or anticomplete to {a, b, c} by Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If x is  complete to {a, b, c}, then G contains an induced W, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So x is anticom-  plete to {a, b, c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This completes the proof of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  For any y ∈ Ri, {y, a, b, c} induces either a P4 or a 2P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  (2)  Let y ∈ Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Note that {y} ∪ S4(i) is triangle-free or else G contains a K5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If y is  not adjacent to a, then y ∼ b, y ∼ c by Claim 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Now G induces a K5, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  So y ∼ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If y ≁ b, y ≁ c, then {y, a, b, c} induces a 2P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If y is adjacent to exact one  vertex of {b, c}, we assume by symmetry that y ∼ b, y ≁ c and so {a, y, b, c} induces  a P4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This completes the proof of (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Next we discuss about S4(i)\\{a, b, c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let x ∈ S1  3(i), z ∈ S4(i)\\{a, b, c}, and  we define Y1 = {y1 ∈ Ri : {y1, a, b, c} induces a P4}, and Y2 = {y2 ∈ Ri :  7  {y2, a, b, c} induces a 2P2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  S1  3(i) is anticomplete to S4(i)\\{a, b, c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  (3)  If z ∼ x, then z is complete to {a, b, c} by (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Now G contains an induced W, a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So z ≁ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This completes the proof of (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  So S1  3(i) is anticomplete to S4(i) by (1) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  For any y1 ∈ Y1, z1 ∈ S4(i)\\{a, b, c}, z1y1, z1c ∈ E, and z1a, z1b /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  (4)  If z1 ≁ y1, then z1 ∼ c by y1c /∈ E and Claim 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So z1 ≁ b by Claim 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If z1 ≁ a,  {y1, a, b, c, z} induces a P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So z1 ∼ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then there is an induced C5 = {a, y1, b, c, z1},  contradicting Claim 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So z1 ∼ y1, then z1 ≁ a and z1 ≁ b since S4(i) is triangle-  free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If z1 ≁ c, {a, y1, b, c} and {z1} induce a chair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So z1 ∼ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This completes the  proof (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  For any y2 ∈ Y2, z2 ∈ S4(i)\\{a, b, c}, z2y2 ∈ E, and z2a, z2b, z2c /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  (5)  If z2 ≁ y2, then z2 ∼ b and z2 ∼ c by y2b, y2c /∈ E and Claim 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then {z2, b, c}  induces a triangle, contradicting Claim 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So z2 ∼ y2 and then z2 ≁ a by the fact that  {y2} ∪ S4(i) is triangle-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If z2 is adjacent to exact one of b, c, then {z2, y2, a, b, c}  induces a P5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So z2 ≁ b and z2 ≁ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This completes the proof (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  We can infer that any vertex in Ri is complete to S4(i)\\{a, b, c} by (4) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Suppose that there exist two vertices z, z′ ∈ S4(i)\\{a, b, c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If Y1 ̸= ∅ and Y2 ̸= ∅,  z is adjacent to c by (4) and is nonadjacent to c by (5), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So Ri =  Y1 or Ri = Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Note that any vertex in Ri is complete to two ends of an edge of  C5 ∩ N(S4(i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Since G is K5-free, z ≁ z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then N(z) = N(z′) by Claim 7,  contradicting to Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So |S4(i)\\{a, b, c}| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then |S4(i)| ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For each i, when S4(i) is complete to S4(i±2) and Ri is empty, |S4(i)| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If S4(i) is disconnected, then there are two components K1, K2 of S4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Every  vertex of S1  3(i) is either complete or anticomplete to K1 ∪ K2 by Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So K1  and K2 are homogeneous components by Claim 7 - Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Moreover, N(K1) =  N(K2) ⊆ Ti ∪ S1  3(i) ∪ S4(i ± 2) ∪ C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' This contradicts Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Therefore, S4(i)  is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Recall that χ(S4(i)) ≤ 2 by Claim 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If χ(S4(i)) = 1, then |S4(i)| = |K1| = 1  and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' When χ(S4(i)) = 2, S4(i) is a bipartite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Let (X, Y ) be the  bipartition of S4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Every vertex s ∈ S1  3(i) is either complete or anticomplete to  X(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Y ) by Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So X(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Y ) is homogeneous with respect to G − Y (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  G − X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' If there are x ∈ X, y ∈ Y with x ≁ y, then every vertex s ∈ S1  3(i) cannot  mix on S4(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then S4(i) is a homogeneous set, and |S4(i)| = |K2| = 2 by Claim 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  If X is complete to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Then X is a homogeneous set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' For any pairwise vertices  x1, x2 ∈ X, we have N(x1) = N(x2), contradicting Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So |X| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' In the  same way, |Y | = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Therefore, |S4(i)| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Claim 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' |S4(i)| ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' It follows from Claim 17 to Claim 19 that |S4(i)| ≤ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  8  Claim 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' |S5| ≤ 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Suppose that |S5| > 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' We know any two vertices in S5 are nonadjacent  by Claim 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' By the pigeonhole principle, there are two vertices u, v ∈ S5 such that  N(u) = N(v), contradicting Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' So |S5| ≤ 25(|S1  3(i)∪S2  3(i)∪S4(i)|)  ≤ 25(2+3+6) = 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  The lemma follows from Claim 11, Claim 15, Claim 20 and Claim 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  4  Appendix  Below we give the adjacency lists of graphs in F other than K5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  Graph W: {0: 1 4 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 1: 0 2 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 2: 1 3 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 3: 2 4 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 4: 0 3 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 5: 0 1 2 3 4  6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 6: 0 1 2 3 4 5}  Graph P: {0: 1 4 5 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' 1: 0 2 7 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
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+page_content=' Discrete Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
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+page_content='  Discrete Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Mathematics, 216:211–232, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
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+page_content=' Moore, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Recoskiez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Sawada, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Vatshelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Constructions  of k-critical P5-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Discrete Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
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+page_content=' Huang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Critical (P5,bull)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='04179 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='CO],  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  [9] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
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+page_content=' Li, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Shi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Critical (P6, banner)-free graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content=' Discrete Applied  Mathematics, 258:143–151, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
+page_content='  [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
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+page_content=' Pstrucha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
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+page_content=' Discrete Applied Mathematics, 261:258–267, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9E0T4oBgHgl3EQfhwGK/content/2301.02436v1.pdf'}
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+arXiv:2301.11636v1  [math.DG]  27 Jan 2023
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF
+CALABI-YAU THREEFOLDS
+FEDERICO GIUSTI AND CRISTIANO SPOTTI
+ABSTRACT. Given a (smoothable) projective nodal Kähler Calabi-Yau threefold, we show, via a
+gluing construction, that all its - possibly non-Kähler - small resolutions admit Chern-Ricci flat
+balanced metrics, which among other things solve the dilatino equation appearing in the Hull-
+Strominger system.
+1. INTRODUCTION
+With the ultimate aim of geometrizing and classifying, one of the most studied problems in com-
+plex geometry is the existence of hermitian metrics that can be regarded as special. Through the
+years, the Kähler case is the one that has been studied and understood the most, however, in the last
+decades the interest towards the non-Kähler world has been increasing more and more, leading to
+the search for special metrics also in this particular context. While in the Kähler case special metrics
+arise naturally, the non-Kähler scenario is too wild to guide us directly towards some central no-
+tion of special metric, nevertheless, one can have indications on the path to follow by watching the
+Kähler world. More specifically, given an n-dimensional complex manifold (M, J), if it is Kähler
+the obvious class of special (on a first level) metrics is given exactly by Kähler metrics - which we
+recall being hermitian metrics h whose fundamental form ω := h(J_, _) is d-closed. In addition,
+this condition can also be combined with the notion of Einstein metric (thanks to the properties
+of Kähler metrics) from the general riemannian case, giving rise to the notion of Kähler-Einstein
+metrics, which are universally regarded as the "most special" in the Kähler world. Likewise, other
+notions of special Kähler metrics have been introduced and studied (some of them are still central
+in the study of Kähler geometry), like constant scalar curvature Kähler (cscK) metrics, or the more
+general class of extremal Kähler metrics (introduced by Calabi in [C]), however they all share the
+fact that they are giving a curvature condition on the metric, thus this suggests that when searching
+for special metrics in the non-Kähler case we shall ask for these metrics to be special under two
+aspects: the cohomological one (satisfying a condition possibly generalizing the Kähler one) and
+the curvature one.
+Regarding the cohomological aspect, several conditions have been introduced that generalize the
+Kähler one, and one of the most studied is given by dωn−1 = 0, identifying the class of balanced
+metrics (originally introduced by Gauduchon in [G] as semi-Kähler metrics, and later on studied
+Date: January 30, 2023.
+2010 Mathematics Subject Classification. 53C55, 53C25,53C07.
+Key words and phrases. complex non-Kähler manifolds, balanced metrics, Chern-Ricci flat metrics, Calabi-Yau
+manifolds, Hull-Strominger system.
+1
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+2
+further by Michelsohn in [M]), which is the class of metrics we are interested in working with.
+Balanced metrics carry many interesting properties such as the coincidence between the Hodge
+laplacian and the Dolbeault laplacian on scalar functions (showed by Gauduchon in [G]), or the
+preservation of the balanced condition for manifolds under holomorphic submersions proved in
+[M] (showing a sort of duality between the Kähler condition and the balanced condition). Also in
+[M], Michelsohn proved a characterization of balanced metrics in terms of currents, which leads to
+the celebrated result from Alessandrini and Bassanelli (see [AB]) showing that the class of com-
+pact balanced manifolds is closed under proper modifications (condition not satisfied by the class
+of Kähler manifolds). Moreover, balanced metrics ended up being central in many interesting cur-
+rently open problems, such as the conjecture from Fino and Vezzoni (see [FV]) and the Gauduchon
+conjecture for balanced metrics (see [STW], in which was solved in its original version for Gaudu-
+chon metrics - identified by the condition ∂∂ωn−1 = 0, which weakens the balanced condition -
+posed by Gauduchon).
+Moving instead on the curvature aspect, there are several known notions of special metrics in the
+non-Kähler world such as Chern-Ricci flat metrics, Bismut-Ricci flat metrics (which in the bal-
+anced case are equivalent to Chern-Ricci flat metrics, see [AI]), Chern-Einstein metrics and many
+more.
+The main goal of this paper is to construct Chern-Ricci flat balanced metrics on the compact small
+resolutions of certain smoothable singular Kähler Calabi-Yau threefolds. More specifically we wish
+to work on smoothable singular threefolds ˜
+M whose singular set is made of ordinary double points
+and are endowed with a Kähler Ricci-flat singular metric ˜ω. Then, using the information on the
+asymptotics of this metric around the singularity given by the results in [HS], together with what
+it is known on the standard conifold (i.e. the local model of ordinary double points on threefolds)
+and its small resolution to build, with a gluing construction (inspired mostly by [BM], but also
+[AP] and [J]), Chern-Ricci flat balanced metrics on the compact small resolutions of the singu-
+lar threefold. The strategy of the proof consists of two main steps: (1) a metric gluing between
+the singular Calabi-Yau metric ˜ω with the (rescaled) Candelas-de la Ossa metrics ωco,a (that are
+Kähler Calabi-Yau metrics on the small resolution of the standard conifold, introduced in [CO]),
+and (2) an Implicit Function Theorem deformation argument, where the deformation is a balanced
+deformation (introduced in [FWW]) and all the analysis is performed in suitable weighted Hölder
+spaces. Our main result is the following.
+Theorem 1.1. Let ( ˜
+M, ˜ω) be a smoothable projective Kähler Calabi-Yau nodal threefold (with ˜ω
+a singular Calabi-Yau metric), and let M be a compact (not necessarily Kähler) small resolution
+of ˜
+M. Then M admits a Chern-Ricci flat balanced metric ˆω such that [ˆω2] = [˜ω2] + ε4[P1], and
+[ω2] converges to a nef class.
+Here by "smoothable" we mean that ˜
+M admits a polarized flat deformation to a smooth pro-
+jective Calabi-Yau threefold (examples of such manifolds are given by nodal quintic threefolds in
+P4), and the reason why we require this condition to be satisfied is to be able to apply a result from
+[HS].
+Our interest towards Chern-Ricci flat balanced metrics comes actually from the realm of Calabi-
+Yau geometry. Indeed, for a not necessarily Kähler Calabi-Yau manifold (i.e. a complex manifold
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+3
+endowed with a holomorphic volume form) it was introduced by Hull and Strominger (respec-
+tively in [Hu] and [S]) a system of four equations coming from superstring theory known as the
+Hull-Strominger system, whose solutions have proved to be extremely hard to construct (see [GF]
+for a full presentation of the system and some known solutions, together with several other ref-
+erences such as [AGF], [Fe], [FuY], [LY3], [P], [TY] and the very recent [CPY2], [FeY] for the
+invariant case and [PPZ] for a flow approach). The problem of solving This system, apart from its
+physical meaning, carries great geometric interest, since it generalizes the Calabi-Yau condition to
+the non-Kähler framework, and it holds a central role in the geometrization conjecture for com-
+pact Calabi-Yau threefolds known as Reid’s Fantasy (see [R]). This last conjecture, in particular,
+states that all compact Kähler Calabi-Yau threefolds can be connected through a finite number of
+conifold transitions (introduced by Clemens and Friedman, see [F]), i.e. a procedure consisting of
+the contraction of a finite family of disjoint (−1, −1)-curves in a compact Calabi-Yau threefold,
+followed by the smoothing of the ordinary double points obtained from the previous step. These
+objects thus show our interest towards singular threefolds with a finite number of ordinary dou-
+ble points and our aim to find special metrics on their small resolution; in particular the interest
+towards Chern-Ricci flat balanced metrics is directly related to one of the equation of the Hull-
+Strominger system, namely the conformally balanced equation, which on a compact Calabi-Yau
+manifold (X, Ω) - where Ω is the holomorphic volume form - is an equation for hermitian metrics
+(actually their fundamental forms) ω given by d(||Ω||ωωn−1) = 0 which is clearly satisfied by
+balanced Chern-Ricci flat hermitian metrics. Moreover, Chern-Ricci flat metrics correspond also
+to Hermite-Einstein metrics on the holomorphic tangent bundle, thus this kind of metric appears as
+particularly suited to be used as a starting point from which possibly build solutions for the Hull-
+Strominger system (we will briefly discuss some ideas at the end of the paper), and also portrays
+our construction (in some sense) as aiming towards "reversing the arrow" of the construction done
+by Fu, Li and Yau in [FLY] and Collins, Picard and Yau in [CPY1].
+It is also interesting to notice that the deformation argument used in Section 4 proves also that on
+this class of manifolds the Gauduchon conjecture for balanced metrics holds for certain classes
+nearby the boundary of the balanced cone.
+The paper is structured as follows. In Section 2 we recall some basic aspects on ordinary dou-
+ble points and their resolutions, both regarding their geometry and their topology, with focus on
+some fundamental results useful for our construction. In Section 3 we present the first step of our
+work, consisting of a gluing construction of a balanced metric made with the objects previously
+introduced, together with the construction of a global "small" Chern-Ricci potential for our new
+balanced metric. In the last section, i.e. Section 4, we apply a deformation argument to obtain a
+Chern-Ricci flat balanced metric and we discuss its associated balanced class, as well as its possi-
+ble applications to the search of solutions for the Hull-Strominger system.
+Acknowledgements. Both the authors are supported by Villum Young Investigator 0019098.
+The authors would like to thank Mario Garcia-Fernandez for useful conversations and remarks.
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+4
+2. ORDINARY DOUBLE POINTS ON THREEFOLDS
+In this preliminary section we shall recall some known facts about a certain type of singularity
+on threefolds, regarding both their topology and their geometry, and also fix some notation for the
+following sections.
+The type of singularities we are interested in studying are called ordinary double points (which
+are the most common kind of singularities in our context), and are described by the model
+X := {z2
+1 + z2
+2 + z2
+3 + z2
+4 = 0} ⊆ C4,
+which is known as the 3-dimensional standard conifold, whose only singular point is the origin.
+Then we have:
+Definition 2.1. A singular point p in a singular threefold Y is called ordinary double point (ODP)
+if we can find a neighborhood p ∈ U ⊆ M and a neighborhood 0 ∈ V ⊆ X such that U and V are
+biholomorphic through a map that sends p to 0.
+2.1. Topological aspects. These singularities arise naturally on threefolds when collapsing (−1, −1)-
+curves, i.e. rational curves biholomorphic to P1 whose normal bundle is isomorphic to OP1(−1)⊕2,
+and actually this procedure to obtain ODPs covers all the possibilities on threefolds. Indeed, the
+standard conifold can be constructed in several ways, one of which is the following: consider the
+rank 2 bundle OP1(−1)⊕2 on P1 and notice that the map
+([X1 : X2], (w1, w2)) �→ (w1X1, w1X2, w2X1, w2X2)
+maps OP1(−1)⊕2 onto X - since X through a change of coordinates is biholomorphic to the set
+{W1W2 − W3W4 = 0} - sending the zero section onto the origin. Moreover this map restricted
+to OP1(−1)⊕2 \ P1 (where P1 is meant as the zero section) gives a biholomorphism with X \
+{0}, proving our previous statement. This shows us that these singularities always admit small
+resolutions (with P1 as the exceptional curve) biholomorphic to ˜X := OP1(−1)⊕2, and it can be
+shown that a singular threefold with n ordinary double points admits exactly 2n small resolutions
+of this type (every singularity can be resolved with a curve in two distinct bimeromorphic ways).
+Since our following work aims to face the case of singular threefolds whith a finite number of
+ODPs which admit a compact small resolution, in particular those who do not admit Kähler metrics,
+before going further in exploring the geometry of these singularities we will quickly show that this
+kind of resolutions are actually a very common situation.
+Remark 2.2. Thanks to a result from Cheltsov (see [Ch]) we know that a hypersurface ˜
+M in P4 of
+degree d with only isolated ODPs is factorial when ˜
+M has at most (d−1)2 −1 singularities, thus is
+in particular Q-factorial. We can then apply the work from Namikawa and Steenbrink (see [NS]) to
+obtain that ˜
+M is smoothable, and hence, thanks to the results from Friedman (see [F]) we have that
+any small resolution M of ˜
+M with exceptional curves C1, ..., Ck, Ci ≃ P1, satisfies necessarily a
+condition
+k
+�
+i=1
+λi[Ci] = 0
+in H2(M, R),
+where each λi ̸= 0,
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+5
+which immediately implies that if ˜
+M has only one ODP, then M can’t be Kähler because it contains
+a homologically trivial curve (note that the generic quintic threefold has exactly one node, hence
+satisfies this situation).
+Moreover, Werner proved in [W] that M is projective if and only if all Cis are homologically
+non-trivial, and since M is Moishezon, projectivity is equivalent to Kählerness. Thus the class of
+examples above lies in a larger one, since every small resolution with at least a homologically
+trivial exceptional curve is non-Kähler.
+2.2. Geometric aspects. The standard conifold X is naturally endowed with a conical structure.
+Indeed, we can introduce the function on C4
+r(z) := ||z||
+2
+3 ,
+which restricted to X yields the conical distance to the singularity, and can be used to define the
+metric
+ωco,0 := 3
+2i∂∂r2,
+on the smooth part of X, which is clearly Kähler. Moreover, it can be seen that ωco,0 is actually
+also Ricci flat, as well as a cone metric over the link L := {r = 1} ⊆ X which can be written as
+gco,0 = 3
+2(dr2 + r2gL),
+with gL a Sasaki-Einstein metric on the link L.
+This metric structure of the standard conifold, with some further work, yields also a Kähler Calabi-
+Yau structure on the small resolution. In fact, Candelas and de la Ossa (see [CO]) constructed a
+family of metrics, depending on the parameter a > 0, of the form
+ωco,a := i∂∂fa(r3) + 4a2π∗
+P1ωF S,
+where ωF S is the Fubini-Study metric on P1, and fa is smooth function satisfying the ODE
+(xf ′
+a(x))3 + 6a2(xf ′
+a(x))2 = x2,
+fa(x) ≥ 0,
+on [0, +∞), which immediately gives fa(x) = a2f1(x/a3). Here the function r is simply the
+conical distance from the singularity re-read on the resolution, hence portraying the conical distance
+from the exceptional curve. Moreover, this family of metrics is such that as a → 0 the metrics
+ωco,a converges, away from the exceptional curve, to the standard cone metric ωco,0, and it is also
+asymptotic (at infinity) to the cone metric ωco,0, and these facts can be seen explicitly with the
+following expansion from [CPY1].
+Lemma 2.3. For x ≫ 1, the function f1(x) has a convergent expansion
+f1(x) = 3
+2x
+2
+3 − 2 log(x) +
++∞
+�
+n=0
+cnx− 2n
+3 .
+This expansion is going to be crucial for our work, hence we will work again with it in the next
+sections.
+Before concluding this preliminary part, is extremely important for our construction (and highly
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+6
+interesting just as a fact) to remark that the standard cone metric ωco,0 is actually the (asymptotic)
+model near the singularities for all singular Kähler Calabi-Yau metrics around ODPs, and this was
+shown by Hein and Sun in the following crucial theorem (Theorem 1.4 and Lemma A.1 from [HS])
+which can be simplified with the following statement (written only for threefolds):
+Theorem 2.4 (Hein-Sun). Let ˜
+M be a smoothable singular threefold whose singular set is a finite
+family of ODPs endowed with a Kähler Calabi-Yau metric ˜ω on its smooth part Mreg. Then for
+every singular point p ∈ ˜
+M \ Mreg there exist a constant λ0 > 0, neighborhoods p ∈ Up ⊆ ˜
+M
+and 0 ∈ Vp ⊆ X, and a biholomorphism P : Vp \ {0} → Up \ {p} such that
+P ∗˜ω − ωco,0 = i∂∂ϕ,
+for some ϕ ∈ C∞
+2+λ0,
+where r is the conical distance from the singularities and C∞
+2+λ0 is the space of smooth functions
+with decay rate at zero of 2 + λ0 (i.e. an f ∈ C∞
+2+λ0 is a smooth function such that nearby zero it
+holds |∇kf| ≤ cr2+λ0−k for all k ≥ 0).
+3. THE GLUING CONSTRUCTION
+We now start presenting the first part of our work, which consists of a gluing construction of
+a family of balanced metrics on the compact small resolution of a smoothable Kähler Calabi-Yau
+singular threefold. This type of gluing constructions are not new in complex geometry, and have
+been used in the past to construct significant examples, e.g. in [J], [AP] and [BM].
+Before going ahead with the details of the construction it is significant to make the following
+remark.
+Remark 3.1. Thanks to the results from Hironaka and Alessandrini-Bassanelli ([Hi] and [AB]),
+we already know that compact small resolutions of smoothable Kähler Calabi-Yau threefolds with
+a finite number of ODPs admit balanced metrics. However, we are interested in finding special
+balanced metrics - in particular, as we will see, with a small Chern-Ricci potential - hence we
+need to perform an explicit construction in order to build metrics satisfying our desired curvature
+properties.
+In what follows we will lay out the initial data and see in details the construction.
+Let ˜
+M be a smoothable complex threefold obtained from the contraction of a finite family of
+disjoint (−1, −1)-curves in a compact complex threefold (thus the singular set of ˜
+M is made of a
+finite number of Ordinary Double Points), let M be a compact small resolution of ˜
+M and suppose
+that the regular part Mreg of ˜
+M is equipped with ˜ω a Kähler Calabi-Yau metric.
+The idea of the gluing construction is to use the ingredients we introduced in the previous sections,
+and notice that they are suitable to be glued together. In particular, we know that around each one
+of the exceptional curves we have the Candelas-de la Ossa metrics ωco,a which are asymptotic, far
+from the exceptional curve, to the cone metric; on the other hand we have that also our background
+metric ˜ω is asymptotic to the cone metric, but this time when approaching the singularity. We
+then want to use the standard conifold as a "bridge" to glue together the metrics ˜ω and ωco,a,
+while maintaining the balanced condition, and in order to be able to do the analysis for this gluing
+construction is going to be crucial to have a "concrete" construction of the small resolution M.
+To do this, we will divide the process into three natural steps, and for simplicity assume that ˜
+M has
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+7
+just one singularity (the process obviously applies analogously to the case in which the singularities
+are any finite number). In addition, since our resolution is known to be a crepant resolution, we are
+also going to compute explicitly a holomorphic volume form for M (starting from the one on ˜
+M),
+since such form is a crucial ingredient for the deformation argument in the following section.
+3.1. Step 1. We first glue together the metrics ˜ω and ωco,0 nearby the singularity. To do this,
+thanks to the nature of ODPs, we can introduce, with an abuse of notation (that we will keep on
+using throughout the whole paper - as it does not generate any confusion), a function r on ˜
+M
+which coincides with the (pullback) conical distance from the singularity in a neighborhood of said
+singularity (induced by the local biholomorphism P from Theorem 2.4), and is identically equal to
+1 far away from them (as a consequence we will do the same with the standard cone metric ωco,0).
+In order to perform this first step of the gluing, we will need to work on the cone X and pull back
+on Mreg the object obtained.
+Let p > 0 and consider ε > 0 sufficiently small, such that, thanks to the result from Hein-Sun
+cited above, on the region {0 < r ≤ 4εp} ⊆ X exists a constant λ0 > 0 and is defined a function
+ϕ ∈ C∞
+2+λ0 such that
+P ∗˜ω = ωco,0 + i∂∂ϕ.
+Hence, we introduce a cut-off function χε(x) := χ1(x/εp) where χ1 is a smooth function on
+[0, +∞) such that
+χ1(x) :=
+
+
+
+
+
+0
+if x ≤ 2,
+Non decreasing
+if x ∈ (2, 4),
+1
+if x ≥ 4,
+and define the smooth real (1, 1)-form
+˜ωε := ωco,0 + i∂∂(χε(r)ϕ),
+which for ε sufficiently small is such that
+ωε := (P −1)∗˜ωε
+defines a Kähler metric on Mreg (extended to the whole Mreg in the obvious way). Indeed, we
+notice that on Gε := {2εp < r ≤ 4εp} it holds
+|ωε − ωco,0|ωco,0 = |i∂∂(χε(r)ϕ)|ωco,0 ≤ c(r−2|ϕ| + r−1|∂ϕ|ωco,0 + |i∂∂ϕ|ωco,0) ≤ crλ0,
+from which on Gε we can write
+|∇k
+ωco,0(ωε − ωco,0)|ωco,0 ≤ Crλ0−k,
+for all k ≥ 0.
+Moreover we notice that on {0 < r ≤ 2εp} the metric ωε is exactly conical, i.e. equal to ωco,0,
+while on {r > 4εp} ωε coincides with the background metric ˜ω, and on Gε is the only region where
+ωε is not Ricci-flat.
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+8
+3.2. Step 2. In this second step we instead perform the gluing between the Candelas-de la Ossa
+metrics ωco,a and the standard cone ωco,0, on the small resolution ˆX of the standard conifold X,
+preserving the balanced condition. To do this we recall from Lemma 2.3 that the Candelas-de la
+Ossa metrics are of the form
+ωco,a := i∂∂fa(r3) + 4a2π∗ωF S,
+which away from the exceptional curve have the expansion
+fa(r3) = 3
+2r2 + a2ψa(r),
+where
+ψa(r) = 3 log r − 3 log a +
++∞
+�
+n=0
+cna2nr−2n.
+This suggests introducing a large parameter R ≫ 1 and a smooth cut-off function χR(x) :=
+χ2(x/R) on [0, +∞) such that
+χ1(x) :=
+
+
+
+
+
+1
+if x ≤ 1
+2,
+Non increasing
+if x ∈
+�1
+2, 1
+�
+,
+0
+if x ≥ 1,
+from which we introduce the family of closed (2, 2)-forms
+ω2
+a,R =
+�
+i∂∂
+�3
+2r2 + a2χR(r)ψa(r)
+��2
++ 2a2i∂∂
+�
+χR(r)fa(r3)
+�
+∧ π∗ωF S.
+The reason why the notation ω2
+a,R makes sense is because the results from Michelsohn ([M]) ensure
+us that every real positive closed (n − 1, n − 1)-form admits a unique real positive (1, 1)-form as
+a (n − 1)th root, and the forms we defined above happen to be positive for sufficiently large R.
+Indeed, positivity is clear when r ≤ R
+2 , where ω2
+a,R = ω2
+co,a, and when r ≥ R, where ω2
+a,R = ω2
+co,0.
+In the gluing region GR := {R/2 ≤ r ≤ R}, instead, we need to do a few estimates to confirm
+the positivity. In order to avoid having to impose limitations on the parameter a we substitute the
+functions ψa with ψa + 3 log a; this won’t alter the metrics ωco,a and any of its properties, thus it
+can be done without problems. For simplicity, we will keep writing ψa, and in the following the
+constant c might vary from line to line. We have
+|ω2
+a,R − ω2
+co,0|ωco,0 ≤2a2|ωco,0 ∧ i∂∂(χR(r)ψa(r))|ωco,0 + 2a2|i∂∂(χR(r)fa(r3)) ∧ π∗ωF S|ωco,0
++ a4|i∂∂(χR(r)ψa(r))|2
+ωco,0
+≤ca2(|ωco,0|ωco,0(r−2|ψa| + r−1|∂ψa|ωco,0 + |i∂∂ψa|ωco,0)
++ |π∗ωF S|ωco,0(r−2|fa(r3)| + r−1|∂fa(r3)|ωco,0 + |i∂∂fa(r3)|ωco,0)
++ a2(r−2|ψa| + r−1|∂ψa|ωco,0 + |i∂∂ψa|ωco,0)2)
+≤ca2(|ωco,0|ωco,0(r−2|ψa| + r−1|∂ψa|ωco,0 + |i∂∂ψa|ωco,0)
++ |π∗ωF S|ωco,0(1 + |ωco,0|ωco,0 + r−2|ψa| + r−1|∂ψa|ωco,0 + |i∂∂ψa|ωco,0)
++ a2(r−2|ψa| + r−1|∂ψa|ωco,0 + |i∂∂ψa|ωco,0)2) ≤ (∗)
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+9
+To conclude the estimate let’s take a look at the single terms, keeping in mind that we are taking
+the norms with respect to the standard cone metric. It holds
+• |ωco,0|ωco,0 = O(1);
+• |π∗ωF S|ωco,0 = O(r−2);
+• |ψa| ≤ c(| log r| + �+∞
+n=0 |cn|a2nr−2n) ≤ c| log r|;
+• |∂ψa|ωco,0 ≤ c(r−1 + �+∞
+n=0 |˜cn|a2nr−2n−1) ≤ cr−1;
+• |i∂∂ψa|ωco,0 ≤ c(r−2 + �+∞
+n=0 |˜˜cn|a2nr−2n−2) ≤ cr−2;
+from which we get
+(∗) ≤ ca2(r−2| log r|+r−2+r−2(1+r−2| log r|+r−2)+a2(r−2| log r|+r−2)2) ≤ ca2r−2| log r|,
+and hence
+|∇k
+ωco,0(ω2
+a,R − ω2
+co,0)|ωco,0 ≤ Ca2r−2−k| log r|
+for all k ≥ 0,
+which clearly implies the positivity also on GR, as long as R is chosen to be sufficiently large.
+3.3. Step 3. In this third and last step we want to glue together the metrics ωε from step 1 with the
+metric ωa,R from step 2 by matching isometrically the exactly conical regions. In order to do this
+we are going to need to rescale by a constant λ > 0 the metric on ˆX, and we will now see that this
+constant is a geometric constant, since it is dictated by the geometries of the two metrics we are
+gluing together.
+In what follows we will denote with z the coordinates on Mreg nearby the singularity and with
+ζ the coordinates on ˆX, both given by the identification with the standard conifold X. We then
+consider the regions
+CR := {R/2 ≤ r(ζ) ≤ 4R} ⊆ ˆX
+and
+Cε := {εp/2 ≤ r(z) ≤ 4εp} ⊆ Mreg
+and define a biholomorphism between them by imposing
+ζ =
+� R
+εp
+� 3
+2
+z.
+From this expression we have that on the identified region the following identity holds
+r(ζ) = r
+�� R
+εp
+� 3
+2
+z
+�
+= R
+εp r(z)
+which yields λ = λ(ε, R) :=
+�εp
+R
+�2. From this follows λr2(ζ) = r2(z), and thus on the identified
+conical regions C′
+R := {R ≤ r(ζ) ≤ 2R} ≃ {εp ≤ r(z) ≤ 2εp} =: C′
+ε holds
+λωco,0(ζ) = ωco,0(z),
+and consequently
+λωa,R = ωε.
+Hence, λ is the needed rescaling factor, which allows us to define the glued family of balanced
+metrics on the small resolution M as
+ωa,ε,R :=
+
+
+
+
+
+λωa,R
+on r(ζ) ≤ R,
+ωco,0
+on εp ≤ r(z) ≤ 2εp,
+ωε
+on r(z) ≥ 2εp.
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+10
+Now we will move to the following part in which we will try to understand better the geometry of
+this new family of metrics.
+3.4. Description of the volume. We are mainly interested in the Chern-Ricci tensor - and in par-
+ticular on its potential - hence, in order to understand it, we are going to need to obtain information
+on the volume of our metrics ω = ωε,R := ω1,ε,R. To do this, we need to estimate the distance
+between our glued metrics and the standard cone metric on the gluing region, since in the other
+parts of the manifold we already know the geometry from the initial data. Moreover, inside the
+gluing region there is also an exactly conical region - whose geometry is also understood - which
+separates the two gluing regions from the first two steps, hence we can just estimate the said dis-
+tance separately on the two regions and then take the maximum.
+Clearly, the metric is unaltered on the gluing region from step 1, thus we still have on Gε that
+|∇k
+ωco,0(ωε − ωco,0)|ωco,o ≤ crλ0−k,
+for all k ≥ 0.
+On the other hand, since in step 3 we had to rescale the metric nearby the exceptional curve, we
+are going to need to repeat the estimate to understand how the rescaling affected the distance from
+the standard cone metric. At some point we are going to link the parameters ε and R. Again the
+constant c might be varying from line to line. We have
+(ω2
+ε,R − ω2
+co,0)(ζ) =λ2(2ωco,0 ∧ i∂∂(χR(r)ψ1(r)) + 2i∂∂(χR(r)f1(r3)) ∧ π∗ωF S
++ i∂∂(χR(r)ψ1(r))2)(ζ) = (∗)
+Let’s now change coordinates from ζ to z, recalling that λr2(ζ) = r2(z) and noticing that, by
+construction, π∗ωF S is invariant under rescalings.
+(1)
+(∗) =λ2
+
+λ−1ωco,0 ∧ i∂∂
+
+χR
+�
+λ− 1
+2 r
+�
+
+3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+
+
+
++ π∗ωF S ∧ i∂∂
+
+χR
+�
+λ− 1
+2 r
+�
+
+λ−1 3
+2r2 + 3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+
+
+
++
+
+i∂∂
+
+χR
+�
+λ− 1
+2 r
+�
+
+3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+
+
+
+
+
+2
+ (z).
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+11
+What we want to do now is estimate the norm (with respect to the cone metric) of this quantity,
+thus we start computing the derivatives that appear in the above expression. We have
+∂∂
+
+χR
+�
+λ− 1
+2 r
+�
+
+3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+
+
+
+= λ−1
+R2 χ′′
+R
+�
+λ−1/2r
+� �
+3 log r − 3
+2 log λ +
+�
+n∈N
+cnλnr−2n
+�
+∂r ∧ ∂r
++ 2λ−1/2
+R
+χ′
+R
+�
+λ−1/2r
+� �
+3r−1 +
+�
+n∈N
+˜cnλnr−2n−1
+�
+∂r ∧ ∂r
++ λ−1/2
+R
+χ′
+R
+�
+λ−1/2r
+�
+
+3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+ ∂∂r
++ χR
+�
+λ−1/2r
+� �
+−3r−2 +
+�
+n∈N
+˜˜cnλnr−2n−2
+�
+∂r ∧ ∂r
++ χR
+�
+λ−1/2r
+� �
+3r−1 +
+�
+n∈N
+˜cnλnr−2n−1
+�
+∂∂r,
+and
+∂∂
+
+χR
+�
+λ− 1
+2 r
+�
+
+λ−1 3
+2r2 + 3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+
+
+
+=
+�
+λ−1
+R2 χ′′
+R
+�
+λ−1/2r
+�
+r2 + 4λ−1/2
+R
+χ′
+R
+�
+λ−1/2r
+�
+r + 2χR
+�
+λ−1/2r
+��
+∂r ∧ ∂r
++
+�
+λ−1/2
+R
+χ′
+R
+�
+λ−1/2r
+�
+r2 + 2χR
+�
+λ−1/2r
+�
+r
+�
+∂∂r
++ ∂∂
+
+χR
+�
+λ− 1
+2 r
+�
+
+3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+
+
+
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+12
+From here we can obtain the estimates
+������
+∂∂
+
+χR
+�
+λ− 1
+2 r
+�
+
+3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+
+
+
+������
+ωco,0
+≤ cλ−1
+R2
+�
+| log r| + | log λ| +
+�
+n∈N
+|cn|λnr−2n
+�
++ cλ−1/2
+R
+r−1
+
+1 +
+�
+n∈N
+|˜cn|λnr−2n + | log r| + | log λ| +
+�
+n≥0
+|cn|λnr−2n
+
+
++ cr−2
+�
+1 +
+�
+n∈N
+|˜˜cn|λnr−2n +
+�
+n∈N
+|˜cn|λnr−2n
+�
+and������
+∂∂
+
+χR
+�
+λ− 1
+2r
+�
+
+λ−1 3
+2r2 + 3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+
+
+
+������
+ωco,0
+≤ c
+�
+λ−1
+R2 r2 + λ−1/2
+R
+(r + 1)
+�
++
+������
+∂∂
+
+χR
+�
+λ− 1
+2 r
+�
+
+3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+
+
+
+������
+ωco,0
+In order to conclude the estimate, we impose R := ε−q, with q > 0 to be chosen, from which
+follows λ = ε2(p+q). Also we are now going to use that we are in the region {2εp ≤ r ≤ 4εp},
+which among the others implies the estimate
+| log r| ≤ | log 4| + p| log ε| ≤ c(1 + | log ε|).
+We thus get
+������
+∂∂
+
+χR
+�
+λ− 1
+2r
+�
+
+3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+
+
+
+������
+ωco,0
+≤ cε−2p
+�
+1 + | log ε| +
+�
+n∈N
+|cn|(1/2)2n
+�
++ cε−2p
+
+1 +
+�
+n∈N
+|˜cn|(1/2)2n + 1 + | log ε| +
+�
+n≥0
+|cn|(1/2)2n
+
+
++ cε−2p
+�
+1 +
+�
+n∈N
+|˜˜cn|(1/2)2n +
+�
+n∈N
+|˜cn|(1/2)2n
+�
+≤ cε−2p| log ε|
+and
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+13
+������
+∂∂
+
+χR
+�
+λ− 1
+2r
+�
+
+λ−1 3
+2r2 + 3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+
+
+
+������
+ωco,0
+≤ c(1 + ε−p(ε + 1)) +
+������
+∂∂
+
+χR
+�
+λ− 1
+2 r
+�
+
+3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+
+
+
+������
+ωco,0
+≤ c(ε−p + ε−2p| log ε|) ≤ cε−2p| log ε|
+Hence, putting together this estimates we can finally obtain
+|ω2
+ε,R − ω2
+co,0|ωco,0
+≤ cε4(p+q)
+
+
+ε−2(p+q)
+������
+i∂∂
+
+χR
+�
+λ− 1
+2r
+�
+
+3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+
+
+
+������
+ωco,0
++ ε−2p
+������
+i∂∂
+
+χR
+�
+λ− 1
+2 r
+�
+
+λ−1 3
+2r2 + 3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+
+
+
+������
+ωco,0
++
+������
+i∂∂
+
+χR
+�
+λ− 1
+2r
+�
+
+3 log r − 3
+2 log λ +
+�
+n≥0
+cnλnr−2n
+
+
+
+
+������
+2
+ωco,0
+
+
+
+≤ cε4(p+q)(ε−4p−2q| log ε| + ε−4p| log ε|2) ≤ cε2q| log ε|
+which implies, on the whole gluing region, that for all k ≥ 0 holds
+|∇k
+ωco,0(ω2
+q,ε − ω2
+co,0)|ωco,0 ≤ crm−k,
+where m = m(λ0, p, q).
+In order to obtain geometric information from this estimate, we are going to need to recall a result
+from Michelsohn ([M]). This result gives an explicit isomorphism between the cone of strictly
+positive (1, 1)-forms and the cone of strictly positive (n − 1, n − 1)-forms. In particular, if Ψ ∈
+�n−1,n−1 and ψ ∈ �1,1 are strictly positive forms such that ψn−1 = Ψ, we can always find a base
+{ei} of (1, 0)-forms that "diagonalizes" simultaneously Ψ and ψ, giving the expressions
+Ψ =
+n
+�
+j=1
+Λj �
+ej ∧ Jej
+and
+ψ =
+n
+�
+j=1
+λjej ∧ Jej,
+where J is the complex structure and
+�
+ej ∧ Jej means the wedge product of all the terms of the
+form ek ∧ Jek except the one corresponding to the index j. Moreover, Michelsohn’s theorem gives
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+14
+us also a formula relating the coefficients of ψ and Ψ in this basis, that is
+λj = (Λ1 · ... · Λn)
+1
+n−1
+Λj
+for all j = 1, ..., n.
+We will then use this to obtain an expansion for ω. In order to have a more compact notation we
+will introduce the notation O(rl) to denote the decay of a function (or of the norm of a tensor) and
+all its derivatives; to be clearer, saying that some tensor θ is such that θ = O(rl) means that for all
+k ≥ 0 holds |∇k
+ωco,0θ| ≤ crl−k.
+Remark 3.2. Notice that we can choose a basis {ej} of (1, 0)-forms diagonalizing simultaneously
+ωco,0 (we can actually assume it to be the identity) and ω; this means that also ω2
+co,0 and ω2 are
+diagonal (in the sense of (n − 1, n − 1)-forms, implying that also the term O(rλ0) is necessarily
+diagonal with respect to this basis. Thus we can write
+ω2 =
+n
+�
+j=1
+(αj + O(rm)) �
+ej ∧ Jej
+and applying Michelson’s theorem with Λj = αj + O(rm), we obtain ω = �n
+j=1 λjej ∧ Jej, with
+λj = ((α1 + O(rm))(α2 + O(rm))(α3 + O(rm)))
+1
+2
+αj + O(rm)
+= (α1α2α3)
+1
+2
+αj
++ O(rm),
+which implies, again thanks to Michelson’s theorem
+ω =
+n
+�
+j=1
+�
+(α1α2α3)
+1
+2
+αj
++ O(rm)
+�
+ej ∧ Jej = ωco,0 + O(rm).
+Hence we can write the volume form as ω3 = ω3
+co,0 + O(rm).
+3.5. The Chern-Ricci potential. In order to use this description of the volume to estimate the
+Chern-Ricci potential on the gluing region we are also going to need to understand how the holo-
+morphic volume form of the resolution is related to the holomorphic volume of our background
+Calabi-Yau singular manifold.
+Before doing it we start by fixing some notation. Denote
+• with Ω the holomorphic volume of Mreg such that
+˜ω3 = iΩ ∧ Ω;
+• with Ω0 the holomorphic volume of the cone X such that
+ω3
+co,0 = iΩ0 ∧ Ω0;
+• with χ the holomorphic volume of the resolution ˆX such that
+(λωco,1)3 = iχ ∧ χ.
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+15
+In order to explain clearly how these forms are related and how we can use them to construct
+a holomorphic volume on M we shall take a step back and watch closer at the topological gluing
+between Mreg and ˆX.
+Call P the biholomorphism given from Theorem 2.4, and we can suppose ε to be sufficiently small
+so that we can assume that P is defined (eventually after a restriction) on the set {0 < r(z) ≤ 4εp}.
+Moreover, if we call Q the biholomorphism introduced at the beginning of Step 3 of the gluing
+construction, we can assume it to be defined not only on the region written before, but actually on
+the whole {0 < r(ζ) ≤ 4R}. This allows us to give a topologically concrete description of M, that
+is
+M = Mreg
+◦� ˆX
+∼
+,
+where the equivalence relation ∼ is defined as
+y ∈ Mreg, x ∈ ˆX, y ∼ x if and only if P −1(y) = Q−1(x).
+Thanks to this explicit description we can observe that on the gluing region holds
+Q∗χ = Ω0
+and there exists a holomorphic function h on the gluing region such
+P ∗Ω = fQ∗χ = hΩ0,
+and h can actually be extended to the whole ˆX. We can however obtain even more information on
+h recalling again Theorem 2.4; indeed said theorem guarantees us that
+P ∗Ω ∧ P ∗Ω = Ω0 ∧ Ω0 + i∂∂Φ
+where Φ is a smooth (2, 2)-form that behaves as = O(r2+λ0) nearby the singularity. From this we
+get that
+(|h|2 − 1)Ω0 ∧ Ω0 = i∂∂Φ,
+and thus
+|h|2 = 1 + O(rλ0) on (a neighborhood of P1 in) ˆX,
+from which, by continuity, we have that |h|2 ≡ 1 on the exceptional curve. Thus we can define the
+holomorphic volume ˆΩ of M by gluing together hχ and Ω, and thus define a global Chern-Ricci
+potential as
+f = fa,q,ε := log
+�
+iˆΩ ∧ ˆΩ
+ω3
+�
+.
+We now conclude this section by describing the behaviour of f in all the regions of M, to show
+that it is suitable to apply a deformation argument similar to the one done in [BM]. We have
+• on {r(z) > 4εp} hold ω = ˜ω and ˆΩ = Ω, thus f ≡ 0;
+• on {2εp ≤ r(z) ≤ 4εp} hold ω = ωco,0 + O(rm) and ˆΩ ∧ ˆΩ = Ω0 ∧ Ω0 + O(rλ0), from
+which we have
+f = log
+�
+ω3
+co,0 + O(rm)
+Ω0 ∧ Ω0 + O(rλ0)
+�
+= log(1 + O(r ˜m)) = O(r ˜m),
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+16
+where ˜m = min{λ0, m};
+• on{εp ≤ r(z) ≤ 2εp} hold ω = ωco,0 and ˆΩ ∧ ˆΩ = i(1 + O(rλ0))Ω0 ∧ Ω0, from which
+follows f = O(rλ0);
+• on {εp/2 ≤ r(z) ≤ εp} hold ω = ωco,0+O(rm) and ˆΩ∧ ˆΩ = Ω0∧Ω0+O(rλ0), implying
+again f = O(r ˜m);
+• on {r(z) < εp/2} hold ω = λωco,1 and ˆΩ∧ ˆΩ = i(1+O(rλ0))Ω0 ∧Ω0, giving once again
+f = O(r ˜m).
+Thus we can write globally (on M) that
+|f| ≤ cr ˜m.
+4. THE DEFORMATION ARGUMENT
+In this last section we will see that what was built in the previous section are exactly the in-
+gredients we need to introduce a deformation argument in the same fashion as [BM], in order to
+obtain a balanced Chern-Ricci flat metric on our small resolution M. We will also analyze the co-
+homology class of the metric obtained and see why said metric is interested in the framework of
+the Hull-Strominger system.
+4.1. The strategy. We will now set up the problem for this section. First of all we need to intro-
+duce a deformation of the metric that preserves the balanced condition, and this can be done using
+the one introduced in [FWW]
+ω2
+ψ := ω2 + i∂∂(ψω),
+ψ ∈ C∞(M, R) such that ω2
+ψ > 0.
+Notice that again, thanks to the results of Michelsohn in [M], writing immediately ω2
+ψ makes sense.
+Thus the problem we are interested in solving, following what was done in [BM], is the equation
+ω3
+ψ = efω3
+for ψ ∈ C∞(M, R) such that ω2
+ψ > 0.
+Remark 4.1. The equation introduced above makes sense, because, as we’ve seen, f = O(r ˜m),
+thus ef = 1 + O(r ˜m), meaning that efω3 is nearby ω3 itself, hence it makes sense to try to obtain
+it as a small deformation of ω.
+For practicality is useful to reformulate our equation as an operator on the space of smooth
+functions, thus we introduce F : C∞(M, R) → C∞(M, R) as
+F(ψ) = Fε(ψ) :=
+ω3
+ψ
+ω3 − ef.
+Our aim is to solve this equation through a fixed point argument, i.e. turning the problem in a new
+one that can be solved by applying Banach’s Lemma. In order to achieve this, the first step to take is
+to start studying the linearization at 0 of the operator F. To do this we shall introduce the notation
+ω′
+0 := d
+dt |t=0ωtu, where ωtu is the curve corresponding to the tangent vector u ∈ C∞(M, R), and
+compute the derivative at zero of ω3
+tu in two different ways:
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+17
+d
+dt |t=0
+ω3
+tu = 3ω2 ∧ ω′
+0;
+d
+dt |t=0
+ω3
+tu = i∂∂(uω) ∧ ω + ω2 ∧ ω′
+0.
+Even though none of these two expressions are explicit, we can put them together to obtain an
+explicit one, that is
+d0F(u) = Lu = Lεu := 3
+2
+i∂∂u ∧ ω2 + ui∂ω ∧ ∂ω
+ω3
+.
+Remark 4.2. The estimates done in the previous sections ensure us that, on the gluing region from
+Step 2 of the gluing construction, hold
+∂ω = O(r2(q/p)−1| log r|)
+and
+∂∂ω = O(r2(q/p)−2| log r|),
+and ∂ω = 0 and i∂∂ω = 0 everywhere else. Thus if q > p, then ∂ω, i∂∂ω → 0 as ε → 0, showing
+that L is a bounded operator, and implying that ˜m = m = λ0 (assuming λ0 ≤ 2).
+Before starting the analytical part, we shall establish once for all a value for p and q, and in light
+of Remark 4.2, we have that a good choice is given by
+p = 2
+5
+and
+q = 3
+5,
+from which we get λ = ε2 and m = 2
+5λ0.
+4.2. Weighted analysis. Our aim is now to study the invertibility of this linear operator, and we
+wish to do this in suitable weighted function spaces. In order to introduce said spaces we shall start
+by introducing a weight function useful in our situation, and for simplicity we may assume that the
+biholomorphism P is defined on the region {r(z) ≤ 1} (this is true up to a rescaling). Define then
+ρ = ρε(z) :=
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ε
+on r(z) ≤ ε,
+non decreasing
+on ε ≤ r(z) ≤ 2ε,
+r(z)
+on 2ε ≤ r(z) ≤ 1/2,
+non decreasing
+on 1/2 ≤ r(z) ≤ 1,
+1
+on r(z) ≥ 1,
+where we recall that z are the "small" coordinates around the singularity in ˜
+M. Using this weight
+function we can introduce the weighted Hölder norm and its corresponding weighted Hölder spaces
+Ck,α
+ε,b (M), where k ≥ 0, α ∈ (0, 1) is the Hölder constant, b ∈ R is the weight and ε indicates the
+dependence on the metric ω obtained by the gluing construction done above. We define
+||u||Ck,α
+ε,b (M) :=
+k
+�
+i=0
+sup
+M
+|ρb+i∇i
+εu|ω
++
+sup
+dε(x,y)<injε
+����min
+�
+ρb+k+α(x), ρb+k+α(y)
+� ∇k
+εu(x) − ∇k
+εu(y)
+dε(x, y)α
+����
+ω
+,
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+18
+where injε is the injectivity radius of the metric ω, and thus interpret F (and L) as operators defined
+as F : C2,α
+ε,b (M) → C0,α
+ε,b+2(M).
+Following now a blended strategy between the one from Székelyhidi (as in Chapter 8 in [Sz]) and
+the one from Tosatti and Weinkove (see [TW]), we consider a variation of our operator by fixing a
+point x ∈ M, where x is away from the gluing region (for example we can assume x to be outside
+of the open set on which the biholomorphism P is defined, and in the region where the weight is
+identically equal to 1), and defining
+˜F(ψ) :=
+ω3
+ψ
+ω3 − ef+evx(ψ)
+(where evx is the evaluation operator at x), with consequent linearization at zero
+˜L = L + efevx.
+Thus the new problem to solve is
+(2)
+˜F(ψ) = 0,
+whose solution clearly still gives a balanced Chern-Ricci flat metric.
+We then wish to prove the following lemma.
+Lemma 4.3. With the same notations as above, there exist b ∈ R and c > 0 such that for sufficiently
+small ε it holds
+||u||C2,α
+ε,b ≤ c||˜Lu||C0,α
+ε,b+2,
+for all u ∈ C2,α
+ε,b .
+Proof. Suppose by contradiction that the above inequality does not hold. This means that for all
+n ∈ N we can find εn > 0 and un ∈ C2,α
+εn,b such that εn → 0 as n → 0, ||un||C2,α
+εn,b = 1 and
+(3)
+||˜Lun||C0,α
+εn,b+2 < 1
+n.
+In the first place we analyze what happens on M \ P1 ≃ Mreg, i.e. away from the exceptional
+curve. The properties of the sequence {un}n∈N guarantee us that up to subsequences we may
+assume un → u∞ uniformly on compact subsets of Mreg in the sense of C0,α
+b
+, with respect to
+the background Calabi-Yau metric ˜ω. Moreover, since for any compact set K ⊆ Mreg there exists
+nK ∈ N such that for all n ≥ nK, on K it holds ω = ˜ω and hence ∇ω = ∇˜ω, we actually have
+C2,α
+b
+-convergence (again uniformly on compact subsets of Mreg). Also, for the same reason, on
+any compact subset of Mreg, for n sufficiently large the Chern-Ricci potential is identically zero,
+thus we get ˜L = L + evx; and still choosing n sufficiently large, we end up with ω = ˜ω on K,
+resulting in
+Lun = 3
+2
+i∂∂un ∧ ˜ω2
+˜ω3
+,
+i.e. a scalar multiple of the laplacian of un with respect to the Kähler metric ˜ω. Thus taking the
+limit for n → +∞ of ˜Lun yields
+(4)
+L∞u∞ + u∞(x) = 0.
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+19
+We shall now prove that u∞ is necessarily identically zero on the whole Mreg. Indeed, take δ > 0
+and Bδ a ball of radius δ around the singularity, and notice that
+0 =
+�
+Mδ:=Mreg\Bδ
+L∞u∞ + u∞(x)dV˜ω = −
+�
+∂Bδ
+i∂u∞ ∧ ˜ω2 + Vol(Mδ)u∞(x).
+Writing then with d ˆV the volume induced on ∂Bδ induced by ˜ω, we notice that
+����
+�
+∂Bδ
+∂u∞ ∧ ˜ω2
+���� ≤
+�
+∂Bδ
+|∂u∞|ωd ˆV ≤ cδ4−b,
+thus assuming b < 4 and taking the limit for δ → 0 in (4) we get u∞(x) = 0, and, consequently,
+that u∞ is harmonic on the whole Mreg. Notice now that, since L∞u∞ = 0, we get
+(5)
+0 =
+�
+Mδ
+u∞L∞u∞˜ω3 =
+�
+Mδ
+u∞i∂∂u∞ ∧ ˜ω2
+and since ˜ω is balanced (actually Kähler) it holds
+d(i∂u∞ ∧ (u∞˜ω2)) = u∞i∂∂u∞ ∧ ˜ω2 + i∂u∞ ∧ ∂u∞ ∧ ˜ω2,
+which integrated on Mδ and combined with (5) gives
+(6)
+0 =
+�
+∂Bδ
+u∞i∂u∞ ∧ ˜ω2 =
+�
+Mδ
+i∂u∞ ∧ ∂u∞ ∧ ˜ω2
+= 2
+�
+Mδ
+i∂u∞ ∧ ∂u∞ ∧ ∗˜ω = 2
+�
+Mδ
+⟨i∂u∞ ∧ ∂u∞, ˜ω⟩˜ω3 = 2
+�
+Mδ
+|∂u∞|2˜ω3.
+But since
+����
+�
+∂Bδ
+u∞i∂u∞ ∧ ˜ω2
+���� ≤
+�
+∂Bδ
+|u∞|ω|∂u∞|ωd ˆV ≤ cδ4−2b,
+thus choosing b < 2 and taking the limit for δ → 0 in (6) we get
+�
+Mreg |∂u∞|2
+ω ˜ω3 = 0, that is
+u∞ ≡ const., and since u∞(x) = 0, necessarily u∞ ≡ 0 on the whole Mreg.
+Let now Mc := {r(z) ≥ 1/2} ⊆ Mreg be the compact set on which un → 0 uniformly in C2,α
+b
+. To
+obtain a contradiction we want to prove that {un}n∈N admits a subsequence uniformly convergent
+to zero in C2,α
+b
+also on A := {r(z) < 1/2}.
+In order to work in this region, it is simpler to shift to the "large" coordinates ζ, i.e. the coordinates
+on ˆX away from the exceptional divisor. It is then useful to recall the relations
+ζ = ε−3/2z
+and
+r(z) = εr(ζ),
+from which we can write down the explicit identification
+�
+r(z) < 1
+2
+�
+= A ≃ ˜A = ˜Aε =
+�
+r(ζ) < 1
+2ε−1
+�
+;
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+20
+this last set ˜A is the one we will be working on.
+The first thing to do is rewrite the weight function in terms of this coordinates on ˜A, resulting in
+ρ =
+
+
+
+
+
+ε
+on r(ζ) ≤ 1,
+non decreasing
+on 1 ≤ r(ζ) ≤ 2,
+εr(ζ)
+on 2 ≤ r(ζ) ≤ 1/2ε−1.
+Notice that the entire gluing region of the metric (from the previous step) is entirely contained
+inside the third region, i.e. {2 ≤ r(ζ) ≤ 1/2ε−1}.
+We now go back to our sequence {un}n∈N. Since ||un||C2,α
+εn,b = 1 for all n ∈ N, we have in
+particular that on all ˜An := ˜Aεn holds
+|ρbun| ≤ c.
+Introducing then the new sequence
+Un := εb
+nun,
+the above weighted estimates for un imply the following ones for this new sequence:
+
+
+
+
+
+|Un| ≤ c
+on r(ζ) ≤ 1,
+|Un| ≤ c
+on 1 ≤ r(ζ) ≤ 2,
+|Un| ≤ cr−b(ζ)
+on 2 ≤ r(ζ) ≤ 1/2ε−1
+n .
+This estimates for Un suggest us to introduce a new weight function ˜ρ = ˜ρn on ˜An given by
+˜ρ(ζ) =
+
+
+
+
+
+1
+on r(ζ) ≤ 1,
+non decreasing
+on 1 ≤ r(ζ) ≤ 2,
+r(ζ)
+on 2 ≤ r(ζ) ≤ 1/2ε−1
+n ,
+,
+with which we get that
+|˜ρbUn| ≤ c,
+and analogous weighted estimates also for ∇Un and ∇2Un, hence again by Ascoli-Arzelà theorem
+we have that Un → U∞ uniformly on compact sets of ˆX (since ˜An → ˆX) in the sense of ˜C2,α
+b
+=
+C2,α
+b
+(˜ρ), where this last space is the weighted Hölder space on ˆX identified by the weight ˜ρ and
+the Candelas-de la Ossa metric ωco,1.
+On the other hand it holds
+ρb+2Lun = ˜ρb+2∆ωco,1Un,
+hence
+(7)
+||Lun||C0,α
+εn,b+2 = ||∆ωco,1Un|| ˜C0,α
+b+2,
+and since 1
+n > ||˜Lun||C0,α
+εn,b+2 and un(x) → 0 hold, taking the limit in (7) we obtain that U∞ is
+harmonic. Moreover, if we assume b > 0, then U∞ is a harmonic function decaying at infinity, thus
+necessarily U∞ ≡ 0, and thus Un
+˜C2,α
+b→ 0 uniformly on compact sets of ˆX.
+If we are now able to prove that Un admits a subsequence converging uniformly to zero on the
+whole ˆX in the sense ˜C0
+b we get our contradiction, and we are done. Indeed, if Un
+˜C0
+b
+→ 0 uniformly
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+21
+(up to subsequences) on ˆX, then scaled Schauder estimates imply that also Un
+˜C2,α
+b→ 0 uniformly,
+which is the same as saying un
+˜C2,α
+εn,b
+→
+0 uniformly on {r(z) < 1/2}. Thus {un}n∈N up to subse-
+quences is uniformly convergent to zero on the whole manifold M, which is a contradiction with
+the fact that ||un||C2,α
+εn,b = 1 for all n ∈ N.
+Now we will prove that the said uniformly convergent subsequence exists. If by contradiction this
+was not the case, since we have the uniform convergence on compact sets, we would be able to find
+γ > 0 and {xn}n∈N ⊆ ˆX, xn ∈ ˜An, such that Rn := r(ζ(xn)) → +∞ and Rb
+nUn(xn) ≥ γ for
+all n ∈ N. But {xn}n∈N is a bounded sequence (since it lies in {r(z) < 1/2}), thus if we name
+rn := r(z(xn)) and recall that rn = εnRn, up to subsequences we can end up into two cases:
+(i) if xn → x∞ such that r(x∞) > 0, then, since un is uniformly convergent on compact sets
+on Mreg, we get that un(xn) is bounded, giving
+0 < γ ≤ Rb
+nUn(xn) = (Rnεn)bun(xn) = rb
+nun(xn) −→
+n→∞ 0,
+which is a contradiction;
+(ii) if rn → 0, we consider the biholomorphisms σn : Bn → A \ {0}, given by
+σn(z′) := r3/2
+n z′,
+where Bn := {0 < r(z′) < r−1
+n
+2 } ⊆ X. Then, if we endow Bn with the metric
+ηn := r−2
+n σ∗
+nω,
+using (1) it is easy to notice that the couple (Bn, ηn) converges to (X, ωco,0), i.e. the stan-
+dard Calabi-Yau cone. If we then introduce the functions
+wn := rb
+nσ∗
+nun
+on Bn, we notice that the pullback of the weight function ρ gives
+ρ′(z′) = σ∗
+nρ(z′) =
+
+
+
+
+
+εn
+on r(z′) < R−1
+n ,
+non decreasing
+on R−1
+n
+≤ r(z) ≤ 2R−1
+n ,
+rnr(z′)
+on 2R−1
+n
+≤ r(z) < r−1
+n
+2 ,
+from which we get (pulling back the inequality ρbun ≤ 1)
+(8)
+rb(z′)wn(z′) ≤ 1
+on each z′ ∈ X (assuming n to be sufficiently large). Hence, this shows that for any
+compact K ⊆ X, we can chose n ∈ N to be sufficiently large in order to have K ⊆ Bn
+and ρ′(z′) = rnr(z′) on the whole K, and get that wn is uniformly bounded on K; and
+since this works for any compact K ⊆ X, we obtain that - up to subsequences - {wn}n∈N
+converges uniformly on compact sets of X to a function w∞, and from (8) we get that
+w∞ is decaying at infinity. Moreover, recalling that Rb
+nUn(xn) ≥ γ for all n ∈ N, if we
+introduce the sequence yn := σ−1
+n (xn), it is straighforward to notice that from its definition
+follows that wn(yn)
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+22
+geqγ and ||yn||ηn = 1 for all n ∈ N, thus implying that - up to subsequences - yn → y∞ ∈
+X, and hence
+(9)
+w∞(y∞) > 0.
+Now, if we recall the definition of the operator ˜L and take the pullback with respect to σn
+of ρb+2 ˜Lun, it is immediate to see that on every compact K ⊆ X we get
+(10)
+σ∗
+n
+�
+ρb+2 ˜Lun
+�
+= 3
+2rb+2(z′)i∂∂wn ∧ η2
+n
+η3n
++ 3
+2
+i∂ηn ∧ ∂ηn
+η3n
++ rb+2
+n
+rb+2(z′)un(x).
+Analyzing then the single terms, we can notice that as n → +∞
+• σ∗
+n
+�
+ρb+2 ˜Lun
+�
+→ 0 from the hypothesis (3);
+•
+i∂∂wn ∧ η2
+n
+η3n
+→ ∆ωco,0w∞
+from the above observations;
+• rb+2
+n
+rb+2(z′)un(x) → 0 trivially;
+•
+i∂ηn ∧ ∂ηn
+η3n
+→ 0,
+since from Remark 4.2 we can pullback with σn the inequality
+|∂ω|ω ≤ cr2| log r|,
+and obtain
+|∂ηn|ηn ≤ cr3
+nr2(z′)(| log rn| + | log r(z′)|) → 0.
+Hence, taking the limit as n → +∞ in (10) we obtain
+∆ωco,0w∞ ≡ 0,
+and since it holds on every compact K subseteqX, we get that w∞ is harmonic on the
+whole cone X. But now, since it holds (8), we can apply Lemma 6.9 from [CPY1] (actually
+just its proof applied to ±w∞, without the hypothesis w∞ ≥ 0), and obtain that necessarily
+w∞ ≡ 0 on X, which is a contradiction, as it holds (9).
+Thus the proof is complete.
+□
+As a direct consequence we get
+Lemma 4.4. The operator ˜L : C2,α
+ε,b (M) → C0,α
+ε,b+2(M) defined above is an isomorphism.
+Proof. Notice that L is elliptic and evx is a compact operator, thus L and ˜L share their index, which
+is zero. Moreover, by Lemma 4.3 we have that ˜L is injective, thus we automatically get that ˜L is
+also surjective and has bounded inverse, thus ˜L is an isomorphism.
+□
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+23
+With this result we can now show how to reformulate the original equation as a fixed point
+problem.
+In order to do this we shall introduce the operators ˆF, E, G : C2,α
+ε,b (M) → C0,α
+ε,b+2(M) defined as
+ˆF(ψ) :=
+ω3
+ψ
+ω3 ,
+E(ψ) := ef+evx(ψ)
+and
+G(ψ) = efevx(ψ),
+from which we can write
+˜F = ˆF − E.
+Now, we can consider the expansion
+ˆF(ψ) = ˆF(0) + L(ψ) + ˆQ(ψ),
+and thus rewrite Equation (2) as
+ˆF(0) + L(ψ) + ˆQ(ψ) − E(ψ) = 0.
+Here, we notice that ˜L = L − G, thus we can rewrite Equation (2) once more and get
+ˆF(0) + ˜L(ψ) + ˆQ(ψ) + G(ψ) − E(ψ) = 0,
+and using now Lemma 4.4, we get that Equation (2) is therefore equivalent to
+(11)
+ψ = ˜L−1(E(ψ) − G(ψ) − ˆF(0) − ˆQ(ψ)) =: N(ψ),
+i.e. the search for a fixed point for the operator N : C2,α
+ε,b (M) → C2,α
+ε,b (M). To do this, we will
+have to identify the open set on which we wish to apply Banach’s Lemma, and show that on said
+open set N can be restricted and gives rise to a contraction.
+The first thing to do, is the following remark.
+Remark 4.5. If C, τ > 0, and ϕ is a function on M such that ||ϕ||C2,α
+ε,−2 ≤ Cετ, thanks to Remark
+4.2 it is straightforward to see that
+||i∂∂(ϕω)||C0,α
+ε,0 ≤ ||ϕ||C2,α
+ε,−2 ≤ Cετ,
+thus we’re guaranteed that, choosing ε to be sufficiently small, ω2
+ϕ > 0, and thus it’s square root ωϕ
+exists and is a balanced metric. Moreover, we can apply again the argument used in Remark 3.2,
+and obtain that if ||ϕ||C2,α
+ε,−2 ≤ Cετ, then
+|ωϕ − ω|ω ≤ c||ϕ||C2,α
+ε,−2 ≤ cετ,
+which also implies that ωϕ → ω, as ε → 0.
+Thanks to this remark, we have a suggestion on how to choose the open set on which apply
+Banach’s Lemma, hence we introduce
+U := {ϕ ∈ C2,α
+ε,b | ||ϕ||C2,α
+ε,b < ˜cεb+2+τ} ⊆ C2,α
+ε,b ,
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+24
+and we notice that for every ϕ ∈ U it holds ||ϕ||C2,α
+ε,−2 ≤ Cετ, with C independent of ϕ and ε.
+We will now prove that on U, the operator N is a contraction. In particular, given ϕ1, ϕ2 ∈ U, we
+want to estimate
+N(ϕ1) − N(ϕ2) = ˜L−1(E(ϕ1) − G(ϕ1) − (E(ϕ2) − G(ϕ2)) − ( ˆQ(ϕ1) − ˆQ(ϕ2))).
+Starting by analyzing the ˆQ-term, we notice that by the Mean Value Theorem we can find t ∈ [0, 1]
+such that
+ˆQ(ϕ1) − ˆQ(ϕ2) = d ˆQη(ϕ1 − ϕ2) = (Lη − L)(ϕ1 − ϕ2),
+where η = tϕ1 + (1 − t)ϕ2 ∈ U, and Lη is the linearization of ˆF at η. With the same strategy used
+to compute L we can easily obtain an expression for Lη, and thus get
+(Lη − L)(ϕ1 − ϕ2) = 3
+2
+(ωη − ω) ∧ i∂∂((ϕ1 − ϕ2)ω)
+ω3
+.
+From here, taking the norms with respect to ω, we can use the fact that η ∈ U together with Remark
+4.5, to obtain
+|(Lη − L)(ϕ1 − ϕ2)| ≤ c|ωη − ω|ω|i∂∂((ϕ1 − ϕ2)ω)|ω ≤ cετ|i∂∂((ϕ1 − ϕ2)ω)|ω,
+and thus, by multiplying the inequality with ρb+2, get
+(12)
+|| ˆQ(ϕ1) − ˆQ(ϕ2)||C0,α
+b+2,ε ≤ cετ||ϕ1 − ϕ2||C2,α
+ε,b .
+Focusing instead on the (E − G)-term, and again using that ϕ1, ϕ2 ∈ U together with Remark 4.5,
+we get
+|E(ϕ1) − G(ϕ1) − (E(ϕ2) − G(ϕ2))| =ef|(ϕ2(x) − ϕ1(x) + eϕ1(x) − eϕ2(x)|
+≤c(|ϕ1(x)| + |ϕ2(x)|)|ϕ1(x) − ϕ2(x)|
+≤cετ|ϕ1(x) − ϕ2(x)|.
+Now, as above, multiplying the inequality with ρb+2 and recalling the choice of x ∈ M and the
+definition of ρ, we get
+ρb+2|E(ϕ1) − G(ϕ1) − (E(ϕ2) − G(ϕ2))| ≤ cετρb+2(x)|ϕ1(x) − ϕ2(x)|
+and hence
+(13)
+||E(ϕ1) − G(ϕ1) − (E(ϕ2) − G(ϕ2))||C0,α
+b+2,ε ≤ cετ||ϕ1 − ϕ2||C2,α
+ε,b .
+Thus, combining estimates (12) and (13) with Lemma 4.3, we get
+(14)
+||N(ϕ1) − N(ϕ2)||C2,α
+ε,b ≤ cετ||ϕ1 − ϕ2||C2,α
+ε,b ,
+hence, choosing ε sufficiently small ensures us that N is a contraction on U.
+We are left with proving that N(U) ⊆ U. To do this, we shall impose a restriction on τ, that is
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+25
+τ < 2
+5λ0, and we will use it in the inequalities to follow. Take then ϕ ∈ U, thanks to estimate (14)
+and Lemma 4.3, we have
+||N(ϕ)|||C2,α
+ε,b ≤||N(ϕ) − N(0)|||C2,α
+ε,b + ||N(0)|||C2,α
+ε,b
+≤cετ||ϕ|||C2,α
+ε,b + ||˜L−1(1 − ef)|||C2,α
+ε,b
+≤cετ||ϕ|||C2,α
+ε,b + ||f||C0,α
+ε,b
+≤c(εb+2+2τ + εb+2+(2/5)λ0)
+≤cεmin{τ,(2/5)λ0−τ}εb+2+τ
+≤˜cεb+2+τ,
+implying that N(U) ⊆ U.
+This shows that everything is into place to apply Banach’s Lemma on the open set U and obtain ˆω
+a Chern-Ricci flat balanced metric ˆω on M, thus proving most of Theorem 1.1.
+Remark 4.6. It is known that for a manifold which is Calabi-Yau with holomorphic volume Ω,
+the existence of a Chern-Ricci flat balanced metric implies that Ω is parallel with respect to the
+Bismut connection associated to said metric. Among the other things, this implies that the restricted
+holonomy of the Bismut connection of Chern-Ricci flat balanced metrics is contained in SU(n).
+Remark 4.7. Recalling Remark 4.5, we have that the couple (M, ˆω) Gromov-Hausdorff converges
+to the singular Calabi-Yau metric on Mreg and, up to rescaling, to the Candelas-de la Ossa metrics
+nearby the exceptional curve.
+Remark 4.8. Notice that en passant we have proved that, on our small resolutions, for certain
+classes nearby the boundary of the balanced cone, the Gauduchon conjecture for balanced metrics
+(see [STW]) holds, i.e. in said balanced classes we can always find a Chern-Ricci flat balanced
+metric.
+4.3. Cohomology classes. We end the discussion on the construction by analyzing the cohomol-
+ogy class naturally associated to the metric ˆω just obtained. Indeed, balanced metrics naturally
+induce an (n − 1, n − 1)-class, thus in our case we get
+[ˆω2] ∈ H2,2
+dR(M).
+We are then interested in understanding if this cohomology class can be described in terms of the
+cohomology classes of the objects we used for the gluing construction. The first thing to observe is
+that the deformation introduced in the previous section preserves the balanced class, thus we have
+that [ω2] = [ˆω2], thus we shall work with ω.
+Let’s then introduce two cut off functions θ1, θ2 : [0, +∞) → [0, 1] defined as follows:
+θ1(x) :=
+
+
+
+
+
+1
+if x ≤ 1
+8ε−3/5
+non increasing
+if 1
+8ε−3/5 ≤ x ≤ 1
+4ε−3/5
+0
+if x ≥ 1
+4ε−3/5
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+26
+and
+θ2(x) :=
+
+
+
+
+
+0
+if x ≤ 8ε−3/5
+non decreasing
+if 8ε−3/5 ≤ x ≤ 16ε−3/5
+1
+if x ≥ 16ε−3/5;
+and since for sufficiently small ε we have that ω is exact on K := {1
+8ε−3/5 ≤ r(ζ) ≤ 8ε−3/5}, it
+exists a 3-form β such that
+ω2 = dβ
+on K.
+Introduce then the form
+Ωc := d(θ1(r(ζ)) + θ2(r(ζ)))β),
+which is a smooth compactly supported form. Moreover, the form
+β − (θ1(r) + θ2(r))β
+can be extended as zero to the whole M, thanks to the definition of the cut-offs, and thus get that
+[ω2] = [Ωc],
+i.e. the class [ω2] admits a compactly supported representative. In addition, the two cut-offs intro-
+duced also allow us to decompose Ωc = Ω′
+c + Ω′′
+c, such that on K hold
+Ω′
+c = d(θ1(r)β)
+and
+Ω′′
+c = d(θ2(r)β),
+and both Ω′
+c and Ω′′
+c are compactly supported and closed; in particular said compact supports are
+respectively contained in ˆX and Mreg (via the obvious identifications), and from their definition it
+is straightforward to see that
+[Ω′
+c] = ε4[ω2
+co,1] ∈ H4
+c ( ˆX)
+and
+[Ω′′
+c] = [˜ω2] ∈ H4
+c (Mreg),
+where Hc denotes the compactly supported cohomology group. Also, recalling that ˆX ≃ OP1(−1)⊕2,
+it is clear that ˆX is homotopically equivalent to P1; hence applying Poincaré duality we get
+H4
+c ( ˆX) ≃ Hc
+2( ˆX) ≃ Hc
+2(P1) = H2(P1) = ⟨[P1]⟩,
+which means that the non-zero class [ω2
+co,1], up to multiplicative constants, is the Poincaré dual of
+the generator of H2(P1) (thus we can "confuse" them with each other), and thus we can write
+[ω2] = [˜ω2] + ε4[P1]
+in H2,2
+dR(M).
+Finally, we also notice that
+�
+P1 ω = ε2
+�
+P1 ωco,1 −→
+ε→0 0,
+hence the balanced class [ω2], as ε → 0, converges to a nef class, i.e. to the boundary of the bal-
+anced cone.
+This concludes the proof of Theorem 1.1.
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+27
+Remark 4.9. Even though this construction is done to address the situation of a non-Kähler small
+resolution, it can also be applied when said small resolution M is instead Kähler. In this case we
+know from Yau’s theorem that M admits a Kähler Calabi-Yau metric ω1, hence together with the
+balanced class induced by our Chern-Ricci flat balanced metric ˆω we also have the one induced by
+ω1. This two balanced classes need not be the same, however, even if they are to coincide, there
+is no uniqueness result that would guarantee that the two metrics have to be the same; moreover,
+the deformation we used in our construction does not cover the whole balanced class, hence in this
+case we are not even guaranteed that the two metrics are linked by our chosen deformation.
+4.4. Relation to the Hull-Strominger system. As a conclusion of the paper, we would like to
+briefly relate our construction to the Hull-Strominger system and how we intend to develop our
+research in this direction, hence we shall first quickly recall the definition of said system (for more
+details we refer to the notes [GF]).
+The framework is given by (X, Ω) a (not necessarily Kähler) Calabi-Yau manifold, and the first
+equation of the system, for ω a hermitian metric, is known as the dilatino equation and is given by
+d∗ω = dc log ||Ω||ω,
+which easily seen to be equivalent to the conformally balanced equation
+d
+�
+||Ω||ωωn−1�
+= 0.
+Hence, this last equation tells us that we need to work with balanced manifolds, and thus we see a
+first relation to our scenario.
+To complete the system we need to pair the dilatino equation with two Hermite-Einstein equations
+for holomorphic vector bundles, thus, in the same fashion as what we had with the dilatino equa-
+tion, the presence of the Hermite-Einstein equation in the Hull-Strominger system will limit us to
+consider only polystable bundles. Finally, adding one last equation, known as the Bianchi identity,
+we can introduce the system.
+Definition 4.10. Given a Calabi-Yau manifold (X, Ω) and a holomorphic vector bundle E on X,
+we say that the triple (ω, h, ∂T ) is a solution of the Hull-Strominger system if it satisfies
+ΛωFh = 0,
+ΛωR = 0,
+d∗ω − dc log ||Ω||ω = 0,
+ddcω − α(trR ∧ R − trFh ∧ Fh) = 0;
+where, α is a non-vanishing constant, ω is a hermitian metric on X, h is a hermitian metric along
+the fibers of E, ∂T is a holomorphic structure on the tangent bundle of X, and R is the Chern
+curvature tensor of ω, read as a hermitian metric on the holomorphic vector bundle (TX, J, ∂T ).
+The Bianchi identity (also known as anomaly cancellation equation), is the hardest and least
+understood equation of the system and also the one we wish to address in the development of our
+research.
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+28
+It is significant to notice that if we choose (E, h) to be the holomorphic tangent bundle with the
+metric ω, and take ω a Kähler Ricci-flat metric, we see that this satisfies the system, thus being
+a solution of the Hull-Strominger system is a condition that generalizes being Kähler Calabi-Yau,
+hence a very promising candidate class of special metrics.
+Also, thanks to the equivalence between Hermite-Einstein metrics and Hermite-Yang-Mills con-
+nections it is possible to rewrite the system from a gauge-theoretical point of view.
+Definition 4.11. Given a Calabi-Yau manifold (X, Ω) and a hermitian vector bundle (E, h) (with
+a fixed holomorphic structure) on M, the triple (ω, A, ∇) is a solution of the Hull-Strominger
+system if it satisfies
+ΛωFA = 0,
+F 0,2
+A
+= 0,
+ΛωR∇ = 0,
+R0,2
+∇ = 0,
+d
+�
+||Ω||ωωn−1�
+= 0,
+ddcω − α(trR ∧ R − trFh ∧ Fh) = 0;
+where α is a non-vanishing constant, ω a hermitian metric on X, A is unitary connection on (E, h)
+and ∇ is a unitary connection on (TM, J, g).
+This second description is useful to notice a series of necessary conditions when X is compact.
+Indeed, as already observed we have that M has to be necessarily balanced, and given the natural
+balanced class τ given by the dilatino equation, we have that E and TM have to be τ-polystable.
+Morever, we have that c1(E) · τ = 0 and c1(M) = 0, and also
+(15)
+c1(E) · τ = 0
+ch2(E) = ch2(X) ∈ H2,2
+BC(X, R),
+where ch2 denotes the second Chern character.
+Although several examples of solutions have been studied over time (several can be found, for
+example, in [GF]), there is still a very poor understanding on the existence of solutions, even on
+threefolds, where however it was conjectured by Yau in [Y] that conditions (15) are not only a
+necessary condition but even a sufficient one to the existence of solutions in the case of threefolds.
+If we then take our construction, and we view it in the system’s scenario, we can make the
+following final remark in which we explain our ideas on how to expand our construction in this
+direction.
+Remark 4.12. Given ˆβ a Chern-Ricci flat balanced metric on a Calabi-Yau threefold (Y, Ψ), it holds
+||Ψ||ˆβ ≡ const.,
+showing that our metric ˆω gives a solution of the conformally balanced/dilatino equation on our
+small resolutions (M, Ω). Thus our construction gives us two solutions of the dilatino equation
+on (M, Ω), that are ˆω and ω′ := ||ˆΩ||−2
+ω ω, where this last one is the dilatino equation solution
+associated to the balanced metric ω obtained in the first part of the gluing construction. From
+here, thanks to the fact that this metrics are nearby a Kähler Ricci-flat metric, our idea (currently
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+29
+in devolopment in the upcoming [Gi]) is to adapt the stretegy in [CPY1] to construct a Hermite-
+Einstein metric on the tangent bundle with respect to the above metrics, and eventually from there
+try and extend it to a whole solution of the Hull-Strominger system, using - for example - some
+version of the approach of [AGF].
+REFERENCES
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+folds, J. Diff. Geom. 90 (2012) 81–129.
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+equation, J. Differential Geom. 78 (2008), no. 3, 369-428.
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+[FWW] J. Fu, Z. Wang, D. Wu, Form-type Calabi-Yau equations, Math. Res. Lett. 17 (2010), no. 05, 887-903.
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+P. Gauduchon, Fibré hermitiens à endomorphisme de Ricci non négativ, Bulletin de la S.M.F. 105, 113–140
+(1977).
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+F. Giusti, Hermite-Einstein metrics on non-Kähler flops, Upcoming
+[GF] M. Garcia-Fernandez, Lectures on the Strominger system, Trav. Math. 24, 7–61 (2016).
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+H. Hironaka, Flattening theorems in complex analytic geometry, Amer. J. Math. 97 (1975) 503-547.
+[HS] H. J. Hein, S. Sun, Calabi-Yau manifolds with isolated conical singularities, Publications mathématiques de
+l’IHES 126.1 (2017), 73-130.
+[Hu] C. Hull, Superstring compactifications with torsion and space-time supersimmetry, In Turin 1985 Proceedings
+"Superunification and Extra Dimensions" (1986) 347-375.
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+D. D. Joyce, Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University
+Press, Oxford, 2000.
+[LY1] J. Li, S.-T. Yau, The existence of supersymmetric string theory with torsion, J. Diff. Geom. 70 (2005) 143-181.
+[LY2] J. Li, S.-T. Yau, Hermitian-Yang-Mills connections on non-Kähler manifiolds, Mathematical aspects of string
+theory (San Diego, Calif., 1986) 560-573, Adv. Ser. Math. Phys., 1, Worlds Sci. Publishing.
+[LY3] J. Li, S.-T. Yau, The existence of supersymmetric string theory with torsion, J. Differential Geom. 70 (2005), no.
+1, 143–181.
+
+CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS
+30
+[M]
+M.L. Michelsohn, On the existence of special metrics in complex geometry, Acta Math. 149, 261–295 (1982).
+[NS] Y. Namikawa, J. H. M. Steenbrink. Global smoothing of Calabi-Yau threefolds, Inventiones Mathematicae 122
+(1995) 403-419.
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+D.H. Phong, Moduli in geometry and physics, in Geometry and Physics in Cracow, Acta Phys. Polon. B Proc.
+Suppl. 4 (2011), no.3, 351–378.
+[PPZ] D. H. Phong, S. Picard, and X. Zhang. Geometric flows and Strominger systems, Mathematische Zeitschrift 288.1
+(2018): 101-113.
+[R]
+M. Reid, The moduli space of 3-folds with K = 0 may nevertheless be irreducible, Math. Ann. 278 (1987), no.
+1-4, 329-334.
+[S]
+A. Strominger, Superstrings with torsion, Nucl. Phys. B 274 (2) (1986) 253-284.
+[STW] G. Székelyhidi, V. Tosatti, B. Weinkove, Gauduchon metrics with prescribed volume form, Acta Math., 219
+(2017), 181-211.
+[Sz]
+G. Székelyhidi, An introduction to extremal Kähler metrics, Graduate Studies in Mathematics. Vol. 152. American
+Mathematical Soc. (2014).
+[TW] V. Tosatti, B. Weinkove, The complex Monge-Ampère equation on compact Hermitian manifolds, J. Amer. Math.
+Soc. 23 (2010), no.4, 1187–1195.
+[TY] L.-S. Tseng, S.-T. Yau, Non-Kähler Calabi-Yau manifolds, in String-Math 2011, 241– 254, Proc. Sympos. Pure
+Math., 85, Amer. Math. Soc., Providence, RI, 2012.
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+J. Werner Kleine Auflösungen spezieller dreidimensionaler Varietäten, Bonn, Univ., Diss., 1987 (Nicht f.d. Aus-
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+DEPARTMENT OF MATHEMATICS, AARHUS UNIVERSITY, NY MUNKEGADE 118, 8000 AARHUS C, DENMARK
+Email address: federico.giusti@math.au.dk
+DEPARTMENT OF MATHEMATICS, AARHUS UNIVERSITY, NY MUNKEGADE 118, 8000 AARHUS C, DENMARK
+Email address: c.spotti@math.au.dk
+
diff --git a/LNFJT4oBgHgl3EQfxy2s/content/tmp_files/load_file.txt b/LNFJT4oBgHgl3EQfxy2s/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,844 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf,len=843
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='11636v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='DG]  27 Jan 2023  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF  CALABI-YAU THREEFOLDS  FEDERICO GIUSTI AND CRISTIANO SPOTTI  ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Given a (smoothable) projective nodal Kähler Calabi-Yau threefold, we show, via a  gluing construction, that all its - possibly non-Kähler - small resolutions admit Chern-Ricci flat  balanced metrics, which among other things solve the dilatino equation appearing in the Hull-  Strominger system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' INTRODUCTION  With the ultimate aim of geometrizing and classifying, one of the most studied problems in com-  plex geometry is the existence of hermitian metrics that can be regarded as special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Through the  years, the Kähler case is the one that has been studied and understood the most, however, in the last  decades the interest towards the non-Kähler world has been increasing more and more, leading to  the search for special metrics also in this particular context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' While in the Kähler case special metrics  arise naturally, the non-Kähler scenario is too wild to guide us directly towards some central no-  tion of special metric, nevertheless, one can have indications on the path to follow by watching the  Kähler world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' More specifically, given an n-dimensional complex manifold (M, J), if it is Kähler  the obvious class of special (on a first level) metrics is given exactly by Kähler metrics - which we  recall being hermitian metrics h whose fundamental form ω := h(J_, _) is d-closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In addition,  this condition can also be combined with the notion of Einstein metric (thanks to the properties  of Kähler metrics) from the general riemannian case, giving rise to the notion of Kähler-Einstein  metrics, which are universally regarded as the "most special" in the Kähler world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Likewise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' other  notions of special Kähler metrics have been introduced and studied (some of them are still central  in the study of Kähler geometry),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' like constant scalar curvature Kähler (cscK) metrics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' or the more  general class of extremal Kähler metrics (introduced by Calabi in [C]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' however they all share the  fact that they are giving a curvature condition on the metric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' thus this suggests that when searching  for special metrics in the non-Kähler case we shall ask for these metrics to be special under two  aspects: the cohomological one (satisfying a condition possibly generalizing the Kähler one) and  the curvature one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Regarding the cohomological aspect, several conditions have been introduced that generalize the  Kähler one, and one of the most studied is given by dωn−1 = 0, identifying the class of balanced  metrics (originally introduced by Gauduchon in [G] as semi-Kähler metrics, and later on studied  Date: January 30, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' 53C55, 53C25,53C07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' complex non-Kähler manifolds, balanced metrics, Chern-Ricci flat metrics, Calabi-Yau  manifolds, Hull-Strominger system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  1  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  2  further by Michelsohn in [M]), which is the class of metrics we are interested in working with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Balanced metrics carry many interesting properties such as the coincidence between the Hodge  laplacian and the Dolbeault laplacian on scalar functions (showed by Gauduchon in [G]), or the  preservation of the balanced condition for manifolds under holomorphic submersions proved in  [M] (showing a sort of duality between the Kähler condition and the balanced condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Also in  [M], Michelsohn proved a characterization of balanced metrics in terms of currents, which leads to  the celebrated result from Alessandrini and Bassanelli (see [AB]) showing that the class of com-  pact balanced manifolds is closed under proper modifications (condition not satisfied by the class  of Kähler manifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Moreover, balanced metrics ended up being central in many interesting cur-  rently open problems, such as the conjecture from Fino and Vezzoni (see [FV]) and the Gauduchon  conjecture for balanced metrics (see [STW], in which was solved in its original version for Gaudu-  chon metrics - identified by the condition ∂∂ωn−1 = 0, which weakens the balanced condition -  posed by Gauduchon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Moving instead on the curvature aspect, there are several known notions of special metrics in the  non-Kähler world such as Chern-Ricci flat metrics, Bismut-Ricci flat metrics (which in the bal-  anced case are equivalent to Chern-Ricci flat metrics, see [AI]), Chern-Einstein metrics and many  more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  The main goal of this paper is to construct Chern-Ricci flat balanced metrics on the compact small  resolutions of certain smoothable singular Kähler Calabi-Yau threefolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' More specifically we wish  to work on smoothable singular threefolds ˜  M whose singular set is made of ordinary double points  and are endowed with a Kähler Ricci-flat singular metric ˜ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Then, using the information on the  asymptotics of this metric around the singularity given by the results in [HS], together with what  it is known on the standard conifold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' the local model of ordinary double points on threefolds)  and its small resolution to build, with a gluing construction (inspired mostly by [BM], but also  [AP] and [J]), Chern-Ricci flat balanced metrics on the compact small resolutions of the singu-  lar threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' The strategy of the proof consists of two main steps: (1) a metric gluing between  the singular Calabi-Yau metric ˜ω with the (rescaled) Candelas-de la Ossa metrics ωco,a (that are  Kähler Calabi-Yau metrics on the small resolution of the standard conifold, introduced in [CO]),  and (2) an Implicit Function Theorem deformation argument, where the deformation is a balanced  deformation (introduced in [FWW]) and all the analysis is performed in suitable weighted Hölder  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Our main result is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Let ( ˜  M, ˜ω) be a smoothable projective Kähler Calabi-Yau nodal threefold (with ˜ω  a singular Calabi-Yau metric), and let M be a compact (not necessarily Kähler) small resolution  of ˜  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Then M admits a Chern-Ricci flat balanced metric ˆω such that [ˆω2] = [˜ω2] + ε4[P1], and  [ω2] converges to a nef class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Here by "smoothable" we mean that ˜  M admits a polarized flat deformation to a smooth pro-  jective Calabi-Yau threefold (examples of such manifolds are given by nodal quintic threefolds in  P4), and the reason why we require this condition to be satisfied is to be able to apply a result from  [HS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Our interest towards Chern-Ricci flat balanced metrics comes actually from the realm of Calabi-  Yau geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Indeed, for a not necessarily Kähler Calabi-Yau manifold (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' a complex manifold  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  3  endowed with a holomorphic volume form) it was introduced by Hull and Strominger (respec-  tively in [Hu] and [S]) a system of four equations coming from superstring theory known as the  Hull-Strominger system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' whose solutions have proved to be extremely hard to construct (see [GF]  for a full presentation of the system and some known solutions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' together with several other ref-  erences such as [AGF],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' [Fe],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' [FuY],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' [LY3],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' [P],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' [TY] and the very recent [CPY2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' [FeY] for the  invariant case and [PPZ] for a flow approach).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' The problem of solving This system, apart from its  physical meaning, carries great geometric interest, since it generalizes the Calabi-Yau condition to  the non-Kähler framework, and it holds a central role in the geometrization conjecture for com-  pact Calabi-Yau threefolds known as Reid’s Fantasy (see [R]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' This last conjecture, in particular,  states that all compact Kähler Calabi-Yau threefolds can be connected through a finite number of  conifold transitions (introduced by Clemens and Friedman, see [F]), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' a procedure consisting of  the contraction of a finite family of disjoint (−1, −1)-curves in a compact Calabi-Yau threefold,  followed by the smoothing of the ordinary double points obtained from the previous step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' These  objects thus show our interest towards singular threefolds with a finite number of ordinary dou-  ble points and our aim to find special metrics on their small resolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' in particular the interest  towards Chern-Ricci flat balanced metrics is directly related to one of the equation of the Hull-  Strominger system, namely the conformally balanced equation, which on a compact Calabi-Yau  manifold (X, Ω) - where Ω is the holomorphic volume form - is an equation for hermitian metrics  (actually their fundamental forms) ω given by d(||Ω||ωωn−1) = 0 which is clearly satisfied by  balanced Chern-Ricci flat hermitian metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Chern-Ricci flat metrics correspond also  to Hermite-Einstein metrics on the holomorphic tangent bundle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' thus this kind of metric appears as  particularly suited to be used as a starting point from which possibly build solutions for the Hull-  Strominger system (we will briefly discuss some ideas at the end of the paper),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' and also portrays  our construction (in some sense) as aiming towards "reversing the arrow" of the construction done  by Fu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Li and Yau in [FLY] and Collins,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Picard and Yau in [CPY1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  It is also interesting to notice that the deformation argument used in Section 4 proves also that on  this class of manifolds the Gauduchon conjecture for balanced metrics holds for certain classes  nearby the boundary of the balanced cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In Section 2 we recall some basic aspects on ordinary dou-  ble points and their resolutions, both regarding their geometry and their topology, with focus on  some fundamental results useful for our construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In Section 3 we present the first step of our  work, consisting of a gluing construction of a balanced metric made with the objects previously  introduced, together with the construction of a global "small" Chern-Ricci potential for our new  balanced metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In the last section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Section 4, we apply a deformation argument to obtain a  Chern-Ricci flat balanced metric and we discuss its associated balanced class, as well as its possi-  ble applications to the search of solutions for the Hull-Strominger system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Both the authors are supported by Villum Young Investigator 0019098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  The authors would like to thank Mario Garcia-Fernandez for useful conversations and remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' ORDINARY DOUBLE POINTS ON THREEFOLDS  In this preliminary section we shall recall some known facts about a certain type of singularity  on threefolds, regarding both their topology and their geometry, and also fix some notation for the  following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  The type of singularities we are interested in studying are called ordinary double points (which  are the most common kind of singularities in our context), and are described by the model  X := {z2  1 + z2  2 + z2  3 + z2  4 = 0} ⊆ C4,  which is known as the 3-dimensional standard conifold, whose only singular point is the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Then we have:  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' A singular point p in a singular threefold Y is called ordinary double point (ODP)  if we can find a neighborhood p ∈ U ⊆ M and a neighborhood 0 ∈ V ⊆ X such that U and V are  biholomorphic through a map that sends p to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Topological aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' These singularities arise naturally on threefolds when collapsing (−1, −1)-  curves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' rational curves biholomorphic to P1 whose normal bundle is isomorphic to OP1(−1)⊕2,  and actually this procedure to obtain ODPs covers all the possibilities on threefolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Indeed, the  standard conifold can be constructed in several ways, one of which is the following: consider the  rank 2 bundle OP1(−1)⊕2 on P1 and notice that the map  ([X1 : X2], (w1, w2)) �→ (w1X1, w1X2, w2X1, w2X2)  maps OP1(−1)⊕2 onto X - since X through a change of coordinates is biholomorphic to the set  {W1W2 − W3W4 = 0} - sending the zero section onto the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Moreover this map restricted  to OP1(−1)⊕2 \\ P1 (where P1 is meant as the zero section) gives a biholomorphism with X \\  {0}, proving our previous statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' This shows us that these singularities always admit small  resolutions (with P1 as the exceptional curve) biholomorphic to ˜X := OP1(−1)⊕2, and it can be  shown that a singular threefold with n ordinary double points admits exactly 2n small resolutions  of this type (every singularity can be resolved with a curve in two distinct bimeromorphic ways).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Since our following work aims to face the case of singular threefolds whith a finite number of  ODPs which admit a compact small resolution, in particular those who do not admit Kähler metrics,  before going further in exploring the geometry of these singularities we will quickly show that this  kind of resolutions are actually a very common situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Thanks to a result from Cheltsov (see [Ch]) we know that a hypersurface ˜  M in P4 of  degree d with only isolated ODPs is factorial when ˜  M has at most (d−1)2 −1 singularities, thus is  in particular Q-factorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' We can then apply the work from Namikawa and Steenbrink (see [NS]) to  obtain that ˜  M is smoothable, and hence, thanks to the results from Friedman (see [F]) we have that  any small resolution M of ˜  M with exceptional curves C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=', Ck, Ci ≃ P1, satisfies necessarily a  condition  k  �  i=1  λi[Ci] = 0  in H2(M, R),  where each λi ̸= 0,  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  5  which immediately implies that if ˜  M has only one ODP, then M can’t be Kähler because it contains  a homologically trivial curve (note that the generic quintic threefold has exactly one node, hence  satisfies this situation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Moreover, Werner proved in [W] that M is projective if and only if all Cis are homologically  non-trivial, and since M is Moishezon, projectivity is equivalent to Kählerness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Thus the class of  examples above lies in a larger one, since every small resolution with at least a homologically  trivial exceptional curve is non-Kähler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Geometric aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' The standard conifold X is naturally endowed with a conical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Indeed, we can introduce the function on C4  r(z) := ||z||  2  3 ,  which restricted to X yields the conical distance to the singularity, and can be used to define the  metric  ωco,0 := 3  2i∂∂r2,  on the smooth part of X, which is clearly Kähler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Moreover, it can be seen that ωco,0 is actually  also Ricci flat, as well as a cone metric over the link L := {r = 1} ⊆ X which can be written as  gco,0 = 3  2(dr2 + r2gL),  with gL a Sasaki-Einstein metric on the link L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  This metric structure of the standard conifold, with some further work, yields also a Kähler Calabi-  Yau structure on the small resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In fact, Candelas and de la Ossa (see [CO]) constructed a  family of metrics, depending on the parameter a > 0, of the form  ωco,a := i∂∂fa(r3) + 4a2π∗  P1ωF S,  where ωF S is the Fubini-Study metric on P1, and fa is smooth function satisfying the ODE  (xf ′  a(x))3 + 6a2(xf ′  a(x))2 = x2,  fa(x) ≥ 0,  on [0, +∞), which immediately gives fa(x) = a2f1(x/a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Here the function r is simply the  conical distance from the singularity re-read on the resolution, hence portraying the conical distance  from the exceptional curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Moreover, this family of metrics is such that as a → 0 the metrics  ωco,a converges, away from the exceptional curve, to the standard cone metric ωco,0, and it is also  asymptotic (at infinity) to the cone metric ωco,0, and these facts can be seen explicitly with the  following expansion from [CPY1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' For x ≫ 1, the function f1(x) has a convergent expansion  f1(x) = 3  2x  2  3 − 2 log(x) +  +∞  �  n=0  cnx− 2n  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  This expansion is going to be crucial for our work, hence we will work again with it in the next  sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Before concluding this preliminary part, is extremely important for our construction (and highly  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  6  interesting just as a fact) to remark that the standard cone metric ωco,0 is actually the (asymptotic)  model near the singularities for all singular Kähler Calabi-Yau metrics around ODPs, and this was  shown by Hein and Sun in the following crucial theorem (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='4 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='1 from [HS])  which can be simplified with the following statement (written only for threefolds):  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='4 (Hein-Sun).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Let ˜  M be a smoothable singular threefold whose singular set is a finite  family of ODPs endowed with a Kähler Calabi-Yau metric ˜ω on its smooth part Mreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Then for  every singular point p ∈ ˜  M \\ Mreg there exist a constant λ0 > 0, neighborhoods p ∈ Up ⊆ ˜  M  and 0 ∈ Vp ⊆ X, and a biholomorphism P : Vp \\ {0} → Up \\ {p} such that  P ∗˜ω − ωco,0 = i∂∂ϕ,  for some ϕ ∈ C∞  2+λ0,  where r is the conical distance from the singularities and C∞  2+λ0 is the space of smooth functions  with decay rate at zero of 2 + λ0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' an f ∈ C∞  2+λ0 is a smooth function such that nearby zero it  holds |∇kf| ≤ cr2+λ0−k for all k ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' THE GLUING CONSTRUCTION  We now start presenting the first part of our work, which consists of a gluing construction of  a family of balanced metrics on the compact small resolution of a smoothable Kähler Calabi-Yau  singular threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' This type of gluing constructions are not new in complex geometry, and have  been used in the past to construct significant examples, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' in [J], [AP] and [BM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Before going ahead with the details of the construction it is significant to make the following  remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Thanks to the results from Hironaka and Alessandrini-Bassanelli ([Hi] and [AB]),  we already know that compact small resolutions of smoothable Kähler Calabi-Yau threefolds with  a finite number of ODPs admit balanced metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' However, we are interested in finding special  balanced metrics - in particular, as we will see, with a small Chern-Ricci potential - hence we  need to perform an explicit construction in order to build metrics satisfying our desired curvature  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  In what follows we will lay out the initial data and see in details the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Let ˜  M be a smoothable complex threefold obtained from the contraction of a finite family of  disjoint (−1, −1)-curves in a compact complex threefold (thus the singular set of ˜  M is made of a  finite number of Ordinary Double Points), let M be a compact small resolution of ˜  M and suppose  that the regular part Mreg of ˜  M is equipped with ˜ω a Kähler Calabi-Yau metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  The idea of the gluing construction is to use the ingredients we introduced in the previous sections,  and notice that they are suitable to be glued together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In particular, we know that around each one  of the exceptional curves we have the Candelas-de la Ossa metrics ωco,a which are asymptotic, far  from the exceptional curve, to the cone metric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' on the other hand we have that also our background  metric ˜ω is asymptotic to the cone metric, but this time when approaching the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' We  then want to use the standard conifold as a "bridge" to glue together the metrics ˜ω and ωco,a,  while maintaining the balanced condition, and in order to be able to do the analysis for this gluing  construction is going to be crucial to have a "concrete" construction of the small resolution M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  To do this, we will divide the process into three natural steps, and for simplicity assume that ˜  M has  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  7  just one singularity (the process obviously applies analogously to the case in which the singularities  are any finite number).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In addition, since our resolution is known to be a crepant resolution, we are  also going to compute explicitly a holomorphic volume form for M (starting from the one on ˜  M),  since such form is a crucial ingredient for the deformation argument in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' We first glue together the metrics ˜ω and ωco,0 nearby the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' To do this,  thanks to the nature of ODPs, we can introduce, with an abuse of notation (that we will keep on  using throughout the whole paper - as it does not generate any confusion), a function r on ˜  M  which coincides with the (pullback) conical distance from the singularity in a neighborhood of said  singularity (induced by the local biholomorphism P from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='4), and is identically equal to  1 far away from them (as a consequence we will do the same with the standard cone metric ωco,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  In order to perform this first step of the gluing, we will need to work on the cone X and pull back  on Mreg the object obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Let p > 0 and consider ε > 0 sufficiently small, such that, thanks to the result from Hein-Sun  cited above, on the region {0 < r ≤ 4εp} ⊆ X exists a constant λ0 > 0 and is defined a function  ϕ ∈ C∞  2+λ0 such that  P ∗˜ω = ωco,0 + i∂∂ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Hence, we introduce a cut-off function χε(x) := χ1(x/εp) where χ1 is a smooth function on  [0, +∞) such that  χ1(x) :=  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  0  if x ≤ 2,  Non decreasing  if x ∈ (2, 4),  1  if x ≥ 4,  and define the smooth real (1, 1)-form  ˜ωε := ωco,0 + i∂∂(χε(r)ϕ),  which for ε sufficiently small is such that  ωε := (P −1)∗˜ωε  defines a Kähler metric on Mreg (extended to the whole Mreg in the obvious way).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Indeed, we  notice that on Gε := {2εp < r ≤ 4εp} it holds  |ωε − ωco,0|ωco,0 = |i∂∂(χε(r)ϕ)|ωco,0 ≤ c(r−2|ϕ| + r−1|∂ϕ|ωco,0 + |i∂∂ϕ|ωco,0) ≤ crλ0,  from which on Gε we can write  |∇k  ωco,0(ωε − ωco,0)|ωco,0 ≤ Crλ0−k,  for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Moreover we notice that on {0 < r ≤ 2εp} the metric ωε is exactly conical, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' equal to ωco,0,  while on {r > 4εp} ωε coincides with the background metric ˜ω, and on Gε is the only region where  ωε is not Ricci-flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In this second step we instead perform the gluing between the Candelas-de la Ossa  metrics ωco,a and the standard cone ωco,0, on the small resolution ˆX of the standard conifold X,  preserving the balanced condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' To do this we recall from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='3 that the Candelas-de la  Ossa metrics are of the form  ωco,a := i∂∂fa(r3) + 4a2π∗ωF S,  which away from the exceptional curve have the expansion  fa(r3) = 3  2r2 + a2ψa(r),  where  ψa(r) = 3 log r − 3 log a +  +∞  �  n=0  cna2nr−2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  This suggests introducing a large parameter R ≫ 1 and a smooth cut-off function χR(x) :=  χ2(x/R) on [0, +∞) such that  χ1(x) :=  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  1  if x ≤ 1  2,  Non increasing  if x ∈  �1  2, 1  �  ,  0  if x ≥ 1,  from which we introduce the family of closed (2, 2)-forms  ω2  a,R =  �  i∂∂  �3  2r2 + a2χR(r)ψa(r)  ��2  + 2a2i∂∂  �  χR(r)fa(r3)  �  ∧ π∗ωF S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  The reason why the notation ω2  a,R makes sense is because the results from Michelsohn ([M]) ensure  us that every real positive closed (n − 1, n − 1)-form admits a unique real positive (1, 1)-form as  a (n − 1)th root, and the forms we defined above happen to be positive for sufficiently large R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Indeed, positivity is clear when r ≤ R  2 , where ω2  a,R = ω2  co,a, and when r ≥ R, where ω2  a,R = ω2  co,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  In the gluing region GR := {R/2 ≤ r ≤ R}, instead, we need to do a few estimates to confirm  the positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In order to avoid having to impose limitations on the parameter a we substitute the  functions ψa with ψa + 3 log a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' this won’t alter the metrics ωco,a and any of its properties, thus it  can be done without problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' For simplicity, we will keep writing ψa, and in the following the  constant c might vary from line to line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' We have  |ω2  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='R − ω2  co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0 ≤2a2|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0 ∧ i∂∂(χR(r)ψa(r))|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0 + 2a2|i∂∂(χR(r)fa(r3)) ∧ π∗ωF S|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0  + a4|i∂∂(χR(r)ψa(r))|2  ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0  ≤ca2(|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0(r−2|ψa| + r−1|∂ψa|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0 + |i∂∂ψa|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0)  + |π∗ωF S|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0(r−2|fa(r3)| + r−1|∂fa(r3)|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0 + |i∂∂fa(r3)|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0)  + a2(r−2|ψa| + r−1|∂ψa|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0 + |i∂∂ψa|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0)2)  ≤ca2(|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0(r−2|ψa| + r−1|∂ψa|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0 + |i∂∂ψa|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0)  + |π∗ωF S|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0(1 + |ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0 + r−2|ψa| + r−1|∂ψa|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0 + |i∂∂ψa|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0)  + a2(r−2|ψa| + r−1|∂ψa|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0 + |i∂∂ψa|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0)2) ≤ (∗)  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  9  To conclude the estimate let’s take a look at the single terms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' keeping in mind that we are taking  the norms with respect to the standard cone metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' It holds  |ωco,0|ωco,0 = O(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  |π∗ωF S|ωco,0 = O(r−2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  |ψa| ≤ c(| log r| + �+∞  n=0 |cn|a2nr−2n) ≤ c| log r|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  |∂ψa|ωco,0 ≤ c(r−1 + �+∞  n=0 |˜cn|a2nr−2n−1) ≤ cr−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  |i∂∂ψa|ωco,0 ≤ c(r−2 + �+∞  n=0 |˜˜cn|a2nr−2n−2) ≤ cr−2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  from which we get  (∗) ≤ ca2(r−2| log r|+r−2+r−2(1+r−2| log r|+r−2)+a2(r−2| log r|+r−2)2) ≤ ca2r−2| log r|,  and hence  |∇k  ωco,0(ω2  a,R − ω2  co,0)|ωco,0 ≤ Ca2r−2−k| log r|  for all k ≥ 0,  which clearly implies the positivity also on GR, as long as R is chosen to be sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In this third and last step we want to glue together the metrics ωε from step 1 with the  metric ωa,R from step 2 by matching isometrically the exactly conical regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In order to do this  we are going to need to rescale by a constant λ > 0 the metric on ˆX, and we will now see that this  constant is a geometric constant, since it is dictated by the geometries of the two metrics we are  gluing together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  In what follows we will denote with z the coordinates on Mreg nearby the singularity and with  ζ the coordinates on ˆX, both given by the identification with the standard conifold X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' We then  consider the regions  CR := {R/2 ≤ r(ζ) ≤ 4R} ⊆ ˆX  and  Cε := {εp/2 ≤ r(z) ≤ 4εp} ⊆ Mreg  and define a biholomorphism between them by imposing  ζ =  � R  εp  � 3  2  z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  From this expression we have that on the identified region the following identity holds  r(ζ) = r  �� R  εp  � 3  2  z  �  = R  εp r(z)  which yields λ = λ(ε, R) :=  �εp  R  �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' From this follows λr2(ζ) = r2(z), and thus on the identified  conical regions C′  R := {R ≤ r(ζ) ≤ 2R} ≃ {εp ≤ r(z) ≤ 2εp} =: C′  ε holds  λωco,0(ζ) = ωco,0(z),  and consequently  λωa,R = ωε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Hence, λ is the needed rescaling factor, which allows us to define the glued family of balanced  metrics on the small resolution M as  ωa,ε,R :=  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  λωa,R  on r(ζ) ≤ R,  ωco,0  on εp ≤ r(z) ≤ 2εp,  ωε  on r(z) ≥ 2εp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  10  Now we will move to the following part in which we will try to understand better the geometry of  this new family of metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Description of the volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' We are mainly interested in the Chern-Ricci tensor - and in par-  ticular on its potential - hence, in order to understand it, we are going to need to obtain information  on the volume of our metrics ω = ωε,R := ω1,ε,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' To do this, we need to estimate the distance  between our glued metrics and the standard cone metric on the gluing region, since in the other  parts of the manifold we already know the geometry from the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Moreover, inside the  gluing region there is also an exactly conical region - whose geometry is also understood - which  separates the two gluing regions from the first two steps, hence we can just estimate the said dis-  tance separately on the two regions and then take the maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Clearly, the metric is unaltered on the gluing region from step 1, thus we still have on Gε that  |∇k  ωco,0(ωε − ωco,0)|ωco,o ≤ crλ0−k,  for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  On the other hand, since in step 3 we had to rescale the metric nearby the exceptional curve, we  are going to need to repeat the estimate to understand how the rescaling affected the distance from  the standard cone metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' At some point we are going to link the parameters ε and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Again the  constant c might be varying from line to line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' We have  (ω2  ε,R − ω2  co,0)(ζ) =λ2(2ωco,0 ∧ i∂∂(χR(r)ψ1(r)) + 2i∂∂(χR(r)f1(r3)) ∧ π∗ωF S  + i∂∂(χR(r)ψ1(r))2)(ζ) = (∗)  Let’s now change coordinates from ζ to z, recalling that λr2(ζ) = r2(z) and noticing that, by  construction, π∗ωF S is invariant under rescalings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  (1)  (∗) =λ2  \uf8eb  \uf8edλ−1ωco,0 ∧ i∂∂  \uf8eb  \uf8edχR  �  λ− 1  2 r  �  \uf8eb  \uf8ed3 log r − 3  2 log λ +  �  n≥0  cnλnr−2n  \uf8f6  \uf8f8  \uf8f6  \uf8f8  + π∗ωF S ∧ i∂∂  \uf8eb  \uf8edχR  �  λ− 1  2 r  �  \uf8eb  \uf8edλ−1 3  2r2 + 3 log r − 3  2 log λ +  �  n≥0  cnλnr−2n  \uf8f6  \uf8f8  \uf8f6  \uf8f8  +  \uf8eb  \uf8edi∂∂  \uf8eb  \uf8edχR  �  λ− 1  2 r  �  \uf8eb  \uf8ed3 log r − 3  2 log λ +  �  n≥0  cnλnr−2n  \uf8f6  \uf8f8  \uf8f6  \uf8f8  \uf8f6  \uf8f8  2\uf8f6  \uf8f8 (z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  11  What we want to do now is estimate the norm (with respect to the cone metric) of this quantity,  thus we start computing the derivatives that appear in the above expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' We have  ∂∂  \uf8eb  \uf8edχR  �  λ− 1  2 r  �  \uf8eb  \uf8ed3 log r − 3  2 log λ +  �  n≥0  cnλnr−2n  \uf8f6  \uf8f8  \uf8f6  \uf8f8  = λ−1  R2 χ′′  R  �  λ−1/2r  � �  3 log r − 3  2 log λ +  �  n∈N  cnλnr−2n  �  ∂r ∧ ∂r  + 2λ−1/2  R  χ′  R  �  λ−1/2r  � �  3r−1 +  �  n∈N  ˜cnλnr−2n−1  �  ∂r ∧ ∂r  + λ−1/2  R  χ′  R  �  λ−1/2r  �  \uf8eb  \uf8ed3 log r − 3  2 log λ +  �  n≥0  cnλnr−2n  \uf8f6  \uf8f8 ∂∂r  + χR  �  λ−1/2r  � �  −3r−2 +  �  n∈N  ˜˜cnλnr−2n−2  �  ∂r ∧ ∂r  + χR  �  λ−1/2r  � �  3r−1 +  �  n∈N  ˜cnλnr−2n−1  �  ∂∂r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='∂∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8edχR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='λ− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2 r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8edλ−1 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2r2 + 3 log r − 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2 log λ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='n≥0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='cnλnr−2n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='λ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='R2 χ′′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='λ−1/2r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='r2 + 4λ−1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='χ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='λ−1/2r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='r + 2χR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='λ−1/2r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='∂r ∧ ∂r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='λ−1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='χ′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='λ−1/2r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='r2 + 2χR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='λ−1/2r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='∂∂r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='+ ∂∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8edχR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='λ− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2 r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8ed3 log r − 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2 log λ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='n≥0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='cnλnr−2n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='From here we can obtain the estimates  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='∂∂  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8edχR  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='λ− 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2 r  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8eb  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8ed3 log r − 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2 log λ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='n≥0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='cnλnr−2n  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8f6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='\uf8f8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0  ≤ cλ−1  R2  �  | log r| + | log λ| +  �  n∈N  |cn|λnr−2n  �  + cλ−1/2  R  r−1  \uf8eb  \uf8ed1 +  �  n∈N  |˜cn|λnr−2n + | log r| + | log λ| +  �  n≥0  |cn|λnr−2n  \uf8f6  \uf8f8  + cr−2  �  1 +  �  n∈N  |˜˜cn|λnr−2n +  �  n∈N  |˜cn|λnr−2n  �  and������  ∂∂  \uf8eb  \uf8edχR  �  λ− 1  2r  �  \uf8eb  \uf8edλ−1 3  2r2 + 3 log r − 3  2 log λ +  �  n≥0  cnλnr−2n  \uf8f6  \uf8f8  \uf8f6  \uf8f8  ������  ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0  ≤ c  �  λ−1  R2 r2 + λ−1/2  R  (r + 1)  �  +  ������  ∂∂  \uf8eb  \uf8edχR  �  λ− 1  2 r  �  \uf8eb  \uf8ed3 log r − 3  2 log λ +  �  n≥0  cnλnr−2n  \uf8f6  \uf8f8  \uf8f6  \uf8f8  ������  ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0  In order to conclude the estimate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' we impose R := ε−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' with q > 0 to be chosen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' from which  follows λ = ε2(p+q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Also we are now going to use that we are in the region {2εp ≤ r ≤ 4εp},  which among the others implies the estimate  | log r| ≤ | log 4| + p| log ε| ≤ c(1 + | log ε|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  We thus get  ������  ∂∂  \uf8eb  \uf8edχR  �  λ− 1  2r  �  \uf8eb  \uf8ed3 log r − 3  2 log λ +  �  n≥0  cnλnr−2n  \uf8f6  \uf8f8  \uf8f6  \uf8f8  ������  ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0  ≤ cε−2p  �  1 + | log ε| +  �  n∈N  |cn|(1/2)2n  �  + cε−2p  \uf8eb  \uf8ed1 +  �  n∈N  |˜cn|(1/2)2n + 1 + | log ε| +  �  n≥0  |cn|(1/2)2n  \uf8f6  \uf8f8  + cε−2p  �  1 +  �  n∈N  |˜˜cn|(1/2)2n +  �  n∈N  |˜cn|(1/2)2n  �  ≤ cε−2p| log ε|  and  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  13  ������  ∂∂  \uf8eb  \uf8edχR  �  λ− 1  2r  �  \uf8eb  \uf8edλ−1 3  2r2 + 3 log r − 3  2 log λ +  �  n≥0  cnλnr−2n  \uf8f6  \uf8f8  \uf8f6  \uf8f8  ������  ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0  ≤ c(1 + ε−p(ε + 1)) +  ������  ∂∂  \uf8eb  \uf8edχR  �  λ− 1  2 r  �  \uf8eb  \uf8ed3 log r − 3  2 log λ +  �  n≥0  cnλnr−2n  \uf8f6  \uf8f8  \uf8f6  \uf8f8  ������  ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0  ≤ c(ε−p + ε−2p| log ε|) ≤ cε−2p| log ε|  Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' putting together this estimates we can finally obtain  |ω2  ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='R − ω2  co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0  ≤ cε4(p+q)  \uf8eb  \uf8ec  \uf8edε−2(p+q)  ������  i∂∂  \uf8eb  \uf8edχR  �  λ− 1  2r  �  \uf8eb  \uf8ed3 log r − 3  2 log λ +  �  n≥0  cnλnr−2n  \uf8f6  \uf8f8  \uf8f6  \uf8f8  ������  ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0  + ε−2p  ������  i∂∂  \uf8eb  \uf8edχR  �  λ− 1  2 r  �  \uf8eb  \uf8edλ−1 3  2r2 + 3 log r − 3  2 log λ +  �  n≥0  cnλnr−2n  \uf8f6  \uf8f8  \uf8f6  \uf8f8  ������  ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0  +  ������  i∂∂  \uf8eb  \uf8edχR  �  λ− 1  2r  �  \uf8eb  \uf8ed3 log r − 3  2 log λ +  �  n≥0  cnλnr−2n  \uf8f6  \uf8f8  \uf8f6  \uf8f8  ������  2  ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0  \uf8f6  \uf8f7  \uf8f8  ≤ cε4(p+q)(ε−4p−2q| log ε| + ε−4p| log ε|2) ≤ cε2q| log ε|  which implies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' on the whole gluing region,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' that for all k ≥ 0 holds  |∇k  ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0(ω2  q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='ε − ω2  co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0)|ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='0 ≤ crm−k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  where m = m(λ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  In order to obtain geometric information from this estimate, we are going to need to recall a result  from Michelsohn ([M]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' This result gives an explicit isomorphism between the cone of strictly  positive (1, 1)-forms and the cone of strictly positive (n − 1, n − 1)-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In particular, if Ψ ∈  �n−1,n−1 and ψ ∈ �1,1 are strictly positive forms such that ψn−1 = Ψ, we can always find a base  {ei} of (1, 0)-forms that "diagonalizes" simultaneously Ψ and ψ, giving the expressions  Ψ =  n  �  j=1  Λj �  ej ∧ Jej  and  ψ =  n  �  j=1  λjej ∧ Jej,  where J is the complex structure and  �  ej ∧ Jej means the wedge product of all the terms of the  form ek ∧ Jek except the one corresponding to the index j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Moreover, Michelsohn’s theorem gives  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  14  us also a formula relating the coefficients of ψ and Ψ in this basis, that is  λj = (Λ1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' · Λn)  1  n−1  Λj  for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  We will then use this to obtain an expansion for ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In order to have a more compact notation we  will introduce the notation O(rl) to denote the decay of a function (or of the norm of a tensor) and  all its derivatives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' to be clearer, saying that some tensor θ is such that θ = O(rl) means that for all  k ≥ 0 holds |∇k  ωco,0θ| ≤ crl−k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Notice that we can choose a basis {ej} of (1, 0)-forms diagonalizing simultaneously  ωco,0 (we can actually assume it to be the identity) and ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' this means that also ω2  co,0 and ω2 are  diagonal (in the sense of (n − 1, n − 1)-forms, implying that also the term O(rλ0) is necessarily  diagonal with respect to this basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Thus we can write  ω2 =  n  �  j=1  (αj + O(rm)) �  ej ∧ Jej  and applying Michelson’s theorem with Λj = αj + O(rm), we obtain ω = �n  j=1 λjej ∧ Jej, with  λj = ((α1 + O(rm))(α2 + O(rm))(α3 + O(rm)))  1  2  αj + O(rm)  = (α1α2α3)  1  2  αj  + O(rm),  which implies, again thanks to Michelson’s theorem  ω =  n  �  j=1  �  (α1α2α3)  1  2  αj  + O(rm)  �  ej ∧ Jej = ωco,0 + O(rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Hence we can write the volume form as ω3 = ω3  co,0 + O(rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' The Chern-Ricci potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In order to use this description of the volume to estimate the  Chern-Ricci potential on the gluing region we are also going to need to understand how the holo-  morphic volume form of the resolution is related to the holomorphic volume of our background  Calabi-Yau singular manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Before doing it we start by fixing some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Denote  with Ω the holomorphic volume of Mreg such that  ˜ω3 = iΩ ∧ Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  with Ω0 the holomorphic volume of the cone X such that  ω3  co,0 = iΩ0 ∧ Ω0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  with χ the holomorphic volume of the resolution ˆX such that  (λωco,1)3 = iχ ∧ χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  15  In order to explain clearly how these forms are related and how we can use them to construct  a holomorphic volume on M we shall take a step back and watch closer at the topological gluing  between Mreg and ˆX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Call P the biholomorphism given from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='4, and we can suppose ε to be sufficiently small  so that we can assume that P is defined (eventually after a restriction) on the set {0 < r(z) ≤ 4εp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Moreover, if we call Q the biholomorphism introduced at the beginning of Step 3 of the gluing  construction, we can assume it to be defined not only on the region written before, but actually on  the whole {0 < r(ζ) ≤ 4R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' This allows us to give a topologically concrete description of M, that  is  M = Mreg  � ˆX  ∼  ,  where the equivalence relation ∼ is defined as  y ∈ Mreg, x ∈ ˆX, y ∼ x if and only if P −1(y) = Q−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Thanks to this explicit description we can observe that on the gluing region holds  Q∗χ = Ω0  and there exists a holomorphic function h on the gluing region such  P ∗Ω = fQ∗χ = hΩ0,  and h can actually be extended to the whole ˆX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' We can however obtain even more information on  h recalling again Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' indeed said theorem guarantees us that  P ∗Ω ∧ P ∗Ω = Ω0 ∧ Ω0 + i∂∂Φ  where Φ is a smooth (2, 2)-form that behaves as = O(r2+λ0) nearby the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' From this we  get that  (|h|2 − 1)Ω0 ∧ Ω0 = i∂∂Φ,  and thus  |h|2 = 1 + O(rλ0) on (a neighborhood of P1 in) ˆX,  from which, by continuity, we have that |h|2 ≡ 1 on the exceptional curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Thus we can define the  holomorphic volume ˆΩ of M by gluing together hχ and Ω, and thus define a global Chern-Ricci  potential as  f = fa,q,ε := log  �  iˆΩ ∧ ˆΩ  ω3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  We now conclude this section by describing the behaviour of f in all the regions of M, to show  that it is suitable to apply a deformation argument similar to the one done in [BM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' We have  on {r(z) > 4εp} hold ω = ˜ω and ˆΩ = Ω, thus f ≡ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  on {2εp ≤ r(z) ≤ 4εp} hold ω = ωco,0 + O(rm) and ˆΩ ∧ ˆΩ = Ω0 ∧ Ω0 + O(rλ0), from  which we have  f = log  �  ω3  co,0 + O(rm)  Ω0 ∧ Ω0 + O(rλ0)  �  = log(1 + O(r ˜m)) = O(r ˜m),  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  16  where ˜m = min{λ0, m};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  on{εp ≤ r(z) ≤ 2εp} hold ω = ωco,0 and ˆΩ ∧ ˆΩ = i(1 + O(rλ0))Ω0 ∧ Ω0, from which  follows f = O(rλ0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  on {εp/2 ≤ r(z) ≤ εp} hold ω = ωco,0+O(rm) and ˆΩ∧ ˆΩ = Ω0∧Ω0+O(rλ0), implying  again f = O(r ˜m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  on {r(z) < εp/2} hold ω = λωco,1 and ˆΩ∧ ˆΩ = i(1+O(rλ0))Ω0 ∧Ω0, giving once again  f = O(r ˜m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Thus we can write globally (on M) that  |f| ≤ cr ˜m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' THE DEFORMATION ARGUMENT  In this last section we will see that what was built in the previous section are exactly the in-  gredients we need to introduce a deformation argument in the same fashion as [BM], in order to  obtain a balanced Chern-Ricci flat metric on our small resolution M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' We will also analyze the co-  homology class of the metric obtained and see why said metric is interested in the framework of  the Hull-Strominger system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' The strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' We will now set up the problem for this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' First of all we need to intro-  duce a deformation of the metric that preserves the balanced condition, and this can be done using  the one introduced in [FWW]  ω2  ψ := ω2 + i∂∂(ψω),  ψ ∈ C∞(M, R) such that ω2  ψ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Notice that again, thanks to the results of Michelsohn in [M], writing immediately ω2  ψ makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Thus the problem we are interested in solving, following what was done in [BM], is the equation  ω3  ψ = efω3  for ψ ∈ C∞(M, R) such that ω2  ψ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' The equation introduced above makes sense, because, as we’ve seen, f = O(r ˜m),  thus ef = 1 + O(r ˜m), meaning that efω3 is nearby ω3 itself, hence it makes sense to try to obtain  it as a small deformation of ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  For practicality is useful to reformulate our equation as an operator on the space of smooth  functions, thus we introduce F : C∞(M, R) → C∞(M, R) as  F(ψ) = Fε(ψ) :=  ω3  ψ  ω3 − ef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Our aim is to solve this equation through a fixed point argument, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' turning the problem in a new  one that can be solved by applying Banach’s Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In order to achieve this, the first step to take is  to start studying the linearization at 0 of the operator F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' To do this we shall introduce the notation  ω′  0 := d  dt |t=0ωtu, where ωtu is the curve corresponding to the tangent vector u ∈ C∞(M, R), and  compute the derivative at zero of ω3  tu in two different ways:  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  17  d  dt |t=0  ω3  tu = 3ω2 ∧ ω′  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  d  dt |t=0  ω3  tu = i∂∂(uω) ∧ ω + ω2 ∧ ω′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Even though none of these two expressions are explicit, we can put them together to obtain an  explicit one, that is  d0F(u) = Lu = Lεu := 3  2  i∂∂u ∧ ω2 + ui∂ω ∧ ∂ω  ω3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' The estimates done in the previous sections ensure us that, on the gluing region from  Step 2 of the gluing construction, hold  ∂ω = O(r2(q/p)−1| log r|)  and  ∂∂ω = O(r2(q/p)−2| log r|),  and ∂ω = 0 and i∂∂ω = 0 everywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Thus if q > p, then ∂ω, i∂∂ω → 0 as ε → 0, showing  that L is a bounded operator, and implying that ˜m = m = λ0 (assuming λ0 ≤ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Before starting the analytical part, we shall establish once for all a value for p and q, and in light  of Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2, we have that a good choice is given by  p = 2  5  and  q = 3  5,  from which we get λ = ε2 and m = 2  5λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Weighted analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Our aim is now to study the invertibility of this linear operator, and we  wish to do this in suitable weighted function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In order to introduce said spaces we shall start  by introducing a weight function useful in our situation, and for simplicity we may assume that the  biholomorphism P is defined on the region {r(z) ≤ 1} (this is true up to a rescaling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Define then  ρ = ρε(z) :=  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ε  on r(z) ≤ ε,  non decreasing  on ε ≤ r(z) ≤ 2ε,  r(z)  on 2ε ≤ r(z) ≤ 1/2,  non decreasing  on 1/2 ≤ r(z) ≤ 1,  1  on r(z) ≥ 1,  where we recall that z are the "small" coordinates around the singularity in ˜  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Using this weight  function we can introduce the weighted Hölder norm and its corresponding weighted Hölder spaces  Ck,α  ε,b (M), where k ≥ 0, α ∈ (0, 1) is the Hölder constant, b ∈ R is the weight and ε indicates the  dependence on the metric ω obtained by the gluing construction done above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' We define  ||u||Ck,α  ε,b (M) :=  k  �  i=0  sup  M  |ρb+i∇i  εu|ω  +  sup  dε(x,y)<injε  ����min  �  ρb+k+α(x), ρb+k+α(y)  � ∇k  εu(x) − ∇k  εu(y)  dε(x, y)α  ����  ω  ,  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  18  where injε is the injectivity radius of the metric ω, and thus interpret F (and L) as operators defined  as F : C2,α  ε,b (M) → C0,α  ε,b+2(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Following now a blended strategy between the one from Székelyhidi (as in Chapter 8 in [Sz]) and  the one from Tosatti and Weinkove (see [TW]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' we consider a variation of our operator by fixing a  point x ∈ M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' where x is away from the gluing region (for example we can assume x to be outside  of the open set on which the biholomorphism P is defined,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' and in the region where the weight is  identically equal to 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' and defining  ˜F(ψ) :=  ω3  ψ  ω3 − ef+evx(ψ)  (where evx is the evaluation operator at x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' with consequent linearization at zero  ˜L = L + efevx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Thus the new problem to solve is  (2)  ˜F(ψ) = 0,  whose solution clearly still gives a balanced Chern-Ricci flat metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  We then wish to prove the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' With the same notations as above, there exist b ∈ R and c > 0 such that for sufficiently  small ε it holds  ||u||C2,α  ε,b ≤ c||˜Lu||C0,α  ε,b+2,  for all u ∈ C2,α  ε,b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Suppose by contradiction that the above inequality does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' This means that for all  n ∈ N we can find εn > 0 and un ∈ C2,α  εn,b such that εn → 0 as n → 0, ||un||C2,α  εn,b = 1 and  (3)  ||˜Lun||C0,α  εn,b+2 < 1  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  In the first place we analyze what happens on M \\ P1 ≃ Mreg, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' away from the exceptional  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' The properties of the sequence {un}n∈N guarantee us that up to subsequences we may  assume un → u∞ uniformly on compact subsets of Mreg in the sense of C0,α  b  , with respect to  the background Calabi-Yau metric ˜ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Moreover, since for any compact set K ⊆ Mreg there exists  nK ∈ N such that for all n ≥ nK, on K it holds ω = ˜ω and hence ∇ω = ∇˜ω, we actually have  C2,α  b  convergence (again uniformly on compact subsets of Mreg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Also, for the same reason, on  any compact subset of Mreg, for n sufficiently large the Chern-Ricci potential is identically zero,  thus we get ˜L = L + evx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' and still choosing n sufficiently large, we end up with ω = ˜ω on K,  resulting in  Lun = 3  2  i∂∂un ∧ ˜ω2  ˜ω3  ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' a scalar multiple of the laplacian of un with respect to the Kähler metric ˜ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Thus taking the  limit for n → +∞ of ˜Lun yields  (4)  L∞u∞ + u∞(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  19  We shall now prove that u∞ is necessarily identically zero on the whole Mreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Indeed, take δ > 0  and Bδ a ball of radius δ around the singularity, and notice that  0 =  �  Mδ:=Mreg\\Bδ  L∞u∞ + u∞(x)dV˜ω = −  �  ∂Bδ  i∂u∞ ∧ ˜ω2 + Vol(Mδ)u∞(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Writing then with d ˆV the volume induced on ∂Bδ induced by ˜ω, we notice that  ����  �  ∂Bδ  ∂u∞ ∧ ˜ω2  ���� ≤  �  ∂Bδ  |∂u∞|ωd ˆV ≤ cδ4−b,  thus assuming b < 4 and taking the limit for δ → 0 in (4) we get u∞(x) = 0, and, consequently,  that u∞ is harmonic on the whole Mreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Notice now that, since L∞u∞ = 0, we get  (5)  0 =  �  Mδ  u∞L∞u∞˜ω3 =  �  Mδ  u∞i∂∂u∞ ∧ ˜ω2  and since ˜ω is balanced (actually Kähler) it holds  d(i∂u∞ ∧ (u∞˜ω2)) = u∞i∂∂u∞ ∧ ˜ω2 + i∂u∞ ∧ ∂u∞ ∧ ˜ω2,  which integrated on Mδ and combined with (5) gives  (6)  0 =  �  ∂Bδ  u∞i∂u∞ ∧ ˜ω2 =  �  Mδ  i∂u∞ ∧ ∂u∞ ∧ ˜ω2  = 2  �  Mδ  i∂u∞ ∧ ∂u∞ ∧ ∗˜ω = 2  �  Mδ  ⟨i∂u∞ ∧ ∂u∞, ˜ω⟩˜ω3 = 2  �  Mδ  |∂u∞|2˜ω3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  But since  ����  �  ∂Bδ  u∞i∂u∞ ∧ ˜ω2  ���� ≤  �  ∂Bδ  |u∞|ω|∂u∞|ωd ˆV ≤ cδ4−2b,  thus choosing b < 2 and taking the limit for δ → 0 in (6) we get  �  Mreg |∂u∞|2  ω ˜ω3 = 0, that is  u∞ ≡ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=', and since u∞(x) = 0, necessarily u∞ ≡ 0 on the whole Mreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Let now Mc := {r(z) ≥ 1/2} ⊆ Mreg be the compact set on which un → 0 uniformly in C2,α  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' To  obtain a contradiction we want to prove that {un}n∈N admits a subsequence uniformly convergent  to zero in C2,α  b  also on A := {r(z) < 1/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  In order to work in this region, it is simpler to shift to the "large" coordinates ζ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' the coordinates  on ˆX away from the exceptional divisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' It is then useful to recall the relations  ζ = ε−3/2z  and  r(z) = εr(ζ),  from which we can write down the explicit identification  �  r(z) < 1  2  �  = A ≃ ˜A = ˜Aε =  �  r(ζ) < 1  2ε−1  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  20  this last set ˜A is the one we will be working on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  The first thing to do is rewrite the weight function in terms of this coordinates on ˜A, resulting in  ρ =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  ε  on r(ζ) ≤ 1,  non decreasing  on 1 ≤ r(ζ) ≤ 2,  εr(ζ)  on 2 ≤ r(ζ) ≤ 1/2ε−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Notice that the entire gluing region of the metric (from the previous step) is entirely contained  inside the third region, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' {2 ≤ r(ζ) ≤ 1/2ε−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  We now go back to our sequence {un}n∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Since ||un||C2,α  εn,b = 1 for all n ∈ N, we have in  particular that on all ˜An := ˜Aεn holds  |ρbun| ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Introducing then the new sequence  Un := εb  nun,  the above weighted estimates for un imply the following ones for this new sequence:  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  |Un| ≤ c  on r(ζ) ≤ 1,  |Un| ≤ c  on 1 ≤ r(ζ) ≤ 2,  |Un| ≤ cr−b(ζ)  on 2 ≤ r(ζ) ≤ 1/2ε−1  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  This estimates for Un suggest us to introduce a new weight function ˜ρ = ˜ρn on ˜An given by  ˜ρ(ζ) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  1  on r(ζ) ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  non decreasing  on 1 ≤ r(ζ) ≤ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  r(ζ)  on 2 ≤ r(ζ) ≤ 1/2ε−1  n ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  with which we get that  |˜ρbUn| ≤ c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  and analogous weighted estimates also for ∇Un and ∇2Un,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' hence again by Ascoli-Arzelà theorem  we have that Un → U∞ uniformly on compact sets of ˆX (since ˜An → ˆX) in the sense of ˜C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='α  b  =  C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='α  b  (˜ρ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' where this last space is the weighted Hölder space on ˆX identified by the weight ˜ρ and  the Candelas-de la Ossa metric ωco,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  On the other hand it holds  ρb+2Lun = ˜ρb+2∆ωco,1Un,  hence  (7)  ||Lun||C0,α  εn,b+2 = ||∆ωco,1Un|| ˜C0,α  b+2,  and since 1  n > ||˜Lun||C0,α  εn,b+2 and un(x) → 0 hold, taking the limit in (7) we obtain that U∞ is  harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Moreover, if we assume b > 0, then U∞ is a harmonic function decaying at infinity, thus  necessarily U∞ ≡ 0, and thus Un  ˜C2,α  b→ 0 uniformly on compact sets of ˆX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  If we are now able to prove that Un admits a subsequence converging uniformly to zero on the  whole ˆX in the sense ˜C0  b we get our contradiction, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Indeed, if Un  ˜C0  b  → 0 uniformly  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  21  (up to subsequences) on ˆX, then scaled Schauder estimates imply that also Un  ˜C2,α  b→ 0 uniformly,  which is the same as saying un  ˜C2,α  εn,b  →  0 uniformly on {r(z) < 1/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Thus {un}n∈N up to subse-  quences is uniformly convergent to zero on the whole manifold M, which is a contradiction with  the fact that ||un||C2,α  εn,b = 1 for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Now we will prove that the said uniformly convergent subsequence exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' If by contradiction this  was not the case, since we have the uniform convergence on compact sets, we would be able to find  γ > 0 and {xn}n∈N ⊆ ˆX, xn ∈ ˜An, such that Rn := r(ζ(xn)) → +∞ and Rb  nUn(xn) ≥ γ for  all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' But {xn}n∈N is a bounded sequence (since it lies in {r(z) < 1/2}), thus if we name  rn := r(z(xn)) and recall that rn = εnRn, up to subsequences we can end up into two cases:  (i) if xn → x∞ such that r(x∞) > 0, then, since un is uniformly convergent on compact sets  on Mreg, we get that un(xn) is bounded, giving  0 < γ ≤ Rb  nUn(xn) = (Rnεn)bun(xn) = rb  nun(xn) −→  n→∞ 0,  which is a contradiction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  (ii) if rn → 0, we consider the biholomorphisms σn : Bn → A \\ {0}, given by  σn(z′) := r3/2  n z′,  where Bn := {0 < r(z′) < r−1  n  2 } ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Then, if we endow Bn with the metric  ηn := r−2  n σ∗  nω,  using (1) it is easy to notice that the couple (Bn, ηn) converges to (X, ωco,0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' the stan-  dard Calabi-Yau cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' If we then introduce the functions  wn := rb  nσ∗  nun  on Bn, we notice that the pullback of the weight function ρ gives  ρ′(z′) = σ∗  nρ(z′) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  εn  on r(z′) < R−1  n ,  non decreasing  on R−1  n  ≤ r(z) ≤ 2R−1  n ,  rnr(z′)  on 2R−1  n  ≤ r(z) < r−1  n  2 ,  from which we get (pulling back the inequality ρbun ≤ 1)  (8)  rb(z′)wn(z′) ≤ 1  on each z′ ∈ X (assuming n to be sufficiently large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Hence, this shows that for any  compact K ⊆ X, we can chose n ∈ N to be sufficiently large in order to have K ⊆ Bn  and ρ′(z′) = rnr(z′) on the whole K, and get that wn is uniformly bounded on K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' and  since this works for any compact K ⊆ X, we obtain that - up to subsequences - {wn}n∈N  converges uniformly on compact sets of X to a function w∞, and from (8) we get that  w∞ is decaying at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Moreover, recalling that Rb  nUn(xn) ≥ γ for all n ∈ N, if we  introduce the sequence yn := σ−1  n (xn), it is straighforward to notice that from its definition  follows that wn(yn)  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  22  geqγ and ||yn||ηn = 1 for all n ∈ N, thus implying that - up to subsequences - yn → y∞ ∈  X, and hence  (9)  w∞(y∞) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Now, if we recall the definition of the operator ˜L and take the pullback with respect to σn  of ρb+2 ˜Lun, it is immediate to see that on every compact K ⊆ X we get  (10)  σ∗  n  �  ρb+2 ˜Lun  �  = 3  2rb+2(z′)i∂∂wn ∧ η2  n  η3n  + 3  2  i∂ηn ∧ ∂ηn  η3n  + rb+2  n  rb+2(z′)un(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Analyzing then the single terms, we can notice that as n → +∞  σ∗  n  �  ρb+2 ˜Lun  �  → 0 from the hypothesis (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' i∂∂wn ∧ η2  n  η3n  → ∆ωco,0w∞  from the above observations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  rb+2  n  rb+2(z′)un(x) → 0 trivially;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' i∂ηn ∧ ∂ηn  η3n  → 0,  since from Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2 we can pullback with σn the inequality  |∂ω|ω ≤ cr2| log r|,  and obtain  |∂ηn|ηn ≤ cr3  nr2(z′)(| log rn| + | log r(z′)|) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Hence, taking the limit as n → +∞ in (10) we obtain  ∆ωco,0w∞ ≡ 0,  and since it holds on every compact K subseteqX, we get that w∞ is harmonic on the  whole cone X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' But now, since it holds (8), we can apply Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='9 from [CPY1] (actually  just its proof applied to ±w∞, without the hypothesis w∞ ≥ 0), and obtain that necessarily  w∞ ≡ 0 on X, which is a contradiction, as it holds (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Thus the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  □  As a direct consequence we get  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' The operator ˜L : C2,α  ε,b (M) → C0,α  ε,b+2(M) defined above is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Notice that L is elliptic and evx is a compact operator, thus L and ˜L share their index, which  is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Moreover, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='3 we have that ˜L is injective, thus we automatically get that ˜L is  also surjective and has bounded inverse, thus ˜L is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  □  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  23  With this result we can now show how to reformulate the original equation as a fixed point  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  In order to do this we shall introduce the operators ˆF, E, G : C2,α  ε,b (M) → C0,α  ε,b+2(M) defined as  ˆF(ψ) :=  ω3  ψ  ω3 ,  E(ψ) := ef+evx(ψ)  and  G(ψ) = efevx(ψ),  from which we can write  ˜F = ˆF − E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Now, we can consider the expansion  ˆF(ψ) = ˆF(0) + L(ψ) + ˆQ(ψ),  and thus rewrite Equation (2) as  ˆF(0) + L(ψ) + ˆQ(ψ) − E(ψ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Here, we notice that ˜L = L − G, thus we can rewrite Equation (2) once more and get  ˆF(0) + ˜L(ψ) + ˆQ(ψ) + G(ψ) − E(ψ) = 0,  and using now Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='4, we get that Equation (2) is therefore equivalent to  (11)  ψ = ˜L−1(E(ψ) − G(ψ) − ˆF(0) − ˆQ(ψ)) =: N(ψ),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' the search for a fixed point for the operator N : C2,α  ε,b (M) → C2,α  ε,b (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' To do this, we will  have to identify the open set on which we wish to apply Banach’s Lemma, and show that on said  open set N can be restricted and gives rise to a contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  The first thing to do, is the following remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' If C, τ > 0, and ϕ is a function on M such that ||ϕ||C2,α  ε,−2 ≤ Cετ, thanks to Remark  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2 it is straightforward to see that  ||i∂∂(ϕω)||C0,α  ε,0 ≤ ||ϕ||C2,α  ε,−2 ≤ Cετ,  thus we’re guaranteed that, choosing ε to be sufficiently small, ω2  ϕ > 0, and thus it’s square root ωϕ  exists and is a balanced metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Moreover, we can apply again the argument used in Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='2,  and obtain that if ||ϕ||C2,α  ε,−2 ≤ Cετ, then  |ωϕ − ω|ω ≤ c||ϕ||C2,α  ε,−2 ≤ cετ,  which also implies that ωϕ → ω, as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Thanks to this remark, we have a suggestion on how to choose the open set on which apply  Banach’s Lemma, hence we introduce  U := {ϕ ∈ C2,α  ε,b | ||ϕ||C2,α  ε,b < ˜cεb+2+τ} ⊆ C2,α  ε,b ,  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  24  and we notice that for every ϕ ∈ U it holds ||ϕ||C2,α  ε,−2 ≤ Cετ, with C independent of ϕ and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  We will now prove that on U, the operator N is a contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In particular, given ϕ1, ϕ2 ∈ U, we  want to estimate  N(ϕ1) − N(ϕ2) = ˜L−1(E(ϕ1) − G(ϕ1) − (E(ϕ2) − G(ϕ2)) − ( ˆQ(ϕ1) − ˆQ(ϕ2))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Starting by analyzing the ˆQ-term, we notice that by the Mean Value Theorem we can find t ∈ [0, 1]  such that  ˆQ(ϕ1) − ˆQ(ϕ2) = d ˆQη(ϕ1 − ϕ2) = (Lη − L)(ϕ1 − ϕ2),  where η = tϕ1 + (1 − t)ϕ2 ∈ U, and Lη is the linearization of ˆF at η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' With the same strategy used  to compute L we can easily obtain an expression for Lη, and thus get  (Lη − L)(ϕ1 − ϕ2) = 3  2  (ωη − ω) ∧ i∂∂((ϕ1 − ϕ2)ω)  ω3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  From here, taking the norms with respect to ω, we can use the fact that η ∈ U together with Remark  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='5, to obtain  |(Lη − L)(ϕ1 − ϕ2)| ≤ c|ωη − ω|ω|i∂∂((ϕ1 − ϕ2)ω)|ω ≤ cετ|i∂∂((ϕ1 − ϕ2)ω)|ω,  and thus, by multiplying the inequality with ρb+2, get  (12)  || ˆQ(ϕ1) − ˆQ(ϕ2)||C0,α  b+2,ε ≤ cετ||ϕ1 − ϕ2||C2,α  ε,b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Focusing instead on the (E − G)-term, and again using that ϕ1, ϕ2 ∈ U together with Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='5,  we get  |E(ϕ1) − G(ϕ1) − (E(ϕ2) − G(ϕ2))| =ef|(ϕ2(x) − ϕ1(x) + eϕ1(x) − eϕ2(x)|  ≤c(|ϕ1(x)| + |ϕ2(x)|)|ϕ1(x) − ϕ2(x)|  ≤cετ|ϕ1(x) − ϕ2(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Now, as above, multiplying the inequality with ρb+2 and recalling the choice of x ∈ M and the  definition of ρ, we get  ρb+2|E(ϕ1) − G(ϕ1) − (E(ϕ2) − G(ϕ2))| ≤ cετρb+2(x)|ϕ1(x) − ϕ2(x)|  and hence  (13)  ||E(ϕ1) − G(ϕ1) − (E(ϕ2) − G(ϕ2))||C0,α  b+2,ε ≤ cετ||ϕ1 − ϕ2||C2,α  ε,b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Thus, combining estimates (12) and (13) with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='3, we get  (14)  ||N(ϕ1) − N(ϕ2)||C2,α  ε,b ≤ cετ||ϕ1 − ϕ2||C2,α  ε,b ,  hence, choosing ε sufficiently small ensures us that N is a contraction on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  We are left with proving that N(U) ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' To do this, we shall impose a restriction on τ, that is  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  25  τ < 2  5λ0, and we will use it in the inequalities to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Take then ϕ ∈ U, thanks to estimate (14)  and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='3, we have  ||N(ϕ)|||C2,α  ε,b ≤||N(ϕ) − N(0)|||C2,α  ε,b + ||N(0)|||C2,α  ε,b  ≤cετ||ϕ|||C2,α  ε,b + ||˜L−1(1 − ef)|||C2,α  ε,b  ≤cετ||ϕ|||C2,α  ε,b + ||f||C0,α  ε,b  ≤c(εb+2+2τ + εb+2+(2/5)λ0)  ≤cεmin{τ,(2/5)λ0−τ}εb+2+τ  ≤˜cεb+2+τ,  implying that N(U) ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  This shows that everything is into place to apply Banach’s Lemma on the open set U and obtain ˆω  a Chern-Ricci flat balanced metric ˆω on M, thus proving most of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' It is known that for a manifold which is Calabi-Yau with holomorphic volume Ω,  the existence of a Chern-Ricci flat balanced metric implies that Ω is parallel with respect to the  Bismut connection associated to said metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Among the other things, this implies that the restricted  holonomy of the Bismut connection of Chern-Ricci flat balanced metrics is contained in SU(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Recalling Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='5, we have that the couple (M, ˆω) Gromov-Hausdorff converges  to the singular Calabi-Yau metric on Mreg and, up to rescaling, to the Candelas-de la Ossa metrics  nearby the exceptional curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Notice that en passant we have proved that, on our small resolutions, for certain  classes nearby the boundary of the balanced cone, the Gauduchon conjecture for balanced metrics  (see [STW]) holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' in said balanced classes we can always find a Chern-Ricci flat balanced  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Cohomology classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' We end the discussion on the construction by analyzing the cohomol-  ogy class naturally associated to the metric ˆω just obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Indeed, balanced metrics naturally  induce an (n − 1, n − 1)-class, thus in our case we get  [ˆω2] ∈ H2,2  dR(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  We are then interested in understanding if this cohomology class can be described in terms of the  cohomology classes of the objects we used for the gluing construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' The first thing to observe is  that the deformation introduced in the previous section preserves the balanced class, thus we have  that [ω2] = [ˆω2], thus we shall work with ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Let’s then introduce two cut off functions θ1, θ2 : [0, +∞) → [0, 1] defined as follows:  θ1(x) :=  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  1  if x ≤ 1  8ε−3/5  non increasing  if 1  8ε−3/5 ≤ x ≤ 1  4ε−3/5  0  if x ≥ 1  4ε−3/5  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  26  and  θ2(x) :=  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  0  if x ≤ 8ε−3/5  non decreasing  if 8ε−3/5 ≤ x ≤ 16ε−3/5  1  if x ≥ 16ε−3/5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  and since for sufficiently small ε we have that ω is exact on K := {1  8ε−3/5 ≤ r(ζ) ≤ 8ε−3/5}, it  exists a 3-form β such that  ω2 = dβ  on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Introduce then the form  Ωc := d(θ1(r(ζ)) + θ2(r(ζ)))β),  which is a smooth compactly supported form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Moreover, the form  β − (θ1(r) + θ2(r))β  can be extended as zero to the whole M, thanks to the definition of the cut-offs, and thus get that  [ω2] = [Ωc],  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' the class [ω2] admits a compactly supported representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In addition, the two cut-offs intro-  duced also allow us to decompose Ωc = Ω′  c + Ω′′  c, such that on K hold  Ω′  c = d(θ1(r)β)  and  Ω′′  c = d(θ2(r)β),  and both Ω′  c and Ω′′  c are compactly supported and closed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' in particular said compact supports are  respectively contained in ˆX and Mreg (via the obvious identifications), and from their definition it  is straightforward to see that  [Ω′  c] = ε4[ω2  co,1] ∈ H4  c ( ˆX)  and  [Ω′′  c] = [˜ω2] ∈ H4  c (Mreg),  where Hc denotes the compactly supported cohomology group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Also, recalling that ˆX ≃ OP1(−1)⊕2,  it is clear that ˆX is homotopically equivalent to P1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' hence applying Poincaré duality we get  H4  c ( ˆX) ≃ Hc  2( ˆX) ≃ Hc  2(P1) = H2(P1) = ⟨[P1]⟩,  which means that the non-zero class [ω2  co,1], up to multiplicative constants, is the Poincaré dual of  the generator of H2(P1) (thus we can "confuse" them with each other), and thus we can write  [ω2] = [˜ω2] + ε4[P1]  in H2,2  dR(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Finally, we also notice that  �  P1 ω = ε2  �  P1 ωco,1 −→  ε→0 0,  hence the balanced class [ω2], as ε → 0, converges to a nef class, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' to the boundary of the bal-  anced cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  This concludes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  27  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Even though this construction is done to address the situation of a non-Kähler small  resolution, it can also be applied when said small resolution M is instead Kähler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' In this case we  know from Yau’s theorem that M admits a Kähler Calabi-Yau metric ω1, hence together with the  balanced class induced by our Chern-Ricci flat balanced metric ˆω we also have the one induced by  ω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' This two balanced classes need not be the same, however, even if they are to coincide, there  is no uniqueness result that would guarantee that the two metrics have to be the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' moreover,  the deformation we used in our construction does not cover the whole balanced class, hence in this  case we are not even guaranteed that the two metrics are linked by our chosen deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Relation to the Hull-Strominger system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' As a conclusion of the paper, we would like to  briefly relate our construction to the Hull-Strominger system and how we intend to develop our  research in this direction, hence we shall first quickly recall the definition of said system (for more  details we refer to the notes [GF]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  The framework is given by (X, Ω) a (not necessarily Kähler) Calabi-Yau manifold, and the first  equation of the system, for ω a hermitian metric, is known as the dilatino equation and is given by  d∗ω = dc log ||Ω||ω,  which easily seen to be equivalent to the conformally balanced equation  d  �  ||Ω||ωωn−1�  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Hence, this last equation tells us that we need to work with balanced manifolds, and thus we see a  first relation to our scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  To complete the system we need to pair the dilatino equation with two Hermite-Einstein equations  for holomorphic vector bundles, thus, in the same fashion as what we had with the dilatino equa-  tion, the presence of the Hermite-Einstein equation in the Hull-Strominger system will limit us to  consider only polystable bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Finally, adding one last equation, known as the Bianchi identity,  we can introduce the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Given a Calabi-Yau manifold (X, Ω) and a holomorphic vector bundle E on X,  we say that the triple (ω, h, ∂T ) is a solution of the Hull-Strominger system if it satisfies  ΛωFh = 0,  ΛωR = 0,  d∗ω − dc log ||Ω||ω = 0,  ddcω − α(trR ∧ R − trFh ∧ Fh) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  where, α is a non-vanishing constant, ω is a hermitian metric on X, h is a hermitian metric along  the fibers of E, ∂T is a holomorphic structure on the tangent bundle of X, and R is the Chern  curvature tensor of ω, read as a hermitian metric on the holomorphic vector bundle (TX, J, ∂T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  The Bianchi identity (also known as anomaly cancellation equation), is the hardest and least  understood equation of the system and also the one we wish to address in the development of our  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  28  It is significant to notice that if we choose (E, h) to be the holomorphic tangent bundle with the  metric ω, and take ω a Kähler Ricci-flat metric, we see that this satisfies the system, thus being  a solution of the Hull-Strominger system is a condition that generalizes being Kähler Calabi-Yau,  hence a very promising candidate class of special metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Also, thanks to the equivalence between Hermite-Einstein metrics and Hermite-Yang-Mills con-  nections it is possible to rewrite the system from a gauge-theoretical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Given a Calabi-Yau manifold (X, Ω) and a hermitian vector bundle (E, h) (with  a fixed holomorphic structure) on M, the triple (ω, A, ∇) is a solution of the Hull-Strominger  system if it satisfies  ΛωFA = 0,  F 0,2  A  = 0,  ΛωR∇ = 0,  R0,2  ∇ = 0,  d  �  ||Ω||ωωn−1�  = 0,  ddcω − α(trR ∧ R − trFh ∧ Fh) = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  where α is a non-vanishing constant, ω a hermitian metric on X, A is unitary connection on (E, h)  and ∇ is a unitary connection on (TM, J, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  This second description is useful to notice a series of necessary conditions when X is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Indeed, as already observed we have that M has to be necessarily balanced, and given the natural  balanced class τ given by the dilatino equation, we have that E and TM have to be τ-polystable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Morever, we have that c1(E) · τ = 0 and c1(M) = 0, and also  (15)  c1(E) · τ = 0  ch2(E) = ch2(X) ∈ H2,2  BC(X, R),  where ch2 denotes the second Chern character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Although several examples of solutions have been studied over time (several can be found, for  example, in [GF]), there is still a very poor understanding on the existence of solutions, even on  threefolds, where however it was conjectured by Yau in [Y] that conditions (15) are not only a  necessary condition but even a sufficient one to the existence of solutions in the case of threefolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  If we then take our construction, and we view it in the system’s scenario, we can make the  following final remark in which we explain our ideas on how to expand our construction in this  direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Given ˆβ a Chern-Ricci flat balanced metric on a Calabi-Yau threefold (Y, Ψ), it holds  ||Ψ||ˆβ ≡ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=',  showing that our metric ˆω gives a solution of the conformally balanced/dilatino equation on our  small resolutions (M, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Thus our construction gives us two solutions of the dilatino equation  on (M, Ω), that are ˆω and ω′ := ||ˆΩ||−2  ω ω, where this last one is the dilatino equation solution  associated to the balanced metric ω obtained in the first part of the gluing construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' From  here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' thanks to the fact that this metrics are nearby a Kähler Ricci-flat metric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' our idea (currently  CHERN-RICCI FLAT BALANCED METRICS ON SMALL RESOLUTIONS OF CALABI-YAU THREEFOLDS  29  in devolopment in the upcoming [Gi]) is to adapt the stretegy in [CPY1] to construct a Hermite-  Einstein metric on the tangent bundle with respect to the above metrics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' and eventually from there  try and extend it to a whole solution of the Hull-Strominger system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
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+page_content='  [TW] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Tosatti, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Weinkove, The complex Monge-Ampère equation on compact Hermitian manifolds, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' 23 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='4, 1187–1195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
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+page_content=' Tseng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Yau, Non-Kähler Calabi-Yau manifolds, in String-Math 2011, 241– 254, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Sympos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Pure  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
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+page_content=' Aus-  tausch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  [Y]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content=' Yau, Metrics on complex manifolds, Science in China Series A Mathematics 53 (2010) 565-572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='  DEPARTMENT OF MATHEMATICS, AARHUS UNIVERSITY, NY MUNKEGADE 118, 8000 AARHUS C, DENMARK  Email address: federico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='giusti@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='dk  DEPARTMENT OF MATHEMATICS, AARHUS UNIVERSITY, NY MUNKEGADE 118, 8000 AARHUS C, DENMARK  Email address: c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='spotti@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
+page_content='dk' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LNFJT4oBgHgl3EQfxy2s/content/2301.11636v1.pdf'}
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+Anomalous triple gauge couplings
+in electroweak dilepton tails at the LHC
+and interference resurrection
+Haeyun Hwang a, Ui Min b, Junghyeon Park b, Minho Son b and Jae Hyeok Yoo a
+a Department of Physics, Korea University, Seoul, Republic of Korea
+b Department of Physics, Korea Advanced Institute of Science and Technology,
+291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
+Abstract
+We study the electroweak dilepton production with two forward jets at the LHC, aiming to measure
+the anomalous triple gauge couplings in the Effective Field Theory (EFT) approach. This process
+has two distinctive advantages compared to typical diboson processes. The EFT cutoff can be
+unambiguously imposed on the dilepton invariant mass. The interference between Standard Model
+(SM) and beyond the SM is resurrected in the inclusive cross section of the full amplitude, including
+two forward jets.
+As a concrete illustration, we perform the detailed analytic and numerical
+study of the interference using a simpler toy process.
+We propose a new kinematic variable,
+VBFhardness, that controls the amount of energy flowing into the dilepton subprocess. We show
+that an appropriate cut on VBFhardness makes the interference resurrection manifest. Finally, we
+use the invariant mass of the dilepton mass system as well as the transverse momentum, as done
+in the literature, to derive the sensitivity to anomalous triple gauge couplings at the LHC and the
+high luminosity LHC.
+arXiv:2301.13663v1  [hep-ph]  31 Jan 2023
+
+Contents
+I. Introduction
+3
+II. EW dilepton production with two associated jets
+6
+III. Toy process for analytic study: Single lepton with an associated jet
+7
+A. Cross section for on-shell W boson
+9
+B. Cross section for off-shell W boson
+10
+C. Numerical calculation of toy process and interference resurrection
+12
+IV. Numerical analysis of EW dilepton with two associated jets
+15
+A. Interference resurrection
+15
+B. Validation against the CMS analysis and BDT analysis
+17
+C. Sensitivity to aTGC at the LHC
+20
+V. Conclusion
+27
+Acknowledgments
+28
+A. Details on simulation
+28
+1. Signal and background generation
+28
+B. Computation detail of qV → q′νℓ
+29
+1. Choice of four momenta and amplitudes
+29
+2. Phase space integration
+32
+3. Interference between SM and BSM amplitudes for coupling λz
+32
+C. Detail of BDT Analysis
+33
+References
+36
+2
+
+I.
+INTRODUCTION
+Although the LHC has been performing great including the discovery of the Higgs
+boson [1, 2], it continuously shows no evidence for the new physics, or beyond the Standard
+Model (BSM), only confirming the Standard Model (SM) to a better precision. It indicates
+that either new particles, if they exist, are very weakly coupled to the SM or they may be
+hidden in the energy scale beyond the LHC reach, especially, if a new physics has a sizeable
+coupling to the SM. Given the strong indication for the mass gap between the electroweak
+and new physics scales, the effective field theory approach makes sense to parametrize
+the possible new physics effects encoded in the higher-dimensional operators. Deviating
+from the SM with the Higgs doublet under the SM gauge symmetry, the effective La-
+grangian, known as the SM Effective Field Theory (SMEFT), below the cutoff Λ is written as
+L = LSM +
+�
+i
+c(6)
+i
+Λ2 O(6)
+i
++
+�
+i
+c(8)
+i
+Λ4 O(8)
+i
++ · · · ,
+(1)
+where the lepton number conservation was assumed and c(d)
+i
+is the Wilson coefficient for the
+dimension-d operator O(d)
+i . The non-vanishing effect from the new physics on the Wilson
+coefficients of higher-dimensional operators will cause a deviation of couplings among SM
+particles from the SM prediction.
+In this work, we focus on the precision measurements of the cubic interaction of the gauge
+bosons at the LHC. Taking into account the property of the SMEFT up to dimension-6
+operators, the deviation of the triple gauge couplings from the SM can be parametrized in
+terms of three anomalous Triple Gauge Couplings (aTGC) as
+Ltgc = ie
+�
+W +
+µνW −
+µ − W −
+µνW +
+µ
+�
+Aν + iecθ
+sθ
+(1 + δg1,z)
+�
+W +
+µνW −
+µ − W −
+µνW +
+µ
+�
+Zν
++ ie(1 + δκγ)Aµν W +
+µ W −
+ν + iecθ
+sθ
+(1 + δκz) Zµν W +
+µ W −
+ν
++ i λze
+m2
+W
+�
+W +
+µνW −
+νρAρµ + cθ
+sθ
+W +
+µνW −
+νρZρµ
+�
+,
+(2)
+where cθ =
+�
+1 − s2
+θ and δκz = δg1,z −
+s2
+θ
+c2
+θ δκγ. Considering only amplitudes with a single
+insertion of aTGC, the cross section is in general a quadratic function of aTGC and it can
+be parametrized as
+σ = σSM + Ciσi
+SM×BSM + CiCjσij
+BSM×BSM ,
+(3)
+where the index i runs over three aTGCs, Ci ≡ {λz, δg1,z, δκz}.
+Typically, measurements of aTGC at the LHC have been performed by using diboson
+processes such as WW, WZ, and Wγ in the lepton-enriched final state channels. Unlike the
+precision measurement in LEP from WW production process with the fixed center of mass
+3
+
+energy around the electroweak scale, the sensitivity on aTGC from the LHC relies on the
+accessibility to the higher energy as long as it does not violate the validity of the EFT [3],
+or one should not use the data at the energy E above the cutoff Λ, or E/Λ ≲ 1. While
+the leptonic channel is clean and thus provides good sensitivity, the accompanying neutrinos
+make it difficult to experimentally extract the exact scale of the hard process. When one
+can not impose an appropriate cut on the scale of the hard process to ensure E/Λ ≲ 1,
+one can only hope for setting a conservative bound in this situation [4, 5]. Another issue
+in the diboson process has been the noninterference between the SM and BSM amplitudes
+which was found to be dictated by the helicity structure of the amplitudes [6, 7]. Including
+only dimension-6 operators, in the absence of interference, the leading BSM contribution to
+the total cross section scales O(Λ−4), and it may invalidate the EFT expansion in terms of
+Λ. This also makes the translation of the data to the SMEFT sensitive to the dimension-
+8 operators as the leading contribution is in the same order of the interference between
+dimension-8 operators and the SM.
+There have been many attempts to resurrect the interference in the diboson process.
+While 2 → 2 diboson processes are subject to the noninterference, unstable vector gauge
+bosons must decay. Once the 2 → 2 diboson amplitude is extended to 2 → 3, 4 by gluing
+with the three point amplitude(s) for a gauge boson decay into two fermions, the total
+helicity of both amplitudes of the dimension-6 and the SM can match and thus interfere.
+The authors in [5, 8] suggested to look into differential angular distributions in the leptonic
+decay channels to resurrect the interference. See [9] for a related discussion. The authors
+in [5, 10] pointed out the partial resurrection of the interference due to the QCD next-
+to-leading order (NLO) effect. The role of off-shellness of the vector gauge bosons in the
+diboson process on the interference has been studied in [11].
+In this work, we newly add the dilepton production process with two associated forward
+jets in the vector boson fusion (VBF) to the list regarding the interference resurrection. This
+process at √s = 13 TeV, using the integrated luminosity of 35.9 fb−1, has been analyzed
+by the CMS collaboration [12]. While the signal rate of the VBF process is smaller than
+the diboson production from the QCD process, it can solve the two aforementioned issues:
+the scale of the hard subprocess can exactly be extracted from the invariant mass of two
+leptons and the helicity of the full 2 → 4 process apparently allows the interference between
+the amplitudes with dimension-6 operators and those from the SM. While the electroweak
+(EW) ℓℓ + jets process which is our main interest in this work may be considered as the
+EW Drell-Yan process, we aim to measure aTGCs via the tree-level process whereas QCD
+Drell-Yan process can access them via one loop effect [13]. One can see [14–19] (and [20, 21]
+for the experiment) for the precision study at the high energy tail of the QCD ℓℓ process
+focusing on the tree-level four-fermion interactions.
+4
+
+A confusion arises due to the usual effective W approximation (EWA) [22–35] which
+takes the gauge boson radiated off the quark line as an on-shell particle and focuses on the
+WW initiated subprocess. If this is the case, the interference will be suppressed again as
+the different total helicities of the SM and BSM amplitudes of WW → ℓℓ do not allow the
+interference in the massless limit [7]. For better understanding, an analytic study of the
+process that takes the full effect of the forward quark current would be highly beneficial. To
+this end, we carry out the full analytic calculation for a simpler process uγ → dνe+ that has
+only one forward quark current and one intermediate gauge boson as a toy process. As will
+be discussed below in detail, we find that the interference cross section of uγ → dνe+ with
+respect to the SM counterpart does resurrect the energy growing behavior, importantly in
+the inclusive cross section, that would have been lost in the W +γ → νe+ subprocess due to
+the helicity structure. The resurrected energy growing interference survives even when the
+apparent conditions for the EWA, namely, small transverse momenta of the forward quarks
+and small gauge boson masses compared to the overall energy scale of the hard subprocess,
+are applied in the full uγ → dνe+ process and thus provides a counter-example to the
+usual EWA assumption (see [35] for a related discussion). Our simpler toy process provides
+the proof of concept example for the resurrected interference in the inclusive cross section,
+and the intuition from it greatly helps for a better qualitative understanding of our EW ℓℓ
+production process with two associated jets.
+It turns out to be crucial that an enough energy must flow into the ℓℓ hard subprocess
+to resurrect the energy growing interference in the inclusive cross section of the full 2 → 4
+process. Unlike the QCD Drell-Yan process where mℓℓ directly controls the fraction of the
+energy that goes into the dilepton system, it becomes ambiguous in our EW ℓℓ with two for-
+ward jets process because some fraction of energy goes to the scattered quarks. In this work,
+we propose a new variable, what we call VBFhardness, that allows to control the fraction of
+energy carried by the ℓℓ subsystem. We demonstrate that the energy growing interference
+with respect to the SM is clearly resurrected with an appropriate cut on VBFhardness.
+In Section II we briefly sketch the (non)interference of the dilepton production with two
+associated jets. In Section III we provide the analytic result of a simpler 2 → 3 (instead of our
+2 → 4) toy process as this simpler example can be analytically calculated to capture the full
+effect of the forward jet from the viewpoint of the interference resurrection and the validity of
+the EWA. In Section IV we perform the numerical simulation of the EW ℓℓ production with
+two associated jets as our main process of interest. In particular, we validate our simulation
+against the CMS cut-and-count analysis. We carry out the multivariate analysis using the
+Boosted Desicion Tree (BDT). We finally derive the sensitivity of aTGC at the LHC and
+high luminosity LHC (HL-LHC).
+5
+
+II.
+EW DILEPTON PRODUCTION WITH TWO ASSOCIATED JETS
+±1
+2
+∓1
+2
+±1
+2
+∓1
+2
+±1
+2
+∓1
+2
+−1
++1
++1
++1
+−1
+−1
+±1
+2
+∓1
+2
+±1
+2
+∓1
+2
+±1
+2
+∓1
+2
+−1
+∓1
+2
+±1
+2
++1
++1
+−1
+±1
+2
+∓1
+2
+∓1
+2
+±1
+2
+∓1
+2
+±1
+2
+−1
+±1
+2
+∓1
+2
+−1
++1
++1
+FIG. 1: Interference between BSM and SM diagrams in the massless limit where only two types
+of SM diagrams are shown. The blob denotes the insertion of the dimension-6 operator tr(W 3
+µν).
+The helicity assignment is displayed as an example.
+Fig. 1 illustrates the subset of diagrams for the 2 → 4 amplitudes, leading to the dilepton
+with two associated jets, and possible helicity assignments which allow the interference
+between SM and BSM amplitudes. There is no similar diagram to the first one in Fig. 1 with
+the SM triple gauge couplings of the transverse modes as the helicity can not be correctly
+assigned. Its non-vanishing diagram can arise via helicity flips along with the Higgs VEV
+insertions and it will be suppressed by O(m2
+W/E2).
+The virtuality of W emitted off the initial quark current induces the energy uncertainty
+of W whose inverse sets the time uncertainty ∆t ∼ E/V 2 where V is the virtuality of W and
+E is the scale of the hard process. As long as ∆t is much longer than the typical interaction
+time t ∼ 1/E, one can not distinguish the virtual W from on-shell one. In this situation, one
+typically computes the partonic cross section of the hard subprocess, treating W as on-shell
+gauge boson, and convolutes it with the probability distribution function of the W gauge
+boson. This is known as the effective W approximation (EWA) [35]. The helicity structure
++1
++1
+−1
++1
+±1
+2
+∓1
+2
++1
+−1
++1
+−1
+±1
+2
+∓1
+2
++1
+−1
+±1
+2
+∓1
+2
+∓1
+2
+±1
+2
+FIG. 2: Noninterference between BSM and SM amlitudes for WW → ℓℓ. The blob denotes the
+insertion of the tr(W 3
+µν) operator. The helicity assignment displayed is an example.
+of the forward quark current and the correlation of its phase space with that of the hard
+subprocess seem to be simply dropped in the EWA. While diagrams that do not have two
+6
+
+±1
+2
+∓1
+2
++1
+−1
++1
+−1
++1
+±1
+2
+∓1
+2
+±1
+2
+∓1
+2
+−1
++1
++1
++1
+−1
+±1
+2
+∓1
+2
+FIG. 3: The interference (noninterfernce) in the 2 → 3 process (2 → 2 subprocess). The blob
+denotes the single insertion of the tr(W 3
+µν) operator. The complete set of diagrams are shown in
+Fig. 4.
+obvious forward quark currents, for instance, third diagram in Fig. 1 and other radiation
+type diagrams, are not included in the EWA assumption, they will be harmless upon the
+forward jet tagging. Apparently, the SM amplitudes of the WW → ℓℓ process in Fig. 2 can
+not interfere with the BSM amplitudes in the massless limit, known as noninterference [7],
+unlike the aforementioned allowed interference in 2 → 4 amplitudes.
+The middle diagram in Fig. 2 can interfere with the BSM amplitude via helicity flips
+in the sub-leading order. It is consistent with that it can extend to the 2 → 4 amplitude
+by attaching two quark currents upon helicity flips. On contrary, the extended amplitude
+with two attached quark currents of the third diagram in Fig. 2 can interfere with the
+corresponding BSM amplitude without any suppression as is evident in Fig. 1. 1 Assuming
+the EWA from the beginning and working on the 2 → 2 process may not be the same as
+what is obtained by applying the EWA limit of the full amplitude.
+III.
+TOY PROCESS FOR ANALYTIC STUDY: SINGLE LEPTON WITH AN AS-
+SOCIATED JET
+The purpose of this section is to analytically investigate (and numerically confirm) the
+helicity structure and related kinematics of simpler 2 → 3 process uγ → dνe+ (see Fig. 3
+and Fig. 4) that captures the full effect of the quark current attached to a vector gauge
+boson. While the analytic calculation of the full 2 → 4 process in Section II is beyond the
+scope of this work, our analytically calculable 2 → 3 toy process
+2 provides the proof of
+1 In the 2 → 4 diboson process decaying into two pairs of fermions, the narrow width approximation allows
+to factorize the phase space of decaying on-shell gauge bosons from that of the hard process, and the
+process is subject to the noninterference. In this situation, the interference can appear, for instance, in
+the differential cross section of the azimuthal angle [5, 8].
+2 A similar explicit computation may be applicable to the 2 → 3 process of qq′ → γW ∗ → γℓνℓ where the
+effect of the off-shell W gauge boson on the interference, for instance, whether the interference between
+7
+
+concept for the resurrected interference and an intuition on the validity of the EWA in the
+SMEFT. We expect this simpler toy process to capture important missing properties when
+simply approximating with 2 → 2 VBF process under the EWA assumption. We consider
+uγ → dνe+ since it involves with the exchange of only the W gauge boson.
+A similar
+discussion is applied to qV → q′ℓℓ although the evaluation is more challenging.
+The helicity assignments of two diagrams in Fig. 3 indicate that the interference between
+the SM and BSM amplitudes in the 2 → 3 process can be allowed. We separately consider
+the kinematic regions for the on-shell and off-shell intermediate W gauge bosons decaying to
+ℓνℓ since they have different qualitative behaviors. For the resonant on-shell W gauge boson,
+the 2 → 3 process is factorized into the production of the on-shell W gauge boson and its
+decay. It is expected that the inclusive cross section is subject to the noninterference and
+the interference at best can be resurrected only in the differential cross section of an angular
+observable (although it is difficult to be reconstructed in the experiment). On contrary, for
+the non-resonant 2 → 3 process, the aforementioned factorization is not possible and the
+interference in principle can appear in the inclusive cross section.
+u
+d
+γ
+W
+W
+p2
+k1
+k2
+ν
+e+
+p1
+k3
+k
+(a)
+u
+d
+W
+γ
+k1
+k2
+ν
+e+
+p1
+k3
+e
+p2
+(b)
+u
+p1
+p2
+k3
+k1
+k2
+ν
+e+
+W
+d
+d
+γ
+(c)
+p1
+p2
+u
+γ
+u
+d
+W
+ν
+e+
+k3
+k1
+k2
+(d)
+FIG. 4: The complete set of SM diagrams for the process uγ → dνe+. These four diagrams are
+required to guarantee the Ward identity and to get the correct high energy behavior.
+The full set of SM diagrams of the EW uγ → dνe+ process are shown in Fig. 4. We
+classify the first two diagrams a and b as the process of interest that probe the hard 2 → 2
+subprocess and the last two diagrams c and d as the radiation type. All four diagrams in
+SM and BSM amplitudes can be resurrected in the inclusive cross section, can be explicitly understood.
+8
+
+Fig. 4 are required to satisfy the Ward identity, namely p2 ·(Ma +Mb +Mc +Md) = 0. For
+the resonant intermediate W, the Ward identity can be shown to be satisfied among three
+diagrams, p2 · (Ma + Mc + Md) = 0, using the narrow width approximation. We postpone
+all the details for the analytic calculation of uγ → dνe+ to Appendix B. In what follows, we
+quote only the final result.
+A.
+Cross section for on-shell W boson
+The diagrams a, c, and d in Fig. 4 mainly contribute to the resonant 2 → 3 process where
+we can restrict the phase space to those in the W mass window, or k2 = (2z − 1)ˆs ≈ m2
+W
+where z = [1/2, 1] is the fraction of the total energy
+√
+ˆs flowing into the νee+ system and k is
+the four-momentum of it. The process can be factorized into the 2 → 2 process of uγ → dW
+and the decay of W to νee+ using the narrow width approximation for the on-shell W boson
+of the width ΓW.
+We evaluate the partonic differential cross section with respect to φ in the limit of ˆs ≫ m2
+W
+where φ is the angle between the planes made out of the forward quark current and the lepton
+current (see Fig. 17). The SM contribution is rather subtle to evaluate due to the forward
+singularity in the massless fermion limit. Its size is roughly given by
+dˆσSM
+dφ
+=
+� cos θmax
+cos θmin
+d cos θ d2ˆσSM
+dφd cos θ ≈
+1
+2 · 2
+1
+512π2
+8πe2g4
+3
+mW
+ΓW
+1
+ˆs
+1
+δ ,
+(4)
+where δ = 2p2
+T min/ˆs assuming δ ≪ 1 and it comes from the integration regularized by the
+pT cut of the forward quark,
+cos θmax/min = ±
+�
+1 −
+p2
+T min
+ˆs(1 − z)2 ≈ ±
+�
+1 − 2 p2
+T min
+ˆs
+�
+for
+ˆs ≫ m2
+W, p2
+T min .
+(5)
+On the other hand, the leading contribution to the partonic differential cross section for the
+interference in the high energy limit, ˆs ≫ m2
+W, is estimated to be
+dˆσSM×BSM
+dφ
+=
+1
+2 · 2
+λz
+512π4
+πe2g4
+3
+2
+mWΓW
+�
+cos(2φ)
+�
+2 − log
+ˆs
+m2
+W
+��
++ O(ˆs−1/2) .
+(6)
+Upon the integration over the angle φ, the interference term vanishes while it is recovered
+in the differential cross section with respect to φ. The individual contributions to the total
+9
+
+cross section for the interference are given by
+dˆσSM×BSM(uLγL → dνe+)
+dφ
+=
+λz
+512π4
+πe2g4
+144
+1
+mWΓW
+×
+�
+9π2 cos φ + 16 cos(2φ)
+�
+5 − 3 log
+ˆs
+m2
+W
+� �
++ O(ˆs−1/2) ,
+dˆσSM×BSM(uLγR → dνe+)
+dφ
+=
+λz
+512π4
+πe2g4
+144
+1
+mWΓW
+×
+�
+− 9π2 cos φ + 16 cos(2φ)
+�
+7 − 3 log
+ˆs
+m2
+W
+� �
++ O(ˆs−1/2) ,
+(7)
+where linear terms in cos φ cancels upon the summation and there is no contribution from
+the right-handed quark in the massless limit. In the same high energy limit, the quadratic
+term in the anomalous coupling λz is approximately estimated to be
+dˆσBSM2
+dφ
+=
+1
+2 · 2
+λ2
+z
+512π4
+πe2g4
+6
+ˆs
+m3
+WΓW
+�
+1 + O(ˆs−1/2)
+�
+,
+(8)
+where φ dependent terms are subdominant. For the quadratic terms in aTGC couplings,
+the leading contributions from both photon polarizations are the same.
+The energy flowing into the on-shell W is z
+√
+ˆs ∼
+√
+ˆs/2 as usual in the high ˆs limit because
+of z ∼ 1/2 + m2
+W/(2ˆs) → 1/2 for ˆs ≫ m2
+W. The produced on-shell W gets boosted with the
+transverse momentum of the order O(
+√
+ˆs). The boosted W boson requires a large recoiling
+against a hard quark jet which likely invalidates the EWA as the jet can not be treated as
+a forward jet anymore. The process is subject to the noninterference, as is seen in Eq. (6),
+since it is basically 2 → 2 process uγ → dW where the W decay can be factorized.
+When generalizing our toy process to the 2 → 4 process with the intermediate resonant
+W by attaching the fermion line to the photon, the situation becomes less obvious. Similarly
+the EWA of either jet or both will not be valid if the boosted W boson is considered. The
+interference can be in principle possible as the helicity structure of the 2 → 3 process for
+the on-shell W production (pp → qq′W) allows the amplitudes to interfere.
+B.
+Cross section for off-shell W boson
+Alternatively, one can probe the high energy behavior of the anomalous coupling by
+directly accessing far off-shell region of W, or k2 = (2z − 1)ˆs ≫ m2
+W. For this case, the
+full matrix element for the 2 → 3 process needs to be considered. One new feature will be
+the resurrection of the interference in the inclusive cross section, and its size is expected to
+be proportional to the off-shellness of the W boson. While the analytic evaluation of the
+differential cross section in terms of φ is challenging due to the diagram b in Fig. 4, we have
+10
+
+managed to get the leading contribution only for the left-handed polarization of the photon
+in the high energy limit. Similarly to the previous Section III A, we will use ˆs to take a high
+energy limit. The cross section for the interference in the limit of ˆs ≫ m2
+W far away from
+the W mass window, k2 ≫ m2
+W, is estimated to be
+dˆσSM×BSM(uLγL → dνe+)
+dφ
+=
+λz
+512π4
+e2g4
+m2
+W
+�
+− 2
+9 − π2
+6 cos φ
++ 1
+18
+�
+π2 − 26 + 22 ln
+ˆs
+m2
+W
+− 6 ln2
+ˆs
+m2
+W
+�
+cos(2φ)
+� �
+1 + O(ˆs−1)
+�
+,
+(9)
+where ΓW dependent terms are not shown as they contribute to the region of the W mass
+window. Although the off-shell contribution is suppressed by the factor of O(ΓW/mW), or
+∼ O(1/m2
+W), compared to the cross section from the W mass window, the interference term
+can survive in the inclusive cross section even after the integration over all angular variables
+(see the first term in Eq. (9)). The cross section for the quadratic term in λz is given by
+dˆσBSM2(uLγL → dνe+)
+dφ
+= λ2
+z e2g4
+512π4
+ˆs
+m4
+W
+� 1
+24
+�
+−9 + 4 ln
+ˆs
+m2
+W
+�
+− π2
+48 cos φ − 1
+12 cos(2φ)
+� �
+1 + O(ˆs−1/2)
+�
+,
+(10)
+where φ dependence only appears in cos φ and cos(2φ) terms. For the quadratic term, the
+analytic expression in the high energy limit outside the W mass window can be obtained for
+both polarizations of the photon, and the summed and averaged cross section over helicities
+is given by
+dˆσBSM2
+dφ
+=
+1
+2 · 2
+λ2
+z e2g4
+512π4
+ˆs
+m4
+W
+� 1
+216
+�
+−143 + 60 ln
+ˆs
+m2
+W
+�
++ π2
+240 cos φ − 1
+12 cos(2φ)
+� �
+1 + O(ˆs−1/2)
+�
+.
+(11)
+While we have shown the evidence of the resurrected interference in the inclusive cross
+section through the computation of the differential cross section in terms of φ only for the left-
+handed photon helicity, the experimental reconstruction of the angle φ will be challenging
+and we may be able to access only to the inclusive cross section summed and averaged
+over helicities. For the direct analytic computation of the inclusive cross section, we have
+managed to get the final result for both helicities of the photon by performing the integration
+over φ first and the remaining variables later. The leading contribution of the summed and
+averaged cross section over helicities is given by
+ˆσSM×BSM =
+1
+2 · 2
+λz
+512π4
+e2g4
+m2
+W
+× π
+3
+�
+13 − 6 ln
+ˆs
+m2
+W
+�
++ · · · ,
+(12)
+11
+
+where · · · denotes the higher order in ˆs and the logarithmic term is due to the contribution
+from the right-handed helicity of the photon. The summed and averaged cross section which
+is quadratic in λz can be easily obtained by integrating Eq. (11) over the angle φ.
+For more off-shell W, more energy flows into the eν system. For this case, the transverse
+momentum of the forward quark and the W mass are small compared to the scale of the
+hard subprocess, or the usual condition for the EWA is satisfied. We can isolate the behavior
+of the corresponding phase space by integrating over only the interval z = [1 − ε, 1] with
+ε ≪ 1,
+ˆσSM×BSM
+512π4
+λz
+m2
+W
+2πe2g4
+= −1
+3ε2 + 1
+3
+m2
+W
+ˆs
+��
+−3 + 2 ln 2εˆs
+m2
+W
+�
+ε +
+�
+−13 + 6 ln 2εˆs
+m2
+W
+�
+ε2 + · · ·
+�
++ · · · ,
+(13)
+where · · · denotes the higher order terms in ε and m2
+W/ˆs. In the high energy limit of ˆs → ∞,
+the first constant term will eventually dominate, and it will appear as the energy growing
+interference in ˆσSM×BSM/ˆσSM assuming ˆσSM ∼ 1/ˆs. The variable meν may be considered to
+be more relevant one to take a high energy limit of the hard subprocess Wγ → νe+ inside
+uγ → dνe+. Simply changing variable from
+√
+ˆs to meν in expressions obtained in the high
+ˆs limit in Eqs. 9, 10 and 11 could be misleading or not well defined, for instance, a wide
+range of
+√
+ˆs can be associated with a small value of meν for z ∼ 1/2 (see the right panel
+of Fig. 5). The analytic computation of the interference in terms of meν, performed at this
+time starting from amplitudes, reveals a similar energy growing behavior to Eq. (13).
+C.
+Numerical calculation of toy process and interference resurrection
+We numerically investigate the analytic behavior discussed in Sections III A and III B.
+To this end, we generate partonic level events for the EW uγ → dνe+ process using Mad-
+Graph5 aMC@NLO v2.6.7 [36] only with the nominal pT cuts of 10 GeV for the final quark,
+neutrino, and electron. As the noninterference is well established for the operator involving
+λz, the events for the interference are generated only for λz coupling. While the separation
+of the off-shell region from the on-shell one in Section III B was done just by dropping out
+all ΓW dependent terms by hand, we numerically control the separation using two variables
+∆mW = meν − mW and z for the purpose of the demonstration.
+The fraction of the energy, z∗ = 1/2 + m2
+W/(2ˆs), carried by the on-shell νee+ system is
+roughly order one for ˆs ∼ m2
+W, and it rapidly drops to 1/2 with increasing
+√
+ˆs as is seen in
+the left panel of Fig. 5. Due to the relation meν =
+�
+(2z − 1)ˆs, two variables meν and
+√
+ˆs
+are comparable to each other only for a low
+√
+ˆs where 2z − 1 is roughly order one and they
+can be very different for a large
+√
+ˆs as 2z − 1 can be almost zero. The region bounded by
+12
+
+100
+200
+500
+1000
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+s [GeV]
+z
+uγ → dνe
+z = z*+ 0.05
+z = 0.9
+z = z*- 0.05
+z = z*
+|meν - mW| = 10 GeV
+100
+200
+500
+1000
+1
+10
+100
+1000
+s [GeV]
+meν [GeV]
+uγ → dνe
+z = z*+ 0.05
+z = 0.9
+z = z*- 0.05
+z = z*
+FIG. 5: Left: The fraction of the energy that flows into the eν system, z = Eeν/
+√
+ˆs, as a function
+of
+√
+ˆs. The band bounded by red lines correspond to the W mass window of 10 GeV. Black line
+denotes the z value for the on-shell W of the mass mW . Right: the correlation between meν and
+√
+ˆs depending on the cut on z.
+red lines in the left panel of Fig. 5 corresponds to the z value for the W mass window of
+10 GeV, or |mℓν(=
+�
+(2z − 1)ˆs) − mW| < 10 GeV. To access the off-shell region, we impose
+the cut on z such as |z − z∗| = |(m2
+eν − m2
+W)/(2ˆs)| > 0.05 and z > 0.9 and they are shown
+by gray lines in Fig. 5. The selected region by z > 0.9 isolates the phase space where most
+of the center of mass energy
+√
+ˆs flows into the νee+ system and meν can be as large as
+√
+ˆs
+as is seen in the right panel of Fig. 5. It is also the phase space where the condition for the
+typical EWA is expected to be satisfied. While the cut of z − z∗ > 0.05 makes it possible
+to access a deeper off-shell region in a large
+√
+ˆs region than that specified by the W mass
+window of 10 GeV (red lines in Fig. 5), most events are still populated near the lower meν
+value than
+√
+ˆs.
+Our numerical simulations of |σSM×BSM|/σSM binned in
+√
+ˆs is illustrated in Fig. 6 3. As
+is evident by the red-colored almost flat distribution in Fig. 6, the noninterference predicted
+for the on-shell W in Section III A is numerically confirmed. The black-colored distribution
+in Fig. 6 demonstrates the resurrected energy growing interference in the inclusive cross
+section for the off-shell W and they agree with our expectation in Section III B.
+The left panel of Fig. 7 illustrates the interference cross section with respect to the SM
+in terms of the meν variable for the same phase space as those in Fig. 6, and the energy-
+growing behavior is clearly seen. In the right panel of Fig. 7, we take a limit where almost
+all energy
+√
+ˆs flows into the eν system (see the solid gray line in the left panel of Fig. 5). For
+3 In this work, we will not explore the sign of the interference and its sensitivity at the collider. The sign
+of the interference depends on the phase space (see Appendix B 3 for the related discussion).
+13
+
+0
+500
+1000
+1500
+0.01
+0.05
+0.10
+0.50
+1
+5
+10
+s [GeV]
+|σSMBSM|/σSM
+uγ → dνe
+|z - z*| > 0.05
+meν > 90 GeV
+|z - z*| < 0.05
+|meν - mW| < 10 GeV
+FIG. 6: The differential distribution of |σSM×BSM|/σSM in
+√
+ˆs for the EW uγ → dνee+ at the
+parton level. Black lines demonstrate the interference at off-shell region specified as |z − z∗| > 0.05
+and meν > 90 GeV. Red lines demonstrate the noninterference for the on-shell W defined by
+|z − z∗| < 0.05 and |meν − mW | < 10 GeV.
+0
+500
+1000
+1500
+0.01
+0.05
+0.10
+0.50
+1
+5
+10
+meν [GeV]
+|σSMBSM|/σSM
+uγ → dνe
+|z - z*| > 0.05
+meν > 90 GeV
+0
+500
+1000
+1500
+0.01
+0.05
+0.10
+0.50
+1
+meν [GeV]
+|σSMBSM|/σSM
+uγ → dνe
+z = [1-ε, 1]
+ε = 0.1
+meν > 90 GeV
+FIG. 7: The differential distribution of |σSM×BSM|/σSM in meν for the EW uγ → dνee+ at the
+parton level. Events are restricted to satisfy |z − z∗| > 0.05 and meν > 90 GeV (left) or z > 0.9
+and meν > 90 GeV (right).
+this situation, the transverse momentum of the forward quark and the W mass are small
+compared to the scale of the hard subprocess, or the EWA condition is satisfied. As is clearly
+seen in the right panel of Fig. 7, the energy growing interference term looks survive in the
+EWA limit of the full 2 → 3 process. However, this energy growing interference term allowed
+by the helicity selection rule of the full 2 → 3 process will get lost if one simply assumes the
+EWA and works on the 2 → 2 hard subprocess. Recall that the helicity selection rule of the
+2 → 2 subprocess does not allow the interference in the massless limit.
+While we have exploited the variable z to distinguish the phase spaces of the on-shell and
+off-shell regions, it can be traded for a combination of experimental variables. Using the
+14
+
+transverse momentum of the forward quark, pT(q) = (1−z)
+√
+ˆs sin θ (with sin θ = 1/ cosh η),
+and meν = √2z − 1ˆs, one can easily derive the relation,
+pT(q) cosh η
+meν
+=
+1 − z
+√2z − 1 ≤
+1 − zmin
+√2zmin − 1 ≡ δmin → pT(q) ≤ δmin
+meν
+cosh η ,
+(14)
+where zmin = {z∗ +0.05, 1−ε} was used in the plots in Fig. 7 and η is the pseudorapidity of
+the outgoing quark. Note that z∗ (thus δmin as well) is still a function of the experimentally
+inaccessible ˆs although its dependence gets mild in the high ˆs limit. For the hard cut on
+z, δmin becomes a constant. We have numerically checked that the cut pT(q) < 0.112 ×
+(meν/ cosh η) is physically equivalent to z > 1 − ε (with ε = 0.1) and reproduces the
+same plot as the right panel of Fig. 7. For a small constant δmin, the cut in Eq. (14), or
+pT/meν ≤ δmin/ cosh η, is consistent with the conditions for the EWA.
+IV.
+NUMERICAL ANALYSIS OF EW DILEPTON WITH TWO ASSOCIATED
+JETS
+In this section we numerically investigate the EW ℓℓ + two jets process at the LHC. We
+take the CMS analysis in [12] as our baseline for both the validation of our analysis and the
+derivation of the sensitivity on aTGCs at the LHC 4. The detail of the event generation can
+be found in Appendix A.
+A.
+Interference resurrection
+We can use the intuition from the EW uγ → dνe+ process in Section III to isolate the
+phase space that reveals the interference resurrection in the EW ℓℓ+ two jets process. In the
+partonic EW ℓℓ + qq′ process, we can treat the ℓℓ (qq′) system effectively as a single particle
+with the energy of z
+√
+ˆs ((1−z)
+√
+ˆs) and the invariant mass of mℓℓ (mqq′). Similarly to our toy
+process in Section III, the variable z represents the fraction of the total energy flowing into
+the dilepton system. Three momentum conservation, ⃗pT(ℓℓ) = −⃗pT(qq′), in the center of
+mass frame of two initial quarks leads to m2
+ℓℓ−m2
+qq′ = (2z−1)ˆs where z varies over the range
+z = [ mℓℓ/
+√
+ˆs, 1 − mqq′/
+√
+ˆs ]. Similarly to the previous section, we start with the variable
+z = 1/2+(m2
+ℓℓ −m2
+qq′)/(2ˆs) to separate the off-shell phase space from the on-shell one where
+z∗ = 1/2 + (m2
+Z − m2
+qq′)/(2ˆs) at the Z pole. An appropriate cut on z such as |z − z∗| =
+|(m2
+ℓℓ − m2
+Z)/(2ˆs)| > ∆z or z > zmin will select the corresponding off-shell region, while
+ensuring a certain correlation between mℓℓ and
+√
+ˆs. Combining m2
+ℓℓ − m2
+qq′ = (2z − 1)ˆs with
+the transverse momentum of the effective qq′ system pT(qq′) =
+�
+(1 − z)2ˆs − m2
+qq′ sin θqq′, the
+4 Similar study by the CMS collaboration for the EW ℓνℓ + two jets process has been made in [37]
+15
+
+500
+1000
+1500
+1
+2
+3
+4
+5
+mℓℓ [GeV]
+|σSMBSM|/σSM
+Partonic EW ℓℓ+qq',
+s =13 TeV
+VBFhardness > 5
+500
+1000
+1500
+0.1
+1
+10
+100
+1000
+mℓℓ [GeV]
+|σSMBSM, BSM2|/σSM
+Partonic EW ℓℓ+qq',
+s =13 TeV
+VBFhardness > 5
+BSM2
+SMBSM
+FIG. 8: The distributions of |σSM×BSM|/σSM in mℓℓ for the partonic EW ℓℓ + qq′ (black lines in
+both panels) for the λz coupling (other couplings are set to zero). Similarly for |σBSM2|/σSM (red
+lines). Events for solid lines are restricted to those with VBFhardness > 5 in Eq. (15) along with
+pT (q) > 25 GeV, pT (ℓ) > 10 GeV, and mqq′ > 120 GeV. For dashed lines in the right panel, the
+VBFhardness cut is removed while others kept the same.
+variable z can be translated into the nontrivial combination of various kinematic variables
+via the relation,
+VBFhardness ≡
+m2
+ℓℓ − m2
+qq′
+p2
+T(qq′) cosh2 ηqq′ + m2
+qq′
+= 2z − 1
+(1 − z)2 ≥ 2zmin − 1
+(1 − zmin)2
+for
+z ≥ zmin ,
+(15)
+where the ratio is the monotonically increasing function, while it can have either sign, and
+sin θqq′ = 1/ cosh ηqq′ was used to express in terms of the pseudorapidity of the qq′ system.
+The positive value of the VBFhardness (or equivalently z > 1/2) corresponds to the case
+where more than half the total energy flows into the dilepton system. Just like the case of
+our toy process in Section III, zmin still has the ˆs dependence if one intends to impose a cut
+on |z − z∗| instead of a constant cut on z itself.
+As is evident in the right panel of Fig. 8 (see black dashed lines), the interference does not
+reveal the energy growing behavior without a cut on the ratio in Eq. (15). As an illustration,
+the resurrected interference in the inclusive cross section for the λz coupling is clearly shown
+in the left panel of Fig. 8 for VBFhardness > 5 that corresponds to z ≥ zmin = 0.71. We
+checked that a similar energy growing interference appears in terms of
+√
+ˆs as well. The same
+interference is displayed again with the quadratic cross section in the right panel of Fig. 8.
+The square of the interference term in this illustrative example in Fig. 8 appears to have a
+milder energy growing behavior than the quadratic term itself. The interference would have
+been lost if one has not included the full effect of the forward quarks or not imposed a cut
+on a proper variable like the one in Eq. (15). In Fig. 9, we show the resurrected interference
+pattern continues to survive at the hadron level where the VBFhardness is constructed out of
+16
+
+500
+1000
+1500
+1
+10
+100
+1000
+104
+mℓℓ [GeV]
+|σX|/σSM
+EW ℓℓ+jets,
+s =13 TeV
+VBFhardness > 5
+SMBSM
+BSM2
+500
+1000
+1500
+1
+2
+5
+10
+mℓℓ [GeV]
+|σSMBSM|/σSM
+EW ℓℓ+jets,
+s =13 TeV
+VBFhardness > 5
+0
+200
+400
+600
+800
+1000
+0.1
+1
+10
+100
+1000
+104
+pT(ℓℓ ) [GeV]
+σX/σSM
+EW ℓℓ+jets,
+s =13 TeV
+|mℓℓ-mZ|<15 GeV
+SMBSM
+BSM2
+FIG. 9: |σX|/σSM where X = SM×BSM (black) or BSM2 (red) for the EW ℓℓ + two jets for the
+coupling Ci = λz (other couplings are set to zero). Plots are made with events at the jet level after
+imposing the loosened cuts, compared to the CMS analysis [12], pT (j) > 30 GeV, pT (ℓ) > 20 GeV,
+|η(j)| < 4.5 , |η(ℓ)| < 2.5, and mjj > 120 GeV.
+two forward jet candidates and lepton pairs. The CMS analysis in [12] derives the sensitivity
+on aTGC using the pT distribution of Z only for the events inside the Z mass window. In
+the bottom panel of Fig. 9, the interference and quadratic terms of the inclusive cross section
+are illustrated in pT(ℓℓ) only for the events in the Z mass window |mℓℓ − mZ| < 15 GeV.
+B.
+Validation against the CMS analysis and BDT analysis
+We adopt the CMS analysis in [12] for the validation of our framework. Events with two
+isolated leptons (electrons or muons) and at least two jets are selected. A lepton is declared
+to be isolated if the ratio of the pT-sum of all particles within the isolation cone Riso = 0.4
+around the lepton to the pT of the lepton is below 15% and 25% for electrons and muons,
+respectively. While two isolated leptons need to satisfy pT > 20 GeV and |η(ℓ)| < 2.4, and
+have the opposite electric charges, the harder lepton must pass the cut pT > 30 GeV as well.
+17
+
+The particles excluding the isolated leptons are clustered into jets by anti-kt algorithm [44]
+with the distance parameter of Rjet = 0.4. Jets are required to satisfy pT(j) > 15 GeV and
+|η(j)| ≤ 4.7. Two hardest jets, called the tagging jets, are required to have pT(j) > 50 GeV
+and pT(j) > 30 GeV for the leading and subleading jets, respectively, and their invariant
+mass should satisfy mjj > 200 GeV. The initial cuts in CMS analysis in [12] are defined as
+pT(ℓ1) > 30 GeV ,
+pT(ℓ2) > 20 GeV ,
+|η(µ)| < 2.4 ,
+|η(e)| < 2.1 ,
+pT(j1) > 50 GeV ,
+pT(j2) > 30 GeV ,
+|η(j)| ≤ 4.7 ,
+|mZ − mℓℓ| < 15 GeV , and
+mjj > 200 GeV
+(16)
+where the subscripts 1 and 2 mean leading and subleading objects, respectively. The event
+yields after imposing the initial cuts are given in Table I where we included only two largest
+backgrounds. The smaller yield of the ee channel is due to the lower selection efficiency of
+Initial
+Sample
+ee
+µµ
+t¯t
+5454 (5363±48)
+13962 (12938±81)
+DY Zjj (pythia8) 146147 (152750±510) 373731 (394640±880)
+EW Zjj (pythia8)
+2639 (2833±10)
+6328 (6665±16)
+TABLE I: Validation of our simulation at √s =13 TeV assuming 35.9 fb−1 of the integrated
+luminosity. The numbers in parenthesis are CMS values for comparison. The k-factor of 1.7 was
+applied for the t¯t process.
+electrons. We adopted the pT-dependent electron selection efficiency [43] in our analysis,
+while setting the selection efficiency for muons to unity. The electron selection efficiency is
+roughly 0.7 − 0.8 for the pT of interest.
+Having our analysis validated with the initial cuts, we move onto the BDT analysis.
+The CMS analysis introduces two additional variables. Event balance variable, R(phard
+T
+), is
+defined as
+R(phard
+T
+) =
+|⃗pTj1 + ⃗pTj2 + ⃗pTZ|
+|⃗pTj1| + |⃗pTj2| + |⃗pTZ|
+(17)
+The z∗ Zeppenfeld variable is defined as
+z∗ =
+y∗
+∆yjj
+,
+(18)
+where y∗ = yZ − 1
+2 (yj1 + yj2). Additionally, the quark-gluon discrimination is applied to two
+tagging jets. Instead of constructing a likelihood function for the q/g discrimination and
+use it in the BDT analysis afterwards as done in the CMS analysis [38], we directly use the
+18
+
+three input variables to the likelihood in our BDT. They are multiplicity, jet shapes, and
+the fragmentation function. The jet shape variable is defined as
+σ =
+�
+σ2
+1 + σ2
+2
+with
+σ1 = (λ1/
+�
+i
+p2
+T,i)1/2 ,
+σ2 = (λ2/
+�
+i
+p2
+T,i)1/2 ,
+(19)
+where the sum runs over the jet constituents. λ1 and λ2 are the two eigenvalues of the
+matrix with the elements, M11 = �
+i p2
+T,i∆η2
+i , M22 = �
+i p2
+T,i∆φ2
+i , and M12 = M21 =
+− �
+i p2
+T,i∆ηi∆φi where ∆ηi and ∆φi are the pseudorapidity and azimuthal distances be-
+tween a constituent and the average direction which is defined as the p2
+T,i-weighted direction
+of jet constituents in η − φ space. The fragmentation function is captured by the variable,
+pTD =
+��
+i p2
+T,i
+�
+i pT,i
+,
+(20)
+where the sum runs over the jet constituents. For the multiplicity we count all charged and
+neutral constituents of a jet whose energy is above 1 GeV, and it is denoted as ntracks(j).
+Similarly to the CMS analysis in [12], we use the following set of the BDT variables to
+train and test our signal and background samples with the initial cuts in Eq. (16):
+mjj ,
+|∆ηjj| ,
+pT(jj) ,
+R(phard
+T
+) ,
+z∗(Z) ,
+ntracks(j1,2) ,
+pTD(j1,2) ,
+σ1(j1,2) ,
+(21)
+where mjj, ηjj, and pT(jj) are the invariant mass, pseudorapidity, and transverse momentum
+of two leading jets system, respectively. To simplify our analysis and at the same time to
+take full advantage of kinematic distribution to efficiently suppress the largest QCD Drell-
+Yan background, we first train and test over the EW ℓℓ + jets in the SM as a signal and
+the remaining samples as the background using the gradient boosting algorithm (BDTG)
+provided by the TMVA package [42]. Since the signal and the dominant background have the
+largest population in the Z mass window with the small transverse momentum, the BSM
+effect is expected to be small. This rejects the QCD Drell-Yan and top pair backgrounds as
+much as possible. We impose an appropriate cut on the BDT variable, that was computed
+in the previous training, for all the samples of EW ℓℓ + jets in the SM and BSM, and
+background processes. While it is nontrivial to exactly reproduce the outcome of the CMS
+BDT analysis, the outcome of our BDT training, illustrated in Fig. 20 in Appendix C, shows
+the clear separation between the signal and background.
+We do not add our newly introduced VBFhardness in Eq. (15) to the BDT variable set
+although it has a correlation with mjj, ηjj, and pT(jj). Since we take the EW ℓℓ + jets in the
+SM as a signal in the training, we expect its effect on the signal/background discrimination
+to be mild as is indicated in Fig. 10. While the VBFhardness variable helps in resurrecting
+19
+
+1
+−
+0.5
+−
+0
+0.5
+1
+VBFhardness
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+0.14
+0.16
+0.18
+0.2
+Normalized unit/0.04
+Z+jets
++jets
+tt
+EW Zjj
+ = 0.04
+z
+λ
+1
+−
+0
+1
+2
+3
+4
+5
+6
+VBFhardness
+5
+−
+10
+4
+−
+10
+3
+−
+10
+2
+−
+10
+1
+−
+10
+Normalized unit/0.4
+Z+jets
++jets
+tt
+EW Zjj
+ = 0.04
+z
+λ
+FIG. 10: The normalized distribution of VBFhardness for the EFT signal for λz = 0.04, EW
+dilepton (denoted by EW Zjj), t¯t+jets, and QCD Drell-Yan backgrounds (denoted by Z+jets)
+after imposing pT (j) > 30 GeV, pT (ℓ) > 20 GeV, |η(j)| < 4.5 , |η(ℓ)| < 2.5, and mjj > 120 GeV.
+Right panel is logarithmic plot of the left panel in a large VBFhardness range.
+the interference, its effect should be small as well in the situation where the sensitivity
+of aTGCs is mainly driven by the quadratic terms. It will be relevant in case where the
+sensitivity is derived by the interference cross section. As is seen in Fig. 10, although a
+proper cut may reduce the signal rate, VBFhardness seems to be a good discriminator for
+the EFT signal as it controls the amount of energy going into the dilepton subsystem. It
+will be important at the HL-LHC or future collider and we leave more dedicated analysis
+for the future study.
+C.
+Sensitivity to aTGC at the LHC
+To evaluate sentivity to aTGC, we construct 1D templates binned either in pT(ℓℓ) and
+mℓℓ. Events are distributed over 20 equal-spaced bins of pT(ℓℓ) between 0 and 1200 GeV
+where the last bin contains events beyond 1200 GeV. ℓ includes both electrons and muons 5.
+We also newly construct templates of mℓℓ with 10 equal-spaced bins between 0 and 2000 GeV
+where the last bin contains events beyond 2000 GeV. The distributions of backgrounds and
+two selected EFT benchmark points (with the SM contribution subtracted) are illustrated in
+5 On the contrary, the CMS analysis in [12] separately distribute events in 15 bins in pT (ℓℓ) = [0, 900] GeV
+and 20 bins in [0, 1200] GeV for electrons and muons, respectively.
+20
+
+0
+200
+400
+600
+800
+1000
+1200
+ (ll) (GeV)
+T
+p
+10
+2
+10
+3
+10
+4
+10
+5
+10
+Events/60 GeV
+ (13 TeV)
+-1
+35.9 fb
+Z+jets
++jets
+tt
+EW Zjj
+ = 0.04
+z
+λ
+ = 0.1
+1,z
+g
+δ
+0
+200 400 600 800 100012001400160018002000
+ (GeV)
+ll
+m
+1
+10
+2
+10
+3
+10
+4
+10
+5
+10
+6
+10
+Events/200 GeV
+ (13 TeV)
+-1
+35.9 fb
+Z+jets
++jets
+tt
+EW Zjj
+ = 0.04
+z
+λ
+ = 0.1
+1,z
+g
+δ
+FIG. 11: The distributions of pT (ℓℓ) (left) and mℓℓ (right) at 13 TeV, using the integrated luminosity
+of 35.9−1, for backgrounds and two selected EFT benchmark signals with the SM contribution
+subtracted. Events are restricted to those satisfying CMS initial cuts in Eq. (16).
+Fig. 11.
+We construct a log likelihood in terms of aTGCs, assuming the Poisson distribution,
+Using the template analysis of pT (ℓℓ) in the Z mass window at 13 TeV, L = 35.9 fb−1
+No BDT cut
+BDT > 0.6
+aTGC
+68% CL
+95% CL
+95% CL (Linear)
+68% CL
+95% CL
+95% CL (Linear)
+λz
+[−0.026, 0.025] [−0.036, 0.036]
+[−0.20, 0.20]
+[−0.015, 0.016] [−0.025, 0.026]
+[−0.099, 0.1]
+δg1,z
+[−0.069, 0.040] [−0.130, 0.068]
+[−0.096, 0.097]
+[−0.029, 0.024] [−0.066, 0.043]
+[−0.051, 0.051]
+δκz
+[−0.18, 0.19]
+[−0.29, 0.32]
+[−0.41, 0.41]
+[−0.089, 0.095]
+[−0.16, 0.18]
+[−0.18, 0.18]
+TABLE II: One-dimensional limits on aTGCs at 68% and 95% CL. Linear denotes the limits
+obtained using only the interference cross section between the SM and BSM amplitudes.
+Using the template analysis of mℓℓ at 13 TeV, L = 35.9 fb−1
+No BDT cut
+BDT > 0.6
+aTGC
+68% CL
+95% CL
+95% CL (Linear)
+68% CL
+95% CL
+95% CL (Linear)
+λz
+[−0.031, 0.029] [−0.045, 0.043]
+[−0.22, 0.22]
+[−0.025, 0.023] [−0.039, 0.035]
+[−0.13, 0.13]
+δg1,z
+[−0.074, 0.056] [−0.13, 0.094]
+[−0.13, 0.13]
+[−0.033, 0.029] [−0.067, 0.052]
+[−0.062, 0.063]
+δκz
+[−0.099, 0.099]
+[−0.14, 0.15]
+[−0.56, 0.56]
+[−0.062, 0.062] [−0.097, 0.098]
+[−0.26, 0.26]
+TABLE III: Similar caption to Table II.
+− 2∆ log L(λz, δg1,z, δκz) ,
+(22)
+where ∆ indicates that the minimum is subtracted. We include only the statistical uncer-
+tainty since the systematic uncertainty in each bin is not reported in [12] and the overall
+size of it in Table I looks subdominant to the statistical one.
+21
+
+-0.2
+-0.1
+0.0
+0.1
+0.2
+-0.06
+-0.04
+-0.02
+0.00
+0.02
+0.04
+0.06
+δg1,z
+λz
+s = 13 TeV , 35.9 fb -1
+95% CL
+68C% CL
+95% CL (BDT >0.6 )
+68% CL (BDT >0.6 )
+-0.2
+-0.1
+0.0
+0.1
+0.2
+-0.15
+-0.10
+-0.05
+0.00
+0.05
+0.10
+0.15
+δg1,z
+λz
+s = 13 TeV , 35.9 fb -1
+95% CL
+68C% CL
+95% CL (BDT >0.6 )
+68% CL (BDT >0.6 )
+-0.20 -0.15 -0.10 -0.05 0.00
+0.05
+0.10
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+δg1,z
+δκz
+s = 13 TeV , 35.9 fb -1
+95% CL
+68% CL
+95% CL (BDT >0.6 )
+68% CL (BDT >0.6 )
+-0.06-0.04-0.02 0.00 0.02 0.04 0.06
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+λz
+δκz
+s = 13 TeV , 35.9 fb -1
+95% CL
+68% CL
+95% CL (BDT >0.6 )
+68% CL (BDT >0.6 )
+FIG. 12: Two-dimensional limits on aTGCs at 68% (dashed) and 95% CL (solid) regions obtained
+using the binned analysis of pT (ℓℓ) in the Z mass window, assuming the integrated luminosity of
+35.9 fb−1 at √s = 13 TeV. Compared to the red solid lines, thin gray lines were obtained only with
+the interference term which is linear in the aTGC coupling for the BDT > 0.6.
+The 68% and 95% CL intervals of an individual aTGC, where two others are set to
+zero without the marginalization, are presented in Table II and III. For the result with
+the BDT cut, we estimated the sensitivity with the incremental BDT cut starting with a
+mild value, and did not find visible improvement with a stronger BDT cut than 0.6. For
+λz, the 95% CL interval from BDT > 0.6 is worse than the expected value of the CMS
+one, or λCMS
+z
+= [ −0.014, 0.014 ] [12] 6. For the δg1,z coupling, our analysis gives roughly
+6 Comparing two distributions of pT (Z) in Fig. 8 of [12] (separately displayed for electrons and muons) and
+Fig. 11 (summed over both leptons), our signal to background ratio looks rather smaller than the CMS
+one in a high pT region where a large statistical power is expected. We suspect that this discrepancy
+could be partly due to the different configuration for simulation of the aTGC signal and lepton selection
+22
+
+-0.2
+-0.1
+0.0
+0.1
+0.2
+-0.06
+-0.04
+-0.02
+0.00
+0.02
+0.04
+0.06
+δg1,z
+λz
+s = 13 TeV , 35.9 fb -1
+95% CL
+68% CL
+95% CL (BDT >0.6 )
+68% CL (BDT >0.6 )
+-1.0
+-0.5
+0.0
+0.5
+1.0
+1.5
+-3
+-2
+-1
+0
+1
+2
+3
+δg1,z
+λz
+s = 13 TeV , 35.9 fb -1
+95% CL
+68% CL
+95% CL (BDT >0.6 )
+68% CL (BDT >0.6 )
+-0.5 -0.4 -0.3 -0.2 -0.1
+0.0
+0.1
+0.2
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+δg1,z
+δκz
+s = 13 TeV , 35.9 fb -1
+95% CL
+68% CL
+95% CL (BDT >0.6 )
+68% CL (BDT >0.6 )
+-0.06 -0.04 -0.02 0.00
+0.02
+0.04
+0.06
+-0.6
+-0.4
+-0.2
+0.0
+0.2
+0.4
+0.6
+λz
+δκz
+s = 13 TeV , 35.9 fb -1
+95% CL
+68% CL
+95% CL (BDT >0.6 )
+68% CL (BDT >0.6 )
+FIG. 13: Two-dimensional limits on aTGCs at 68% (dashed) and 95% CL (solid) regions obtained
+using the binned analysis of mℓℓ, assuming the integrated luminosity of 35.9 fb−1 at √s = 13 TeV.
+Compared to the red solid lines, thin gray lines were obtained only with the interference term
+which is linear in the aTGC coupling for the BDT > 0.6. No cuts on VBFhardness was imposed.
+comparable with the CMS one, δgCMS
+1,z
+= [−0.053, 0.061 ] [12]. The two-dimensional exclusion
+regions from the binned analysis of pT(ℓℓ) in the Z mass window are illustrated in Fig. 12
+where the remaining coupling is set to zero without the marginalization. The gray lines in
+Fig. 12 illustrate the exclusion region at 95% CL using only linear terms in aTGCs in our
+parametrization of the cross section (see Eq. (3)). It indicates that the sensitivity of λz is
+dominantly driven by the quadratic term whereas the effect of the quadratic term is less
+pronounced for two other aTGC couplings.
+We newly derive the sensitivity using the binned analysis of mℓℓ. As discussed in Sec-
+tion IV A, the invariant mass of the dilepton system has the relation m2
+ℓℓ − m2
+jj = (2z − 1)ˆs,
+efficiency and so on. As our estimation is conservative, we leave it as-is.
+23
+
+-0.3
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.3
+-0.10
+-0.05
+0.00
+0.05
+0.10
+δg1,z
+λz
+s = 13 TeV , 35.9 fb -1
+No BDT cut
+pT(ℓℓ)=[0,
+∞] GeV
+pT(ℓℓ)=[0, 1140 ] GeV
+pT(ℓℓ)=[0, 600 ] GeV
+pT(ℓℓ)=[0, 300 ] GeV
+-0.3
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.3
+-0.10
+-0.05
+0.00
+0.05
+0.10
+δg1,z
+λz
+s = 13 TeV , 35.9 fb -1
+BDT >0.6
+pT(ℓℓ)=[0,
+∞] GeV
+pT(ℓℓ)=[0, 1140 ] GeV
+pT(ℓℓ)=[0, 600 ] GeV
+pT(ℓℓ)=[0, 300 ] GeV
+-0.4
+-0.2
+0.0
+0.2
+0.4
+-0.10
+-0.05
+0.00
+0.05
+0.10
+δg1,z
+λz
+s = 13 TeV , 35.9 fb -1
+No BDT cut
+mℓℓ=[0,
+∞] GeV
+mℓℓ=[0,1800 ] GeV
+mℓℓ=[0,1200 ] GeV
+mℓℓ=[0, 600 ] GeV
+-0.3
+-0.2
+-0.1
+0.0
+0.1
+0.2
+0.3
+-0.10
+-0.05
+0.00
+0.05
+0.10
+δg1,z
+λz
+s = 13 TeV , 35.9 fb -1
+BDT >0.6
+mℓℓ=[0,
+∞] GeV
+mℓℓ=[0,1800 ] GeV
+mℓℓ=[0,1200 ] GeV
+mℓℓ=[0, 600 ] GeV
+FIG. 14: Breakdown of pT (ℓℓ) (top) and mℓℓ (bottom) categories in the plane (λz, δg1,z), assuming
+the integrated luminosity of 35.9 fb−1 at √s = 13 TeV. Curves of various styles indicate the 95%
+CL contours.
+where mjj is the invariant mass of two forward jets, z is the fraction of the total energy of
+the partonic system carried by the ℓℓ system, and mℓℓ alone does not guarantee the hardness
+of the ℓℓ subsystem. However, while a nominal cut on the VBFhardness (see Eq. (15) for the
+definition) ensures that at least some amount of the total energy goes into the ℓℓ subsystem
+and greatly helps recovering the interference, as is clearly seen in Fig. 8, it may not improve
+the situation for the case where the sensitivity is dominantly driven by the quadratic terms.
+For this reason, we have not exploited VBFhardness. The 68% and 95% CL intervals of an
+individual aTGC are presented in Table III. From the comparison between Tables II and III,
+we observe that δκz is better constrained by the binned analysis of mℓℓ whereas λz and δg1,z
+are better constrained by the analysis using the distribution of pT(ℓℓ).
+The two-dimensional exclusion regions from the binned analysis of mℓℓ are illustrated in
+Fig. 13 where similarly the remaining coupling was set to zero without the marginalization.
+24
+
+13 TeV, L = 300 fb−1
+Using the template analysis of pT (ℓℓ) in the Z mass
+No BDT cut
+BDT > 0.6
+aTGC
+68% CL
+95% CL
+95% CL (Linear)
+68% CL
+95% CL
+95% CL (Linear)
+λz
+[−0.017, 0.017]
+[−0.025, 0.024]
+[−0.070, 0.070]
+[−0.0076, 0.0081]
+[−0.012, 0.012]
+[−0.035, 0.035]
+δg1,z
+[−0.019, 0.016]
+[−0.042, 0.029]
+[−0.033, 0.033]
+[−0.0093, 0.0087]
+[−0.019, 0.017]
+[−0.018, 0.018]
+δκz
+[−0.069, 0.072]
+[−0.13, 0.14]
+[−0.14, 0.14]
+[−0.032, 0.033]
+[−0.062, 0.065]
+[−0.064, 0.064]
+Using the template analysis of mℓℓ
+λz
+[−0.017, 0.016]
+[−0.025, 0.023]
+[−0.075, 0.075]
+[−0.013, 0.012]
+[−0.022, 0.018]
+[−0.045, 0.046]
+δg1,z
+[−0.025, 0.022]
+[−0.051, 0.040]
+[−0.047, 0.047]
+[−0.011, 0.011]
+[−0.023, 0.020]
+[−0.022, 0.022]
+δκz
+[−0.054, 0.054]
+[−0.080, 0.080]
+[−0.19, 0.19]
+[−0.031, 0.030]
+[−0.049, 0.048]
+[−0.089, 0.089]
+13 TeV, L = 3000 fb−1
+Using the template analysis of pT (ℓℓ) in the Z mass
+λz
+[−0.0077, 0.0072] [−0.011, 0.011]
+[−0.022, 0.022]
+[−0.0036, 0.0039] [−0.0056, 0.0060]
+[−0.011, 0.011]
+δg1,z
+[−0.0055, 0.0052] [−0.011, 0.010]
+[−0.011, 0.011]
+[−0.0029, 0.0028] [−0.0057, 0.0055]
+[−0.0057, 0.0057]
+δκz
+[−0.023, 0.023]
+[−0.044, 0.045]
+[−0.045, 0.045]
+[−0.010, 0.010]
+[−0.020, 0.020]
+[−0.020, 0.020]
+Using the template analysis of mℓℓ
+λz
+[−0.0090, 0.0077] [−0.013, 0.012]
+[−0.024, 0.024]
+[−0.0060, 0.0053] [−0.0096, 0.0085]
+[−0.014, 0.014]
+δg1,z
+[−0.0076, 0.0077] [−0.015, 0.014]
+[−0.015, 0.015]
+[−0.0035, 0.0034] [−0.0070, 0.0067]
+[−0.0069, 0.0069]
+δκz
+[−0.025, 0.025]
+[−0.040, 0.040]
+[−0.062, 0.062]
+[−0.013, 0.013]
+[−0.022, 0.022]
+[−0.028, 0.028]
+TABLE IV: One-dimensional limits on aTGCs at 68% and 95% CL at 13 TeV using the integrated
+luminosity of L = 300 fb−1 and L = 3000 fb−1. No cut on VBFhardness was imposed.
+Unlike the case using pT(ℓℓ) in Fig. 12, the sensitivity, for instance, of λz is significantly
+weakened (see upper right panel of Fig. 13) when the quadratic term is removed. This is
+due to the interference suppression as illustrated by the black dashed line in the right panel
+of Fig. 8.
+The situation is contrasted to those obtained using the binned analysis with
+pT(ℓℓ). As observed in the bottom panel of Fig. 9, the discrepancy between the interference
+and quadratic terms in the pT(ℓℓ) distribution is less pronounced, compared to the current
+case, and the interference term itself also shows the pT-growing behavior.
+Fig. 14 illustrates how the sensitivity in the plane (λz, δg1,z) changes as some of the
+higher bins are removed in the binned analysis of pT(ℓℓ) and mℓℓ, respectively, for two
+cases without (left panels of Fig. 14) and with the BDT cut (right panels of Fig. 14). This
+practice is meaningful especially for mℓℓ as the EFT cutoff can be directly imposed on the
+mℓℓ variable. For the case with the BDT cut, sensitivity to δg1,z mostly comes from the
+first small number of bins, corresponding to the well below sub-TeV in both pT(ℓℓ) and mℓℓ
+whereas a wider range of the energy contributes to the sensitivity to λz. On the contrary,
+for the case without the BDT cut, δg1,z becomes sensitive to the wide range of the energy.
+We derive the sensitivity at the LHC and HL-LHC, assuming an integrated luminosity
+of 300 fb−1 and 3 ab−1, respectively. We assume that the systematic errors remain to be
+25
+
+-0.10 -0.05
+0.00
+0.05
+0.10
+0.15
+-0.15
+-0.10
+-0.05
+0.00
+0.05
+0.10
+0.15
+δg1,z
+λz
+s = 13 TeV , BDT >0.6
+35.9 fb -1
+300 fb -1
+35.9 fb -1 (Linear )
+300 fb -1 (Linear )
+-0.15 -0.10 -0.05
+0.00
+0.05
+0.10
+-0.4
+-0.2
+0.0
+0.2
+0.4
+δg1,z
+δκz
+s = 13 TeV , BDT >0.6
+35.9 fb -1
+300 fb -1
+35.9 fb -1 (Linear )
+300 fb -1 (Linear )
+-0.04
+-0.02
+0.00
+0.02
+0.04
+-0.4
+-0.2
+0.0
+0.2
+0.4
+λz
+δκz
+s = 13 TeV , BDT >0.6
+35.9 fb -1
+300 fb -1
+35.9 fb -1 (Linear )
+300 fb -1 (Linear )
+-0.10 -0.05
+0.00
+0.05
+0.10
+0.15
+-0.15
+-0.10
+-0.05
+0.00
+0.05
+0.10
+0.15
+δg1,z
+λz
+s = 13 TeV , BDT >0.6
+35.9 fb -1
+300 fb -1
+35.9 fb -1 (Linear )
+300 fb -1 (Linear )
+-0.3
+-0.2
+-0.1
+0.0
+0.1
+-0.4
+-0.2
+0.0
+0.2
+0.4
+δg1,z
+δκz
+s = 13 TeV , BDT >0.6
+35.9 fb -1
+300 fb -1
+35.9 fb -1 (Linear )
+300 fb -1 (Linear )
+-0.04
+-0.02
+0.00
+0.02
+0.04
+-0.4
+-0.2
+0.0
+0.2
+0.4
+λz
+δκz
+s = 13 TeV , BDT >0.6
+35.9 fb -1
+300 fb -1
+35.9 fb -1 (Linear )
+300 fb -1 (Linear )
+FIG. 15: The two-dimensional contours at 95% CL, obtained using the binned analysis of pT (ℓℓ)
+(upper) and mℓℓ (bottom), assuming the integrated luminosities of 35.9 fb−1 and 300 fb−1 at
+√s = 13 TeV. The dashed lines were obtained only with the interference term which is linear in
+the aTGC coupling.
+negligible, and we include only the statistical uncertainty. Our projection for the LHC and
+the HL-LHC is illustrated in Table IV. The 95% CL contours in the two-dimensional plane
+are shown in Fig. 15 where upper two plots were obtained by the template analysis of pT(ℓℓ)
+and the bottom ones using mℓℓ. The comparison between two analyses for δg1,z and δκz,
+namely, one by total cross section up to the quadratic order in aTGC and the other only with
+the interference cross section, indicates that the sensitivity is mainly driven by the linear term
+for the case of pT(ℓℓ). While, for the case of mℓℓ, the role of the interference hardly becomes
+important except for δg1,z where the other two couplings were set to zero, the VBFhardness
+may help making the interference more important. Although, as is evident in Fig. 10 a cut
+on VBFhardness may reduce the signal rate, loosening other cuts may compensate it and it
+can be an important variable at the HL-LHC regarding the interference.
+26
+
+Bounds on aTGCs
+Binned Anal. of pT(ℓℓ)
+Binned Anal. of m(ℓℓ)
+BDT>0.6
+No BDT
+No BDT
+BDT>0.6
+68%, 95% CL
+68%, 95% CL
+68%, 95% CL
+68%, 95% CL
+-x.xx +x.xx
+68% CL bound
+-x.xx +x.xx
+95% CL bound
+Tick size for λz/δg1,z/δκz
+0.06/0.013/0.04
+0
+λz
+LHC
+35.9 fb-1
+LHC
+300 fb-1
+HL-LHC
++0.026
+-0.025
++0.016
+-0.015
++0.036
+-0.036
++0.025
+-0.026
++0.035
+-0.039
++0.023
+-0.025
++0.043
+-0.045
++0.029
+-0.031
++0.012
+-0.012
++0.0081
+-0.0076
++0.024
+-0.025
++0.017
+-0.017
++0.018
+-0.022
++0.012
+-0.013
++0.023
+-0.025
++0.016
+-0.017
++0.0060
+-0.0056
++0.0039
+-0.0036
++0.011
+-0.011
++0.0072
+-0.0077
++0.0085
+-0.0096
++0.0053
+-0.0060
++0.012
+-0.013
++0.0077
+-0.0090
+0
+δg1,z
++0.043
+-0.066
++0.024
+-0.029
++0.068
+-0.130
++0.040
+-0.069
++0.052
+-0.067
++0.029
+-0.033
++0.094
+-0.130
++0.056
+-0.074
++0.017
+-0.019
++0.0087
+-0.0093
++0.029
+-0.042
++0.016
+-0.019
++0.020
+-0.023
++0.011
+-0.011
++0.040
+-0.051
++0.022
+-0.025
++0.0055
+-0.0057
++0.0028
+-0.0029
++0.010
+-0.011
++0.0052
+-0.0055
++0.0067
+-0.0070
++0.0034
+-0.0035
++0.014
+-0.015
++0.0077
+-0.0076
+0
+δκz
++0.18
+-0.16
++0.095
+-0.089
++0.32
+-0.29
++0.19
+-0.18
++0.098
+-0.097
++0.062
+-0.062
++0.150
+-0.140
++0.099
+-0.099
++0.065
+-0.062
++0.033
+-0.032
++0.14
+-0.13
++0.072
+-0.069
++0.048
+-0.049
++0.03
+-0.031
++0.08
+-0.08
++0.054
+-0.054
++0.02
+-0.02
++0.01
+-0.01
++0.045
+-0.044
++0.023
+-0.023
++0.022
+-0.022
++0.013
+-0.013
++0.04
+-0.04
++0.025
+-0.025
+FIG. 16: The visual presentation of the sensitivity of aTGCs at 13 TeV, assuming three different
+luminosities, given in Tables II, III, and IV.
+V.
+CONCLUSION
+We have explored the EW dilepton production with two associated jets for the precision
+measurement of aTGC couplings. This process has a few advantages compared to the typical
+diboson production that has been a major process for studying aTGCs. The EFT cutoff
+can be unambiguously imposed on the dilepton invariant mass. As was explicitly shown
+(both analytically and numerically) in this work, the full amplitude, including the forward
+quarks that radiate off the vector gauge bosons, exhibits the interference, importantly, in
+the inclusive cross section. It contrasts with the interference resurrected in the differential
+angular distributions of the diboson production process. For the purpose of the interference
+27
+
+resurrection in our dilepton production in vector boson fusion, we have introduced a new
+variable, VBFhardness, that can control the amount of energy flowing into the dilepton
+system. Using this variable, we have demonstrated that the interference clearly appears
+when an appropriate cut is applied. It is also indicated that the usual practice of using
+the EWA, namely, restricting ourselves to the subprocess initiated by gauge bosons and
+convoluting its cross section with the probability distribution functions of the gauge bosons,
+can be ill-defined. As a proof-of-concept example for the interference resurrection in the
+inclusive cross section, we have performed the analytic study using the simpler toy process,
+or uγ → dνe+, which was numerically confirmed as well.
+We have derived the sensitivity to aTGCs for three scenarios of the LHC and HL-LHC,
+assuming the integrated luminosity of 35.9 fb−1, 300 fb−1, and 3000 fb−1. In addition to the
+template analysis using the transverse momentum of the dilepton, we also carried out the
+template analysis using the invariant mass of the dilepton in this work. While the bounds
+on λz and δg1,z from the dilepton invariant mass are rather weaker than those from the
+transverse momentum of the dilepton system, the situation is opposite for δκz. The final
+result of the one-dimensional bounds at 68% and 95% CL is summarized in Fig. 16. Our
+analysis using the dilepton invariant mass may further be optimized. Vetoing b-jets could
+help suppress top-enriched backgrounds. Exploiting VBFhardness may help in enhancing
+the role of the interference with respect to the quadratic terms in aTGCs.
+Acknowledgments
+MS thanks A. Azatov and D. Marzocca for valuable discussions. Especially, MS thanks
+A. Azatov for the explanation of his previous work regarding the interference resurec-
+tion. JP, MS, and MU were supported by National Research Foundation of Korea under
+Grant Number NRF-2021R1A2C1095430. JY and HH were supported by the National Re-
+search Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.
+2020R1C1C1005916).
+Appendix A: Details on simulation
+1.
+Signal and background generation
+The aTGC interaction in Eq. (2) is implemented in FeynRules [39] from which we
+generate the UFO output for the MadGraph. Electroweak ℓ+ℓ−jj samples were simulated
+at leading order (LO) by MadGraph5 aMC@NLO v2.6.7 [36] (QED=4, QCD=0) with the
+default factorization and renormalization scales, interfaced with the Pythia8 v8.306 for
+the parton shower and hadronization. For the parton distribution function, the NNPDF30
+28
+
+(lo as0130) [40] is used. The linear (or interference) and quadratic terms in aTGC in our
+parametrization of the cross section in Eq. (3) were separately simulated by using flags TGC2
+= 1 and TGC2 = 2, respectively 7, where TGC denotes the order of aTGC interaction. The
+phase space was restricted to those satisfying mℓℓ > 50 GeV, pT(j) > 25 GeV, and mjj > 120
+GeV at the generation level 8.
+All background samples were similarly simulated at leading order (LO) by Mad-
+Graph5 aMC@NLO v2.6.7 [36] with the default factorization and renormalization scales,
+interfaced with the Pythia8. The NNPDF30 (lo as0130) was used. The QCD Drell-Yan
+process γ∗/Z(ℓ+ℓ−)+jets samples where jets arise from QCD interaction were matched us-
+ing kT-jet MLM matching at LO up to three extra jets in 5-flavor. k-factor of 1.23 was
+applied [12]. The t¯t samples were matched using kT-jet MLM matching (QCUT = 45 GeV)
+at LO up to two extra jets in 5-flavor and the total cross section was rescaled to match the
+NLO value from Powheg [41] by applying the k-factor of 1.7.
+Appendix B: Computation detail of qV → q′νℓ
+1.
+Choice of four momenta and amplitudes
+The polarization vectors of the photon are obtained by rotating ϵL/R =
+1
+√
+2(0, 1, ±i, 0)
+(for the massless momenta moving to −z axis) with angle θ about y-axis (similarly angle φ
+about z-axis).
+ϵµ
+L/R(p2) = 1
+√
+2 (0, cos θ cos φ ∓ i sin φ, cos θ sin φ ± i cos φ, − sin θ) .
+(B1)
+The spinor solutions in our coordinate system are
+¯uL(k1) = ˆs1/4
+�
+0, 0, −
+√
+2z − 1 sin ψ
+2 , cos ψ
+2
+�
+,
+vL(k2) = ˆs1/4
+�√
+2z − 1 cos ψ
+2 , sin ψ
+2 , 0, 0
+�T
+,
+uL(p1) = ˆs1/4
+�
+− sin θ
+2, eiφ cos θ
+2, 0, 0
+�T
+,
+¯uL(k3) = ˆs1/4�
+2(1 − z) (0, 0, −1, 0) ,
+(B2)
+7 On the other hand, the CMS analysis [12] generated aTGC signal samples (differently from ours) effectively
+over 5 × 5 × 5 grid of cW W W /Λ2 × cW /Λ2 × cB/Λ2 which were equivalent to our aTGCs. We suspect that
+this could be partly responsible for the discrepancy between our sensitivity of aTGCs and that in [12].
+8 To guarantee enough statistics and the smoothness of the differential distribution in the high invariant
+mass tail, events were generated separately for multiple intervals of mℓℓ and combined. Similarly for the
+EW ℓℓjj samples in the SM.
+29
+
+where T denotes the transpose. We choose the following four momenta of the particles in
+d(k3)
+γ(p2)
+u(p1)
+k
+ντ(k1)
+e+(k2)
+ψ
+φ
+θ
+FIG. 17: The angular configuration of the illustrative toy process, uγ → dνe+.
+our 2 → 3 process, uγ → dνe+ and they are illustrated in Fig. 17.
+pµ
+1 =
+√
+ˆs
+2 (1, sin θ cos φ, sin θ sin φ, cos θ) ,
+pµ
+2 =
+√
+ˆs
+2 (1, − sin θ cos φ, − sin θ sin φ, − cos θ) ,
+kµ
+1 =
+√
+ˆs
+2
+�
+z + (1 − z) cos ψ,
+�
+(2z − 1) sin ψ, 0, (1 − z) + z cos ψ
+�
+,
+kµ
+2 =
+√
+ˆs
+2
+�
+z − (1 − z) cos ψ, −
+�
+(2z − 1) sin ψ, 0, (1 − z) − z cos ψ
+�
+,
+kµ
+3 =
+√
+ˆs (1 − z, 0, 0, −(1 − z)) ,
+kµ =
+√
+ˆs (z, 0, 0, (1 − z)) ,
+(B3)
+where the momentum k has the invariant mass of m2
+k = (2z − 1)ˆs. Note that the 2 → 3
+process can be effectively factorized into 2 → 2 and 1 → 2 via an intermediate momentum
+k. The momenta k1 and k2 in Eq. (B3) are obtained by boosting those in the νe rest frame,
+kµ
+1 = mk
+2 (1, sin ψ, 0, cos ψ ) ,
+kµ
+2 = mk
+2 (1, − sin ψ, 0, − cos ψ) ,
+(B4)
+along the z-axis with the boosting factor,
+kz = γz mkβz → γz = k0
+mk
+=
+z
+√2z − 1 .
+(B5)
+When the intermediate W emitted from the quark line is produced nearly on shell, z is
+nearly fixed to be
+z ∼ 1
+2
+�
+1 + m2
+W
+ˆs
+�
+.
+(B6)
+30
+
+The helicity amplitudes for four diagrams in Fig. 4 are given by
+iϵ · Ma = ¯uL(k3)
+�
+i g
+√
+2γρ�
+uL(p1) −iηρν
+q2 − m2
+W
+× ϵλ(p2) i e
+��
+ηµν(q − k)λ − (2 + δκγ)(pµ
+2ηνλ − pν
+2ηµλ) + ηνλkµ − ηµλqν�
++ λz
+m2
+W
+�
+(pµ
+2ηνλ − pν
+2ηµλ)(k · q) + (qληµν − qµηνλ)(k · p2)
++ (kνηµλ − kληµν)(q · p2) − kνqλpµ
+2 + kλqµpν
+2
+��
+×
+−iηµσ
+k2 − m2
+W + imWΓW
+¯uL(k1)
+�
+i g
+√
+2γσ�
+vL(k2)
+=
+�
+i g
+√
+2
+�2
+(ie)
+(−i)2
+q2 − m2
+W
+1
+k2 − m2
+W + imWΓW
+ϵλjν
+q jµ
+l V λνµ .
+(B7)
+where q = p2 − k = k3 − p1.
+iϵ · Mb = ϵµ(p2)¯uL(k3)
+�
+i g
+√
+2γρ�
+uL(p1) −iηρσ
+q2 − m2
+W
+× ¯uL(k1)
+�
+i g
+√
+2γσ�i( /p2 − /k2)
+(p2 − k2)2 (−ieγµ) vL(k2)
+=
+�
+i g
+√
+2
+�2
+(−ie)
+(−i)i
+q2 − m2
+W
+1
+(p2 − k2)2 ¯uL(k1)/jq( /p2 − /k2)/ϵvL(k2) ,
+(B8)
+where q = k3 − p1.
+iϵ · Mc = ϵµ(p2)¯uL(k3)
+�
+− i
+3eγµ
+� i( /k3 − /p2)
+(k3 − p2)2
+�
+i g
+√
+2γρ�
+uL(p1)
+×
+−iηρσ
+k2 − m2
+W + imWΓW
+¯uL(k1)
+�
+i g
+√
+2γσ�
+vL(k2)
+=
+�
+i g
+√
+2
+�2 �
+− i
+3e
+�
+(−i)i
+k2 − m2
+W + imWΓW
+1
+(k3 − p2)2 ¯uL(k3)/ϵ( /k3 − /p2)/jluL(p1) ,
+iϵ · Md = ϵµ(p2)¯uL(k3)
+�
+i g
+√
+2γρ�i( /p1 + /p2)
+(p1 + p2)2
+�2i
+3 eγµ
+�
+uL(p1)
+×
+−iηρσ
+k2 − m2
+W + imWΓW
+¯uL(k1)
+�
+i g
+√
+2γσ�
+vL(k2)
+=
+�
+i g
+√
+2
+�2 �2i
+3 e
+�
+(−i)i
+k2 − m2
+W + imWΓW
+1
+(p1 + p2)2 ¯uL(k3)/jl( /p1 + /p2)/ϵuL(p1)
+(B9)
+where jµ
+q = ¯uL(k3)γµuL(p1) and jµ
+l = ¯uL(k1)γµvL(k2).
+31
+
+2.
+Phase space integration
+The partonic cross section of 2 → 3 process in our coordinate system is obtained by the
+following phase space integration,
+ˆσ =
+1
+512π4
+� 1
+1/2
+dz(1 − z)
+� 1
+−1
+d cos θ
+� 1
+−1
+d cos ψ
+� 2π
+0
+dφ
+��M
+��2 ,
+(B10)
+where
+��M
+��2 is the summed and averaged amplitude-squared over polarizations of the initial
+partons and
+��M
+�� has a negative mass dimension of one.
+3.
+Interference between SM and BSM amplitudes for coupling λz
+2000
+4000
+6000
+8000
+-0.0001
+0.0000
+0.0001
+0.0002
+s [GeV]
+σ
+SMBSM
+Total
+σ
+SMBSM
+hard
+σ
+SMBSM
+rad
+1000
+2000
+3000
+4000
+5000
+-0.0005
+0.0000
+0.0005
+0.0010
+0.0015
+0.0020
+s [GeV]
+σ
+SMBSM
+σ
+SMBSM
+aa
+σ
+SMBSM
+ab
+σ
+SMBSM
+ac
+σ
+SMBSM
+ad
+FIG. 18: The partonic inclusive cross section in an arbitrary rate for the interference between the
+SM and BSM, ˆσSM×BSM(uLγL → dνe+), integrated over the entire phase space.
+In our 2 → 3 toy process, diagrams a and b in Fig. 4 are those of interest that probe the
+hard subprocess and diagrams c and d belong to the radiation type where W decaying to e+ν
+is attached to either incoming or outgoing quark line. Restricting only to the interference,
+we split the contribution into two categories.
+ˆσhard
+SM×BSM ≡ ˆσaa
+SM×BSM + ˆσab
+SM×BSM ,
+ˆσrad
+SM×BSM ≡ ˆσac
+SM×BSM + ˆσad
+SM×BSM ,
+(B11)
+where ˆσij
+SM×BSM refers to the partonic cross section from the product of two diagrams i and
+j in Fig. 4. The relative difference between two categories is purely due to the SM as the
+λz dependence comes from the common diagram a. The left panel of Fig. 18 shows that
+ˆσhard
+SM×BSM and ˆσrad
+SM×BSM are comparable. While the magnitude of each ˆσaa
+SM×BSM and ˆσab
+SM×BSM
+is bigger than both ˆσac
+SM×BSM and ˆσad
+SM×BSM, there is a cancellation between two contributions
+from the hard subprocess, dictated by the gauge symmetry. It should be an artifact due
+32
+
+2000
+4000
+6000
+8000
+-1.×10-6
+-5.×10-7
+0
+s [GeV]
+σ
+SMBSM
+Total
+σ
+SMBSM
+hard
+σ
+SMBSM
+rad
+1000
+2000
+3000
+4000
+-0.00004
+-0.00002
+0.00000
+0.00002
+0.00004
+s [GeV]
+σ
+SMBSM
+σ
+SMBSM
+aa
+σ
+SMBSM
+ab
+σ
+SMBSM
+ac
+σ
+SMBSM
+ad
+FIG. 19: The partonic inclusive cross section in an arbitrary rate for the interference between the
+SM and BSM, ˆσSM×BSM(uLγL → dνe+), integrated over the restricted phase space z = [1 − ε, 1]
+where ε = 0.1 was chosen.
+to the gauge choice in the photon polarization. One may choose a particular gauge for the
+photon polarization to suppress the contribution from the radiation type diagrams. The
+observed property is more pronounced when the phase space is restricted to z = [1 − ε, 1]
+with ε = 0.1 where the usual conditions for the EWA are satisfied. As is clearly seen in
+Fig. 19, an individual contribution from the hard subprocess becomes much bigger than
+those involving the radiation type diagrams, and the cancellation is more dramatic. The
+gauge dependence may not be a problem in the 2 → 4 process where all gauge bosons
+including the photon are attached to the fermion currents.
+Another interesting observation is that the sign of interference is
+√
+ˆs-dependent. For
+instance, in Fig. 18, the interference stays positive until around
+√
+ˆs ∼ 4 TeV whereas, in the
+situation corresponding to Fig. 19, the interference becomes negative well before TeV.
+Appendix C: Detail of BDT Analysis
+For the purpose of the training and testing, we made separate inclusive EW ℓℓjj and QCD
+Drell-Yan samples over the entire mℓℓ range whereas the samples (for the same processes) for
+the actual BDT analysis were generated in multiple mℓℓ bins to guarantee the smoothness
+with enough statistics up to the high invariant mass tail.
+The ratio of samples for the
+training and testing to those for the actual analysis is 1 to 4. For t¯t+jets samples, we used
+30% for the training and testing and the remaining 70% for the analysis. We trained and
+tested over the EW ℓℓjj in the SM as a signal and the remaining as the background using
+the gradient boosting algorithm (called BDTG) provided in TMVA package. Our validation
+of the BDT analysis is illustrated in Fig. 20 which shows the clear separation of the EW ℓℓ
+33
+
+0.8
+−
+0.6
+−
+0.4
+−
+0.2
+−
+0
+0.2
+0.4
+0.6
+0.8
+BDTG response
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+dx
+ / 
+(1/N) dN
+Signal (test sample)
+Background (test sample)
+Signal (training sample)
+Background (training sample)
+FIG. 20: Our validation of the BDT analysis with the variable set in Eq. (21), using the gradient
+boosting algorithm in TMVA package.
++ jets events from the QCD Drell-Yan and top pair backgrounds.
+Training and testing by taking EFT benchmark points as signals and the remaining as
+backgrounds may help in boosting the discrimination of the EFT signals from the back-
+ground, and VBFhardness may play a role in that situation. We also have not included any
+top-related variables, including b-jets, which may be important in the binned analysis of mℓℓ
+as top backgrounds remain significant up to a higher energy tail (see right panel of Fig. 11).
+The distributions for part of the BDT variables, given in Eq. (21), after imposing pT and
+η cuts on jets and leptons are illustrated in Fig. 21 where we also added one selected EFT
+benchmark point for λz = 0.04 as an illustration.
+34
+
+0
+200 400 600 800 1000 1200 1400 1600 1800 2000
+ (GeV)
+jj
+m
+4
+−
+10
+3
+−
+10
+2
+−
+10
+1
+−
+10
+Normalized unit/50 GeV
+Z+jets
++jets
+tt
+EW Zjj
+ = 0.04
+z
+λ
+0
+100
+200
+300
+400
+500
+600
+700
+800
+(jj) (GeV)
+T
+p
+2
+−
+10
+1
+−
+10
+Normalized unit/20 GeV
+Z+jets
++jets
+tt
+EW Zjj
+ = 0.04
+z
+λ
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+(jj)|
+η|
+6
+−
+10
+5
+−
+10
+4
+−
+10
+3
+−
+10
+2
+−
+10
+1
+−
+10
+Normalized unit/0.3
+Z+jets
++jets
+tt
+EW Zjj
+ = 0.04
+z
+λ
+0
+0.5
+1
+1.5
+2
+2.5
+3
+z*
+2
+−
+10
+1
+−
+10
+Normalized unit/0.2
+Z+jets
++jets
+tt
+EW Zjj
+ = 0.04
+z
+λ
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+)
+hard
+T
+R(p
+5
+−
+10
+4
+−
+10
+3
+−
+10
+2
+−
+10
+1
+−
+10
+Normalized unit/0.04
+Z+jets
++jets
+tt
+EW Zjj
+ = 0.04
+z
+λ
+FIG. 21: The normalized distribution of BDT variables for the EFT signal for λz = 0.04 and
+backgrounds after imposing pT (j1) > 50 GeV, pT (j2) > 30 GeV, pT (ℓ1) > 30 GeV, pT (ℓ2) > 20
+GeV, |η(j)| < 4.5 , |η(ℓ)| < 2.5. Recall that EW Zjj and EFT samples were generated with
+mjj > 120 GeV at the generation level.
+35
+
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diff --git a/MtFRT4oBgHgl3EQf3Di-/content/tmp_files/load_file.txt b/MtFRT4oBgHgl3EQf3Di-/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..cef1217aad21a8839078e08ffcd5e8d2e1f7e3f0
--- /dev/null
+++ b/MtFRT4oBgHgl3EQf3Di-/content/tmp_files/load_file.txt
@@ -0,0 +1,1628 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf,len=1627
+page_content='Anomalous triple gauge couplings  in electroweak dilepton tails at the LHC  and interference resurrection  Haeyun Hwang a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Ui Min b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Junghyeon Park b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Minho Son b and Jae Hyeok Yoo a  a Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Korea University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Seoul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Republic of Korea  b Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Korea Advanced Institute of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  291 Daehak-ro,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Yuseong-gu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Daejeon 34141,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Republic of Korea  Abstract  We study the electroweak dilepton production with two forward jets at the LHC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' aiming to measure  the anomalous triple gauge couplings in the Effective Field Theory (EFT) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' This process  has two distinctive advantages compared to typical diboson processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The EFT cutoff can be  unambiguously imposed on the dilepton invariant mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The interference between Standard Model  (SM) and beyond the SM is resurrected in the inclusive cross section of the full amplitude, including  two forward jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  As a concrete illustration, we perform the detailed analytic and numerical  study of the interference using a simpler toy process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  We propose a new kinematic variable,  VBFhardness, that controls the amount of energy flowing into the dilepton subprocess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We show  that an appropriate cut on VBFhardness makes the interference resurrection manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Finally, we  use the invariant mass of the dilepton mass system as well as the transverse momentum, as done  in the literature, to derive the sensitivity to anomalous triple gauge couplings at the LHC and the  high luminosity LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='13663v1  [hep-ph]  31 Jan 2023  Contents  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Introduction  3  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' EW dilepton production with two associated jets  6  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Toy process for analytic study: Single lepton with an associated jet  7  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Cross section for on-shell W boson  9  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Cross section for off-shell W boson  10  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Numerical calculation of toy process and interference resurrection  12  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Numerical analysis of EW dilepton with two associated jets  15  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Interference resurrection  15  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Validation against the CMS analysis and BDT analysis  17  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Sensitivity to aTGC at the LHC  20  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Conclusion  27  Acknowledgments  28  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Details on simulation  28  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Signal and background generation  28  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Computation detail of qV → q′νℓ  29  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Choice of four momenta and amplitudes  29  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Phase space integration  32  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Interference between SM and BSM amplitudes for coupling λz  32  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Detail of BDT Analysis  33  References  36  2  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  INTRODUCTION  Although the LHC has been performing great including the discovery of the Higgs  boson [1, 2], it continuously shows no evidence for the new physics, or beyond the Standard  Model (BSM), only confirming the Standard Model (SM) to a better precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' It indicates  that either new particles, if they exist, are very weakly coupled to the SM or they may be  hidden in the energy scale beyond the LHC reach, especially, if a new physics has a sizeable  coupling to the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Given the strong indication for the mass gap between the electroweak  and new physics scales, the effective field theory approach makes sense to parametrize  the possible new physics effects encoded in the higher-dimensional operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Deviating  from the SM with the Higgs doublet under the SM gauge symmetry, the effective La-  grangian, known as the SM Effective Field Theory (SMEFT), below the cutoff Λ is written as  L = LSM +  �  i  c(6)  i  Λ2 O(6)  i  +  �  i  c(8)  i  Λ4 O(8)  i  + · · · ,  (1)  where the lepton number conservation was assumed and c(d)  i  is the Wilson coefficient for the  dimension-d operator O(d)  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The non-vanishing effect from the new physics on the Wilson  coefficients of higher-dimensional operators will cause a deviation of couplings among SM  particles from the SM prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  In this work, we focus on the precision measurements of the cubic interaction of the gauge  bosons at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Taking into account the property of the SMEFT up to dimension-6  operators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' the deviation of the triple gauge couplings from the SM can be parametrized in  terms of three anomalous Triple Gauge Couplings (aTGC) as  Ltgc = ie  �  W +  µνW −  µ − W −  µνW +  µ  �  Aν + iecθ  sθ  (1 + δg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='z)  �  W +  µνW −  µ − W −  µνW +  µ  �  Zν  + ie(1 + δκγ)Aµν W +  µ W −  ν + iecθ  sθ  (1 + δκz) Zµν W +  µ W −  ν  + i λze  m2  W  �  W +  µνW −  νρAρµ + cθ  sθ  W +  µνW −  νρZρµ  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  (2)  where cθ =  �  1 − s2  θ and δκz = δg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='z −  s2  θ  c2  θ δκγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Considering only amplitudes with a single  insertion of aTGC, the cross section is in general a quadratic function of aTGC and it can  be parametrized as  σ = σSM + Ciσi  SM×BSM + CiCjσij  BSM×BSM ,  (3)  where the index i runs over three aTGCs, Ci ≡ {λz, δg1,z, δκz}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Typically, measurements of aTGC at the LHC have been performed by using diboson  processes such as WW, WZ, and Wγ in the lepton-enriched final state channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Unlike the  precision measurement in LEP from WW production process with the fixed center of mass  3  energy around the electroweak scale, the sensitivity on aTGC from the LHC relies on the  accessibility to the higher energy as long as it does not violate the validity of the EFT [3],  or one should not use the data at the energy E above the cutoff Λ, or E/Λ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' While  the leptonic channel is clean and thus provides good sensitivity, the accompanying neutrinos  make it difficult to experimentally extract the exact scale of the hard process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' When one  can not impose an appropriate cut on the scale of the hard process to ensure E/Λ ≲ 1,  one can only hope for setting a conservative bound in this situation [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Another issue  in the diboson process has been the noninterference between the SM and BSM amplitudes  which was found to be dictated by the helicity structure of the amplitudes [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Including  only dimension-6 operators, in the absence of interference, the leading BSM contribution to  the total cross section scales O(Λ−4), and it may invalidate the EFT expansion in terms of  Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' This also makes the translation of the data to the SMEFT sensitive to the dimension-  8 operators as the leading contribution is in the same order of the interference between  dimension-8 operators and the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  There have been many attempts to resurrect the interference in the diboson process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  While 2 → 2 diboson processes are subject to the noninterference, unstable vector gauge  bosons must decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Once the 2 → 2 diboson amplitude is extended to 2 → 3, 4 by gluing  with the three point amplitude(s) for a gauge boson decay into two fermions, the total  helicity of both amplitudes of the dimension-6 and the SM can match and thus interfere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The authors in [5, 8] suggested to look into differential angular distributions in the leptonic  decay channels to resurrect the interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' See [9] for a related discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The authors  in [5, 10] pointed out the partial resurrection of the interference due to the QCD next-  to-leading order (NLO) effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The role of off-shellness of the vector gauge bosons in the  diboson process on the interference has been studied in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  In this work, we newly add the dilepton production process with two associated forward  jets in the vector boson fusion (VBF) to the list regarding the interference resurrection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' This  process at √s = 13 TeV, using the integrated luminosity of 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9 fb−1, has been analyzed  by the CMS collaboration [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' While the signal rate of the VBF process is smaller than  the diboson production from the QCD process, it can solve the two aforementioned issues:  the scale of the hard subprocess can exactly be extracted from the invariant mass of two  leptons and the helicity of the full 2 → 4 process apparently allows the interference between  the amplitudes with dimension-6 operators and those from the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' While the electroweak  (EW) ℓℓ + jets process which is our main interest in this work may be considered as the  EW Drell-Yan process, we aim to measure aTGCs via the tree-level process whereas QCD  Drell-Yan process can access them via one loop effect [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' One can see [14–19] (and [20, 21]  for the experiment) for the precision study at the high energy tail of the QCD ℓℓ process  focusing on the tree-level four-fermion interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  4  A confusion arises due to the usual effective W approximation (EWA) [22–35] which  takes the gauge boson radiated off the quark line as an on-shell particle and focuses on the  WW initiated subprocess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' If this is the case, the interference will be suppressed again as  the different total helicities of the SM and BSM amplitudes of WW → ℓℓ do not allow the  interference in the massless limit [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For better understanding, an analytic study of the  process that takes the full effect of the forward quark current would be highly beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' To  this end, we carry out the full analytic calculation for a simpler process uγ → dνe+ that has  only one forward quark current and one intermediate gauge boson as a toy process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' As will  be discussed below in detail, we find that the interference cross section of uγ → dνe+ with  respect to the SM counterpart does resurrect the energy growing behavior, importantly in  the inclusive cross section, that would have been lost in the W +γ → νe+ subprocess due to  the helicity structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The resurrected energy growing interference survives even when the  apparent conditions for the EWA, namely, small transverse momenta of the forward quarks  and small gauge boson masses compared to the overall energy scale of the hard subprocess,  are applied in the full uγ → dνe+ process and thus provides a counter-example to the  usual EWA assumption (see [35] for a related discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Our simpler toy process provides  the proof of concept example for the resurrected interference in the inclusive cross section,  and the intuition from it greatly helps for a better qualitative understanding of our EW ℓℓ  production process with two associated jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  It turns out to be crucial that an enough energy must flow into the ℓℓ hard subprocess  to resurrect the energy growing interference in the inclusive cross section of the full 2 → 4  process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Unlike the QCD Drell-Yan process where mℓℓ directly controls the fraction of the  energy that goes into the dilepton system, it becomes ambiguous in our EW ℓℓ with two for-  ward jets process because some fraction of energy goes to the scattered quarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' In this work,  we propose a new variable, what we call VBFhardness, that allows to control the fraction of  energy carried by the ℓℓ subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We demonstrate that the energy growing interference  with respect to the SM is clearly resurrected with an appropriate cut on VBFhardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  In Section II we briefly sketch the (non)interference of the dilepton production with two  associated jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' In Section III we provide the analytic result of a simpler 2 → 3 (instead of our  2 → 4) toy process as this simpler example can be analytically calculated to capture the full  effect of the forward jet from the viewpoint of the interference resurrection and the validity of  the EWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' In Section IV we perform the numerical simulation of the EW ℓℓ production with  two associated jets as our main process of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' In particular, we validate our simulation  against the CMS cut-and-count analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We carry out the multivariate analysis using the  Boosted Desicion Tree (BDT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We finally derive the sensitivity of aTGC at the LHC and  high luminosity LHC (HL-LHC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  5  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  EW DILEPTON PRODUCTION WITH TWO ASSOCIATED JETS  ±1  2  ∓1  2  ±1  2  ∓1  2  ±1  2  ∓1  2  −1  +1  +1  +1  −1  −1  ±1  2  ∓1  2  ±1  2  ∓1  2  ±1  2  ∓1  2  −1  ∓1  2  ±1  2  +1  +1  −1  ±1  2  ∓1  2  ∓1  2  ±1  2  ∓1  2  ±1  2  −1  ±1  2  ∓1  2  −1  +1  +1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 1: Interference between BSM and SM diagrams in the massless limit where only two types  of SM diagrams are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The blob denotes the insertion of the dimension-6 operator tr(W 3  µν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The helicity assignment is displayed as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 1 illustrates the subset of diagrams for the 2 → 4 amplitudes, leading to the dilepton  with two associated jets, and possible helicity assignments which allow the interference  between SM and BSM amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' There is no similar diagram to the first one in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 1 with  the SM triple gauge couplings of the transverse modes as the helicity can not be correctly  assigned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Its non-vanishing diagram can arise via helicity flips along with the Higgs VEV  insertions and it will be suppressed by O(m2  W/E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The virtuality of W emitted off the initial quark current induces the energy uncertainty  of W whose inverse sets the time uncertainty ∆t ∼ E/V 2 where V is the virtuality of W and  E is the scale of the hard process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' As long as ∆t is much longer than the typical interaction  time t ∼ 1/E, one can not distinguish the virtual W from on-shell one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' In this situation, one  typically computes the partonic cross section of the hard subprocess, treating W as on-shell  gauge boson, and convolutes it with the probability distribution function of the W gauge  boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' This is known as the effective W approximation (EWA) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The helicity structure  +1  +1  −1  +1  ±1  2  ∓1  2  +1  −1  +1  −1  ±1  2  ∓1  2  +1  −1  ±1  2  ∓1  2  ∓1  2  ±1  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 2: Noninterference between BSM and SM amlitudes for WW → ℓℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The blob denotes the  insertion of the tr(W 3  µν) operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The helicity assignment displayed is an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  of the forward quark current and the correlation of its phase space with that of the hard  subprocess seem to be simply dropped in the EWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' While diagrams that do not have two  6  ±1  2  ∓1  2  +1  −1  +1  −1  +1  ±1  2  ∓1  2  ±1  2  ∓1  2  −1  +1  +1  +1  −1  ±1  2  ∓1  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 3: The interference (noninterfernce) in the 2 → 3 process (2 → 2 subprocess).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The blob  denotes the single insertion of the tr(W 3  µν) operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The complete set of diagrams are shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  obvious forward quark currents, for instance, third diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 1 and other radiation  type diagrams, are not included in the EWA assumption, they will be harmless upon the  forward jet tagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Apparently, the SM amplitudes of the WW → ℓℓ process in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 2 can  not interfere with the BSM amplitudes in the massless limit, known as noninterference [7],  unlike the aforementioned allowed interference in 2 → 4 amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The middle diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 2 can interfere with the BSM amplitude via helicity flips  in the sub-leading order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' It is consistent with that it can extend to the 2 → 4 amplitude  by attaching two quark currents upon helicity flips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' On contrary, the extended amplitude  with two attached quark currents of the third diagram in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 2 can interfere with the  corresponding BSM amplitude without any suppression as is evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 1 Assuming  the EWA from the beginning and working on the 2 → 2 process may not be the same as  what is obtained by applying the EWA limit of the full amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  TOY PROCESS FOR ANALYTIC STUDY: SINGLE LEPTON WITH AN AS-  SOCIATED JET  The purpose of this section is to analytically investigate (and numerically confirm) the  helicity structure and related kinematics of simpler 2 → 3 process uγ → dνe+ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 3  and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 4) that captures the full effect of the quark current attached to a vector gauge  boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' While the analytic calculation of the full 2 → 4 process in Section II is beyond the  scope of this work, our analytically calculable 2 → 3 toy process  2 provides the proof of  1 In the 2 → 4 diboson process decaying into two pairs of fermions, the narrow width approximation allows  to factorize the phase space of decaying on-shell gauge bosons from that of the hard process, and the  process is subject to the noninterference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' In this situation, the interference can appear, for instance, in  the differential cross section of the azimuthal angle [5, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  2 A similar explicit computation may be applicable to the 2 → 3 process of qq′ → γW ∗ → γℓνℓ where the  effect of the off-shell W gauge boson on the interference, for instance, whether the interference between  7  concept for the resurrected interference and an intuition on the validity of the EWA in the  SMEFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We expect this simpler toy process to capture important missing properties when  simply approximating with 2 → 2 VBF process under the EWA assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We consider  uγ → dνe+ since it involves with the exchange of only the W gauge boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  A similar  discussion is applied to qV → q′ℓℓ although the evaluation is more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The helicity assignments of two diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 3 indicate that the interference between  the SM and BSM amplitudes in the 2 → 3 process can be allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We separately consider  the kinematic regions for the on-shell and off-shell intermediate W gauge bosons decaying to  ℓνℓ since they have different qualitative behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For the resonant on-shell W gauge boson,  the 2 → 3 process is factorized into the production of the on-shell W gauge boson and its  decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' It is expected that the inclusive cross section is subject to the noninterference and  the interference at best can be resurrected only in the differential cross section of an angular  observable (although it is difficult to be reconstructed in the experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' On contrary, for  the non-resonant 2 → 3 process, the aforementioned factorization is not possible and the  interference in principle can appear in the inclusive cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  u  d  γ  W  W  p2  k1  k2  ν  e+  p1  k3  k  (a)  u  d  W  γ  k1  k2  ν  e+  p1  k3  e  p2  (b)  u  p1  p2  k3  k1  k2  ν  e+  W  d  d  γ  (c)  p1  p2  u  γ  u  d  W  ν  e+  k3  k1  k2  (d)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 4: The complete set of SM diagrams for the process uγ → dνe+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' These four diagrams are  required to guarantee the Ward identity and to get the correct high energy behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The full set of SM diagrams of the EW uγ → dνe+ process are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We  classify the first two diagrams a and b as the process of interest that probe the hard 2 → 2  subprocess and the last two diagrams c and d as the radiation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' All four diagrams in  SM and BSM amplitudes can be resurrected in the inclusive cross section, can be explicitly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  8  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 4 are required to satisfy the Ward identity, namely p2 ·(Ma +Mb +Mc +Md) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For  the resonant intermediate W, the Ward identity can be shown to be satisfied among three  diagrams, p2 · (Ma + Mc + Md) = 0, using the narrow width approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We postpone  all the details for the analytic calculation of uγ → dνe+ to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' In what follows, we  quote only the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Cross section for on-shell W boson  The diagrams a, c, and d in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 4 mainly contribute to the resonant 2 → 3 process where  we can restrict the phase space to those in the W mass window, or k2 = (2z − 1)ˆs ≈ m2  W  where z = [1/2, 1] is the fraction of the total energy  √  ˆs flowing into the νee+ system and k is  the four-momentum of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The process can be factorized into the 2 → 2 process of uγ → dW  and the decay of W to νee+ using the narrow width approximation for the on-shell W boson  of the width ΓW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  We evaluate the partonic differential cross section with respect to φ in the limit of ˆs ≫ m2  W  where φ is the angle between the planes made out of the forward quark current and the lepton  current (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The SM contribution is rather subtle to evaluate due to the forward  singularity in the massless fermion limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Its size is roughly given by  dˆσSM  dφ  =  � cos θmax  cos θmin  d cos θ d2ˆσSM  dφd cos θ ≈  1  2 · 2  1  512π2  8πe2g4  3  mW  ΓW  1  ˆs  1  δ ,  (4)  where δ = 2p2  T min/ˆs assuming δ ≪ 1 and it comes from the integration regularized by the  pT cut of the forward quark,  cos θmax/min = ±  �  1 −  p2  T min  ˆs(1 − z)2 ≈ ±  �  1 − 2 p2  T min  ˆs  �  for  ˆs ≫ m2  W, p2  T min .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  (5)  On the other hand, the leading contribution to the partonic differential cross section for the  interference in the high energy limit, ˆs ≫ m2  W, is estimated to be  dˆσSM×BSM  dφ  =  1  2 · 2  λz  512π4  πe2g4  3  2  mWΓW  �  cos(2φ)  �  2 − log  ˆs  m2  W  ��  + O(ˆs−1/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  (6)  Upon the integration over the angle φ, the interference term vanishes while it is recovered  in the differential cross section with respect to φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The individual contributions to the total  9  cross section for the interference are given by  dˆσSM×BSM(uLγL → dνe+)  dφ  =  λz  512π4  πe2g4  144  1  mWΓW  ×  �  9π2 cos φ + 16 cos(2φ)  �  5 − 3 log  ˆs  m2  W  � �  + O(ˆs−1/2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  dˆσSM×BSM(uLγR → dνe+)  dφ  =  λz  512π4  πe2g4  144  1  mWΓW  ×  �  − 9π2 cos φ + 16 cos(2φ)  �  7 − 3 log  ˆs  m2  W  � �  + O(ˆs−1/2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  (7)  where linear terms in cos φ cancels upon the summation and there is no contribution from  the right-handed quark in the massless limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' In the same high energy limit, the quadratic  term in the anomalous coupling λz is approximately estimated to be  dˆσBSM2  dφ  =  1  2 · 2  λ2  z  512π4  πe2g4  6  ˆs  m3  WΓW  �  1 + O(ˆs−1/2)  �  ,  (8)  where φ dependent terms are subdominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For the quadratic terms in aTGC couplings,  the leading contributions from both photon polarizations are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The energy flowing into the on-shell W is z  √  ˆs ∼  √  ˆs/2 as usual in the high ˆs limit because  of z ∼ 1/2 + m2  W/(2ˆs) → 1/2 for ˆs ≫ m2  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The produced on-shell W gets boosted with the  transverse momentum of the order O(  √  ˆs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The boosted W boson requires a large recoiling  against a hard quark jet which likely invalidates the EWA as the jet can not be treated as  a forward jet anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The process is subject to the noninterference, as is seen in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (6),  since it is basically 2 → 2 process uγ → dW where the W decay can be factorized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  When generalizing our toy process to the 2 → 4 process with the intermediate resonant  W by attaching the fermion line to the photon, the situation becomes less obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Similarly  the EWA of either jet or both will not be valid if the boosted W boson is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The  interference can be in principle possible as the helicity structure of the 2 → 3 process for  the on-shell W production (pp → qq′W) allows the amplitudes to interfere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Cross section for off-shell W boson  Alternatively, one can probe the high energy behavior of the anomalous coupling by  directly accessing far off-shell region of W, or k2 = (2z − 1)ˆs ≫ m2  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For this case, the  full matrix element for the 2 → 3 process needs to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' One new feature will be  the resurrection of the interference in the inclusive cross section, and its size is expected to  be proportional to the off-shellness of the W boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' While the analytic evaluation of the  differential cross section in terms of φ is challenging due to the diagram b in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 4, we have  10  managed to get the leading contribution only for the left-handed polarization of the photon  in the high energy limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Similarly to the previous Section III A, we will use ˆs to take a high  energy limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The cross section for the interference in the limit of ˆs ≫ m2  W far away from  the W mass window, k2 ≫ m2  W, is estimated to be  dˆσSM×BSM(uLγL → dνe+)  dφ  =  λz  512π4  e2g4  m2  W  �  − 2  9 − π2  6 cos φ  + 1  18  �  π2 − 26 + 22 ln  ˆs  m2  W  − 6 ln2  ˆs  m2  W  �  cos(2φ)  � �  1 + O(ˆs−1)  �  ,  (9)  where ΓW dependent terms are not shown as they contribute to the region of the W mass  window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Although the off-shell contribution is suppressed by the factor of O(ΓW/mW), or  ∼ O(1/m2  W), compared to the cross section from the W mass window, the interference term  can survive in the inclusive cross section even after the integration over all angular variables  (see the first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The cross section for the quadratic term in λz is given by  dˆσBSM2(uLγL → dνe+)  dφ  = λ2  z e2g4  512π4  ˆs  m4  W  � 1  24  �  −9 + 4 ln  ˆs  m2  W  �  − π2  48 cos φ − 1  12 cos(2φ)  � �  1 + O(ˆs−1/2)  �  ,  (10)  where φ dependence only appears in cos φ and cos(2φ) terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For the quadratic term, the  analytic expression in the high energy limit outside the W mass window can be obtained for  both polarizations of the photon, and the summed and averaged cross section over helicities  is given by  dˆσBSM2  dφ  =  1  2 · 2  λ2  z e2g4  512π4  ˆs  m4  W  � 1  216  �  −143 + 60 ln  ˆs  m2  W  �  + π2  240 cos φ − 1  12 cos(2φ)  � �  1 + O(ˆs−1/2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  (11)  While we have shown the evidence of the resurrected interference in the inclusive cross  section through the computation of the differential cross section in terms of φ only for the left-  handed photon helicity, the experimental reconstruction of the angle φ will be challenging  and we may be able to access only to the inclusive cross section summed and averaged  over helicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For the direct analytic computation of the inclusive cross section, we have  managed to get the final result for both helicities of the photon by performing the integration  over φ first and the remaining variables later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The leading contribution of the summed and  averaged cross section over helicities is given by  ˆσSM×BSM =  1  2 · 2  λz  512π4  e2g4  m2  W  × π  3  �  13 − 6 ln  ˆs  m2  W  �  + · · · ,  (12)  11  where · · · denotes the higher order in ˆs and the logarithmic term is due to the contribution  from the right-handed helicity of the photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The summed and averaged cross section which  is quadratic in λz can be easily obtained by integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (11) over the angle φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  For more off-shell W, more energy flows into the eν system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For this case, the transverse  momentum of the forward quark and the W mass are small compared to the scale of the  hard subprocess, or the usual condition for the EWA is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We can isolate the behavior  of the corresponding phase space by integrating over only the interval z = [1 − ε, 1] with  ε ≪ 1,  ˆσSM×BSM  512π4  λz  m2  W  2πe2g4  = −1  3ε2 + 1  3  m2  W  ˆs  ��  −3 + 2 ln 2εˆs  m2  W  �  ε +  �  −13 + 6 ln 2εˆs  m2  W  �  ε2 + · · ·  �  + · · · ,  (13)  where · · · denotes the higher order terms in ε and m2  W/ˆs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' In the high energy limit of ˆs → ∞,  the first constant term will eventually dominate, and it will appear as the energy growing  interference in ˆσSM×BSM/ˆσSM assuming ˆσSM ∼ 1/ˆs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The variable meν may be considered to  be more relevant one to take a high energy limit of the hard subprocess Wγ → νe+ inside  uγ → dνe+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Simply changing variable from  √  ˆs to meν in expressions obtained in the high  ˆs limit in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 9, 10 and 11 could be misleading or not well defined, for instance, a wide  range of  √  ˆs can be associated with a small value of meν for z ∼ 1/2 (see the right panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The analytic computation of the interference in terms of meν, performed at this  time starting from amplitudes, reveals a similar energy growing behavior to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Numerical calculation of toy process and interference resurrection  We numerically investigate the analytic behavior discussed in Sections III A and III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  To this end, we generate partonic level events for the EW uγ → dνe+ process using Mad-  Graph5 aMC@NLO v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='7 [36] only with the nominal pT cuts of 10 GeV for the final quark,  neutrino, and electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' As the noninterference is well established for the operator involving  λz, the events for the interference are generated only for λz coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' While the separation  of the off-shell region from the on-shell one in Section III B was done just by dropping out  all ΓW dependent terms by hand, we numerically control the separation using two variables  ∆mW = meν − mW and z for the purpose of the demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The fraction of the energy, z∗ = 1/2 + m2  W/(2ˆs), carried by the on-shell νee+ system is  roughly order one for ˆs ∼ m2  W, and it rapidly drops to 1/2 with increasing  √  ˆs as is seen in  the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Due to the relation meν =  �  (2z − 1)ˆs, two variables meν and  √  ˆs  are comparable to each other only for a low  √  ˆs where 2z − 1 is roughly order one and they  can be very different for a large  √  ˆs as 2z − 1 can be almost zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The region bounded by  12  100  200  500  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='0  s\uf111 [GeV]  z  uγ → dνe  z = z*+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9  z = z*- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05  z = z*  |meν - mW| = 10 GeV  100  200  500  1000  1  10  100  1000  s\uf111 [GeV]  meν [GeV]  uγ → dνe  z = z*+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05  z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9  z = z*- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05  z = z*  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 5: Left: The fraction of the energy that flows into the eν system, z = Eeν/  √  ˆs, as a function  of  √  ˆs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The band bounded by red lines correspond to the W mass window of 10 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Black line  denotes the z value for the on-shell W of the mass mW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Right: the correlation between meν and  √  ˆs depending on the cut on z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  red lines in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 5 corresponds to the z value for the W mass window of  10 GeV, or |mℓν(=  �  (2z − 1)ˆs) − mW| < 10 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' To access the off-shell region, we impose  the cut on z such as |z − z∗| = |(m2  eν − m2  W)/(2ˆs)| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05 and z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9 and they are shown  by gray lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The selected region by z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9 isolates the phase space where most  of the center of mass energy  √  ˆs flows into the νee+ system and meν can be as large as  √  ˆs  as is seen in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' It is also the phase space where the condition for the  typical EWA is expected to be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' While the cut of z − z∗ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05 makes it possible  to access a deeper off-shell region in a large  √  ˆs region than that specified by the W mass  window of 10 GeV (red lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 5), most events are still populated near the lower meν  value than  √  ˆs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Our numerical simulations of |σSM×BSM|/σSM binned in  √  ˆs is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' As  is evident by the red-colored almost flat distribution in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 6, the noninterference predicted  for the on-shell W in Section III A is numerically confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The black-colored distribution  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 6 demonstrates the resurrected energy growing interference in the inclusive cross  section for the off-shell W and they agree with our expectation in Section III B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 7 illustrates the interference cross section with respect to the SM  in terms of the meν variable for the same phase space as those in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 6, and the energy-  growing behavior is clearly seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' In the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 7, we take a limit where almost  all energy  √  ˆs flows into the eν system (see the solid gray line in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For  3 In this work, we will not explore the sign of the interference and its sensitivity at the collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The sign  of the interference depends on the phase space (see Appendix B 3 for the related discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  13  0  500  1000  1500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='50  1  5  10  s\uf111 [GeV]  |σSM\uf4a0BSM|/σSM  uγ → dνe  |z - z*| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05  meν > 90 GeV  |z - z*| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05  |meν - mW| < 10 GeV  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 6: The differential distribution of |σSM×BSM|/σSM in  √  ˆs for the EW uγ → dνee+ at the  parton level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Black lines demonstrate the interference at off-shell region specified as |z − z∗| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05  and meν > 90 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Red lines demonstrate the noninterference for the on-shell W defined by  |z − z∗| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05 and |meν − mW | < 10 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  0  500  1000  1500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='50  1  5  10  meν [GeV]  |σSM\uf4a0BSM|/σSM  uγ → dνe  |z - z*| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05  meν > 90 GeV  0  500  1000  1500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='50  1  meν [GeV]  |σSM\uf4a0BSM|/σSM  uγ → dνe  z = [1-ε, 1]  ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='1  meν > 90 GeV  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 7: The differential distribution of |σSM×BSM|/σSM in meν for the EW uγ → dνee+ at the  parton level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Events are restricted to satisfy |z − z∗| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05 and meν > 90 GeV (left) or z > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9  and meν > 90 GeV (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  this situation, the transverse momentum of the forward quark and the W mass are small  compared to the scale of the hard subprocess, or the EWA condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' As is clearly  seen in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 7, the energy growing interference term looks survive in the  EWA limit of the full 2 → 3 process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' However, this energy growing interference term allowed  by the helicity selection rule of the full 2 → 3 process will get lost if one simply assumes the  EWA and works on the 2 → 2 hard subprocess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Recall that the helicity selection rule of the  2 → 2 subprocess does not allow the interference in the massless limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  While we have exploited the variable z to distinguish the phase spaces of the on-shell and  off-shell regions, it can be traded for a combination of experimental variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Using the  14  transverse momentum of the forward quark, pT(q) = (1−z)  √  ˆs sin θ (with sin θ = 1/ cosh η),  and meν = √2z − 1ˆs, one can easily derive the relation,  pT(q) cosh η  meν  =  1 − z  √2z − 1 ≤  1 − zmin  √2zmin − 1 ≡ δmin → pT(q) ≤ δmin  meν  cosh η ,  (14)  where zmin = {z∗ +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='05, 1−ε} was used in the plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 7 and η is the pseudorapidity of  the outgoing quark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Note that z∗ (thus δmin as well) is still a function of the experimentally  inaccessible ˆs although its dependence gets mild in the high ˆs limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For the hard cut on  z, δmin becomes a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We have numerically checked that the cut pT(q) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='112 ×  (meν/ cosh η) is physically equivalent to z > 1 − ε (with ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='1) and reproduces the  same plot as the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For a small constant δmin, the cut in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (14), or  pT/meν ≤ δmin/ cosh η, is consistent with the conditions for the EWA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  NUMERICAL ANALYSIS OF EW DILEPTON WITH TWO ASSOCIATED  JETS  In this section we numerically investigate the EW ℓℓ + two jets process at the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We  take the CMS analysis in [12] as our baseline for both the validation of our analysis and the  derivation of the sensitivity on aTGCs at the LHC 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The detail of the event generation can  be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Interference resurrection  We can use the intuition from the EW uγ → dνe+ process in Section III to isolate the  phase space that reveals the interference resurrection in the EW ℓℓ+ two jets process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' In the  partonic EW ℓℓ + qq′ process, we can treat the ℓℓ (qq′) system effectively as a single particle  with the energy of z  √  ˆs ((1−z)  √  ˆs) and the invariant mass of mℓℓ (mqq′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Similarly to our toy  process in Section III, the variable z represents the fraction of the total energy flowing into  the dilepton system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Three momentum conservation, ⃗pT(ℓℓ) = −⃗pT(qq′), in the center of  mass frame of two initial quarks leads to m2  ℓℓ−m2  qq′ = (2z−1)ˆs where z varies over the range  z = [ mℓℓ/  √  ˆs, 1 − mqq′/  √  ˆs ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Similarly to the previous section, we start with the variable  z = 1/2+(m2  ℓℓ −m2  qq′)/(2ˆs) to separate the off-shell phase space from the on-shell one where  z∗ = 1/2 + (m2  Z − m2  qq′)/(2ˆs) at the Z pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' An appropriate cut on z such as |z − z∗| =  |(m2  ℓℓ − m2  Z)/(2ˆs)| > ∆z or z > zmin will select the corresponding off-shell region, while  ensuring a certain correlation between mℓℓ and  √  ˆs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=" Combining m2  ℓℓ − m2  qq′ = (2z − 1)ˆs with  the transverse momentum of the effective qq′ system pT(qq′) =  �  (1 − z)2ˆs − m2  qq′ sin θqq′, the  4 Similar study by the CMS collaboration for the EW ℓνℓ + two jets process has been made in [37]  15  500  1000  1500  1  2  3  4  5  mℓℓ [GeV]  |σSM\uf4a0BSM|/σSM  Partonic EW ℓℓ+qq',  s =13 TeV  VBFhardness > 5  500  1000  1500  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content="1  1  10  100  1000  mℓℓ [GeV]  |σSM\uf4a0BSM, BSM2|/σSM  Partonic EW ℓℓ+qq',  s =13 TeV  VBFhardness > 5  BSM2  SM\uf4a0BSM  FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 8: The distributions of |σSM×BSM|/σSM in mℓℓ for the partonic EW ℓℓ + qq′ (black lines in  both panels) for the λz coupling (other couplings are set to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Similarly for |σBSM2|/σSM (red  lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Events for solid lines are restricted to those with VBFhardness > 5 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (15) along with  pT (q) > 25 GeV, pT (ℓ) > 10 GeV, and mqq′ > 120 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For dashed lines in the right panel, the  VBFhardness cut is removed while others kept the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  variable z can be translated into the nontrivial combination of various kinematic variables  via the relation,  VBFhardness ≡  m2  ℓℓ − m2  qq′  p2  T(qq′) cosh2 ηqq′ + m2  qq′  = 2z − 1  (1 − z)2 ≥ 2zmin − 1  (1 − zmin)2  for  z ≥ zmin ,  (15)  where the ratio is the monotonically increasing function, while it can have either sign, and  sin θqq′ = 1/ cosh ηqq′ was used to express in terms of the pseudorapidity of the qq′ system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The positive value of the VBFhardness (or equivalently z > 1/2) corresponds to the case  where more than half the total energy flows into the dilepton system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Just like the case of  our toy process in Section III, zmin still has the ˆs dependence if one intends to impose a cut  on |z − z∗| instead of a constant cut on z itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  As is evident in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 8 (see black dashed lines), the interference does not  reveal the energy growing behavior without a cut on the ratio in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' As an illustration,  the resurrected interference in the inclusive cross section for the λz coupling is clearly shown  in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 8 for VBFhardness > 5 that corresponds to z ≥ zmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We  checked that a similar energy growing interference appears in terms of  √  ˆs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The same  interference is displayed again with the quadratic cross section in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The square of the interference term in this illustrative example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 8 appears to have a  milder energy growing behavior than the quadratic term itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The interference would have  been lost if one has not included the full effect of the forward quarks or not imposed a cut  on a proper variable like the one in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 9, we show the resurrected interference  pattern continues to survive at the hadron level where the VBFhardness is constructed out of  16  500  1000  1500  1  10  100  1000  104  mℓℓ [GeV]  |σX|/σSM  EW ℓℓ+jets,  s =13 TeV  VBFhardness > 5  SM\uf4a0BSM  BSM2  500  1000  1500  1  2  5  10  mℓℓ [GeV]  |σSM\uf4a0BSM|/σSM  EW ℓℓ+jets,  s =13 TeV  VBFhardness > 5  0  200  400  600  800  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='1  1  10  100  1000  104  pT(ℓℓ ) [GeV]  σX/σSM  EW ℓℓ+jets,  s =13 TeV  |mℓℓ-mZ|<15 GeV  SM\uf4a0BSM  BSM2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 9: |σX|/σSM where X = SM×BSM (black) or BSM2 (red) for the EW ℓℓ + two jets for the  coupling Ci = λz (other couplings are set to zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Plots are made with events at the jet level after  imposing the loosened cuts, compared to the CMS analysis [12], pT (j) > 30 GeV, pT (ℓ) > 20 GeV,  |η(j)| < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='5 , |η(ℓ)| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='5, and mjj > 120 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  two forward jet candidates and lepton pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The CMS analysis in [12] derives the sensitivity  on aTGC using the pT distribution of Z only for the events inside the Z mass window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' In  the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 9, the interference and quadratic terms of the inclusive cross section  are illustrated in pT(ℓℓ) only for the events in the Z mass window |mℓℓ − mZ| < 15 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Validation against the CMS analysis and BDT analysis  We adopt the CMS analysis in [12] for the validation of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Events with two  isolated leptons (electrons or muons) and at least two jets are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' A lepton is declared  to be isolated if the ratio of the pT-sum of all particles within the isolation cone Riso = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='4  around the lepton to the pT of the lepton is below 15% and 25% for electrons and muons,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' While two isolated leptons need to satisfy pT > 20 GeV and |η(ℓ)| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='4, and  have the opposite electric charges, the harder lepton must pass the cut pT > 30 GeV as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  17  The particles excluding the isolated leptons are clustered into jets by anti-kt algorithm [44]  with the distance parameter of Rjet = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Jets are required to satisfy pT(j) > 15 GeV and  |η(j)| ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Two hardest jets, called the tagging jets, are required to have pT(j) > 50 GeV  and pT(j) > 30 GeV for the leading and subleading jets, respectively, and their invariant  mass should satisfy mjj > 200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The initial cuts in CMS analysis in [12] are defined as  pT(ℓ1) > 30 GeV ,  pT(ℓ2) > 20 GeV ,  |η(µ)| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='4 ,  |η(e)| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='1 ,  pT(j1) > 50 GeV ,  pT(j2) > 30 GeV ,  |η(j)| ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='7 ,  |mZ − mℓℓ| < 15 GeV , and  mjj > 200 GeV  (16)  where the subscripts 1 and 2 mean leading and subleading objects, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The event  yields after imposing the initial cuts are given in Table I where we included only two largest  backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The smaller yield of the ee channel is due to the lower selection efficiency of  Initial  Sample  ee  µµ  t¯t  5454 (5363±48)  13962 (12938±81)  DY Zjj (pythia8) 146147 (152750±510) 373731 (394640±880)  EW Zjj (pythia8)  2639 (2833±10)  6328 (6665±16)  TABLE I: Validation of our simulation at √s =13 TeV assuming 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9 fb−1 of the integrated  luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The numbers in parenthesis are CMS values for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The k-factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='7 was  applied for the t¯t process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We adopted the pT-dependent electron selection efficiency [43] in our analysis,  while setting the selection efficiency for muons to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The electron selection efficiency is  roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='7 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='8 for the pT of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Having our analysis validated with the initial cuts, we move onto the BDT analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The CMS analysis introduces two additional variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Event balance variable, R(phard  T  ), is  defined as  R(phard  T  ) =  |⃗pTj1 + ⃗pTj2 + ⃗pTZ|  |⃗pTj1| + |⃗pTj2| + |⃗pTZ|  (17)  The z∗ Zeppenfeld variable is defined as  z∗ =  y∗  ∆yjj  ,  (18)  where y∗ = yZ − 1  2 (yj1 + yj2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Additionally, the quark-gluon discrimination is applied to two  tagging jets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Instead of constructing a likelihood function for the q/g discrimination and  use it in the BDT analysis afterwards as done in the CMS analysis [38], we directly use the  18  three input variables to the likelihood in our BDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' They are multiplicity, jet shapes, and  the fragmentation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The jet shape variable is defined as  σ =  �  σ2  1 + σ2  2  with  σ1 = (λ1/  �  i  p2  T,i)1/2 ,  σ2 = (λ2/  �  i  p2  T,i)1/2 ,  (19)  where the sum runs over the jet constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' λ1 and λ2 are the two eigenvalues of the  matrix with the elements, M11 = �  i p2  T,i∆η2  i , M22 = �  i p2  T,i∆φ2  i , and M12 = M21 =  − �  i p2  T,i∆ηi∆φi where ∆ηi and ∆φi are the pseudorapidity and azimuthal distances be-  tween a constituent and the average direction which is defined as the p2  T,i-weighted direction  of jet constituents in η − φ space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The fragmentation function is captured by the variable,  pTD =  ��  i p2  T,i  �  i pT,i  ,  (20)  where the sum runs over the jet constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For the multiplicity we count all charged and  neutral constituents of a jet whose energy is above 1 GeV, and it is denoted as ntracks(j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Similarly to the CMS analysis in [12], we use the following set of the BDT variables to  train and test our signal and background samples with the initial cuts in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (16):  mjj ,  |∆ηjj| ,  pT(jj) ,  R(phard  T  ) ,  z∗(Z) ,  ntracks(j1,2) ,  pTD(j1,2) ,  σ1(j1,2) ,  (21)  where mjj, ηjj, and pT(jj) are the invariant mass, pseudorapidity, and transverse momentum  of two leading jets system, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' To simplify our analysis and at the same time to  take full advantage of kinematic distribution to efficiently suppress the largest QCD Drell-  Yan background, we first train and test over the EW ℓℓ + jets in the SM as a signal and  the remaining samples as the background using the gradient boosting algorithm (BDTG)  provided by the TMVA package [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Since the signal and the dominant background have the  largest population in the Z mass window with the small transverse momentum, the BSM  effect is expected to be small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' This rejects the QCD Drell-Yan and top pair backgrounds as  much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We impose an appropriate cut on the BDT variable, that was computed  in the previous training, for all the samples of EW ℓℓ + jets in the SM and BSM, and  background processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' While it is nontrivial to exactly reproduce the outcome of the CMS  BDT analysis, the outcome of our BDT training, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 20 in Appendix C, shows  the clear separation between the signal and background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  We do not add our newly introduced VBFhardness in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (15) to the BDT variable set  although it has a correlation with mjj, ηjj, and pT(jj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Since we take the EW ℓℓ + jets in the  SM as a signal in the training, we expect its effect on the signal/background discrimination  to be mild as is indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' While the VBFhardness variable helps in resurrecting  19  1  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='5  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='5  1  VBFhardness  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='2  Normalized unit/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='04  Z+jets  +jets  tt  EW Zjj  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='04  z  λ  1  −  0  1  2  3  4  5  6  VBFhardness  5  −  10  4  −  10  3  −  10  2  −  10  1  −  10  Normalized unit/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='4  Z+jets  +jets  tt  EW Zjj  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='04  z  λ  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 10: The normalized distribution of VBFhardness for the EFT signal for λz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='04, EW  dilepton (denoted by EW Zjj), t¯t+jets, and QCD Drell-Yan backgrounds (denoted by Z+jets)  after imposing pT (j) > 30 GeV, pT (ℓ) > 20 GeV, |η(j)| < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='5 , |η(ℓ)| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='5, and mjj > 120 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Right panel is logarithmic plot of the left panel in a large VBFhardness range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  the interference, its effect should be small as well in the situation where the sensitivity  of aTGCs is mainly driven by the quadratic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' It will be relevant in case where the  sensitivity is derived by the interference cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' As is seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 10, although a  proper cut may reduce the signal rate, VBFhardness seems to be a good discriminator for  the EFT signal as it controls the amount of energy going into the dilepton subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' It  will be important at the HL-LHC or future collider and we leave more dedicated analysis  for the future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Sensitivity to aTGC at the LHC  To evaluate sentivity to aTGC, we construct 1D templates binned either in pT(ℓℓ) and  mℓℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Events are distributed over 20 equal-spaced bins of pT(ℓℓ) between 0 and 1200 GeV  where the last bin contains events beyond 1200 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' ℓ includes both electrons and muons 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  We also newly construct templates of mℓℓ with 10 equal-spaced bins between 0 and 2000 GeV  where the last bin contains events beyond 2000 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The distributions of backgrounds and  two selected EFT benchmark points (with the SM contribution subtracted) are illustrated in  5 On the contrary, the CMS analysis in [12] separately distribute events in 15 bins in pT (ℓℓ) = [0, 900] GeV  and 20 bins in [0, 1200] GeV for electrons and muons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  20  0  200  400  600  800  1000  1200  (ll) (GeV)  T  p  10  2  10  3  10  4  10  5  10  Events/60 GeV  (13 TeV)  1  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9 fb  Z+jets  +jets  tt  EW Zjj  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='04  z  λ  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='1  1,z  g  δ  0  200 400 600 800 100012001400160018002000  (GeV)  ll  m  1  10  2  10  3  10  4  10  5  10  6  10  Events/200 GeV  (13 TeV)  1  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9 fb  Z+jets  +jets  tt  EW Zjj  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='04  z  λ  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='1  1,z  g  δ  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 11: The distributions of pT (ℓℓ) (left) and mℓℓ (right) at 13 TeV, using the integrated luminosity  of 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9−1, for backgrounds and two selected EFT benchmark signals with the SM contribution  subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Events are restricted to those satisfying CMS initial cuts in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  We construct a log likelihood in terms of aTGCs, assuming the Poisson distribution,  Using the template analysis of pT (ℓℓ) in the Z mass window at 13 TeV, L = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9 fb−1  No BDT cut  BDT > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6  aTGC  68% CL  95% CL  95% CL (Linear)  68% CL  95% CL  95% CL (Linear)  λz  [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
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+page_content='26]  TABLE III: Similar caption to Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  − 2∆ log L(λz, δg1,z, δκz) ,  (22)  where ∆ indicates that the minimum is subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
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+page_content=' Compared to the red solid lines, thin gray lines were obtained only with  the interference term which is linear in the aTGC coupling for the BDT > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The 68% and 95% CL intervals of an individual aTGC, where two others are set to  zero without the marginalization, are presented in Table II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For the result with  the BDT cut, we estimated the sensitivity with the incremental BDT cut starting with a  mild value, and did not find visible improvement with a stronger BDT cut than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
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+page_content=' For  λz, the 95% CL interval from BDT > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6 is worse than the expected value of the CMS  one, or λCMS  z  = [ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='014, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='014 ] [12] 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For the δg1,z coupling, our analysis gives roughly  6 Comparing two distributions of pT (Z) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 8 of [12] (separately displayed for electrons and muons) and  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 11 (summed over both leptons), our signal to background ratio looks rather smaller than the CMS  one in a high pT region where a large statistical power is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
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+page_content='6 )  68% CL (BDT >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6 )  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 13: Two-dimensional limits on aTGCs at 68% (dashed) and 95% CL (solid) regions obtained  using the binned analysis of mℓℓ, assuming the integrated luminosity of 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9 fb−1 at √s = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Compared to the red solid lines, thin gray lines were obtained only with the interference term  which is linear in the aTGC coupling for the BDT > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' No cuts on VBFhardness was imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  comparable with the CMS one, δgCMS  1,z  = [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='053, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='061 ] [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The two-dimensional exclusion  regions from the binned analysis of pT(ℓℓ) in the Z mass window are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 12  where the remaining coupling is set to zero without the marginalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The gray lines in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 12 illustrate the exclusion region at 95% CL using only linear terms in aTGCs in our  parametrization of the cross section (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' It indicates that the sensitivity of λz is  dominantly driven by the quadratic term whereas the effect of the quadratic term is less  pronounced for two other aTGC couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  We newly derive the sensitivity using the binned analysis of mℓℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' As discussed in Sec-  tion IV A, the invariant mass of the dilepton system has the relation m2  ℓℓ − m2  jj = (2z − 1)ˆs,  efficiency and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
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+page_content='10  δg1,z  λz  s = 13 TeV , 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9 fb -1  BDT >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6  pT(ℓℓ)=[0,  ∞] GeV  pT(ℓℓ)=[0, 1140 ] GeV  pT(ℓℓ)=[0, 600 ] GeV  pT(ℓℓ)=[0, 300 ] GeV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
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+page_content='9 fb -1  No BDT cut  mℓℓ=[0,  ∞] GeV  mℓℓ=[0,1800 ] GeV  mℓℓ=[0,1200 ] GeV  mℓℓ=[0, 600 ] GeV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
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+page_content='10  δg1,z  λz  s = 13 TeV , 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9 fb -1  BDT >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6  mℓℓ=[0,  ∞] GeV  mℓℓ=[0,1800 ] GeV  mℓℓ=[0,1200 ] GeV  mℓℓ=[0, 600 ] GeV  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 14: Breakdown of pT (ℓℓ) (top) and mℓℓ (bottom) categories in the plane (λz, δg1,z), assuming  the integrated luminosity of 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9 fb−1 at √s = 13 TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Curves of various styles indicate the 95%  CL contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  where mjj is the invariant mass of two forward jets, z is the fraction of the total energy of  the partonic system carried by the ℓℓ system, and mℓℓ alone does not guarantee the hardness  of the ℓℓ subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' However, while a nominal cut on the VBFhardness (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (15) for the  definition) ensures that at least some amount of the total energy goes into the ℓℓ subsystem  and greatly helps recovering the interference, as is clearly seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 8, it may not improve  the situation for the case where the sensitivity is dominantly driven by the quadratic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  For this reason, we have not exploited VBFhardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The 68% and 95% CL intervals of an  individual aTGC are presented in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' From the comparison between Tables II and III,  we observe that δκz is better constrained by the binned analysis of mℓℓ whereas λz and δg1,z  are better constrained by the analysis using the distribution of pT(ℓℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The two-dimensional exclusion regions from the binned analysis of mℓℓ are illustrated in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 13 where similarly the remaining coupling was set to zero without the marginalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
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+page_content='028]  TABLE IV: One-dimensional limits on aTGCs at 68% and 95% CL at 13 TeV using the integrated  luminosity of L = 300 fb−1 and L = 3000 fb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' No cut on VBFhardness was imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Unlike the case using pT(ℓℓ) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 12, the sensitivity, for instance, of λz is significantly  weakened (see upper right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 13) when the quadratic term is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' This is  due to the interference suppression as illustrated by the black dashed line in the right panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The situation is contrasted to those obtained using the binned analysis with  pT(ℓℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' As observed in the bottom panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 9, the discrepancy between the interference  and quadratic terms in the pT(ℓℓ) distribution is less pronounced, compared to the current  case, and the interference term itself also shows the pT-growing behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 14 illustrates how the sensitivity in the plane (λz, δg1,z) changes as some of the  higher bins are removed in the binned analysis of pT(ℓℓ) and mℓℓ, respectively, for two  cases without (left panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 14) and with the BDT cut (right panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' This  practice is meaningful especially for mℓℓ as the EFT cutoff can be directly imposed on the  mℓℓ variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For the case with the BDT cut, sensitivity to δg1,z mostly comes from the  first small number of bins, corresponding to the well below sub-TeV in both pT(ℓℓ) and mℓℓ  whereas a wider range of the energy contributes to the sensitivity to λz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' On the contrary,  for the case without the BDT cut, δg1,z becomes sensitive to the wide range of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  We derive the sensitivity at the LHC and HL-LHC, assuming an integrated luminosity  of 300 fb−1 and 3 ab−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
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+page_content=' 15: The two-dimensional contours at 95% CL, obtained using the binned analysis of pT (ℓℓ)  (upper) and mℓℓ (bottom), assuming the integrated luminosities of 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
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+page_content='  negligible, and we include only the statistical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Our projection for the LHC and  the HL-LHC is illustrated in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The 95% CL contours in the two-dimensional plane  are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 15 where upper two plots were obtained by the template analysis of pT(ℓℓ)  and the bottom ones using mℓℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The comparison between two analyses for δg1,z and δκz,  namely, one by total cross section up to the quadratic order in aTGC and the other only with  the interference cross section, indicates that the sensitivity is mainly driven by the linear term  for the case of pT(ℓℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
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+page_content=' Although, as is evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 10 a cut  on VBFhardness may reduce the signal rate, loosening other cuts may compensate it and it  can be an important variable at the HL-LHC regarding the interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  26  Bounds on aTGCs  Binned Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
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+page_content=' 16: The visual presentation of the sensitivity of aTGCs at 13 TeV, assuming three different  luminosities, given in Tables II, III, and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  CONCLUSION  We have explored the EW dilepton production with two associated jets for the precision  measurement of aTGC couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' This process has a few advantages compared to the typical  diboson production that has been a major process for studying aTGCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The EFT cutoff  can be unambiguously imposed on the dilepton invariant mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' As was explicitly shown  (both analytically and numerically) in this work, the full amplitude, including the forward  quarks that radiate off the vector gauge bosons, exhibits the interference, importantly, in  the inclusive cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' It contrasts with the interference resurrected in the differential  angular distributions of the diboson production process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For the purpose of the interference  27  resurrection in our dilepton production in vector boson fusion, we have introduced a new  variable, VBFhardness, that can control the amount of energy flowing into the dilepton  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Using this variable, we have demonstrated that the interference clearly appears  when an appropriate cut is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' It is also indicated that the usual practice of using  the EWA, namely, restricting ourselves to the subprocess initiated by gauge bosons and  convoluting its cross section with the probability distribution functions of the gauge bosons,  can be ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' As a proof-of-concept example for the interference resurrection in the  inclusive cross section, we have performed the analytic study using the simpler toy process,  or uγ → dνe+, which was numerically confirmed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  We have derived the sensitivity to aTGCs for three scenarios of the LHC and HL-LHC,  assuming the integrated luminosity of 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9 fb−1, 300 fb−1, and 3000 fb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' In addition to the  template analysis using the transverse momentum of the dilepton, we also carried out the  template analysis using the invariant mass of the dilepton in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' While the bounds  on λz and δg1,z from the dilepton invariant mass are rather weaker than those from the  transverse momentum of the dilepton system, the situation is opposite for δκz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The final  result of the one-dimensional bounds at 68% and 95% CL is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Our  analysis using the dilepton invariant mass may further be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Vetoing b-jets could  help suppress top-enriched backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Exploiting VBFhardness may help in enhancing  the role of the interference with respect to the quadratic terms in aTGCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Acknowledgments  MS thanks A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Azatov and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Marzocca for valuable discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Especially, MS thanks  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Azatov for the explanation of his previous work regarding the interference resurec-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' JP, MS, and MU were supported by National Research Foundation of Korea under  Grant Number NRF-2021R1A2C1095430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' JY and HH were supported by the National Re-  search Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  2020R1C1C1005916).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Appendix A: Details on simulation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Signal and background generation  The aTGC interaction in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (2) is implemented in FeynRules [39] from which we  generate the UFO output for the MadGraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Electroweak ℓ+ℓ−jj samples were simulated  at leading order (LO) by MadGraph5 aMC@NLO v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='7 [36] (QED=4, QCD=0) with the  default factorization and renormalization scales, interfaced with the Pythia8 v8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='306 for  the parton shower and hadronization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For the parton distribution function, the NNPDF30  28  (lo as0130) [40] is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The linear (or interference) and quadratic terms in aTGC in our  parametrization of the cross section in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (3) were separately simulated by using flags TGC2  = 1 and TGC2 = 2, respectively 7, where TGC denotes the order of aTGC interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The  phase space was restricted to those satisfying mℓℓ > 50 GeV, pT(j) > 25 GeV, and mjj > 120  GeV at the generation level 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  All background samples were similarly simulated at leading order (LO) by Mad-  Graph5 aMC@NLO v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='7 [36] with the default factorization and renormalization scales,  interfaced with the Pythia8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The NNPDF30 (lo as0130) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The QCD Drell-Yan  process γ∗/Z(ℓ+ℓ−)+jets samples where jets arise from QCD interaction were matched us-  ing kT-jet MLM matching at LO up to three extra jets in 5-flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' k-factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='23 was  applied [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The t¯t samples were matched using kT-jet MLM matching (QCUT = 45 GeV)  at LO up to two extra jets in 5-flavor and the total cross section was rescaled to match the  NLO value from Powheg [41] by applying the k-factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Appendix B: Computation detail of qV → q′νℓ  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Choice of four momenta and amplitudes  The polarization vectors of the photon are obtained by rotating ϵL/R =  1  √  2(0, 1, ±i, 0)  (for the massless momenta moving to −z axis) with angle θ about y-axis (similarly angle φ  about z-axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  ϵµ  L/R(p2) = 1  √  2 (0, cos θ cos φ ∓ i sin φ, cos θ sin φ ± i cos φ, − sin θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  (B1)  The spinor solutions in our coordinate system are  ¯uL(k1) = ˆs1/4  �  0, 0, −  √  2z − 1 sin ψ  2 , cos ψ  2  �  ,  vL(k2) = ˆs1/4  �√  2z − 1 cos ψ  2 , sin ψ  2 , 0, 0  �T  ,  uL(p1) = ˆs1/4  �  − sin θ  2, eiφ cos θ  2, 0, 0  �T  ,  ¯uL(k3) = ˆs1/4�  2(1 − z) (0, 0, −1, 0) ,  (B2)  7 On the other hand, the CMS analysis [12] generated aTGC signal samples (differently from ours) effectively  over 5 × 5 × 5 grid of cW W W /Λ2 × cW /Λ2 × cB/Λ2 which were equivalent to our aTGCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We suspect that  this could be partly responsible for the discrepancy between our sensitivity of aTGCs and that in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  8 To guarantee enough statistics and the smoothness of the differential distribution in the high invariant  mass tail, events were generated separately for multiple intervals of mℓℓ and combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Similarly for the  EW ℓℓjj samples in the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  29  where T denotes the transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We choose the following four momenta of the particles in  d(k3)  γ(p2)  u(p1)  k  ντ(k1)  e+(k2)  ψ  φ  θ  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 17: The angular configuration of the illustrative toy process, uγ → dνe+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  our 2 → 3 process, uγ → dνe+ and they are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  pµ  1 =  √  ˆs  2 (1, sin θ cos φ, sin θ sin φ, cos θ) ,  pµ  2 =  √  ˆs  2 (1, − sin θ cos φ, − sin θ sin φ, − cos θ) ,  kµ  1 =  √  ˆs  2  �  z + (1 − z) cos ψ,  �  (2z − 1) sin ψ, 0, (1 − z) + z cos ψ  �  ,  kµ  2 =  √  ˆs  2  �  z − (1 − z) cos ψ, −  �  (2z − 1) sin ψ, 0, (1 − z) − z cos ψ  �  ,  kµ  3 =  √  ˆs (1 − z, 0, 0, −(1 − z)) ,  kµ =  √  ˆs (z, 0, 0, (1 − z)) ,  (B3)  where the momentum k has the invariant mass of m2  k = (2z − 1)ˆs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Note that the 2 → 3  process can be effectively factorized into 2 → 2 and 1 → 2 via an intermediate momentum  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The momenta k1 and k2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (B3) are obtained by boosting those in the νe rest frame,  kµ  1 = mk  2 (1, sin ψ, 0, cos ψ ) ,  kµ  2 = mk  2 (1, − sin ψ, 0, − cos ψ) ,  (B4)  along the z-axis with the boosting factor,  kz = γz mkβz → γz = k0  mk  =  z  √2z − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  (B5)  When the intermediate W emitted from the quark line is produced nearly on shell, z is  nearly fixed to be  z ∼ 1  2  �  1 + m2  W  ˆs  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  (B6)  30  The helicity amplitudes for four diagrams in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 4 are given by  iϵ · Ma = ¯uL(k3)  �  i g  √  2γρ�  uL(p1) −iηρν  q2 − m2  W  × ϵλ(p2) i e  ��  ηµν(q − k)λ − (2 + δκγ)(pµ  2ηνλ − pν  2ηµλ) + ηνλkµ − ηµλqν�  + λz  m2  W  �  (pµ  2ηνλ − pν  2ηµλ)(k · q) + (qληµν − qµηνλ)(k · p2)  + (kνηµλ − kληµν)(q · p2) − kνqλpµ  2 + kλqµpν  2  ��  ×  −iηµσ  k2 − m2  W + imWΓW  ¯uL(k1)  �  i g  √  2γσ�  vL(k2)  =  �  i g  √  2  �2  (ie)  (−i)2  q2 − m2  W  1  k2 − m2  W + imWΓW  ϵλjν  q jµ  l V λνµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  (B7)  where q = p2 − k = k3 − p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  iϵ · Mb = ϵµ(p2)¯uL(k3)  �  i g  √  2γρ�  uL(p1) −iηρσ  q2 − m2  W  × ¯uL(k1)  �  i g  √  2γσ�i( /p2 − /k2)  (p2 − k2)2 (−ieγµ) vL(k2)  =  �  i g  √  2  �2  (−ie)  (−i)i  q2 − m2  W  1  (p2 − k2)2 ¯uL(k1)/jq( /p2 − /k2)/ϵvL(k2) ,  (B8)  where q = k3 − p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  iϵ · Mc = ϵµ(p2)¯uL(k3)  �  − i  3eγµ  � i( /k3 − /p2)  (k3 − p2)2  �  i g  √  2γρ�  uL(p1)  ×  −iηρσ  k2 − m2  W + imWΓW  ¯uL(k1)  �  i g  √  2γσ�  vL(k2)  =  �  i g  √  2  �2 �  − i  3e  �  (−i)i  k2 − m2  W + imWΓW  1  (k3 − p2)2 ¯uL(k3)/ϵ( /k3 − /p2)/jluL(p1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  iϵ · Md = ϵµ(p2)¯uL(k3)  �  i g  √  2γρ�i( /p1 + /p2)  (p1 + p2)2  �2i  3 eγµ  �  uL(p1)  ×  −iηρσ  k2 − m2  W + imWΓW  ¯uL(k1)  �  i g  √  2γσ�  vL(k2)  =  �  i g  √  2  �2 �2i  3 e  �  (−i)i  k2 − m2  W + imWΓW  1  (p1 + p2)2 ¯uL(k3)/jl( /p1 + /p2)/ϵuL(p1)  (B9)  where jµ  q = ¯uL(k3)γµuL(p1) and jµ  l = ¯uL(k1)γµvL(k2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  31  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Phase space integration  The partonic cross section of 2 → 3 process in our coordinate system is obtained by the  following phase space integration,  ˆσ =  1  512π4  � 1  1/2  dz(1 − z)  � 1  −1  d cos θ  � 1  −1  d cos ψ  � 2π  0  dφ  ��M  ��2 ,  (B10)  where  ��M  ��2 is the summed and averaged amplitude-squared over polarizations of the initial  partons and  ��M  �� has a negative mass dimension of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Interference between SM and BSM amplitudes for coupling λz  2000  4000  6000  8000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='0001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='0002  s\uf111 [GeV]  σ\uf111  SM\uf4a0BSM  Total  σ\uf111  SM\uf4a0BSM  hard  σ\uf111  SM\uf4a0BSM  rad  1000  2000  3000  4000  5000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='0015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='0020  s\uf111 [GeV]  σ\uf111  SM\uf4a0BSM  σ\uf111  SM\uf4a0BSM  aa  σ\uf111  SM\uf4a0BSM  ab  σ\uf111  SM\uf4a0BSM  ac  σ\uf111  SM\uf4a0BSM  ad  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 18: The partonic inclusive cross section in an arbitrary rate for the interference between the  SM and BSM, ˆσSM×BSM(uLγL → dνe+), integrated over the entire phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  In our 2 → 3 toy process, diagrams a and b in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 4 are those of interest that probe the  hard subprocess and diagrams c and d belong to the radiation type where W decaying to e+ν  is attached to either incoming or outgoing quark line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Restricting only to the interference,  we split the contribution into two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  ˆσhard  SM×BSM ≡ ˆσaa  SM×BSM + ˆσab  SM×BSM ,  ˆσrad  SM×BSM ≡ ˆσac  SM×BSM + ˆσad  SM×BSM ,  (B11)  where ˆσij  SM×BSM refers to the partonic cross section from the product of two diagrams i and  j in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The relative difference between two categories is purely due to the SM as the  λz dependence comes from the common diagram a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 18 shows that  ˆσhard  SM×BSM and ˆσrad  SM×BSM are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' While the magnitude of each ˆσaa  SM×BSM and ˆσab  SM×BSM  is bigger than both ˆσac  SM×BSM and ˆσad  SM×BSM, there is a cancellation between two contributions  from the hard subprocess, dictated by the gauge symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' It should be an artifact due  32  2000  4000  6000  8000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='×10-6  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='×10-7  0  s\uf111 [GeV]  σ\uf111  SM\uf4a0BSM  Total  σ\uf111  SM\uf4a0BSM  hard  σ\uf111  SM\uf4a0BSM  rad  1000  2000  3000  4000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='00004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='00002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='00000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='00002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='00004  s\uf111 [GeV]  σ\uf111  SM\uf4a0BSM  σ\uf111  SM\uf4a0BSM  aa  σ\uf111  SM\uf4a0BSM  ab  σ\uf111  SM\uf4a0BSM  ac  σ\uf111  SM\uf4a0BSM  ad  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 19: The partonic inclusive cross section in an arbitrary rate for the interference between the  SM and BSM, ˆσSM×BSM(uLγL → dνe+), integrated over the restricted phase space z = [1 − ε, 1]  where ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='1 was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  to the gauge choice in the photon polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' One may choose a particular gauge for the  photon polarization to suppress the contribution from the radiation type diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The  observed property is more pronounced when the phase space is restricted to z = [1 − ε, 1]  with ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='1 where the usual conditions for the EWA are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' As is clearly seen in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 19, an individual contribution from the hard subprocess becomes much bigger than  those involving the radiation type diagrams, and the cancellation is more dramatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' The  gauge dependence may not be a problem in the 2 → 4 process where all gauge bosons  including the photon are attached to the fermion currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Another interesting observation is that the sign of interference is  √  ˆs-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For  instance, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 18, the interference stays positive until around  √  ˆs ∼ 4 TeV whereas, in the  situation corresponding to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 19, the interference becomes negative well before TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Appendix C: Detail of BDT Analysis  For the purpose of the training and testing, we made separate inclusive EW ℓℓjj and QCD  Drell-Yan samples over the entire mℓℓ range whereas the samples (for the same processes) for  the actual BDT analysis were generated in multiple mℓℓ bins to guarantee the smoothness  with enough statistics up to the high invariant mass tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The ratio of samples for the  training and testing to those for the actual analysis is 1 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' For t¯t+jets samples, we used  30% for the training and testing and the remaining 70% for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We trained and  tested over the EW ℓℓjj in the SM as a signal and the remaining as the background using  the gradient boosting algorithm (called BDTG) provided in TMVA package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Our validation  of the BDT analysis is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 20 which shows the clear separation of the EW ℓℓ  33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='8  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='4  −  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='2  −  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='8  BDTG response  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='8  dx  /  (1/N) dN  Signal (test sample)  Background (test sample)  Signal (training sample)  Background (training sample)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 20: Our validation of the BDT analysis with the variable set in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (21), using the gradient  boosting algorithm in TMVA package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  + jets events from the QCD Drell-Yan and top pair backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  Training and testing by taking EFT benchmark points as signals and the remaining as  backgrounds may help in boosting the discrimination of the EFT signals from the back-  ground, and VBFhardness may play a role in that situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' We also have not included any  top-related variables, including b-jets, which may be important in the binned analysis of mℓℓ  as top backgrounds remain significant up to a higher energy tail (see right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  The distributions for part of the BDT variables, given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' (21), after imposing pT and  η cuts on jets and leptons are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 21 where we also added one selected EFT  benchmark point for λz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='04 as an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  34  0  200 400 600 800 1000 1200 1400 1600 1800 2000  (GeV)  jj  m  4  −  10  3  −  10  2  −  10  1  −  10  Normalized unit/50 GeV  Z+jets  +jets  tt  EW Zjj  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='04  z  λ  0  100  200  300  400  500  600  700  800  (jj) (GeV)  T  p  2  −  10  1  −  10  Normalized unit/20 GeV  Z+jets  +jets  tt  EW Zjj  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='04  z  λ  0  1  2  3  4  5  6  7  8  9  (jj)|  η|  6  −  10  5  −  10  4  −  10  3  −  10  2  −  10  1  −  10  Normalized unit/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='3  Z+jets  +jets  tt  EW Zjj  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='04  z  λ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='5  3  z*  2  −  10  1  −  10  Normalized unit/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='2  Z+jets  +jets  tt  EW Zjj  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='04  z  λ  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='9  1  )  hard  T  R(p  5  −  10  4  −  10  3  −  10  2  −  10  1  −  10  Normalized unit/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='04  Z+jets  +jets  tt  EW Zjj  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='04  z  λ  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' 21: The normalized distribution of BDT variables for the EFT signal for λz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='04 and  backgrounds after imposing pT (j1) > 50 GeV, pT (j2) > 30 GeV, pT (ℓ1) > 30 GeV, pT (ℓ2) > 20  GeV, |η(j)| < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='5 , |η(ℓ)| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Recall that EW Zjj and EFT samples were generated with  mjj > 120 GeV at the generation level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  35  [1] ATLAS Collaboration, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Aad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=', Observation of a new particle in the search for the  Standard Model Higgs boson with the ATLAS detector at the LHC, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' B 716 (2012)  1–29, [arXiv:1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='7214].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content='  [2] CMS Collaboration, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Chatrchyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=', Observation of a New Boson at a Mass of 125  GeV with the CMS Experiment at the LHC, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFRT4oBgHgl3EQf3Di-/content/2301.13663v1.pdf'}
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+arXiv:2301.05354v1  [math.PR]  13 Jan 2023
+Maximum Likelihood Estimation for Maximal Distribution
+under Sublinear Expectation ∗
+Xinpeng Li† Yue Liu
+Jiaquan Lu
+Research Center for Mathematics and Interdisciplinary Sciences
+Shandong University, 266237, Qingdao, China
+Abstract
+Maximum likelihood estimation is a common method of estimating the parameters of the prob-
+ability distribution from a given sample. This paper aims to introduce the maximum likelihood
+estimation in the framework of sublinear expectation. We find the maximum likelihood estimator
+for the parameters of the maximal distribution via the solution of the associated minimax prob-
+lem, which coincides with the optimal unbiased estimation given by Jin and Peng [8]. A general
+estimation method for samples with dependent structure is also provided. This result provides a
+theoretical foundation for the estimator of upper and lower variances, which is widely used in the
+G-VaR prediction model in finance.
+Keywords: Law of large numbers; Maximal distribution; Maximum likelihood estimation;
+Sublinear expectation
+1
+Introduction
+The sublinear expectation theory established by Peng [14] is a powerful tool to deal with prob-
+lems involving model uncertainties in many fields, especially to solve dynamic problems with
+uncertainty in finance (see, for example, Epstein and Ji [3]), in which the number of underlying
+probability measures may be infinite.
+One typical distribution in sublinear expectation theory is the maximal distribution.
+It is
+usually used to characterize the worst case risk in finance, especially the uncertainty of returns
+of financial assets (see Li et al. [10] and Pei et al. [12]). The primary advantage of maximally
+distributed random variables for modelling purposes in applications is the simplicity of its calcu-
+lation. For example, considering one-dimensional case, the distribution of maximally distributed
+random variable X under the sublinear expectation ˆE can be determined by two parameters µ
+and µ, i.e.,
+ˆE[ϕ(X)] = max
+µ≤µ≤µ ϕ(µ),
+∀ϕ ∈ Cb(R).
+It describes many real phenomena due to the law of large numbers with uncertainty, which is
+initialled by Peng [14] (see Theorem 2.10).
+A fundamental problem is how to choose suitable estimators of upper mean µ = ˆE[X] and
+lower mean µ = −ˆE[−X] for the maximally distributed random variable X? Recently, Jin and
+∗This work was supported by NSF of Shandong Provence (No.ZR2021MA018), NSF of China (No.11601281), National
+Key R&D Program of China (No.2018YFA0703900) and the Young Scholars Program of Shandong University.
+†Corresponding author. Email: lixinpeng@sdu.edu.cn
+1
+
+Peng [8] finds that the largest unbiased estimator for µ and the smallest unbiased estimator for µ
+based on the independent maximally distributed samples {Xi}n
+i=1 can be calculated respectively
+by
+ˆµ = max{X1, · · · , Xn},
+ˆµ = min{X1, · · · , Xn}.
+(1)
+Based on these estimators, Peng et al. [16] and Peng and Yang [15] do extensive experiments
+on both the NASDAQ Composite Index and S&P 500 Index and demonstrate the excellent per-
+formance of the G-VaR predictor, which is a non-trivial generalization of classical normal VaR
+model.
+This paper provides a new perspective of these estimators based on the principle of maximum
+likelihood estimation (MLE) in the classical statistics theory (see, for example, Lehmann and
+Casella [9]). We propose a minimax problem in accordance with the essence of classical MLE. We
+maximize the “probability” of the samples with the smallest uncertainty, in which the additional
+minimum problem aims to reduce the uncertainty in the model. We find that our MLE for µ and µ
+coincides with Jin and Peng’s optimal unbiased estimation (1). In addition, our estimators are also
+valid for the dependent structure, and can be applied to approximate the samples unnecessarily
+maximally distributed. This new result provides the theoretical foundation for the estimator of
+upper and lower variances which is widely used in the G-VaR predictor model.
+The remainder of this paper is organized as follows: in Section 2, we present some basic
+notions and results of sublinear expectation theory and the properties of maximal distribution.
+The detailed MLE of parameters for maximal distribution is provided in Section 3. In Section 4,
+we study the general estimator for the non-maximally distributed samples.
+2
+Preliminaries of Sublinear Expectation Theory
+Let Ω be a Polish space and H be a linear space of real functions defined on Ω such that if
+X1, · · · , Xn ∈ H for each n ∈ N, then ϕ(X1, · · · , Xn) ∈ H, ∀ϕ ∈ CLip(Rn), where CLip(Rn) is the
+space of all Lipschitz functions on Rn.
+Definition 2.1 A sublinear expectation ˆE on H is a functional ˆE : H → R satisfying the following
+conditions: ∀X, Y ∈ H, we have
+(1) Monotonicity: if X ≥ Y , then ˆE[X] ≥ ˆE[Y ];
+(2) Constant preserving: ˆE[c] = c, ∀c ∈ R;
+(3) Sub-additivity: ˆE[X + Y ] ≤ ˆE[X] + ˆE[Y ];
+(4) Positive homogeneity: ˆE[λX] = λˆE[X], ∀λ ≥ 0.
+The triple (Ω, H, ˆE) is called the sublinear expectation space, which is analogous to the prob-
+ability space (Ω, F, P).
+One typical example of sublinear expectation is the upper expectation represented by
+ˆE[X] = sup
+P ∈P
+EP [X],
+∀X ∈ H,
+(2)
+where P is some set of probability measures on (Ω, B(Ω)) and EP is the linear expectation in-
+troduced by P. The size of P is used to characterize the uncertainty of model. In this case, the
+corresponding capacity introduced by P can be defined as
+V (A) := sup
+P ∈P
+P(A),
+∀A ∈ B(Ω).
+The notions of identical distribution and independence are important in the classical proba-
+bility theory and can also be non-trivially generalized to the framework of sublinear expectation
+theory in Peng [13, 14].
+2
+
+Definition 2.2 Given an n-dimensional random vector X = (X1, · · · , Xn) on a sublinear expec-
+tation space (Ω, H, ˆE), where Xi ∈ H, 1 ≤ i ≤ n, we define a functional on CLip(Rn) by
+ˆFX[ϕ] := ˆE[ϕ(X)],
+∀ ϕ ∈ CLip(Rn).
+We call ˆFX[ϕ] the sublinear distribution of X under ˆE.
+It is easy to see that (Rn, CLip(Rn), FX) forms a sublinear expectation space.
+Remark 2.3 Given an integrable random variable X on the classical probability space (Ω, F, P),
+we recall that the distribution function of X is defined by
+FX(x) = P(X ≤ x),
+∀x ∈ R.
+For each ϕ ∈ CLip(R), we can easily calculate
+ˆFX[ϕ] = EP [ϕ(X)] =
+ˆ
+R
+ϕ(x)dFX(x).
+Conversely, if we know the value of ˆFX[ϕ] for every ϕ ∈ CLip(R), then for each x ∈ R, there exists
+a sequence of bounded and Lipschitz functions
+ϕn(y) =
+1
+1 + n(y − x)+
+such that
+ϕn(y) ↓ 1(−∞,x](y),
+∀y ∈ R.
+Then we obtain ˆFX[ϕn(X)] ↓ FX(x). Thus the distribution function FX is determined by ˆFX
+in the linear case. But for the sublinear case, in particular, the sublinear expectation ˆE admits
+representation (2), we emphasize that the following capacity
+V (X ≤ x) := sup
+P ∈P
+P(X ≤ x),
+∀x ∈ R,
+can not always determine the value of ˆE[ϕ(X)]. So we directly define ˆE[ϕ(X)] for each ϕ ∈ CLip(R)
+as the distribution of X.
+Definition 2.4 Let X and Y be two n-dimensional random vectors defined on sublinear expecta-
+tion spaces (Ω, H, ˆE). They are called identically distributed if
+ˆE[ϕ(X)] = ˆE[ϕ(Y )],
+∀ ϕ ∈ CLip(Rn),
+denoted by X
+d= Y .
+The following notion of independence provides a simple model of joint distribution ˆE[ϕ(X, Y )]
+provided the marginal distributions.
+Definition 2.5 Let (Ω, H, ˆE) be a sublinear expectation space, an n-dimensional random vec-
+tor Y is said to be independent of another m-dimensional random vector X under the sublinear
+expectation ˆE, if ∀ ϕ ∈ CLip(Rm+n),
+ˆE[ϕ(X, Y )] = ˆE[ˆE[ϕ(x, Y )]x=X].
+(3)
+Moreover, the sequence of random variables {Xi}∞
+i=1 is said to be independent, if for each i ≥ 1,
+Xi+1 is independent of (X1, · · · , Xi).
+3
+
+Remark 2.6 In order to explain the equation (3), for simplicity, we only consider two random
+variables X and Y , which are defined on the probability space (Ω, F, P) with the joint distribution
+function F(x, y) = P(X ≤ x, Y ≤ y). If they are independent, then F(x, y) = FX(x)FY (y) for all
+x, y ∈ R, where FX and FY are distribution functions of X and Y respectively, we further have
+EP[ϕ(X, Y )] =
+ˆ
+R2 ϕ(x, y)dF(x, y)
+=
+ˆ
+R2 ϕ(x, y)dFX(x)dFY (y)
+=
+ˆ
+R
+dFX(x)
+ˆ
+R
+ϕ(x, y)dFY (y)
+=
+ˆ
+R
+[EP [ϕ(x, Y )]dFX(x)
+= EP [EP[ϕ(x, Y )]|x=X].
+Thus Definition 2.5 is the natural generalization of classical notion of independence. By Fubini’s
+theorem, we obtain
+EP [ϕ(X, Y )] = EP[EP [ϕ(x, Y )]|x=X] = EP [EP [ϕ(X, y)]|y=Y ].
+But it does not hold for sublinear expectation in general, the notion of independence under sublinear
+expectation is usually not symmetric, i.e., Y being independent of X can not automatically imply
+that X is independent of Y . An interesting example can be found in Example 1.3.15 of Peng [14].
+More properties of such independence under sublinear expectation and its relations with classical
+conditional expectations is referred to Guo et al. [6].
+Remark 2.7 We note that f(x) := ˆE[ϕ(x, Y )] may be not continuous (resp. measurable) even if
+ϕ(x, y) is continuous (resp. measurable). Thus the joint distribution ˆE[ϕ(X, Y )] := ˆE[f(X)] is not
+well-defined by the marginal distributions, since ˆE is defined on the domain of continuous (resp.
+measurable) functions. So we consider the Lipschitz functions in the sublinear expectation theory.
+Remark 2.8 By (3), it is obvious that for independent random variables {Xi}n
+i=1 and bounded
+Lipschitz functions ϕi ≥ 0, 1 ≤ i ≤ n, we have
+ˆE[Πn
+i=1ϕi(Xi)] = Πn
+i=1ˆE[ϕi(Xi)],
+(4)
+which is equivalent to the classical independence when ˆE is the linear expectation.
+We also note that (4) is weaker than (3). Moreover, (4) is an important property to obtain
+likelihood function in Section 3.
+Now we introduce the notion of maximal distribution, one of the fundamental sublinear dis-
+tributions in the sublinear expectation theory.
+Definition 2.9 Let (Ω, H, ˆE) be a sublinear expectation space, an n-dimensional random vector
+X is said to be maximally distributed if there exists a bounded, closed and convex subset Λ ⊂ Rn
+such that
+ˆE[ϕ(X)] = max
+x∈Λ ϕ(x),
+∀ ϕ ∈ CLip(Rn).
+For simplicity, we only consider one-dimensional case in this paper. More details about the
+maximal distribution, especially, the related maximally distributed random fields, can be found
+in Li and Peng [11].
+The sublinear distribution of one-dimensional maximally distributed random variable X is
+defined simply as
+ˆFX[ϕ] = ˆE[ϕ(X)] = max
+µ≤x≤µ ϕ(x), ∀ ϕ ∈ CLip(Rn),
+(5)
+4
+
+where µ := ˆE[X] and µ := −ˆE[−X], denoted maximally distributed random variable X by
+X
+d= M[µ,µ]. The interval [µ, µ] describes the uncertainty of the sublinear distribution of X. Since
+such interval is bounded, (5) still holds for all continuous function ϕ.
+The following law of large numbers in Peng [14] plays an important role in the sublinear
+expectation theory.
+Theorem 2.10 (Law of large numbers) Let {Xi}∞
+i=1 be an independent and identically dis-
+tributed (i.i.d.) sequence of random variables defined on (Ω, H, ˆE) and we further assume that X1
+is uniformly integrable under ˆE, i.e.,
+lim
+λ→∞
+ˆE[(|X1| − λ)+] = 0.
+(6)
+Then for all ϕ ∈ CLip(R), we have
+lim
+n→∞
+ˆE
+�
+ϕ
+�
+1
+n
+n
+�
+i=1
+Xi
+��
+= max
+µ∈[µ,µ] ϕ(µ).
+(7)
+Remark 2.11 Recently, Fang et al. [4], Song [17] and Hu et al. [7] obtained the convergence
+rate of (7) under higher moment conditions. If we further assume that ˆE[X2
+1] < ∞ in Theorem
+2.10, then we have
+����ˆE
+�
+d2
+[µ,µ]
+��n
+i=1 Xi
+n
+������ ≤
+ˆE[X2
+1]
+n
+,
+(8)
+where d[µ,µ](x) = infy∈[µ,µ] |x − y|.
+In addition, Chen [1] and Zhang [18] established the corresponding strong law of large numbers.
+Proposition 2.12 Let X be a random variable defined on sublinear expectation spaces (Ω, H, ˆE),
+we further assume that X is uniformly integrable under ˆE. Then X is maximally distributed if
+and only if
+aX + b ¯
+X
+d=(a + b)X,
+∀a, b ≥ 0.
+(9)
+where ¯
+X is an independent copy of X, i.e., ¯
+X is independent of X and ¯
+X
+d= X.
+Proof. Let X
+d= M[µ,µ] and Λ = [µ, µ], then we have, ∀ϕ ∈ CLip(R),
+ˆE[ϕ(aX + b ¯
+X)] = ˆE[ˆE[ϕ(ax + b ¯
+X)]x=X]
+= max
+x∈Λ max
+¯x∈Λ ϕ(ax + b¯x)
+= max
+µ∈Λ ϕ[(a + b)µ]
+= ˆE[ϕ((a + b)X)],
+thus (9) holds.
+Conversely, we construct an i.i.d. sequences {Xi}∞
+i=1 with X1
+d= X and define
+ηn := 1
+2n (X1 + X2 + · · · + X2n),
+∀n ∈ N
+In particular, taking a = b = 1 in (9), we have
+X1 + X2
+d= 2X1,
+· · ·
+X2n−1 + X2n d= 2X2n−1.
+5
+
+By induction, we obtain, for each n ∈ N,
+ηn
+d= 1
+2n (2X1 + 2X3 + · · · + X2n−1)
+d=
+1
+2n−1 (X1 + X3 + · · · + X2n−1)
+d= · · ·
+d= X1
+d= X.
+By Theorem 2.10, we have
+lim
+n→∞
+ˆE[ϕ(ηn)] = max
+µ∈Λ ϕ(µ),
+which implies that
+ˆE[ϕ(X)] = max
+µ∈Λ ϕ(µ).
+Hence X is maximally distributed.
+Corollary 2.13 An uniformly integrable random variable X on (Ω, H, ˆE) is maximally distributed
+if and only if
+X + ¯
+X
+d= 2X,
+where ¯
+X is the independent copy of X.
+Remark 2.14 A counterexample in Guo and Li [5] shows that the law of large numbers fails
+without the uniformly integrable condition (6). It is still open that whether Proposition 2.12 still
+holds without such integrable condition.
+3
+Maximum Likelihood Estimation for Independent Sam-
+ples
+The idea of MLE is to find proper parameters to maximize the probability P(X1 = x1, · · · , Xn =
+xn) of realized samples {xi}n
+i=1 of population X which has prescribed probability measure P.
+Analogously, for the maximal distribution, we hope to maximize the following capacity and call
+it the likelihood function,
+V (x1, · · · , xn) := ˆE[I{X1=x1,··· ,Xn=xn}].
+It is worth pointing out that I{X1=x1,··· ,Xn=xn} /∈ H since indicator function is not continuous.
+But it can be well-defined by the fact that such indicator function can be approximated by the
+Lipschitz functions (see Theorem 3.3).
+Moreover, if ˆE can be represent as
+ˆE[·] = sup
+P ∈P
+EP[·],
+then it is natural to define
+V (x1, · · · , xn) := sup
+P ∈P
+P(X1 = x1, · · · , Xn = xn).
+In particular, for the maximally distributed population X with parameters µ and µ, we denote
+the corresponding likelihood function as
+V (x1, · · · , xn; µ, µ).
+Obviously, the value of likelihood function is increasing when the interval [µ, µ] is enlarging.
+6
+
+Let ∆ = µ − µ be the degree of uncertainty of maximal distribution, and we also hope to
+deduce the uncertainty when we maximize the likelihood function, thus the MLE of parameters
+µ and µ is to solve the following minimax problem:
+min
+∆ max
+µ,µ V (x1, · · · , xn; µ, µ).
+(10)
+In order to solve such minimax problem, we firstly establish a representation theorem for
+maximal distribution by the Dirac measures, where the Dirac measure on a point µ ∈ R is
+denoted by δµ satisfying
+δµ(A) =
+�
+1,
+µ ∈ A,
+0,
+µ /∈ A,
+∀A ∈ B(R).
+Theorem 3.1 Let (Ω, H, ˆE) be a sublinear expectation space and X
+d= M[µ, µ], then for each
+ϕ ∈ CLip(R),
+ˆE[ϕ(X)] = max
+µ≤x≤µ ϕ(x)
+= max
+µ≤µ≤µ
+ˆ µ
+µ
+ϕ(y)δµ(dy).
+Proof. On one hand, it is clear that
+max
+µ≤µ≤µ
+ˆ µ
+µ
+ϕ(y)δµ(dy) ≤ max
+µ≤µ≤µ max
+µ≤y≤µ ϕ(y)
+ˆ µ
+µ
+δµ(dy)
+= max
+µ≤y≤µ ϕ(y).
+On the other hand, there exists µ∗ ∈ [µ, µ] such that ϕ(µ∗) = max
+µ≤x≤µ ϕ(x). Then we have
+maxµ≤µ≤µ
+ˆ µ
+µ
+ϕ(y)δµ(dy) ≥
+ˆ µ
+µ
+ϕ(y)δµ∗(dy)
+= ϕ(µ∗) = max
+µ≤x≤µ ϕ(x).
+Remark 3.2 Let {δµn}∞
+n=1 be a sequence of Dirac measures with µn ∈ [µ, µ]. Then there exists
+a point µ∗ ∈ [µ, µ] and subsequence {δµni }∞
+i=1 such that δµni weakly converges to δµ∗ provided
+µni → µ∗. Thus the set of Dirac measures P = {δµ, µ ∈ [µ, µ]} is weakly compact.
+Secondly, by the independence of samples, we can simply calculate the likelihood function.
+Theorem 3.3 Let {Xi}n
+i=1 be a sequence of independent random variables on sublinear expecta-
+tion space (Ω, H, ˆE). We further assume that ˆE can be represented by
+ˆE[X] = max
+P ∈P EP[X],
+∀X ∈ H,
+where P is a weakly compact set of probability measures on (Ω, B(Ω)).
+Then the terms ˆE[I{X1=x1,··· ,Xn=xn}] and ˆE[I{Xi=xi}], 1 ≤ i ≤ n are well-defined.
+Furthermore, ∀xi ∈ R, 1 ≤ i ≤ n.
+ˆE[I{X1=x1,··· ,Xn=xn}] = Πn
+i=1ˆE[I{Xi=xi}].
+(11)
+7
+
+Proof.
+The indicator function Ix∗(·), x∗ ∈ R is not continuous, thus I{Xi=xi} /∈ H.
+But it
+can be approximated by the Lipschitz functions, thus ˆE[I{Xi=xi}] can be well-defined, so does
+ˆE[I{X1=x1,··· ,Xn=xn}].
+Indeed, for fixed x∗ ∈ R, consider the function
+ϕx∗
+k (x) =
+1
+1 + k|x − x∗|.
+It is easily seen that ϕx∗
+k (·) is a non-negative bounded continuous function and ϕx∗
+k (·) ↓ Ix∗(·).
+For each k ∈ N, we also have
+|ϕx∗
+k (x1) − ϕx∗
+k (x2)| =
+����
+1 + k|x2 − x∗| − 1 − k|x1 − x∗|
+(1 + k|x1 − x∗|)(1 + k|x2 − x∗|)
+����
+≤
+k|x2 − x1|
+(1 + k|x1 − x∗|)(1 + k|x2 − x∗|)
+≤ k|x2 − x1|.
+Thus ϕx∗
+k
+is a Lipschitz function.
+Then by Theorem 31 in Denis et al. [2], we obtain
+ˆE[ϕx∗
+k (Xi)] ↓ ˆE[I{Xi=x∗}].
+Similarly, the term ˆE[I{X1=x1,··· ,Xn=xn}] is well-defined for each n ∈ N, which can be approximated
+as
+ˆE[Πn
+i=1ϕxi
+k (Xi)] ↓ ˆE[I{X1=x1,··· ,Xn=xn}].
+By the independence of {Xi}n
+i=1, we obtain, ∀xi ∈ R, 1 ≤ i ≤ n,
+ˆE[I{X1=x1,··· ,Xn=xn}] = lim
+k→∞
+ˆE[Πn
+i=1ϕxi
+k (Xi)]
+= lim
+k→∞ Πn
+i=1ˆE[ϕxi
+k (Xi)]
+= Πn
+i=1 lim
+k→∞
+ˆE[ϕxi
+k (Xi)]
+= Πn
+i=1ˆE[I{Xi=xi}].
+By Theorem 3.3 and 3.1, we can see that if X
+d= M[µ, µ], then
+ˆE[I{X=x∗}] = max
+µ≤µ≤µ δµ(x∗).
+Now we can solve the minimax problem (10).
+Theorem 3.4 Let {Xi}n
+i=1 be an i.i.d. sequence with X1
+d= M[µ,µ]. Then the MLE of parameters
+µ and µ is given by
+ˆµ = min{X1, · · · , Xn};
+ˆµ = max{X1, · · · , Xn}.
+(12)
+Proof. Let {xi}n
+i=1 be the samples of population {Xi}n
+i=1.
+By Theorem 3.1 and 3.3, we have
+V (x1, · · · , xn; µ, µ) = ˆE[I{X1=x1,··· ,Xn=xn}]
+= Πn
+i=1ˆE[I{Xi=xi}]
+= Πn
+i=1 max
+µ≤µ≤µ δµ(xi).
+8
+
+It is clear that
+V (x1, · · · , xn; µ, µ) =
+�
+1,
+µ ≤ x ≤ x ≤ µ,
+0,
+otherwise,
+(13)
+where x = min{x1, · · · , xn} and x = max{x1, · · · , xn}.
+Thus the solution of minimax problem (10) is given by
+ˆµ = min{X1, · · · , Xn},
+ˆµ = max{X1, · · · , Xn}.
+Remark 3.5 The MLE in (12) is the same as the largest unbiased estimator for µ and smallest
+unbiased estimator for µ obtained in Jin and Peng [8]. In which, a statistic Tn = fn(X1, · · · , Xn)
+is called an unbiased estimator of parameter µ if ˆE[fn(X1, · · · , Xn)] = µ, where fn is continuous
+on Rn.
+4
+General Maximum Likelihood Estimator
+In this section, we firstly prove that Theorem 3.4 still holds without the assumption of inde-
+pendence, which indicates that our results can also be applied to the samples with dependent
+structure.
+Theorem 4.1 Let {Xi}n
+i=1 be identically distributed sequence with X1
+d= M[µ,µ]. Then the MLE
+of parameters µ and µ is given by
+ˆµ = min{X1, · · · , Xn};
+ˆµ = max{X1, · · · , Xn}.
+(14)
+Proof. Let {xi}n
+i=1 be the sample of population {Xi}n
+i=1. Obviously, ∀ 0 ≤ ϕi ∈ CLip(R), 1 ≤
+i ≤ n, we have
+ˆE[Πn
+i=1ϕi(Xi)] ≤ Πn
+i=1
+max
+µ≤µi≤µ ϕi(µi).
+By the similar argument in the previous proofs, we obtain
+ˆE[Πn
+i=1I{Xi=xi}] ≤ Πn
+i=1ˆE[I{Xi=xi}].
+Then we have
+¯V (x1, · · · , xn; µ, µ) : = ˆE[I{X1=x1,··· ,Xn=xn}]
+= ˆE[Πn
+i=1I{Xi=xi}]
+≤ Πn
+i=1ˆE[I{Xi=xi}]
+= V (x1, · · · , xn; µ, µ),
+where V (x1, · · · , xn; µ, µ) is defined as in (13).
+Since the sample {xi}n
+i=1 is realized, we have
+0 < ¯V (x1, · · · , xn; µ, µ) ≤ 1,
+thus
+V (x1, · · · , xn; µ, µ) = 1,
+which implies that
+ˆµ = min{x1, · · · , xn},
+ˆµ = max{x1, · · · , xn}.
+9
+
+In many practical situations, we often have i.i.d. condition for samples {Xi}n
+i=1 of population
+X which is not maximally distributed, and we hope to estimate the value of ˆE[ϕ(X)] for some ϕ
+based on the i.i.d. samples {Xi}n
+i=1. Thanks to the law of large numbers, our results can also be
+applied to approximate the non-maximally distributed population.
+One typical application is to estimate the upper and lower variance of population Z which has
+model uncertainty based on the historical time series {Zt−i}i≥1 (see Li et al. [10], Peng et al. [16]
+and Peng and Yang [15]).
+We assume that Z has mean-zero, i.e.,
+ˆE[Z] = ˆE[−Z] = 0.
+The upper and lower variance of Z, defined as σ2 = ˆE[Z2] and σ2 = −ˆE[−Z2], can be estimated
+as follows:
+Taking L ∈ N to be large enough, and we calculate the local sample variance with window L
+by
+σ2
+j =
+�L
+i=1(Zt−L+i−j − µj)2
+L − 1
+, 1 ≤ j ≤ K,
+where µj =
+�L
+i=1 Zt−L+i−j
+L
+is the sample mean under the same window L and the number K ∈ N
+is fixed with prior knowledge. The value of K can be used to characterize the uncertainty of
+model. The larger K means that we prefer more uncertainty in the model.
+We note that the term µj is closed to 0 when L is large enough, since ˆE[Z] = ˆE[−Z] = 0. Thus
+we have
+σ2
+j ≈
+�L
+i=1 Z2
+t−L+i−j
+L − 1
+,
+1 ≤ j ≤ K.
+For each j ∈ {1, · · · , K}, by the law of large numbers (Theorem 2.10), σ2
+j can be regarded as the
+maximally distributed random variable on [σ2, σ2] when L is large enough. In fact, the error term
+of such approximation can be estimated by (8).
+By Theorem 4.1, we obtain the MLE of σ2 and σ2 by
+ˆσ
+2 = max
+1≤j≤K σ2
+j ;
+ˆσ2 =
+min
+1≤j≤K σ2
+j .
+Based on such estimations of upper and lower variance, extensive experiments on both NASDAQ
+Composite Index and S&P 500 Index demonstrate the excellent performances of G-VaR model, a
+new benchmark predictor for value-at-risk based on the so-called G-normal distribution, which is
+superior to most existing benchmark VaR predictors (see [16] and [15]).
+Remark 4.2 Theorem 3.4 or Jin and Peng’s optimal unbiased estimator can not be applied to
+this situation since the sequence {σ2
+j }K
+j=1 is not independent. In fact, it is L-dependent in sublinear
+case. Our results provide the theoretical foundation for the widely used estimator of upper and
+lower variances in finance.
+References
+[1] Chen, Z. (2016), Strong laws of large numbers for sub-linear expectations. Science in China-
+Mathematics, 59(5): 945–954.
+[2] Denis, L., Hu, M. and Peng, S. (2011) Function spaces and capacity related to a sublinear
+expectation: application to G-Brownian motion paths. Potential analysis, 34(2): 139–161.
+[3] Epstein, L. and Ji, S. (2013) Ambiguous volatility and asset pricing in continuous time.
+Review of Financial Studies 26(7): 1740C-1786.
+10
+
+[4] Fang, X., Peng, S., Shao, Q. and Yong, S. (2019) Limit theorems with rate of convergence
+under sublinear expectations. Bernoulli, 25(4A): 2564–2596.
+[5] Guo, X. and Li, X. (2021) On the laws of large numbers for pseudo-independent random
+variables under sublinear expectation. Statistics and Probability Letters, 172: No.109042.
+[6] Guo, X., Li, S. and Li, X. (2023) Notes on Peng’s independence in sublinear expectation
+theory. Statistics and Probability Letters, 193: No.109719.
+[7] Hu, M., Li, X. and Li, X. (2021) Convergence rate of Peng’s law of large numbers under
+sublinear expectations. Probability, Uncertainty and Quantitative Risk, 6(3): 261–266.
+[8] Jin, H. and Peng, S. (2021) Optimal unbiased estimation for maximal distribution. Proba-
+bility, Uncertainty and Quantitative Risk, 6(3): 189–198.
+[9] Lehmann, E. L. and Casella, G. (1998) Theory of point estimation. Second edition. Springer
+Texts in Statistics. Springer-Verlag, New York.
+[10] Li, S., Li, X. and Yang, X. (2022) Upper and lower variances under model uncertainty
+and their applications in finance. International Journal of Financial Engineering 9(1): No.
+2250007.
+[11] Li, X. and Peng, S. (2022) Maximally distributed random fields under sublinear expectation.
+In Stochastic Analysis, Filtering, and Stochastic Optimization, 339–356, Springer.
+[12] Pei, Z., Wang, X., Xu, Y. and Yue, X. (2021) A worst-case risk measure by G-VaR. Acta
+Mathematicae Applicatae Sinica, English Series, 37(2): 421–440.
+[13] Peng, S. (2019) Law of large numbers and central limit theorem under nonlinear expectations.
+Probability, Uncertainty and Quantitative Risk, 4, No. 4.
+[14] Peng, S. (2019) Nonlinear expectations and stochastic calculus under uncertainty: with robust
+CLT and G-Brownian motion. Springer.
+[15] Peng, S. and Yang, S. (2022) Distributional uncertainty of the financial time series measured
+by G-expectation. Theory of Probability and its Applications, 66(4): 729–741.
+[16] Peng, S., Yang, S. and Yao, J. (2020) Improving Value-at-Risk prediction under model un-
+certainty. Journal of Financial Econometrics, nbaa022(online).
+[17] Song, Y. (2021) Stein’s method for the law of large numbers under sublinear expectations.
+Probability, Uncertainty and Quantitative Risk, 6(3):199–212.
+[18] Zhang, L. (2021) The sufficient and necessary conditions of the strong law of large numbers
+under the sub-linear expectations, arXiv:2104.08471.
+11
+
diff --git a/NdE4T4oBgHgl3EQf9Q6K/content/tmp_files/load_file.txt b/NdE4T4oBgHgl3EQf9Q6K/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1116897e6878db3cbd96fcdf40e55e48df924914
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@@ -0,0 +1,338 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf,len=337
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='05354v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='PR]  13 Jan 2023  Maximum Likelihood Estimation for Maximal Distribution  under Sublinear Expectation ∗  Xinpeng Li† Yue Liu  Jiaquan Lu  Research Center for Mathematics and Interdisciplinary Sciences  Shandong University, 266237, Qingdao, China  Abstract  Maximum likelihood estimation is a common method of estimating the parameters of the prob-  ability distribution from a given sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' This paper aims to introduce the maximum likelihood  estimation in the framework of sublinear expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' We find the maximum likelihood estimator  for the parameters of the maximal distribution via the solution of the associated minimax prob-  lem, which coincides with the optimal unbiased estimation given by Jin and Peng [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' A general  estimation method for samples with dependent structure is also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' This result provides a  theoretical foundation for the estimator of upper and lower variances, which is widely used in the  G-VaR prediction model in finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Keywords: Law of large numbers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Maximal distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Maximum likelihood estimation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Sublinear expectation  1  Introduction  The sublinear expectation theory established by Peng [14] is a powerful tool to deal with prob-  lems involving model uncertainties in many fields, especially to solve dynamic problems with  uncertainty in finance (see, for example, Epstein and Ji [3]), in which the number of underlying  probability measures may be infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  One typical distribution in sublinear expectation theory is the maximal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  It is  usually used to characterize the worst case risk in finance, especially the uncertainty of returns  of financial assets (see Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' [10] and Pei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' The primary advantage of maximally  distributed random variables for modelling purposes in applications is the simplicity of its calcu-  lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' For example, considering one-dimensional case, the distribution of maximally distributed  random variable X under the sublinear expectation ˆE can be determined by two parameters µ  and µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=',  ˆE[ϕ(X)] = max  µ≤µ≤µ ϕ(µ),  ∀ϕ ∈ Cb(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  It describes many real phenomena due to the law of large numbers with uncertainty, which is  initialled by Peng [14] (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  A fundamental problem is how to choose suitable estimators of upper mean µ = ˆE[X] and  lower mean µ = −ˆE[−X] for the maximally distributed random variable X?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Recently, Jin and  ∗This work was supported by NSF of Shandong Provence (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='ZR2021MA018), NSF of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='11601281), National  Key R&D Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='2018YFA0703900) and the Young Scholars Program of Shandong University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  †Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Email: lixinpeng@sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='cn  1  Peng [8] finds that the largest unbiased estimator for µ and the smallest unbiased estimator for µ  based on the independent maximally distributed samples {Xi}n  i=1 can be calculated respectively  by  ˆµ = max{X1, · · · , Xn},  ˆµ = min{X1, · · · , Xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  (1)  Based on these estimators, Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' [16] and Peng and Yang [15] do extensive experiments  on both the NASDAQ Composite Index and S&P 500 Index and demonstrate the excellent per-  formance of the G-VaR predictor, which is a non-trivial generalization of classical normal VaR  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  This paper provides a new perspective of these estimators based on the principle of maximum  likelihood estimation (MLE) in the classical statistics theory (see, for example, Lehmann and  Casella [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' We propose a minimax problem in accordance with the essence of classical MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' We  maximize the “probability” of the samples with the smallest uncertainty, in which the additional  minimum problem aims to reduce the uncertainty in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' We find that our MLE for µ and µ  coincides with Jin and Peng’s optimal unbiased estimation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' In addition, our estimators are also  valid for the dependent structure, and can be applied to approximate the samples unnecessarily  maximally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' This new result provides the theoretical foundation for the estimator of  upper and lower variances which is widely used in the G-VaR predictor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  The remainder of this paper is organized as follows: in Section 2, we present some basic  notions and results of sublinear expectation theory and the properties of maximal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  The detailed MLE of parameters for maximal distribution is provided in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' In Section 4,  we study the general estimator for the non-maximally distributed samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  2  Preliminaries of Sublinear Expectation Theory  Let Ω be a Polish space and H be a linear space of real functions defined on Ω such that if  X1, · · · , Xn ∈ H for each n ∈ N, then ϕ(X1, · · · , Xn) ∈ H, ∀ϕ ∈ CLip(Rn), where CLip(Rn) is the  space of all Lipschitz functions on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='1 A sublinear expectation ˆE on H is a functional ˆE : H → R satisfying the following  conditions: ∀X, Y ∈ H, we have  (1) Monotonicity: if X ≥ Y , then ˆE[X] ≥ ˆE[Y ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  (2) Constant preserving: ˆE[c] = c, ∀c ∈ R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  (3) Sub-additivity: ˆE[X + Y ] ≤ ˆE[X] + ˆE[Y ];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  (4) Positive homogeneity: ˆE[λX] = λˆE[X], ∀λ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  The triple (Ω, H, ˆE) is called the sublinear expectation space, which is analogous to the prob-  ability space (Ω, F, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  One typical example of sublinear expectation is the upper expectation represented by  ˆE[X] = sup  P ∈P  EP [X],  ∀X ∈ H,  (2)  where P is some set of probability measures on (Ω, B(Ω)) and EP is the linear expectation in-  troduced by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' The size of P is used to characterize the uncertainty of model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' In this case, the  corresponding capacity introduced by P can be defined as  V (A) := sup  P ∈P  P(A),  ∀A ∈ B(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  The notions of identical distribution and independence are important in the classical proba-  bility theory and can also be non-trivially generalized to the framework of sublinear expectation  theory in Peng [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  2  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='2 Given an n-dimensional random vector X = (X1, · · · , Xn) on a sublinear expec-  tation space (Ω, H, ˆE), where Xi ∈ H, 1 ≤ i ≤ n, we define a functional on CLip(Rn) by  ˆFX[ϕ] := ˆE[ϕ(X)],  ∀ ϕ ∈ CLip(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  We call ˆFX[ϕ] the sublinear distribution of X under ˆE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  It is easy to see that (Rn, CLip(Rn), FX) forms a sublinear expectation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='3 Given an integrable random variable X on the classical probability space (Ω, F, P),  we recall that the distribution function of X is defined by  FX(x) = P(X ≤ x),  ∀x ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  For each ϕ ∈ CLip(R), we can easily calculate  ˆFX[ϕ] = EP [ϕ(X)] =  ˆ  R  ϕ(x)dFX(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Conversely, if we know the value of ˆFX[ϕ] for every ϕ ∈ CLip(R), then for each x ∈ R, there exists  a sequence of bounded and Lipschitz functions  ϕn(y) =  1  1 + n(y − x)+  such that  ϕn(y) ↓ 1(−∞,x](y),  ∀y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Then we obtain ˆFX[ϕn(X)] ↓ FX(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Thus the distribution function FX is determined by ˆFX  in the linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' But for the sublinear case, in particular, the sublinear expectation ˆE admits  representation (2), we emphasize that the following capacity  V (X ≤ x) := sup  P ∈P  P(X ≤ x),  ∀x ∈ R,  can not always determine the value of ˆE[ϕ(X)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' So we directly define ˆE[ϕ(X)] for each ϕ ∈ CLip(R)  as the distribution of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='4 Let X and Y be two n-dimensional random vectors defined on sublinear expecta-  tion spaces (Ω, H, ˆE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' They are called identically distributed if  ˆE[ϕ(X)] = ˆE[ϕ(Y )],  ∀ ϕ ∈ CLip(Rn),  denoted by X  d= Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  The following notion of independence provides a simple model of joint distribution ˆE[ϕ(X, Y )]  provided the marginal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='5 Let (Ω, H, ˆE) be a sublinear expectation space, an n-dimensional random vec-  tor Y is said to be independent of another m-dimensional random vector X under the sublinear  expectation ˆE, if ∀ ϕ ∈ CLip(Rm+n),  ˆE[ϕ(X, Y )] = ˆE[ˆE[ϕ(x, Y )]x=X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  (3)  Moreover, the sequence of random variables {Xi}∞  i=1 is said to be independent, if for each i ≥ 1,  Xi+1 is independent of (X1, · · · , Xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  3  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='6 In order to explain the equation (3), for simplicity, we only consider two random  variables X and Y , which are defined on the probability space (Ω, F, P) with the joint distribution  function F(x, y) = P(X ≤ x, Y ≤ y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' If they are independent, then F(x, y) = FX(x)FY (y) for all  x, y ∈ R, where FX and FY are distribution functions of X and Y respectively, we further have  EP[ϕ(X, Y )] =  ˆ  R2 ϕ(x, y)dF(x, y)  =  ˆ  R2 ϕ(x, y)dFX(x)dFY (y)  =  ˆ  R  dFX(x)  ˆ  R  ϕ(x, y)dFY (y)  =  ˆ  R  [EP [ϕ(x, Y )]dFX(x)  = EP [EP[ϕ(x, Y )]|x=X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Thus Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='5 is the natural generalization of classical notion of independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' By Fubini’s  theorem, we obtain  EP [ϕ(X, Y )] = EP[EP [ϕ(x, Y )]|x=X] = EP [EP [ϕ(X, y)]|y=Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  But it does not hold for sublinear expectation in general, the notion of independence under sublinear  expectation is usually not symmetric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=', Y being independent of X can not automatically imply  that X is independent of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' An interesting example can be found in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='15 of Peng [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  More properties of such independence under sublinear expectation and its relations with classical  conditional expectations is referred to Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='7 We note that f(x) := ˆE[ϕ(x, Y )] may be not continuous (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' measurable) even if  ϕ(x, y) is continuous (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' measurable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Thus the joint distribution ˆE[ϕ(X, Y )] := ˆE[f(X)] is not  well-defined by the marginal distributions, since ˆE is defined on the domain of continuous (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  measurable) functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' So we consider the Lipschitz functions in the sublinear expectation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='8 By (3), it is obvious that for independent random variables {Xi}n  i=1 and bounded  Lipschitz functions ϕi ≥ 0, 1 ≤ i ≤ n, we have  ˆE[Πn  i=1ϕi(Xi)] = Πn  i=1ˆE[ϕi(Xi)],  (4)  which is equivalent to the classical independence when ˆE is the linear expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  We also note that (4) is weaker than (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Moreover, (4) is an important property to obtain  likelihood function in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Now we introduce the notion of maximal distribution, one of the fundamental sublinear dis-  tributions in the sublinear expectation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='9 Let (Ω, H, ˆE) be a sublinear expectation space, an n-dimensional random vector  X is said to be maximally distributed if there exists a bounded, closed and convex subset Λ ⊂ Rn  such that  ˆE[ϕ(X)] = max  x∈Λ ϕ(x),  ∀ ϕ ∈ CLip(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  For simplicity, we only consider one-dimensional case in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' More details about the  maximal distribution, especially, the related maximally distributed random fields, can be found  in Li and Peng [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  The sublinear distribution of one-dimensional maximally distributed random variable X is  defined simply as  ˆFX[ϕ] = ˆE[ϕ(X)] = max  µ≤x≤µ ϕ(x), ∀ ϕ ∈ CLip(Rn),  (5)  4  where µ := ˆE[X] and µ := −ˆE[−X], denoted maximally distributed random variable X by  X  d= M[µ,µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' The interval [µ, µ] describes the uncertainty of the sublinear distribution of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Since  such interval is bounded, (5) still holds for all continuous function ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  The following law of large numbers in Peng [14] plays an important role in the sublinear  expectation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='10 (Law of large numbers) Let {Xi}∞  i=1 be an independent and identically dis-  tributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=') sequence of random variables defined on (Ω, H, ˆE) and we further assume that X1  is uniformly integrable under ˆE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=',  lim  λ→∞  ˆE[(|X1| − λ)+] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  (6)  Then for all ϕ ∈ CLip(R), we have  lim  n→∞  ˆE  �  ϕ  �  1  n  n  �  i=1  Xi  ��  = max  µ∈[µ,µ] ϕ(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  (7)  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='11 Recently, Fang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' [4], Song [17] and Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' [7] obtained the convergence  rate of (7) under higher moment conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' If we further assume that ˆE[X2  1] < ∞ in Theorem  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='10, then we have  ����ˆE  �  d2  [µ,µ]  ��n  i=1 Xi  n  ������ ≤  ˆE[X2  1]  n  ,  (8)  where d[µ,µ](x) = infy∈[µ,µ] |x − y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  In addition, Chen [1] and Zhang [18] established the corresponding strong law of large numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='12 Let X be a random variable defined on sublinear expectation spaces (Ω, H, ˆE),  we further assume that X is uniformly integrable under ˆE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Then X is maximally distributed if  and only if  aX + b ¯  X  d=(a + b)X,  ∀a, b ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  (9)  where ¯  X is an independent copy of X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=', ¯  X is independent of X and ¯  X  d= X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Let X  d= M[µ,µ] and Λ = [µ, µ], then we have, ∀ϕ ∈ CLip(R),  ˆE[ϕ(aX + b ¯  X)] = ˆE[ˆE[ϕ(ax + b ¯  X)]x=X]  = max  x∈Λ max  ¯x∈Λ ϕ(ax + b¯x)  = max  µ∈Λ ϕ[(a + b)µ]  = ˆE[ϕ((a + b)X)],  thus (9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Conversely, we construct an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' sequences {Xi}∞  i=1 with X1  d= X and define  ηn := 1  2n (X1 + X2 + · · · + X2n),  ∀n ∈ N  In particular, taking a = b = 1 in (9), we have  X1 + X2  d= 2X1,  · ·  X2n−1 + X2n d= 2X2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  5  By induction, we obtain, for each n ∈ N,  ηn  d= 1  2n (2X1 + 2X3 + · · · + X2n−1)  d=  1  2n−1 (X1 + X3 + · · · + X2n−1)  d= · · ·  d= X1  d= X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='10, we have  lim  n→∞  ˆE[ϕ(ηn)] = max  µ∈Λ ϕ(µ),  which implies that  ˆE[ϕ(X)] = max  µ∈Λ ϕ(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Hence X is maximally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='13 An uniformly integrable random variable X on (Ω, H, ˆE) is maximally distributed  if and only if  X + ¯  X  d= 2X,  where ¯  X is the independent copy of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='14 A counterexample in Guo and Li [5] shows that the law of large numbers fails  without the uniformly integrable condition (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' It is still open that whether Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='12 still  holds without such integrable condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  3  Maximum Likelihood Estimation for Independent Sam-  ples  The idea of MLE is to find proper parameters to maximize the probability P(X1 = x1, · · · , Xn =  xn) of realized samples {xi}n  i=1 of population X which has prescribed probability measure P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Analogously, for the maximal distribution, we hope to maximize the following capacity and call  it the likelihood function,  V (x1, · · · , xn) := ˆE[I{X1=x1,··· ,Xn=xn}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  It is worth pointing out that I{X1=x1,··· ,Xn=xn} /∈ H since indicator function is not continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  But it can be well-defined by the fact that such indicator function can be approximated by the  Lipschitz functions (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Moreover, if ˆE can be represent as  ˆE[·] = sup  P ∈P  EP[·],  then it is natural to define  V (x1, · · · , xn) := sup  P ∈P  P(X1 = x1, · · · , Xn = xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  In particular, for the maximally distributed population X with parameters µ and µ, we denote  the corresponding likelihood function as  V (x1, · · · , xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' µ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Obviously, the value of likelihood function is increasing when the interval [µ, µ] is enlarging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  6  Let ∆ = µ − µ be the degree of uncertainty of maximal distribution, and we also hope to  deduce the uncertainty when we maximize the likelihood function, thus the MLE of parameters  µ and µ is to solve the following minimax problem:  min  ∆ max  µ,µ V (x1, · · · , xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' µ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  (10)  In order to solve such minimax problem, we firstly establish a representation theorem for  maximal distribution by the Dirac measures, where the Dirac measure on a point µ ∈ R is  denoted by δµ satisfying  δµ(A) =  �  1,  µ ∈ A,  0,  µ /∈ A,  ∀A ∈ B(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='1 Let (Ω, H, ˆE) be a sublinear expectation space and X  d= M[µ, µ], then for each  ϕ ∈ CLip(R),  ˆE[ϕ(X)] = max  µ≤x≤µ ϕ(x)  = max  µ≤µ≤µ  ˆ µ  µ  ϕ(y)δµ(dy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' On one hand, it is clear that  max  µ≤µ≤µ  ˆ µ  µ  ϕ(y)δµ(dy) ≤ max  µ≤µ≤µ max  µ≤y≤µ ϕ(y)  ˆ µ  µ  δµ(dy)  = max  µ≤y≤µ ϕ(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  On the other hand, there exists µ∗ ∈ [µ, µ] such that ϕ(µ∗) = max  µ≤x≤µ ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Then we have  maxµ≤µ≤µ  ˆ µ  µ  ϕ(y)δµ(dy) ≥  ˆ µ  µ  ϕ(y)δµ∗(dy)  = ϕ(µ∗) = max  µ≤x≤µ ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='2 Let {δµn}∞  n=1 be a sequence of Dirac measures with µn ∈ [µ, µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Then there exists  a point µ∗ ∈ [µ, µ] and subsequence {δµni }∞  i=1 such that δµni weakly converges to δµ∗ provided  µni → µ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Thus the set of Dirac measures P = {δµ, µ ∈ [µ, µ]} is weakly compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Secondly, by the independence of samples, we can simply calculate the likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='3 Let {Xi}n  i=1 be a sequence of independent random variables on sublinear expecta-  tion space (Ω, H, ˆE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' We further assume that ˆE can be represented by  ˆE[X] = max  P ∈P EP[X],  ∀X ∈ H,  where P is a weakly compact set of probability measures on (Ω, B(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Then the terms ˆE[I{X1=x1,··· ,Xn=xn}] and ˆE[I{Xi=xi}], 1 ≤ i ≤ n are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Furthermore, ∀xi ∈ R, 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  ˆE[I{X1=x1,··· ,Xn=xn}] = Πn  i=1ˆE[I{Xi=xi}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  (11)  7  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  The indicator function Ix∗(·), x∗ ∈ R is not continuous, thus I{Xi=xi} /∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  But it  can be approximated by the Lipschitz functions, thus ˆE[I{Xi=xi}] can be well-defined, so does  ˆE[I{X1=x1,··· ,Xn=xn}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Indeed, for fixed x∗ ∈ R, consider the function  ϕx∗  k (x) =  1  1 + k|x − x∗|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  It is easily seen that ϕx∗  k (·) is a non-negative bounded continuous function and ϕx∗  k (·) ↓ Ix∗(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  For each k ∈ N, we also have  |ϕx∗  k (x1) − ϕx∗  k (x2)| =  ����  1 + k|x2 − x∗| − 1 − k|x1 − x∗|  (1 + k|x1 − x∗|)(1 + k|x2 − x∗|)  ����  ≤  k|x2 − x1|  (1 + k|x1 − x∗|)(1 + k|x2 − x∗|)  ≤ k|x2 − x1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Thus ϕx∗  k  is a Lipschitz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Then by Theorem 31 in Denis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' [2], we obtain  ˆE[ϕx∗  k (Xi)] ↓ ˆE[I{Xi=x∗}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Similarly, the term ˆE[I{X1=x1,··· ,Xn=xn}] is well-defined for each n ∈ N, which can be approximated  as  ˆE[Πn  i=1ϕxi  k (Xi)] ↓ ˆE[I{X1=x1,··· ,Xn=xn}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  By the independence of {Xi}n  i=1, we obtain, ∀xi ∈ R, 1 ≤ i ≤ n,  ˆE[I{X1=x1,··· ,Xn=xn}] = lim  k→∞  ˆE[Πn  i=1ϕxi  k (Xi)]  = lim  k→∞ Πn  i=1ˆE[ϕxi  k (Xi)]  = Πn  i=1 lim  k→∞  ˆE[ϕxi  k (Xi)]  = Πn  i=1ˆE[I{Xi=xi}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='1, we can see that if X  d= M[µ, µ], then  ˆE[I{X=x∗}] = max  µ≤µ≤µ δµ(x∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Now we can solve the minimax problem (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='4 Let {Xi}n  i=1 be an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' sequence with X1  d= M[µ,µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Then the MLE of parameters  µ and µ is given by  ˆµ = min{X1, · · · , Xn};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  ˆµ = max{X1, · · · , Xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  (12)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Let {xi}n  i=1 be the samples of population {Xi}n  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='3, we have  V (x1, · · · , xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' µ, µ) = ˆE[I{X1=x1,··· ,Xn=xn}]  = Πn  i=1ˆE[I{Xi=xi}]  = Πn  i=1 max  µ≤µ≤µ δµ(xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  8  It is clear that  V (x1, · · · , xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' µ, µ) =  �  1,  µ ≤ x ≤ x ≤ µ,  0,  otherwise,  (13)  where x = min{x1, · · · , xn} and x = max{x1, · · · , xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Thus the solution of minimax problem (10) is given by  ˆµ = min{X1, · · · , Xn},  ˆµ = max{X1, · · · , Xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='5 The MLE in (12) is the same as the largest unbiased estimator for µ and smallest  unbiased estimator for µ obtained in Jin and Peng [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' In which, a statistic Tn = fn(X1, · · · , Xn)  is called an unbiased estimator of parameter µ if ˆE[fn(X1, · · · , Xn)] = µ, where fn is continuous  on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  4  General Maximum Likelihood Estimator  In this section, we firstly prove that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='4 still holds without the assumption of inde-  pendence, which indicates that our results can also be applied to the samples with dependent  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='1 Let {Xi}n  i=1 be identically distributed sequence with X1  d= M[µ,µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Then the MLE  of parameters µ and µ is given by  ˆµ = min{X1, · · · , Xn};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  ˆµ = max{X1, · · · , Xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  (14)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Let {xi}n  i=1 be the sample of population {Xi}n  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Obviously, ∀ 0 ≤ ϕi ∈ CLip(R), 1 ≤  i ≤ n, we have  ˆE[Πn  i=1ϕi(Xi)] ≤ Πn  i=1  max  µ≤µi≤µ ϕi(µi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  By the similar argument in the previous proofs, we obtain  ˆE[Πn  i=1I{Xi=xi}] ≤ Πn  i=1ˆE[I{Xi=xi}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Then we have  ¯V (x1, · · · , xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' µ, µ) : = ˆE[I{X1=x1,··· ,Xn=xn}]  = ˆE[Πn  i=1I{Xi=xi}]  ≤ Πn  i=1ˆE[I{Xi=xi}]  = V (x1, · · · , xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' µ, µ),  where V (x1, · · · , xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' µ, µ) is defined as in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Since the sample {xi}n  i=1 is realized, we have  0 < ¯V (x1, · · · , xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' µ, µ) ≤ 1,  thus  V (x1, · · · , xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' µ, µ) = 1,  which implies that  ˆµ = min{x1, · · · , xn},  ˆµ = max{x1, · · · , xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  9  In many practical situations, we often have i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' condition for samples {Xi}n  i=1 of population  X which is not maximally distributed, and we hope to estimate the value of ˆE[ϕ(X)] for some ϕ  based on the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' samples {Xi}n  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Thanks to the law of large numbers, our results can also be  applied to approximate the non-maximally distributed population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  One typical application is to estimate the upper and lower variance of population Z which has  model uncertainty based on the historical time series {Zt−i}i≥1 (see Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' [10], Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' [16]  and Peng and Yang [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  We assume that Z has mean-zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=',  ˆE[Z] = ˆE[−Z] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  The upper and lower variance of Z, defined as σ2 = ˆE[Z2] and σ2 = −ˆE[−Z2], can be estimated  as follows:  Taking L ∈ N to be large enough, and we calculate the local sample variance with window L  by  σ2  j =  �L  i=1(Zt−L+i−j − µj)2  L − 1  , 1 ≤ j ≤ K,  where µj =  �L  i=1 Zt−L+i−j  L  is the sample mean under the same window L and the number K ∈ N  is fixed with prior knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' The value of K can be used to characterize the uncertainty of  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' The larger K means that we prefer more uncertainty in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  We note that the term µj is closed to 0 when L is large enough, since ˆE[Z] = ˆE[−Z] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Thus  we have  σ2  j ≈  �L  i=1 Z2  t−L+i−j  L − 1  ,  1 ≤ j ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  For each j ∈ {1, · · · , K}, by the law of large numbers (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='10), σ2  j can be regarded as the  maximally distributed random variable on [σ2, σ2] when L is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' In fact, the error term  of such approximation can be estimated by (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='1, we obtain the MLE of σ2 and σ2 by  ˆσ  2 = max  1≤j≤K σ2  j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  ˆσ2 =  min  1≤j≤K σ2  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Based on such estimations of upper and lower variance, extensive experiments on both NASDAQ  Composite Index and S&P 500 Index demonstrate the excellent performances of G-VaR model, a  new benchmark predictor for value-at-risk based on the so-called G-normal distribution, which is  superior to most existing benchmark VaR predictors (see [16] and [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='2 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='4 or Jin and Peng’s optimal unbiased estimator can not be applied to  this situation since the sequence {σ2  j }K  j=1 is not independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' In fact, it is L-dependent in sublinear  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Our results provide the theoretical foundation for the widely used estimator of upper and  lower variances in finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  References  [1] Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
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+page_content='109042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
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+page_content=' Statistics and Probability Letters, 193: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='109719.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  [7] Hu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=', Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' and Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' (2021) Convergence rate of Peng’s law of large numbers under  sublinear expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Probability, Uncertainty and Quantitative Risk, 6(3): 261–266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
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+page_content=' and Peng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
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+page_content=' Proba-  bility, Uncertainty and Quantitative Risk, 6(3): 189–198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  [9] Lehmann, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
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+page_content=' (1998) Theory of point estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Second edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Springer  Texts in Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' Springer-Verlag, New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  [10] Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
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+page_content=' and Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' (2022) Upper and lower variances under model uncertainty  and their applications in finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content=' International Journal of Financial Engineering 9(1): No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  2250007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
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+page_content=' (2022) Maximally distributed random fields under sublinear expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  In Stochastic Analysis, Filtering, and Stochastic Optimization, 339–356, Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
+page_content='  [12] Pei, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
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+page_content=' Acta  Mathematicae Applicatae Sinica, English Series, 37(2): 421–440.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
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+page_content='  Probability, Uncertainty and Quantitative Risk, 4, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
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+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
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+page_content='  Probability, Uncertainty and Quantitative Risk, 6(3):199–212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
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+page_content='08471.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NdE4T4oBgHgl3EQf9Q6K/content/2301.05354v1.pdf'}
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+arXiv:2301.11271v1  [physics.hist-ph]  14 Jan 2023
+A "network of networks" (from history to algebra)
+Daniel Parrochia
+University of Lyon (France)
+Abstract.
+Recall first the algebraic treatment of flows or tensions in a transportation network
+N, i.e. a connected antisymmetric 1-graph G(X, U). Assume that, unusually, we
+take the values of flows (resp. tensions) in C. So the algebraic lattices Γ of flow
+(resp. tension) values associated to G(X, U) are lattices of C. These lattices are
+congruent modulo the action of the special linear group SL(2, C). Then, it is well
+known one can define a lattice function Gk(Γ), as a modular function of weight 2k,
+on the set R of all lattices of C. Let now N1, N2, ..., Np be connected antisymmetric
+1-graphs and Cn, the set of hermitian symmetric matrices n × n. Let also R′ be the
+set of all the lattices of Cn. The previous structure can be transposed to any n × n
+symmetric hermitian matrices of flow (or tension) values of the Gi. In this case,
+the Siegel space Sn = Cn replaces the Poincaré half-plane, and the symplectic group
+Sp(2n, R) takes the place of the special linear group SL(2, C). We get now the new
+lattice function as a function of all the lattices of Sn, i.e. a model of the "network of
+networks" R′. In the end, we study the tree of minimal length of R′.
+Key words. network, flows, tensions, lattices, lattice functions, modular functions,
+network algebra.
+1
+Introduction
+Consider a connected 1-graph G whose arcs are denoted by 1, 2, ..., m and let some
+quantities bi, ci be such that
+−∞ ≤ bi ≤ ci ≤ +∞
+1
+
+with the conditions:
+1) bi = 0
+(i = 1, 2, ..., m);
+2) ci ≥ 0
+for all i, and ci = +∞;
+3) Arc i = 1 is the arc (b, a) which connects a point b named the output with a point
+a named the input, these two points verifying:
+ω−(a) = (1, 0, 0, ..., 0),
+ω+(a) = (1, 0, 0, ..., 0);
+4) G is an antisymmetric 1-graph.
+The arc 1 = (b, a), that will not be drawn, is named the return arc and is just intro-
+duced to maintain the Kirchoff law at the vertices a and b.
+Definition 1.1. A graph G, with a capacity ci associated to any arc i, and which
+satisfies all these conditions, is called a transportation network (see [3])1 and it will
+be denoted by:
+N = (X, U, c(u)).
+In the following, as we will not pay attention to capacities, the previous network will
+be reduced to a connected 1-graph G.
+2
+Flows and tensions in networks
+Definition 2.1. A flow in a connected graph G is usually defined as a vector φ =
+(φ1, φ2, ..., φm) ∈ Zm such that:
+(1) φi ∈ Z for i = 1, 2, ..., m. (The integer φi is called an arc flow and may be
+regarded as the number of vehicules (signals, etc.) travelling through arc i along its
+direction if φi ≥ 0 or against its direction if φi < 0.)
+1Historically, before Ford and Fulkerson, it seems that interest for combinatorial optimization
+may be found in an article of A. N. Tolsto˘ı from 1930, in which the transportation problem is
+studied, as well as an, until recently secret, RAND report of T. E. Harris and F. S. Ross from 1955,
+that Ford and Fulkerson mention as motivation to study the maximum flow problem. These papers
+have in common that they both apply their methods to the Soviet railway network. As Schrijver
+recalled, the transportation problem was formulated by ([8], and a cycle criterion for optimality
+was considered by [9], [10], [12], [13], [19], [20] [5], [6], [14], [4] and [11]. On all that, see [21].
+2
+
+(2) For each vertex x, the sum of the arc flows entering x equals the sum of the arc
+flows leaving x (Kirchoff law), i.e.,
+�
+i∈ω−(x)
+φi =
+�
+j∈ω+(x)
+φj
+(x ∈ X).
+According to Berge (see [1], 85), it is possible to develop an algebraic study of flows
+in such a graph.
+Firstable, as Zm is a module on Z (not a vector space, because Z is not a field), the
+set Φ of all flows in the graph G constitutes a submodule of Zm, i.e we have:
+φ1, φ2 ∈ Φ ⇒ φ1 + φ2 ∈ Φ,
+s ∈ Z, φ ∈ Φ ⇒ sφ ∈ Φ.
+Berge proves the following theorem:
+Theorem 2.1. Let G = (X, U) a connected graph; H = (X, V ) an arbitrary tree of
+G; 1, 2, ..., k, the arcs of U − V ; µ1, µ2, ..., µk the cycles associated with H. A flow
+φ is uniquely defined by its values φ1, φ2, ..., φk ∈ U − V by:
+φ = φ1⃗µ1 + φ2⃗µ2 + ... + φk⃗µk,
+where the φi are scalars and the ⃗µi are vectors associated with independent elementary
+cycles.
+This means that a flow is uniquely defined by its components on a cotree of G.
+Let now come to tensions.
+Definition 2.2. A tension (or potential difference) in a connected graph G is defined
+to be a vector θ = (θ1, θ2, ..., θm) ∈ Zm such that, for each elementary cycle µ,
+�
+i∈µ+
+θi =
+�
+i∈µ−
+θi.
+For every arc i, we have: θi = t (terminal end of arc i) - t (initial end of arc i).
+Let Θ denote the set of all tensions. Note that Θ is also a submodule of Zm, i.e.,
+θ1, θ2 ∈ Θ ⇒ θ1 + θ2 ∈ Θ,
+3
+
+s ∈ Z, θ ∈ Θ ⇒ sθ ∈ Θ.
+Here again, Berge proves the following theorem:
+Theorem 2.2. Let G = (X, U) a connected graph; H = (X, V ) an arbitrary tree of
+G; 1, 2, ..., k, the arcs of this tree; ⃗ω1, ⃗ω2, ..., ⃗ωℓ the cocycles associated with H. A
+tension θ is uniquely defined by its values θ1, θ2, ..., θℓ on the arcs of the tree by:
+θ = θ1⃗ω1 + θ2⃗ω2 + ... + θℓ⃗ωℓ,
+where the θi are scalars and the ⃗ωi are vectors associated with independent elementary
+cocycles.
+This means that a tension is uniquely defined by its components on a tree of G.
+We can easily see that Θ and Φ are two orthogonal submodules of Zm, which means
+that, for every elementary cycle µ, we have:
+⟨φ, θ⟩ =
+m
+�
+i=1
+φiθi = 0.
+3
+Algebraic lattices
+We propose to extend the previous model. Let us consider now the set of all possible
+values of tensions or flows in some network N. We will prove that this set can be
+associated to a metanetwork Gk(Γ) which satisfies good properties. Recall first the
+following definition:
+Definition 3.1. A lattice Γ, in an R-vector space V of finite dimension, is a subgroup
+of V verifying one of the following equivalent conditions enumerated by Serre (see
+[22], 133):
+1) Γ is discrete and V/Γ is compact;
+2) Γ is discrete and generates the R-vector space V ;
+3) There exists an R-basis {e1, ...en} of V , which is a Z-basis of Γ and Γ = Ze1 ⊕
+... ⊕ Zen.
+Now, let us choose values of flows (or tensions) in an R-vector space V = Rn.
+4
+
+Theorem 3.1. The set of all possible flow (resp. tension) values of the network N
+is a lattice in Rn.
+Proof. Let ǫ = (ǫ1, ǫ2, ..., ǫn), a flow (resp.
+a tension) in some arc(s) of G.
+By
+definition, ǫ belongs to Rn, viewed as a vector space on R. Moreover, according to
+the definition of flows (Def. 2.1) and of tensions (Def. 2.2), the set Γ, of all flow
+(resp. tension) values in the graph G, is the subgroup of all linear combinations with
+integer coefficients of the basis vectors of Rn (cycles, resp. cocyles). So it is such
+that:
+Γ = Zǫ1 ⊕ ... ⊕ Zǫn,
+for any basis of Rn. In other words, it forms a lattice in Rn.
+4
+The lattices of C
+Assume now that the flow (resp. tension) values of G are in C, and consider only
+two-valued flows (resp. tensions).
+Let us call R the set of lattices of C, considered as an R-vector space, and let us now
+choose a pair of flow (resp. tension) values (α1, α2) ∈ C∗ so that Im(α1/α2) > 0. M
+will be the set of these pairs.
+To such a pair (α1, α2), we associate the lattice:
+Γ(α1, α2) = Zα1 ⊕ Zα2.
+with basis {α1, α2}.
+Thus we get a map M → R, which is clearly surjective.
+Now let:
+g =
+�
+a
+b
+c
+d
+�
+∈ SL(2, Z)
+the special linear group of square matrices 2 × 2 with relative coefficients, and let
+(α1, α2) ∈ M. One proves (see [22], 134) the following theorem:
+Theorem 4.1. For two elements of M to define the same lattice, it is necessary and
+sufficient that they are congruent modulo SL(2, Z).
+5
+
+Proof. (Serre) The condition is sufficient. Let us put:
+α′
+1 = aα1 + bα2 and α′
+2 = cα1 + dα2.
+Il is clear that {α′
+1, α′
+2} is a basis of Γ(α1, α2). Moreover, if the set z = α1/α2 and
+z′ = {α′
+1/α′
+2}, we have:
+z′ = az + b
+cz + d = gz.
+This shows that Im(z′) > 0, hence that (α′
+1, α′
+2) belongs to M.
+Conversely, if (α1, α2) and (α′
+1, α′
+2) are two elements of M which define the same
+lattice, there exists an integer matrix
+g =
+�
+a
+b
+c
+d
+�
+of determinant ±1 which transforms the first basis into the second. If det(g) was
+< 0, the sign of Im(α′
+1/α′
+2) would be the opposite of Im(α1/α2) as one sees by an
+immediate computation. The two signs being the same, we have necessarily det(g)
+= 1, which proves the theorem.
+Hence we can identify the set R of all the lattices of C (which are, for us, sets of flow
+(or tension) values associated to connected 1-graphs (or networks) with the quotient
+of M by the action of SL(2, Z).
+5
+Modular functions
+Let now F be a function on R, with complex values, and let k ∈ √Z. We say (with
+Serre) that F is of weight 2k if:
+F(λΓ) = λ−2kF(Γ),
+(1)
+for all lattices Γ and all λ ∈ C∗.
+Let F be such a function. If (α1, α2) ∈ M, we denote by F(α1, α2) the value of F on
+the lattice Γ(α1, α2). The formula (1) translates to:
+F(λα1, λα2) = λ−2kF(α1, α2).
+(2)
+Writing that F is invariant by SL(2, Z), we can see that it satisfies the identity:
+6
+
+F(z) = (cz + d)−2kF(az + b
+cz + d),
+(3)
+for all:
+�a
+b
+c
+d
+�
+∈ SL(2, Z).
+Conversely, if F verifies (3), F is a function on R which is of weight 2k. We can
+thus identify modular functions of weight 2k with some lattice functions of weight
+2k.
+Then we know that some lattice functions, that are modular functions, can be iden-
+tified with Eisenstein series, which are themselves convergent. Serre (1973) proves
+the following lemma:
+Lemma 5.1. Let Γ be a lattice in C. The series:
+′
+�
+γ∈Γ
+1/|γ|σ
+is convergent for σ > 2.
+(The symbol �′ signifies that the summation runs over the nonzero elements of
+Γ.)
+Now let k be an integer >1. If Γ is a lattice of C, put:
+Gk(Γ) =
+′
+�
+γ∈Γ
+1/γ2k.
+This series converges absolutely thanks to the preceding lemma. It proves the exis-
+tence of a lattice function on the set R of lattices of C.
+In other words, all the lattices of C, which represent sets of flow (or tension) val-
+ues in connected 1-graphs (or networks) are themselves connected by this lattice
+function.
+Let now k be an integer >1. Like all the Eisentein series of the type Gk(z):
+1) Gk(Γ), which is a modular form of weight 2k, is holomorphic everywhere (including
+at the infinite);
+7
+
+2) Gk(∞) = 2ζ(2k);
+3) Gk has a limit for Im(z) → ∞, z being the value for which Γ vanishes at one and
+only one point.
+6
+Siegel space
+We can still extend the previous construction.
+Let N1, N2, ..., Nm be some finite connected 1-graphs and consider, for each of them,
+their associated matrices of flow (or tension) values. Let Z1, Z2, ..., Zm be such ma-
+trices with complex coefficients.
+Let L be the set of all n × n complex symmetric matrices and Cn the set of matrices
+Z of L such that the hermitian matrix I − Z ¯Z is strictly positive.
+Let now Sn (the Siegel space) be the set of matrices Z of L whose imaginary part
+Im Z = (1/2i)(Z − ¯Z) is strictly positive. It is well known that the so-called "Cayley
+transformations" apply Cn to Sn and vice versa (see [2], 437-438).
+Hence, the real symplectic group Sp(2n, R) plays the same role, with respect to the
+Siegel space Sn, than the group Sp(2, R) = SL(2, R) with respect to the upper half-
+plane of the complex plane. When the group SL(2, R) operates in C by the Poincaré
+Fuchsian transformations, the group Sp(2n, R) now operates in the Siegel space Sn
+by the transformations:
+g′ =
+�A
+B
+C
+D,
+�
+∈ Sp(2n, R).
+(4)
+So we have:
+g′Z = (AZ + B)(CZ + D)−1.
+Now let us call R′ the set of all the matrix lattices of Cn, and let M′ be the set of pairs
+(A1, A2) ∈ Cn, such that Im(A1, A−
+2 ) > 0, which supposes that A2 is inversible.
+To such a pair (A1, A2), we associate now the lattice:
+Γ′(A1, A2) = ZA1 ⊕ ZA2.
+with basis {A1, A2}. Thus, we get a map M′ → R′, which is clearly surjective.
+8
+
+One gets the following theorem:
+Theorem 6.1. So that two elements of M′ define the same lattice, it is necessary
+and sufficient that they are congruent modulo Sp(2n, R).
+Proof. The condition is sufficient. Let A1, A2 ∈ M′. Then, put :
+A′
+1 = aA1 + bA2 and A′
+2 = cA1 + dA2.
+Il is clear that {A′
+1, A′
+2} is a basis of Γ(A1, A2). Moreover, if Z = A1A−
+2 and Z′ =
+A′
+1A′−
+2 ,
+Z′ = (AZ + B)(CZ + D)− = g′Z.
+This shows that Im(Z′) > 0, hence that (A′
+1, A′
+2) belongs to M′.
+Conversely, if (A1, A2) and (A′
+1, A′
+2) are two elements of M′ which define the same
+lattice, there exists an integer matrix
+g′ =
+�A
+B
+C
+D
+�
+of determinant >0 which transforms the first basis into the second. If det(g′) was
+<0, the sign of Im(A′
+1A′−
+2 ) would be the opposite of Im(A1A−
+2 ) as one sees by an
+immediate computation. The two signs being the same, we have necessarily det(g′)
+>0, which proves the theorem.
+Thus, we can identify the set M′ of all the lattice matrices of Cn with the quotient
+of Sn by the action of Sp(2n, R).
+For the same reasons, we can also define, as previously, a lattice function of weight
+2k.
+Let F ′ be such a function. If (A1, A2) ∈ M′, we denote by F ′(A1, A2) the value of F ′
+on the lattice Γ′(A1, A2). The formula (2) translates to:
+F ′(λA1, λA2) = λ−2kF ′(A1, A2).
+(5)
+Writing now that F ′ is invariant by Sp(2n, R), we can see that this function satisfies
+the identity:
+9
+
+F ′(Z) = (XZ + D)−2kf(AZ + B
+CZ + D),
+(6)
+for all:
+�A
+B
+C
+D
+�
+∈ Sp(2n, R).
+As previously, this function can be identified with an Eisenstein series G′
+k(Γ′) on the
+set R′ of the matrix lattices of Cn, which is absolutely convergent.
+In other words, all the n × n matrix lattices of Cn, which represent sets of subsets
+of flows (or tensions) in connected 1-graphs (or networks), are linked by this lattice
+function.
+If we associate networks with subsets of flow (or tension) values, this proves the
+existence of a "network of networks".
+7
+The tree of minimal length
+Let Gk(Γ) be the graph associated with the set of all subsets of flows and A0, the
+minimal tree of Gk(Γ). If U is the set of arcs of Gk(Γ), U − A0 = A′
+0 is the maximal
+cotree of Gk(Γ)2. Now, it is easy to see that :
+(1) The smallest arc of all cocycles (tensions) is in A0;
+(2) The greatest arc of all cycles (flows) is in U − A0 = A′
+0.
+We finally obtain a set of arcs without a maximal cycle and we can always find an
+optimal flow in the graph because any flow does not circulate in all the arcs of the
+whole graph but only in those of a tree whose capacities, which do not admit a higher
+bound, are infinite.
+Let us now precise the form of the tree of minimal length A0. Let n be the number
+of vertices of A0, A the set of its arcs, a an arc of A, d the distance between two
+vertices s and s′. We have:
+(1) Card(A) = n(n − 1)/2;
+(2) a = {s, s′} : d(a) = d(s, s′).
+2On trees and co-trees, see ([7], 103-128.
+10
+
+Now let P be a polygon, i.e. a set of edges which is a subset of A0. The support
+of P will be the union of the edges of A0, that is, the set of vertices of Gk(Γ) which
+are ends of at least one edge of P. One can speak of polygon P on Gk(Γ) (resp. in
+Gk(Γ)) according to whether the support of P is Gk(Γ) or a subset of Gk(Γ) distinct
+from itself.
+A0, which is the set of all possible edges on Gk(Γ), is a complete polygon of Gk(Γ).
+A graph being the conjunction of a polygon and its support, a chain C will be a
+polygon in Gk(Γ) whose vertices that form its support can be ordered in a sequence
+(s0, s1, ...sp). We have:
+(1) For every i ∈ ]P[, {si−1, si} ∈ C;
+(2) For every i, j ∈ [P], i ̸= j ⇒ si ̸= sj.
+A cycle is a chain where condition (2) holds for all the points of its support except
+s0 and sp which are merged (the ends of C).
+To exhibit Gk(Γ), we need the following complementary considerations:
+A) A tree is a connected polygon that does not contain a loop.
+B) The length of a polygon is the sum of the lengths of all its edges.
+C) The width of a polygon is the length of its longest edge.
+Suppose that the polygon reduced to the edge (s, s′) represents the chain of A with
+minimum width joining s to s′, then {s, s′} is an element of the tree of minimal
+length T on Gk(Γ) and there exists at least one such edge on A, the edge of minimal
+length.
+Conversely, if {r, s} is an element of the tree T on Gk(Γ), then {r, s} is the chain of
+A having the smallest width and joining r to s.
+In this context, Gk(Γ) can be identified with a classification of classifications. This
+would amount to doing a factor analysis on all parts of the representative tree. Such
+a classification would correspond to all the axes of a factor analysis, with an original
+calculation on the first axis.
+8
+Construction of Gk(Γ)
+In order to construct Gk(Γ), we must first look at the lattice Γ = Zα1 ⊕ Zα2, which
+makes possible to distinguish a lattice function and a modular function. It must be
+11
+
+assumed that the minimal bases of this lattice suppose a matroid M. If B is the set
+of these bases, then C, the set of cycles of M (resp. D the set of cocycles of M) is
+the set of subsets which are not included in any basis (resp. which have a non-empty
+intersection with any basis) and minimal for inclusion with this property.
+Let now B ∈ B, b ∈ B, c ∈ X − B. Let D(b, B) be the unique cocycle satisfying
+B ∩ D(b, B) = {b} and C(c, B), the unique cycle satisfying C(c, B) − B = {c}. We
+then have:
+B ∈ C(c, B) ⇐⇒ c ∈ D(b, B) ⇐⇒ B − b ∪ c ∈ B.
+C and D are the sets of cycles and minimal cocycles for the inclusion of the graph.
+The minimum tree of a graph is the set of minimum edges of a cocycle, its complement
+being the set of the maximum edges of a cycle.
+If we consider the lattice (F, ∪, ∩), a sublattice of M, the algebraic properties of F
+(distributivity) are stronger than those of M (semi-modularity). It is thus possible
+to construct Gk(Γ), the super-lattice, by defining it as the set of distributive sub-
+lattices of any geometric lattice, that is to say, a sub-lattice of the semi-geometric
+lattice associated with M.
+9
+Possible applications
+Let’s finish with some more epistemological considerations: after all, mathematical
+physics and philosophy are not so far apart (see [18]).
+The space associated with this "network of networks", that is, the zeros and poles
+of the modular function of all networks, has been studied in hard proof theorems,
+because one does not define a structure of complex analytic variety on the single com-
+pactified network. (A natural way of proceeding would be to define a compactified
+isomorphism on the Riemann sphere S = C ∪ {∞}.)
+Whatever the difficulties of study, it is proved that this network function exists, and
+we have thus proved also that the set of all sets of possible flows exists as a modular
+function of all networks in the algebraic sense of the term.
+Let us now consider some possible applications of the previous formalism.
+1. Since the old work of [Von Neuman 1946], quantum mechanics represents all the
+physical states of the universe by a vector space of infinite dimension called "Hilbert
+space". However, the separability property and the convergence condition make it
+12
+
+possible to reduce to closed subspaces.
+In this case, the complex vectors form a
+finite dimensional subspace and their mathematics is identical to that of flows or
+tensions on a graph, except that their coefficients can take on complex values. This
+situation makes it possible, as we have seen, to apply known theorems of arithmetic
+to them.
+2. Because of the flow-tension duality, the network function defines as well the set of
+all the sets of possible tensions, and hence it specifies the shortest path in the total
+set of all possible paths, as well as the most rational scheduling of tasks in the set of
+all possible actions. Here we have a theorem of the existence of an optimal behavior,
+whatever the field we consider.
+3. Moreover, the problem of the shortest path in a graph is related to the question
+of the tree of minimum length, which itself formalizes the notion of classification. A
+"network of networks" with a maximum voltage would thus make it possible both:
+to confirm the existence of a tree of minimum length of the network of all networks,
+and hence, of a classification of classifications (see [17]).
+4. In general, the variable "weights" can receive different meanings (reliability, econ-
+omy, etc.) on a tree, other than the length of the arcs. So the network of networks
+Gk(Γ) can still make it possible to calculate the maximum reliability path, or the
+most economical route, etc., in the set of all possible paths.
+5. I will say a final word about the aim of this construction : though the world
+may be multiple and chaotic, circulations and actions can be ordered in relation
+to the same structure, which is expressed - in the linear case - through the form
+of this remarkable holomorphic function which has been here constructed. Doing
+that, we tried in fact to formalize the intuition of a "network of networks", as it
+is expressed in the conclusion of our book on networks (see [15], 265-286). This is
+also the achievement of what we called elsewhere a "rationalité réticulaire" (reticular
+rationality)(see [16]).
+References
+[1] Berge, C., Graphes et hypergraphes, Dunod, Paris, 1970.
+[2] Deheuvels, R., Formes quadratiques et groupes classiques, P.U.F., Paris, 1981.
+[3] Ford, L. R., Fulkerson, D. R., Flows in Networks, Rand Corporation, Santa
+Monica, 1962.
+13
+
+[4] Fulkerson, D. R., "An out-of-kilter method for minimal-cost flow problems",
+Journal of the Society for Industrial and Applied Mathematics 9, 18-27, 1961.
+[5] Gallai,
+T.,
+"Gráfokkal
+kapcsolatos
+maximum-minimum
+tételek,
+I
+rész",[Hungarian:
+Maximum-minimum theorems for networks (part I)] A
+Magyar Tudományos Akadémia Matematikai és Fizikai Tudományok Osztályának
+Közleményei, 7, S. 305-338, 1957.
+[6] Gallai, T., "Maximum-minimum Sätze Über Graphen", Acta Mathematica
+Academiae Scientiarum Hungaricae 9, 395-434, 1958.
+[7] Gondran, M., Minoux, M., Graphes et Algorithmes, Eyrolles, Paris, 1979.
+[8] Hitchcock, F. L., "The distribution of a product from several sources to numerous
+localities", Journal of Mathematics and Physics 20, 224-230, 1941.
+[9] Kantorovich, L. V., "O peremeshchenii mass" [Russian], Doklady Akademii Nauk
+SSSR 37, 7-8, 227-230, 1942. [English translation:
+"On the translocation of
+masses", Comptes Rendus (Doklady) de l’Académie des Sciences de l’U.R.S.S,
+37, 199-201,1942, [reprinted: Management Science 5, 1-4, 1958].
+[10] Kantorovich L. V., Gavurin, M. K., "Primenenie matematicheskikh metodov
+v voprosakh analiza gruzopotokov" [Russian; "The application of mathematical
+methods to freight flow analysis"], in Problemy povysheniya effectivnosti raboty
+transporta [Russian; Collection of Problems of Raising the Efficiency of Transport
+Performance], Akademiia Nauk SSSR, Moscow-Leningrad, 1949, pp. 110-138.
+[11] Klein, M., "A primal method for minimal cost flows with applications to the
+assignment and transportation problems", Management Science 14, 205-220, 1967.
+[12] Koopmans, "Optimum utilization of the transportation system", in: The Econo-
+metric Society Meeting Washington, D.C., September 6-18, 1947; D.H. Leavens,
+(ed.), Proceedings of the International Statistical Conferences - Volume V, 1948,
+pp. 136-146; reprinted in Econometrica 17 (Supplement) 136-146, 1949; reprinted
+in Scientific Papers of Tjalling C. Koopmans, Springer, Berlin, 184-193.
+[13] Koopmans, Tj.C., Reiter, S., "A model of transportation", Activity Analysis of
+Production and Allocation - Proceedings of a Conference, Tj.C. Koopmans (ed.),
+222-259, Wiley, New York, 1951.
+[14] Lur’e, A. L., "Methods of establishing the shortest running distances for freights
+on setting up transportation systems" [in Russian], in Primenenie matematiki
+èkonomicheskikh issle-dovaniyakh [Russian; V.S. Nemchinov (ed.), Application of
+14
+
+Mathematics in Economical Studies, Izdatel’stvo Sotsial’no-Èkonomichesk˘i Liter-
+atury, 249-382, Moscow, 1959. English translation in: V.S Nemchinov (ed), The
+Use of Mathematics in Economics, 322-355, Oliver and Boyd, Edinburgh, 1964.
+[15] Parrochia, D., Philosophie des réseaux, P.U.F., Paris, 1993.
+[16] Parrochia, D., "La rationalité réticulaire", in D. Parrochia (ed.), Penser les
+réaux, 7-23, Champ Vallon, Seyssel, 2001.
+[17] Parrochia D., Neuville, P., Towards a general theory of classifications, Basel,
+Birkhäuser, 2013.
+[18] Parrochia, D., Mathematics and Philosophy, Wiley-Iste, London, 2018.
+[19] Robinson, J., "On the Hamiltonian Game (A Traveling Salesman Problem)",
+Research Memorandum RM-303, The RAND Corporation, Santa Monica, Califor-
+nia, 1949.
+[20] Robinson, J., "A Note on the Hitchcock-Koopmans Problem", Research Mem-
+orandum RM-407, The RAND Corporation, Santa Monica, California, 1950.
+[21] Schrijver, A., "On the history of the transportation and maximum flow prob-
+lems", Math. Program., 91, 437-445, 2002.
+[22] Serre, J.-P., Course in Arithmetics, Springer-Verlag, Berlin, 1973.
+15
+
diff --git a/ONFIT4oBgHgl3EQfdyse/content/tmp_files/load_file.txt b/ONFIT4oBgHgl3EQfdyse/content/tmp_files/load_file.txt
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@@ -0,0 +1,394 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf,len=393
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='11271v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='hist-ph]  14 Jan 2023  A "network of networks" (from history to algebra)  Daniel Parrochia  University of Lyon (France)  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Recall first the algebraic treatment of flows or tensions in a transportation network  N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' a connected antisymmetric 1-graph G(X, U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Assume that, unusually, we  take the values of flows (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' tensions) in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' So the algebraic lattices Γ of flow  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' tension) values associated to G(X, U) are lattices of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' These lattices are  congruent modulo the action of the special linear group SL(2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Then, it is well  known one can define a lattice function Gk(Γ), as a modular function of weight 2k,  on the set R of all lattices of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Let now N1, N2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', Np be connected antisymmetric  1-graphs and Cn, the set of hermitian symmetric matrices n × n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Let also R′ be the  set of all the lattices of Cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' The previous structure can be transposed to any n × n  symmetric hermitian matrices of flow (or tension) values of the Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' In this case,  the Siegel space Sn = Cn replaces the Poincaré half-plane, and the symplectic group  Sp(2n, R) takes the place of the special linear group SL(2, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' We get now the new  lattice function as a function of all the lattices of Sn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' a model of the "network of  networks" R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' In the end, we study the tree of minimal length of R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' network, flows, tensions, lattices, lattice functions, modular functions,  network algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  1  Introduction  Consider a connected 1-graph G whose arcs are denoted by 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', m and let some  quantities bi, ci be such that  −∞ ≤ bi ≤ ci ≤ +∞  1  with the conditions:  1) bi = 0  (i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  2) ci ≥ 0  for all i, and ci = +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  3) Arc i = 1 is the arc (b, a) which connects a point b named the output with a point  a named the input, these two points verifying:  ω−(a) = (1, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', 0),  ω+(a) = (1, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  4) G is an antisymmetric 1-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  The arc 1 = (b, a), that will not be drawn, is named the return arc and is just intro-  duced to maintain the Kirchoff law at the vertices a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' A graph G, with a capacity ci associated to any arc i, and which  satisfies all these conditions, is called a transportation network (see [3])1 and it will  be denoted by:  N = (X, U, c(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  In the following, as we will not pay attention to capacities, the previous network will  be reduced to a connected 1-graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  2  Flows and tensions in networks  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' A flow in a connected graph G is usually defined as a vector φ =  (φ1, φ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', φm) ∈ Zm such that:  (1) φi ∈ Z for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' (The integer φi is called an arc flow and may be  regarded as the number of vehicules (signals, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=') travelling through arc i along its  direction if φi ≥ 0 or against its direction if φi < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=')  1Historically, before Ford and Fulkerson, it seems that interest for combinatorial optimization  may be found in an article of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Tolsto˘ı from 1930, in which the transportation problem is  studied, as well as an, until recently secret, RAND report of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Harris and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Ross from 1955,  that Ford and Fulkerson mention as motivation to study the maximum flow problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' These papers  have in common that they both apply their methods to the Soviet railway network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' As Schrijver  recalled, the transportation problem was formulated by ([8], and a cycle criterion for optimality  was considered by [9], [10], [12], [13], [19], [20] [5], [6], [14], [4] and [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' On all that, see [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  2  (2) For each vertex x, the sum of the arc flows entering x equals the sum of the arc  flows leaving x (Kirchoff law), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=',  �  i∈ω−(x)  φi =  �  j∈ω+(x)  φj  (x ∈ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  According to Berge (see [1], 85), it is possible to develop an algebraic study of flows  in such a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Firstable, as Zm is a module on Z (not a vector space, because Z is not a field), the  set Φ of all flows in the graph G constitutes a submodule of Zm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='e we have:  φ1, φ2 ∈ Φ ⇒ φ1 + φ2 ∈ Φ,  s ∈ Z, φ ∈ Φ ⇒ sφ ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Berge proves the following theorem:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Let G = (X, U) a connected graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' H = (X, V ) an arbitrary tree of  G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', k, the arcs of U − V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', µk the cycles associated with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' A flow  φ is uniquely defined by its values φ1, φ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', φk ∈ U − V by:  φ = φ1⃗µ1 + φ2⃗µ2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' + φk⃗µk,  where the φi are scalars and the ⃗µi are vectors associated with independent elementary  cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  This means that a flow is uniquely defined by its components on a cotree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Let now come to tensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' A tension (or potential difference) in a connected graph G is defined  to be a vector θ = (θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', θm) ∈ Zm such that, for each elementary cycle µ,  �  i∈µ+  θi =  �  i∈µ−  θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  For every arc i, we have: θi = t (terminal end of arc i) - t (initial end of arc i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Let Θ denote the set of all tensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Note that Θ is also a submodule of Zm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=',  θ1, θ2 ∈ Θ ⇒ θ1 + θ2 ∈ Θ,  3  s ∈ Z, θ ∈ Θ ⇒ sθ ∈ Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Here again, Berge proves the following theorem:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Let G = (X, U) a connected graph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' H = (X, V ) an arbitrary tree of  G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', k, the arcs of this tree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' ⃗ω1, ⃗ω2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', ⃗ωℓ the cocycles associated with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' A  tension θ is uniquely defined by its values θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', θℓ on the arcs of the tree by:  θ = θ1⃗ω1 + θ2⃗ω2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' + θℓ⃗ωℓ,  where the θi are scalars and the ⃗ωi are vectors associated with independent elementary  cocycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  This means that a tension is uniquely defined by its components on a tree of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  We can easily see that Θ and Φ are two orthogonal submodules of Zm, which means  that, for every elementary cycle µ, we have:  ⟨φ, θ⟩ =  m  �  i=1  φiθi = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  3  Algebraic lattices  We propose to extend the previous model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Let us consider now the set of all possible  values of tensions or flows in some network N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' We will prove that this set can be  associated to a metanetwork Gk(Γ) which satisfies good properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Recall first the  following definition:  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' A lattice Γ, in an R-vector space V of finite dimension, is a subgroup  of V verifying one of the following equivalent conditions enumerated by Serre (see  [22], 133):  1) Γ is discrete and V/Γ is compact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  2) Γ is discrete and generates the R-vector space V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  3) There exists an R-basis {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='en} of V , which is a Z-basis of Γ and Γ = Ze1 ⊕  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' ⊕ Zen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Now, let us choose values of flows (or tensions) in an R-vector space V = Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  4  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' The set of all possible flow (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' tension) values of the network N  is a lattice in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Let ǫ = (ǫ1, ǫ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', ǫn), a flow (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  a tension) in some arc(s) of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  By  definition, ǫ belongs to Rn, viewed as a vector space on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Moreover, according to  the definition of flows (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='1) and of tensions (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='2), the set Γ, of all flow  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' tension) values in the graph G, is the subgroup of all linear combinations with  integer coefficients of the basis vectors of Rn (cycles, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' cocyles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' So it is such  that:  Γ = Zǫ1 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' ⊕ Zǫn,  for any basis of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' In other words, it forms a lattice in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  4  The lattices of C  Assume now that the flow (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' tension) values of G are in C, and consider only  two-valued flows (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' tensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Let us call R the set of lattices of C, considered as an R-vector space, and let us now  choose a pair of flow (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' tension) values (α1, α2) ∈ C∗ so that Im(α1/α2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' M  will be the set of these pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  To such a pair (α1, α2), we associate the lattice:  Γ(α1, α2) = Zα1 ⊕ Zα2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  with basis {α1, α2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Thus we get a map M → R, which is clearly surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Now let:  g =  �  a  b  c  d  �  ∈ SL(2, Z)  the special linear group of square matrices 2 × 2 with relative coefficients, and let  (α1, α2) ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' One proves (see [22], 134) the following theorem:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' For two elements of M to define the same lattice, it is necessary and  sufficient that they are congruent modulo SL(2, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  5  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' (Serre) The condition is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Let us put:  α′  1 = aα1 + bα2 and α′  2 = cα1 + dα2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Il is clear that {α′  1, α′  2} is a basis of Γ(α1, α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Moreover, if the set z = α1/α2 and  z′ = {α′  1/α′  2}, we have:  z′ = az + b  cz + d = gz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  This shows that Im(z′) > 0, hence that (α′  1, α′  2) belongs to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Conversely, if (α1, α2) and (α′  1, α′  2) are two elements of M which define the same  lattice, there exists an integer matrix  g =  �  a  b  c  d  �  of determinant ±1 which transforms the first basis into the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' If det(g) was  < 0, the sign of Im(α′  1/α′  2) would be the opposite of Im(α1/α2) as one sees by an  immediate computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' The two signs being the same, we have necessarily det(g)  = 1, which proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Hence we can identify the set R of all the lattices of C (which are, for us, sets of flow  (or tension) values associated to connected 1-graphs (or networks) with the quotient  of M by the action of SL(2, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  5  Modular functions  Let now F be a function on R, with complex values, and let k ∈ √Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' We say (with  Serre) that F is of weight 2k if:  F(λΓ) = λ−2kF(Γ),  (1)  for all lattices Γ and all λ ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Let F be such a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' If (α1, α2) ∈ M, we denote by F(α1, α2) the value of F on  the lattice Γ(α1, α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' The formula (1) translates to:  F(λα1, λα2) = λ−2kF(α1, α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  (2)  Writing that F is invariant by SL(2, Z), we can see that it satisfies the identity:  6  F(z) = (cz + d)−2kF(az + b  cz + d),  (3)  for all:  �a  b  c  d  �  ∈ SL(2, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Conversely, if F verifies (3), F is a function on R which is of weight 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' We can  thus identify modular functions of weight 2k with some lattice functions of weight  2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Then we know that some lattice functions, that are modular functions, can be iden-  tified with Eisenstein series, which are themselves convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Serre (1973) proves  the following lemma:  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Let Γ be a lattice in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' The series:  ′  �  γ∈Γ  1/|γ|σ  is convergent for σ > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  (The symbol �′ signifies that the summation runs over the nonzero elements of  Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=')  Now let k be an integer >1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' If Γ is a lattice of C, put:  Gk(Γ) =  ′  �  γ∈Γ  1/γ2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  This series converges absolutely thanks to the preceding lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' It proves the exis-  tence of a lattice function on the set R of lattices of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  In other words, all the lattices of C, which represent sets of flow (or tension) val-  ues in connected 1-graphs (or networks) are themselves connected by this lattice  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Let now k be an integer >1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Like all the Eisentein series of the type Gk(z):  1) Gk(Γ), which is a modular form of weight 2k, is holomorphic everywhere (including  at the infinite);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  7  2) Gk(∞) = 2ζ(2k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  3) Gk has a limit for Im(z) → ∞, z being the value for which Γ vanishes at one and  only one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  6  Siegel space  We can still extend the previous construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Let N1, N2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', Nm be some finite connected 1-graphs and consider, for each of them,  their associated matrices of flow (or tension) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Let Z1, Z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', Zm be such ma-  trices with complex coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Let L be the set of all n × n complex symmetric matrices and Cn the set of matrices  Z of L such that the hermitian matrix I − Z ¯Z is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Let now Sn (the Siegel space) be the set of matrices Z of L whose imaginary part  Im Z = (1/2i)(Z − ¯Z) is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' It is well known that the so-called "Cayley  transformations" apply Cn to Sn and vice versa (see [2], 437-438).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Hence, the real symplectic group Sp(2n, R) plays the same role, with respect to the  Siegel space Sn, than the group Sp(2, R) = SL(2, R) with respect to the upper half-  plane of the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' When the group SL(2, R) operates in C by the Poincaré  Fuchsian transformations, the group Sp(2n, R) now operates in the Siegel space Sn  by the transformations:  g′ =  �A  B  C  D,  �  ∈ Sp(2n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  (4)  So we have:  g′Z = (AZ + B)(CZ + D)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Now let us call R′ the set of all the matrix lattices of Cn, and let M′ be the set of pairs  (A1, A2) ∈ Cn, such that Im(A1, A−  2 ) > 0, which supposes that A2 is inversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  To such a pair (A1, A2), we associate now the lattice:  Γ′(A1, A2) = ZA1 ⊕ ZA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  with basis {A1, A2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Thus, we get a map M′ → R′, which is clearly surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  8  One gets the following theorem:  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' So that two elements of M′ define the same lattice, it is necessary  and sufficient that they are congruent modulo Sp(2n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' The condition is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Let A1, A2 ∈ M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Then, put :  A′  1 = aA1 + bA2 and A′  2 = cA1 + dA2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Il is clear that {A′  1, A′  2} is a basis of Γ(A1, A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Moreover, if Z = A1A−  2 and Z′ =  A′  1A′−  2 ,  Z′ = (AZ + B)(CZ + D)− = g′Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  This shows that Im(Z′) > 0, hence that (A′  1, A′  2) belongs to M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Conversely, if (A1, A2) and (A′  1, A′  2) are two elements of M′ which define the same  lattice, there exists an integer matrix  g′ =  �A  B  C  D  �  of determinant >0 which transforms the first basis into the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' If det(g′) was  <0, the sign of Im(A′  1A′−  2 ) would be the opposite of Im(A1A−  2 ) as one sees by an  immediate computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' The two signs being the same, we have necessarily det(g′)  >0, which proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Thus, we can identify the set M′ of all the lattice matrices of Cn with the quotient  of Sn by the action of Sp(2n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  For the same reasons, we can also define, as previously, a lattice function of weight  2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Let F ′ be such a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' If (A1, A2) ∈ M′, we denote by F ′(A1, A2) the value of F ′  on the lattice Γ′(A1, A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' The formula (2) translates to:  F ′(λA1, λA2) = λ−2kF ′(A1, A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  (5)  Writing now that F ′ is invariant by Sp(2n, R), we can see that this function satisfies  the identity:  9  F ′(Z) = (XZ + D)−2kf(AZ + B  CZ + D),  (6)  for all:  �A  B  C  D  �  ∈ Sp(2n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  As previously, this function can be identified with an Eisenstein series G′  k(Γ′) on the  set R′ of the matrix lattices of Cn, which is absolutely convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  In other words, all the n × n matrix lattices of Cn, which represent sets of subsets  of flows (or tensions) in connected 1-graphs (or networks), are linked by this lattice  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  If we associate networks with subsets of flow (or tension) values, this proves the  existence of a "network of networks".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  7  The tree of minimal length  Let Gk(Γ) be the graph associated with the set of all subsets of flows and A0, the  minimal tree of Gk(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' If U is the set of arcs of Gk(Γ), U − A0 = A′  0 is the maximal  cotree of Gk(Γ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Now, it is easy to see that :  (1) The smallest arc of all cocycles (tensions) is in A0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  (2) The greatest arc of all cycles (flows) is in U − A0 = A′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  We finally obtain a set of arcs without a maximal cycle and we can always find an  optimal flow in the graph because any flow does not circulate in all the arcs of the  whole graph but only in those of a tree whose capacities, which do not admit a higher  bound, are infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Let us now precise the form of the tree of minimal length A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Let n be the number  of vertices of A0, A the set of its arcs, a an arc of A, d the distance between two  vertices s and s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' We have:  (1) Card(A) = n(n − 1)/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  (2) a = {s, s′} : d(a) = d(s, s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  2On trees and co-trees, see ([7], 103-128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  10  Now let P be a polygon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' a set of edges which is a subset of A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' The support  of P will be the union of the edges of A0, that is, the set of vertices of Gk(Γ) which  are ends of at least one edge of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' One can speak of polygon P on Gk(Γ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' in  Gk(Γ)) according to whether the support of P is Gk(Γ) or a subset of Gk(Γ) distinct  from itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  A0, which is the set of all possible edges on Gk(Γ), is a complete polygon of Gk(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  A graph being the conjunction of a polygon and its support, a chain C will be a  polygon in Gk(Γ) whose vertices that form its support can be ordered in a sequence  (s0, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='sp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' We have:  (1) For every i ∈ ]P[, {si−1, si} ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  (2) For every i, j ∈ [P], i ̸= j ⇒ si ̸= sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  A cycle is a chain where condition (2) holds for all the points of its support except  s0 and sp which are merged (the ends of C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  To exhibit Gk(Γ), we need the following complementary considerations:  A) A tree is a connected polygon that does not contain a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  B) The length of a polygon is the sum of the lengths of all its edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  C) The width of a polygon is the length of its longest edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Suppose that the polygon reduced to the edge (s, s′) represents the chain of A with  minimum width joining s to s′, then {s, s′} is an element of the tree of minimal  length T on Gk(Γ) and there exists at least one such edge on A, the edge of minimal  length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Conversely, if {r, s} is an element of the tree T on Gk(Γ), then {r, s} is the chain of  A having the smallest width and joining r to s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  In this context, Gk(Γ) can be identified with a classification of classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' This  would amount to doing a factor analysis on all parts of the representative tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Such  a classification would correspond to all the axes of a factor analysis, with an original  calculation on the first axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  8  Construction of Gk(Γ)  In order to construct Gk(Γ), we must first look at the lattice Γ = Zα1 ⊕ Zα2, which  makes possible to distinguish a lattice function and a modular function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' It must be  11  assumed that the minimal bases of this lattice suppose a matroid M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' If B is the set  of these bases, then C, the set of cycles of M (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' D the set of cocycles of M) is  the set of subsets which are not included in any basis (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' which have a non-empty  intersection with any basis) and minimal for inclusion with this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Let now B ∈ B, b ∈ B, c ∈ X − B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Let D(b, B) be the unique cocycle satisfying  B ∩ D(b, B) = {b} and C(c, B), the unique cycle satisfying C(c, B) − B = {c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' We  then have:  B ∈ C(c, B) ⇐⇒ c ∈ D(b, B) ⇐⇒ B − b ∪ c ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  C and D are the sets of cycles and minimal cocycles for the inclusion of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  The minimum tree of a graph is the set of minimum edges of a cocycle, its complement  being the set of the maximum edges of a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  If we consider the lattice (F, ∪, ∩), a sublattice of M, the algebraic properties of F  (distributivity) are stronger than those of M (semi-modularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' It is thus possible  to construct Gk(Γ), the super-lattice, by defining it as the set of distributive sub-  lattices of any geometric lattice, that is to say, a sub-lattice of the semi-geometric  lattice associated with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  9  Possible applications  Let’s finish with some more epistemological considerations: after all, mathematical  physics and philosophy are not so far apart (see [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  The space associated with this "network of networks", that is, the zeros and poles  of the modular function of all networks, has been studied in hard proof theorems,  because one does not define a structure of complex analytic variety on the single com-  pactified network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' (A natural way of proceeding would be to define a compactified  isomorphism on the Riemann sphere S = C ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=')  Whatever the difficulties of study, it is proved that this network function exists, and  we have thus proved also that the set of all sets of possible flows exists as a modular  function of all networks in the algebraic sense of the term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  Let us now consider some possible applications of the previous formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Since the old work of [Von Neuman 1946], quantum mechanics represents all the  physical states of the universe by a vector space of infinite dimension called "Hilbert  space".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' However, the separability property and the convergence condition make it  12  possible to reduce to closed subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  In this case, the complex vectors form a  finite dimensional subspace and their mathematics is identical to that of flows or  tensions on a graph, except that their coefficients can take on complex values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' This  situation makes it possible, as we have seen, to apply known theorems of arithmetic  to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Because of the flow-tension duality, the network function defines as well the set of  all the sets of possible tensions, and hence it specifies the shortest path in the total  set of all possible paths, as well as the most rational scheduling of tasks in the set of  all possible actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Here we have a theorem of the existence of an optimal behavior,  whatever the field we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Moreover, the problem of the shortest path in a graph is related to the question  of the tree of minimum length, which itself formalizes the notion of classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' A  "network of networks" with a maximum voltage would thus make it possible both:  to confirm the existence of a tree of minimum length of the network of all networks,  and hence, of a classification of classifications (see [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' In general, the variable "weights" can receive different meanings (reliability, econ-  omy, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=') on a tree, other than the length of the arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' So the network of networks  Gk(Γ) can still make it possible to calculate the maximum reliability path, or the  most economical route, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', in the set of all possible paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' I will say a final word about the aim of this construction : though the world  may be multiple and chaotic, circulations and actions can be ordered in relation  to the same structure, which is expressed - in the linear case - through the form  of this remarkable holomorphic function which has been here constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Doing  that, we tried in fact to formalize the intuition of a "network of networks", as it  is expressed in the conclusion of our book on networks (see [15], 265-286).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' This is  also the achievement of what we called elsewhere a "rationalité réticulaire" (reticular  rationality)(see [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  References  [1] Berge, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
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+page_content='  [2] Deheuvels, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
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+page_content=' [English translation:  "On the translocation of  masses", Comptes Rendus (Doklady) de l’Académie des Sciences de l’U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
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+page_content='  [11] Klein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
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+page_content='  [12] Koopmans, "Optimum utilization of the transportation system", in: The Econo-  metric Society Meeting Washington, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', September 6-18, 1947;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Leavens,  (ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' ), Proceedings of the International Statistical Conferences - Volume V, 1948,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' 136-146;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' reprinted in Econometrica 17 (Supplement) 136-146, 1949;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' reprinted  in Scientific Papers of Tjalling C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Koopmans, Springer, Berlin, 184-193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  [13] Koopmans, Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', Reiter, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', "A model of transportation", Activity Analysis of  Production and Allocation - Proceedings of a Conference, Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Koopmans (ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' ),  222-259, Wiley, New York, 1951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  [14] Lur’e, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', "Methods of establishing the shortest running distances for freights  on setting up transportation systems" [in Russian], in Primenenie matematiki  èkonomicheskikh issle-dovaniyakh [Russian;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Nemchinov (ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' ), Application of  14  Mathematics in Economical Studies, Izdatel’stvo Sotsial’no-Èkonomichesk˘i Liter-  atury, 249-382, Moscow, 1959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' English translation in: V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='S Nemchinov (ed), The  Use of Mathematics in Economics, 322-355, Oliver and Boyd, Edinburgh, 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  [15] Parrochia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', Philosophie des réseaux, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', Paris, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  [16] Parrochia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', "La rationalité réticulaire", in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Parrochia (ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' ), Penser les  réaux, 7-23, Champ Vallon, Seyssel, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  [17] Parrochia D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', Neuville, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', Towards a general theory of classifications, Basel,  Birkhäuser, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  [18] Parrochia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', Mathematics and Philosophy, Wiley-Iste, London, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  [19] Robinson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', "On the Hamiltonian Game (A Traveling Salesman Problem)",  Research Memorandum RM-303, The RAND Corporation, Santa Monica, Califor-  nia, 1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  [20] Robinson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', "A Note on the Hitchcock-Koopmans Problem", Research Mem-  orandum RM-407, The RAND Corporation, Santa Monica, California, 1950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content='  [21] Schrijver, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', "On the history of the transportation and maximum flow prob-  lems", Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=' Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
+page_content=', 91, 437-445, 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
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+page_content='  15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONFIT4oBgHgl3EQfdyse/content/2301.11271v1.pdf'}
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+arXiv:2301.02245v1  [quant-ph]  5 Jan 2023
+Comments on the Bell-Clauser-Horne-Shimony-Holt
+inequality
+S. P. Sorella∗
+UERJ – State University of Rio de Janeiro,
+Physics Institute – Department of Theoretical Physics – Rua São Francisco Xavier 524,
+20550-013, Maracanã, Rio de Janeiro, Brazil
+Abstract
+We point out that the violation of the Bell-CHSH inequality in Quantum Mechanics exhibits a simple un-
+derstanding when the entangled spin singlet states are thought as the vacuum states of suitable Hamiltonians.
+The construction of the four bounded operators entering the Bell-CHSH inequality can be worked out in an
+elementary way. The inequality acquires a form in which its violation can be traced back to the vacuum prop-
+erties, a feature which enables us to make a bridge among a large class of models, whose vacuum state can be
+described by a Bogoliubov transformation as, for example: superfluids, superconductors and Quantum Field
+Theories. The examples of a pair of entangled spin 1 particles in Quantum Mechanics and of the scalar field
+in relativistic Quantum Field Theory are discussed. In the latter case, we rely on the relation expressing the
+Minkowski vacuum in terms of left and right Rindler modes. As such, the Bell-CHSH inequality turns out to be
+parametrized by the Unruh temperature.
+1
+Introduction
+The aim of this letter is that of pointing out that the Bell-Clauser-Horne-Shimony-Holt inequality [1, 2] can be
+ascribed to the vacuum state |Ω⟩, taking the form
+|⟨Ω|CCHSH|Ω⟩| = |⟨Ω|(A1 + A2)B1 + (A1 − A2)B2|Ω⟩| > 2 ,
+(1)
+where the four Hermitian operators Ai, Bi, i = 1, 2 are such that [3]
+A2
+i = B2
+i = 1 ,
+[Ai, Bk] = 0 .
+(2)
+We underline that, when rewritten in the form (1), the Bell-CHSH inequality applies to a variety of models, ranging
+from Quantum Mechanics to more sophisticated examples such as: relativistic Quantum Field Theories.
+To grasp the meaning of eq.(1), let us discuss two examples.
+∗silvio.sorella@gmail.com
+1
+
+2
+Entangled pair of spin 1 particles
+Let us start by considering a pair of entangled spin 1 particles. The singlet state reads
+|Ω⟩ =
+�|1⟩A| − 1⟩B − |0⟩A|0⟩B + | − 1⟩A|1⟩B
+√
+3
+�
+.
+(3)
+It is easily checked that expression (3) can be thought as the vacuum state of the spin Hamiltonian
+H = ⃗SA · ⃗SB = 1
+2
+�
+⃗SA + ⃗SB
+�2
+− 2 .
+(4)
+with
+�
+⃗SA, ⃗SB
+�
+denoting the spin 1 matrices. For the operators (Ai, Bi) we write
+Ai| − 1⟩A
+=
+eiαi|0⟩A ,
+Ai|0⟩A = e−iαi| − 1⟩A ,
+Ai|1⟩A = |1⟩A ,
+Bi|1⟩B
+=
+eiβi|0⟩B ,
+Bi|0⟩B = e−iβi|1⟩B ,
+Bi| − 1⟩B = | − 1⟩B ,
+(5)
+where (αi, βi) are arbitrary real coefficients. The Bell-CHSH correlator is easily evaluated, yielding
+⟨Ω|CCHSH|Ω⟩ = 2
+3 (1 − cos(α1 + β1) − cos(α2 + β1) − cos(α1 + β2) + cos(α2 + β2)) .
+(6)
+Choosing, for example, α2 = β2 = 0, α1 = π
+2 , β1 = 3π
+4 , one gets the violation1
+|⟨Ω|CCHSH|Ω⟩| = 2(2 +
+√
+2)
+3
+≈ 2.27 ,
+(9)
+which compares well with the value reported by [4].
+3
+Relativistic scalar Quantum Field Theory
+As second example, we consider a free scalar field ϕ in relativistic Quantum Field Theory. The violation of the
+Bell-CHSH has been established by [5] using Algebraic Quantum Field Theory techniques. Also here, the violation
+of the Bell-CHSH inequality can be understood in a simple way as a consequence of the entanglement properties of
+the vacuum. To that end, we make use of the expansion of the Minkowski vacuum in terms of left and right Rindler
+modes [6, 7]:
+|Ω⟩ =
+�
+i
+�
+(1 − e− 2πωi
+a )
+1
+2
+∞
+�
+ni=0
+e− πniωi
+a
+|ni⟩L|ni⟩R
+�
+,
+(10)
+1The same reasoning applies to the Bell spin 1/2 singlet state
+|Ω⟩ =
+� |+⟩A|−⟩B − |−⟩A|+⟩B
+√
+2
+�
+.
+(7)
+For the operators (Ai, Bi) we have now
+Ai|+⟩A
+=
+eiαi|−⟩A ,
+Ai|−⟩A = e−iαi|+⟩A , ,
+Bi|+⟩B
+=
+eiβi|−⟩B ,
+Bi|−⟩B = e−iβi|+⟩B .
+(8)
+Setting α1 = 0, α2 = π
+2 , β1 = π
+4 , β2 = − π
+4 , one recovers Tsirelson’s bound [3], namely |⟨Ω|CCHSH|Ω⟩| = 2
+√
+2.
+2
+
+where (|ni⟩L, |ni⟩R) are the left and right Rindler modes and T =
+a
+2π is the Unruh temperature. The relation (10)
+follows from the use of a Bogoliubov transformation applied to the the quantization of the scalar field in the Rindler
+wedges [6, 7]. Proceeding as in the case of the spin 1 example, for the four operators (Ak, Bk), k = 1, 2, we have
+now
+Ak|2ni⟩L
+=
+eiαk|2ni + 1⟩L ,
+Ak|2ni + 1⟩L = e−iαk|2ni⟩L ,
+Bk|2ni⟩R
+=
+eiβk|2ni + 1⟩R ,
+Bk|2ni + 1⟩R = e−iβk|2ni⟩R
+(11)
+; .
+(12)
+As a consequence, choosing [5]
+α1 = 0 ,
+α2 = π
+2 ,
+β1 = −π
+4 ,
+β2 = π
+4 ,
+(13)
+the Bell-CHSH inequality can be parametrized as
+|⟨Ω|CCHSH|Ω⟩| = 2
+√
+2 τ(T ) ,
+(14)
+where the form factor τ reads
+τ(T ) = 2
+�
+i
+�
+e
+πωi
+a − e− πωi
+a
+�
+�
+e
+2πωi
+a
+− e− 2πωi
+a
+� =
+�
+i
+1
+cosh( ωi
+2T ) ,
+(15)
+from which it follows that the violation of the Bell-CHSH inequality in relativistic Quantum Field Theory can be
+anlysed in terms of the Unruh temperature [8].
+4
+Conclusion
+In this note we have pointed out that the Bell-CHSH inequality acquires a general meaning when entanglement is
+seen as being encrypted into the vacuum state !Ω⟩. Thinking in that way, enables us to make a bridge among a
+large class of systems, ranging from Quantum Mechanics to Quantum Field Theory. We also underline that the
+construction of the four operators (Ai, Bi), i = 1, 2 entering the Bell-CHSH inequality becomes very simple and
+elegant, besides of having large applicability.
+Acknowledgements
+The authors would like to thank the Brazilian agencies CNPq and FAPERJ for financial support. S.P. Sorella is a
+level 1 CNPq researcher under the contract 301030/2019-7.
+References
+[1] J. S. Bell, Physics Physique Fizika 1, 195-200 (1964) doi:10.1103/PhysicsPhysiqueFizika.1.195
+3
+
+[2] J. F. Clauser,
+M. A. Horne,
+A. Shimony and R. A. Holt,
+Phys. Rev. Lett. 23,
+880-884 (1969)
+doi:10.1103/PhysRevLett.23.880
+[3] B .S . Cirelson, Lett. Math. Phys. 4, 93-100, (1980)
+[4] J . C. Howell, A . Lamas-Linares, D . Bouwmeester, Phys. Rev. Lett. 88, 030401 (2002)
+[5] Stephen J. Summers and Reinhard Werner, J. Math. Phys. 28, 2448 (1987); doi: 10.1063/1.527734
+[6] L.
+C.
+B.
+Crispino,
+A.
+Higuchi
+and
+G.
+E.
+A.
+Matsas,
+Rev.
+Mod.
+Phys.
+80,
+787-838
+(2008)
+doi:10.1103/RevModPhys.80.787 [arXiv:0710.5373 [gr-qc]].
+[7] D. Harlow, Rev. Mod. Phys. 88, 015002 (2016) doi:10.1103/RevModPhys.88.015002 [arXiv:1409.1231 [hep-th]].
+[8] In preparation
+4
+
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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='02245v1  [quant-ph]  5 Jan 2023  Comments on the Bell-Clauser-Horne-Shimony-Holt  inequality  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' Sorella∗  UERJ – State University of Rio de Janeiro,  Physics Institute – Department of Theoretical Physics – Rua São Francisco Xavier 524,  20550-013, Maracanã, Rio de Janeiro, Brazil  Abstract  We point out that the violation of the Bell-CHSH inequality in Quantum Mechanics exhibits a simple un-  derstanding when the entangled spin singlet states are thought as the vacuum states of suitable Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  The construction of the four bounded operators entering the Bell-CHSH inequality can be worked out in an  elementary way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' The inequality acquires a form in which its violation can be traced back to the vacuum prop-  erties, a feature which enables us to make a bridge among a large class of models, whose vacuum state can be  described by a Bogoliubov transformation as, for example: superfluids, superconductors and Quantum Field  Theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' The examples of a pair of entangled spin 1 particles in Quantum Mechanics and of the scalar field  in relativistic Quantum Field Theory are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' In the latter case, we rely on the relation expressing the  Minkowski vacuum in terms of left and right Rindler modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' As such, the Bell-CHSH inequality turns out to be  parametrized by the Unruh temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  1  Introduction  The aim of this letter is that of pointing out that the Bell-Clauser-Horne-Shimony-Holt inequality [1, 2] can be  ascribed to the vacuum state |Ω⟩, taking the form  |⟨Ω|CCHSH|Ω⟩| = |⟨Ω|(A1 + A2)B1 + (A1 − A2)B2|Ω⟩| > 2 ,  (1)  where the four Hermitian operators Ai, Bi, i = 1, 2 are such that [3]  A2  i = B2  i = 1 ,  [Ai, Bk] = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  (2)  We underline that, when rewritten in the form (1), the Bell-CHSH inequality applies to a variety of models, ranging  from Quantum Mechanics to more sophisticated examples such as: relativistic Quantum Field Theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  To grasp the meaning of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' (1), let us discuss two examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  ∗silvio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='sorella@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='com  1  2  Entangled pair of spin 1 particles  Let us start by considering a pair of entangled spin 1 particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' The singlet state reads  |Ω⟩ =  �|1⟩A| − 1⟩B − |0⟩A|0⟩B + | − 1⟩A|1⟩B  √  3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  (3)  It is easily checked that expression (3) can be thought as the vacuum state of the spin Hamiltonian  H = ⃗SA · ⃗SB = 1  2  �  ⃗SA + ⃗SB  �2  − 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  (4)  with  �  ⃗SA, ⃗SB  �  denoting the spin 1 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' For the operators (Ai, Bi) we write  Ai| − 1⟩A  =  eiαi|0⟩A ,  Ai|0⟩A = e−iαi| − 1⟩A ,  Ai|1⟩A = |1⟩A ,  Bi|1⟩B  =  eiβi|0⟩B ,  Bi|0⟩B = e−iβi|1⟩B ,  Bi| − 1⟩B = | − 1⟩B ,  (5)  where (αi, βi) are arbitrary real coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' The Bell-CHSH correlator is easily evaluated, yielding  ⟨Ω|CCHSH|Ω⟩ = 2  3 (1 − cos(α1 + β1) − cos(α2 + β1) − cos(α1 + β2) + cos(α2 + β2)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  (6)  Choosing, for example, α2 = β2 = 0, α1 = π  2 , β1 = 3π  4 , one gets the violation1  |⟨Ω|CCHSH|Ω⟩| = 2(2 +  √  2)  3  ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='27 ,  (9)  which compares well with the value reported by [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  3  Relativistic scalar Quantum Field Theory  As second example, we consider a free scalar field ϕ in relativistic Quantum Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' The violation of the  Bell-CHSH has been established by [5] using Algebraic Quantum Field Theory techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' Also here, the violation  of the Bell-CHSH inequality can be understood in a simple way as a consequence of the entanglement properties of  the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' To that end, we make use of the expansion of the Minkowski vacuum in terms of left and right Rindler  modes [6, 7]:  |Ω⟩ =  �  i  �  (1 − e− 2πωi  a )  1  2  ∞  �  ni=0  e− πniωi  a  |ni⟩L|ni⟩R  �  ,  (10)  1The same reasoning applies to the Bell spin 1/2 singlet state  |Ω⟩ =  � |+⟩A|−⟩B − |−⟩A|+⟩B  √  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  (7)  For the operators (Ai, Bi) we have now  Ai|+⟩A  =  eiαi|−⟩A ,  Ai|−⟩A = e−iαi|+⟩A , ,  Bi|+⟩B  =  eiβi|−⟩B ,  Bi|−⟩B = e−iβi|+⟩B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  (8)  Setting α1 = 0, α2 = π  2 , β1 = π  4 , β2 = − π  4 , one recovers Tsirelson’s bound [3], namely |⟨Ω|CCHSH|Ω⟩| = 2  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  2  where (|ni⟩L, |ni⟩R) are the left and right Rindler modes and T =  a  2π is the Unruh temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' The relation (10)  follows from the use of a Bogoliubov transformation applied to the the quantization of the scalar field in the Rindler  wedges [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' Proceeding as in the case of the spin 1 example, for the four operators (Ak, Bk), k = 1, 2, we have  now  Ak|2ni⟩L  =  eiαk|2ni + 1⟩L ,  Ak|2ni + 1⟩L = e−iαk|2ni⟩L ,  Bk|2ni⟩R  =  eiβk|2ni + 1⟩R ,  Bk|2ni + 1⟩R = e−iβk|2ni⟩R  (11)  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  (12)  As a consequence, choosing [5]  α1 = 0 ,  α2 = π  2 ,  β1 = −π  4 ,  β2 = π  4 ,  (13)  the Bell-CHSH inequality can be parametrized as  |⟨Ω|CCHSH|Ω⟩| = 2  √  2 τ(T ) ,  (14)  where the form factor τ reads  τ(T ) = 2  �  i  �  e  πωi  a − e− πωi  a  �  �  e  2πωi  a  − e− 2πωi  a  � =  �  i  1  cosh( ωi  2T ) ,  (15)  from which it follows that the violation of the Bell-CHSH inequality in relativistic Quantum Field Theory can be  anlysed in terms of the Unruh temperature [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  4  Conclusion  In this note we have pointed out that the Bell-CHSH inequality acquires a general meaning when entanglement is  seen as being encrypted into the vacuum state !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='Ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' Thinking in that way, enables us to make a bridge among a  large class of systems, ranging from Quantum Mechanics to Quantum Field Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' We also underline that the  construction of the four operators (Ai, Bi), i = 1, 2 entering the Bell-CHSH inequality becomes very simple and  elegant, besides of having large applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  Acknowledgements  The authors would like to thank the Brazilian agencies CNPq and FAPERJ for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
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+page_content=' Sorella is a  level 1 CNPq researcher under the contract 301030/2019-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
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+page_content='5373 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  [7] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' Harlow, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content=' 88, 015002 (2016) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='1103/RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='015002 [arXiv:1409.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='1231 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
+page_content='  [8] In preparation  4' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QtE0T4oBgHgl3EQfUADF/content/2301.02245v1.pdf'}
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+1 
+ 
+Surprisingly large anomalous Hall effect and giant negative magnetoresistance in half-
+topological semimetals 
+Yanglin Zhu1+, Cheng-Yi Huang2+, Yu Wang1, David Graf 3, Hsin Lin4, Seng Huat Lee1, John Singleton5, 
+Lujin Min1, Johanna C. Palmstrom5, Arun Bansil2, Bahadur Singh6*, and Zhiqiang Mao1* 
+1 Department of Physics, Pennsylvania State University, University Park, PA, 16802 
+2 Department of Physics, Northeastern University, Boston, USA, 02115 
+3National High Magnetic Field Laboratory, Tallahassee, FL, 32310 
+       4 Institute of Physics, Academia Sinica, Taipei 11529, Taiwan 
+5 National High Magnetic Field Laboratory, Pulse Field Facility, Los Alamos National Laboratory,   
+  Los Alamos, NM, 87545 
+6 Department of Condensed Matter Physics and Materials Science, Tata Institute of Fundamental   
+Research, Mumbai 400005, India    
+ 
+Abstract 
+Large intrinsic anomalous Hall effect (AHE) due to the Berry curvature in magnetic 
+topological semimetals is attracting enormous interest due to its fundamental importance and 
+technological relevance. Mechanisms resulting in large intrinsic AHE include diverging Berry 
+curvature in Weyl semimetals, anticrossing nodal rings or points of non-trivial bands, and 
+noncollinear spin structures. Here we show that a half-topological semimetal (HTS) state near a 
+topological critical point can provide a new mechanism for driving an exceptionally large AHE. 
+We reveal this through a systematic experimental and theoretical study of the antiferromagnetic 
+(AFM) half-Heusler compound TbPdBi. We not only observed an unusual AHE with a 
+surprisingly large anomalous Hall angle H (tanH  2, the largest among the antiferromagnets) 
+in its field-driven ferromagnetic (FM) phase, but also found a distinct Hall resistivity peak in the 
+canted AFM phase within a low field range, where its isothermal magnetization is nearly linearly 
+
+2 
+ 
+dependent on the field. Moreover, we observed a nearly isotropic, giant negative 
+magnetoresistance with a magnitude of ~98%. Our in-depth theoretical modelling demonstrates 
+that these exotic transport properties originate from the HTS state. A minimal Berry curvature 
+cancellation between the trivial spin-up and nontrivial spin-down bands results not only in an 
+extremely large AHE, but it also enhances the spin polarization of the spin-down bands 
+substantially and thus leads to a giant negative magnetoresistance. Our study advances the 
+understanding of the interplay between band topology and magnetism and offers new clues for 
+materials design for spintronics and other applications. 
+ 
+ 
++These two authors equally contribute to this work 
+*emails: bahadur.singh@tifr.res.in, zim1@psu.edu 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+3 
+ 
+ 
+Large intrinsic anomalous Hall effect (AHE) in topological semimetals has been a subject 
+under intensive studies. Unlike the conventional Hall effect, which is caused by the Lorentz force 
+under an external magnetic field, the intrinsic AHE stems from the net Berry curvature  (k) of 
+the band structure [1-3].  (k), which describes the geometry of the Bloch wavefunctions, is 
+determined by the topology of the band structure. When a longitudinal electric field is applied, 
+ (k) imparts transverse velocity [ E   (k)] to Bloch electrons and thus results in the AHE. To 
+observe intrinsic AHE, time-reversal symmetry (TRS) is required to be broken. Therefore, large 
+intrinsic AHE is usually expected in magnetic materials with Berry curvature hot spots near the 
+Fermi level. Prior studies have found that several types of magnetic topological materials can 
+exhibit large intrinsic AHE. These include ferromagnetic (FM) Weyl semimetals such as Co3Sn2S2 
+[4,5] and Co2MnGa [6,7], which host Weyl nodal crossings that lie close to the Fermi level and 
+carry a diverging Berry curvature. With the TRS broken by ferromagnetism, the Berry curvature 
+contribution from the Weyl nodes with opposite chirality will not cancel out, resulting in a large 
+AHE. Strength of the AHE is usually characterized by the intrinsic anomalous Hall angle (AHA) 
+H, where H = tan-1( 𝜎𝑦𝑥
+𝐴𝐻/𝜎𝑥𝑥), and 𝜎𝑦𝑥
+𝐴𝐻 and 𝜎𝑥𝑥 are the anomalous Hall conductivity and 
+longitudinal conductivity, respectively. Co3Sn2S2 and Co2MnGa both exhibit large AHA with 
+tanH = 0.2 (Co3Sn2S2) [4] and 0.12 (Co2MnGa) [8]. Antiferromagnetic (AFM) half-Heusler 
+materials such as GdPtBi [9] and TbPtBi [10] also display a large intrinsic AHE with tanH values 
+as large as 0.16-0.76 [9-12]. Although these materials harbor Weyl nodes induced by magnetic 
+fields, their large AHE does not originate from Weyl nodes but arises from the large net Berry 
+curvature produced by the anticrossing of spin-split bands near the Fermi level  [9,12]. Besides the 
+large AHE caused by the presence of anticrossing Weyl nodes or bands, recent studies show that 
+
+4 
+ 
+gapped nodal rings can also generate extremely large AHE [13,14]. This was experimentally 
+demonstrated in FM Heusler Co2MnAl [14], whose AHA reaches a record value with tanH = 0.21 
+at room temperature. In addition, the non-collinear AFM structure can lead to a large Berry 
+curvature, resulting in a large intrinsic AHE [15,16]; this has been seen in a range of AFM 
+topological materials with broken TRS, such as Mn3Sn [17] and Mn3Ge [18].   
+In this paper, we report a surprisingly large AHE in the Pd-based half-Heusler compound 
+TbPdBi. Its tanH value reaches ~ 2, which is the largest among all known magnetic topological 
+materials. Our detailed theoretical analysis suggests that such an extremely large AHE originates 
+from a half-topological semimetal (HTS) state. HTS is a long-sought topological state in materials 
+and can be viewed as a topological version of the half-metallic state in which electrons conduct in 
+only one spin channel, while the other spin channel is insulating [19-21]. Since such a state could 
+generate low power consuming spin current, it holds great promise for applications in topological 
+quantum spintronic devices [22]. Moreover, it has been theoretically shown that a gap opening at 
+the non-trivial band crossing points transforms the HTS into a quantum anomalous Hall insulator 
+[23,24]. These exotic properties have inspired extensive interest and a variety of material systems 
+have been predicted to be HTS, such as two-dimensional(2D) MnN [25], PrOBr [26], and PtCl3 
+[24]; quasi-1D X2RhF6 (X=K, Rb, Cs) [27] and XYZ3 (X=Cs, Rb, Y=Cr, Cu, Z=Cl, I) [28]; 3D 
+MF3 (M=Pd, Mn) [29], LiV2F6 [30], etc. The Dirac/Weyl points or nodal rings in all these materials 
+are comprised of spin-polarized bands [21,30,31]. However, these theoretical predictions are still 
+awaiting experimental verifications. Our work here demonstrates that TbPdBi hosts a unique HTS 
+in proximity to a topological critical point. Such a peculiar HTS in TbPdBi results in not only an 
+unusually large AHE but also leads to a giant negative magnetoresistance. Additionally, we find 
+that the Hall resistivity of TbPdBi exhibits a distinct anomalous peak in the low field range, where 
+
+5 
+ 
+its isothermal magnetization nearly linearly depends on field and this behavior can be understood 
+in terms of the Berry curvature enhancement induced by spin canting.   
+Single crystals of TbPdBi were grown using the Bi-flux method (see Methods). They 
+exhibit semi-metallic behavior, manifested by the broad peak in the temperature dependence of 
+resistivity xx around 50K (see supplemental Fig. S1a). The magnetic susceptibility ( ) 
+measurements (Fig. S1a) show its AFM state with TN = 5.2 K. xx exhibits a steeper drop below 
+TN, indicating electronic transport coupled with magnetism. We performed a Curie-Weiss (CW) 
+fit for its temperature dependence of susceptibility (T); the best fit was obtained in the 
+temperature range of 100-300K (Fig. S1b) which yields the effective magnetic moment (eff) of 
+9.4 B/Tb, consistent with a prior report [32].    
+From Hall resistivity xy measurements under high magnetic fields, we observed 
+exceptionally strong AHE in TbPdBi. Figure 1a shows the xy data measured up to 31T at various 
+temperatures. The most significant feature of the xy data of TbPdBi is that it exhibits a striking 
+peak. This peak not only occurs below TN (= 5.2K) but also extends to temperatures above TN. The 
+peak field is ~ 5T below TN, and slightly shifts to high field with increasing temperature. xy(B) 
+gradually evolves into a linear field dependence after exhibiting a peak and this occurs above 15T 
+for T < TN, at higher temperatures (e.g. 15K & 20K), due to the shift of the peak to higher field, 
+the linear trend develops at higher fields, with the linear slope remaining similar to those of low 
+temperatures. Such a distinct peak in xy has never been observed in any other half-Heusler 
+compounds, full-Heusler antiferromagnets, or conventional ferromagnets. As noted above, prior 
+work has shown isostructural half Heusler compounds (Gd/Tb)PtBi also exhibits large AHE [9-
+12]; their xy data also display anomalous peaks near 4.5T where their AHAs achieve maxima with 
+
+6 
+ 
+tanH = 0.16-0.76  [9-12]. However, the xy anomalous peak of (Tb/Dy)PtBi is far weaker than 
+that of TbPdBi. For comparison, we have added the xy data of TbPtBi at 1.7K to Fig. 1a. While 
+its weak xy peak near 4.5T can be clearly resolved when the data is zoomed into the field range 
+of 0-9T (see Fig. 1(c) in ref. [12]), it is hardly discernible when this data is plotted together with 
+the data of TbPdBi in the field range up to 31T as shown in Fig. 1a, suggesting that the replacement 
+of Pt by Pd leads to essential changes to the band structure, as discussed below.  In the field regime 
+where xy exhibits an anomalous peak, the longitudinal resistivity xx displays a drastic decrease, 
+followed by a saturation trend in the high field range where xy evolves into a linear field 
+dependence, as shown in Fig. 2b which presents representative xx and xy data at 4K. Note that 
+the anomalous xy peak near 5T is not caused by the xx component which might not be removed 
+from the xy data’s anti-symmetrizing process; this can be seen clearly from the raw data of xy and 
+xx shown in supplementary Fig. S2 which shows the xx component in xy is negligibly small. To 
+find whether the unusual field dependences of xy and xy arise from a magnetic transition, we 
+measured the magnetization of TbPdBi up to 35T at various temperatures (see Methods) and added 
+the data from these measurements to Fig. 1b.  From these data, we find a spin-flop transition near 
+15T and the saturated magnetic moment of Tb3+ in the FM phase is ~8.5B/f.u. at 0.57K, 
+comparable with the effective magnetic moment extracted from the CW fit (eff ~ 9.4 B/Tb).  The 
+unusual field dependences of xx and xy are indeed coupled with such a magnetic spin-flop 
+transition. The xy peak as well as the sharp drop of xx are present in the canted AFM state, while 
+the linear field dependence of xy and the xx saturation behavior occur in the polarized FM phase. 
+These observations imply that the spin-flop transition leads to an electronic structure transition. 
+
+7 
+ 
+As to be shown below, our theoretical calculations shows that such a spin-flop transition gives rise 
+to a HTS state.  
+Given that the remarkable ρxy peak is present in the CAFM state, its origin is most likely 
+associated with Berry curvature induced by noncolinear spin structure. In general, ρxy of magnetic 
+systems can be described by  ρxy= R0B + RSM +ρxy
+T  , where the first term is normal Hall contribution 
+(R0, the normal Hall coefficient), the second term represents the anomalous Hall resistivity linearly 
+coupled with magnetization M (RS, the anomalous Hall coefficient), and the last term ρxy
+T  is the 
+anomalous Hall resistivity arising from Berry curvature effects. The remarkable ρxy peak of 
+TbPdBi present in the canted AFM state suggests it involves a significant component of anomalous 
+Hall resistivity Δρxy
+AH(=ρxy- R0B).  To find how the magnetization contributes to Δρxy
+AH, we have 
+plotted Δρxy
+AH as a function of magnetization in supplementary Fig. S3 (Note that Δρxy
+AH is obtained 
+by subtracting the normal Hall contribution, i.e. the dashed line in Fig. 1a which is inferred from 
+the high-field linear field dependence of  ρxy(B)). We find Δρxy
+AH strongly deviates from linear 
+dependence on magnetization and exhibits a striking peak, indicating the anomalous Hall 
+resistivity of TbPdBi involves a significant contribution of ρxy
+T . The maximal Δρxy
+AH  near 5T is  ~ 
+0.67 mΩ cm (supplementary Fig. S4a), almost 5-15 times larger than that in (Gd/Tb)PtBi [9,12]. 
+From Δρxy
+AH, we derive anomalous Hall conductivity σyx
+AH using σyx
+AH = Δρxy
+AH/(ρxy2 +ρxx2) and AHA 
+as shown in supplementary Fig. S4b and Fig. 1c. The maximal value of tanH is ~ 2 at about 20T 
+and 10K (Fig.1c), which is surprisingly large as compared to other magnetic topological materials, 
+as shown in Fig. 1d, which plots tanH versus σyx
+AH for TbPdBi and other magnetic topological 
+materials showing large AHE. Such an extremely large AHE seen in TbPdBi is unlikely induced 
+
+8 
+ 
+by extrinsic mechanisms such as skew scattering or side jump, since their induced Hall angle is 
+usually a few percent [33,34]. The intrinsic mechanisms due to Berry curvature should play a key 
+role in generating such an unusually large AHE in TbPdBi as discussed below. Previous studies 
+have shown Skyrmion magnetic lattices could give rise to an anomalous peak in ρxy  (i.e. 
+topological Hall effect [35]). Compared to the topological Hall effect of a prototype Skyrmion 
+system MnSi where anomalous Hall resistivity jump (i.e. ρxy
+T ) induced by the spin-texture is less 
+than 0.04 µΩ cm [36], our observed maximal Δρxy
+AH value of ~0.67 mΩ cm in TbPdBi is four orders 
+of magnitude larger. This implies that the extremely large AHE in TbPdBi should have a unique 
+mechanism. As to be shown below, it is indeed associated with a peculiar HTS state.  
+The steep decrease of xx in field regions where xy exhibits an anomalous peak (Fig. 1b) 
+suggests that TbPdBi exhibits large negative magnetoresistance. Although we observed large 
+negative magnetoresistance in our earlier low-field ( 9T)  measurements on TbPdBi [37], its 
+origin remained mysterious; it was also unclear whether its magnetoresistance saturates in high 
+field range. With the advantage of high-field measurements and theoretical analyses, we have an 
+opportunity to address these issues in this work. Figures 2a-2b present the transverse and 
+longitudinal magnetoresistivity MR (= 
+𝑥𝑥(𝐵)− 𝑥𝑥(0)
+𝑥𝑥(0)
+) of TbPdBi, measured with the current 
+applied perpendicular and parallel to the magnetic field respectively at various temperatures (see 
+the insets to Fig. 2a & 2b) (Note that these data were measured on the identical sample which was 
+used for the Hall resistivity measurements shown in Fig. 1a). Both transverse and longitudinal MR 
+decreases steeply with increasing magnetic field, and then tend to saturate above 15T; the 
+magnitude of the MR’s drop reaches ~98% as the field rises to 15T. Such a giant negative MR is 
+observed at both T  TN and T >TN. The magnitude of MR remains nearly temperature independent 
+
+9 
+ 
+below 20K, but gradually decreases with increasing temperature above 20K (Fig. 2c). Even as the 
+temperature rises to 200K, the negative MR remains significant, with its magnitude being ~ 20% 
+at 9T (Fig. 2c). The MR changes from negative to positive only at room temperature, with the 
+magnitude of positive MR being much smaller than that of negative MR at lower temperatures. 
+Since TbPdBi is a superconductor with Tc = 1.7K, its MR data measured at 1.7K (the base 
+temperature of the measurement system) first shows a steep increase as the superconducting state 
+is suppressed by magnetic field, then followed by the steep decrease as discussed above (see 
+supplementary Fig. S5). We chose to not include this data in Fig. 2a-2b to avoid complications.  It 
+is worth pointing out that giant negative MR observed in TbPdBi does not occur to isostructural 
+compounds (Gd/Tb/Dy)PtBi [10-12,38,39], which again suggests TbPdBi has a distinct electronic 
+state from (Gd/Tb/Dy)PtBi.  
+Negative MR has been observed in various material systems and originates from several 
+different mechanisms. In topological Weyl semimetals, the topological current induced by the 
+chiral anomaly could lead to large negative MR when the magnetic field is parallel to the current, 
+as observed in GdPtBi [11,38] and TbPtBi [10,12]. When the magnetic field is rotated away from 
+the current, the chiral anomaly is gradually suppressed so that the sign of MR could change from 
+negative to positive above a certain rotation angle. Prior studies have also shown materials with 
+macroscopic disorders may also cause negative longitudinal MR, e.g. polycrystalline Ag2+δSe [40], 
+gallium arsenide Quantum wells [41], and disordered topological insulator TlBi0.15Sb0.85Te2 [42]. 
+Another mechanism is the spin-scattering suppression driven by spin flip/flop transition; for 
+instance, the colossal MR in Ca3Ru2O7 can be attributed to this mechanism [43].  Apparently, none 
+of the above mechanisms is relevant to our observation of giant negative MR in TbPdBi. This is 
+because that (i) our observed negative MR is nearly independent of field orientation and tends to 
+
+10 
+ 
+saturate in the high field regime, which excludes the chiral anomaly effect, (ii) our samples are 
+high-quality single crystals and do not involve macroscopic disorders, and (iii) the giant negative 
+MR of TbPdBi occurs at both T  TN and T >TN. If its giant negative MR was due to the spin-
+scattering suppression in the spin flop transition, we would expect its MR’s magnitude should 
+significantly drop as the spin-flop transition is suppressed at T >TN, which is inconsistent with the 
+observation of the temperature independence of MR between TN and 20K and the survival of large 
+negative MR in a wide temperature region above 20K (Fig. 2c).  Additionally, we note that large 
+negative  transverse/longitudinal MR was recently reported in EuMnSb2 [44] and EuB6 [45]. The 
+origin of large negative MR in EuMnSb2 is ascribed to the field-induced metal-insulator transition, 
+while the large negative MR of EuB6 is suggested to be from the charge carriers’ high spin 
+polarization arising from a half Weyl semimetal [45]. As to be shown below, our band structure 
+calculations for TbPdBi also reveals a high spin-polarization state, which is caused by a unique 
+FM HTS state close to the topological critical point. Our observed giant negative MR in TbPdBi 
+provides strong support for such a HTS state.  
+To understand the unusual transport properties of TbPdBi discussed above, we now present 
+our detailed theoretical analysis of the electronic and transport properties of TbPdBi. Figures 3a-c 
+show the arrangement of atoms in nonmagnetic, ferromagnetic (FM) with ordering vector [100], 
+and A-type AFM state with propagation vector [111]. The high-symmetry nonmagnetic state of 
+TbPdBi is described by the inversion asymmetric space group 𝑇𝑑
+2 (𝐹4̅3𝑚 , No. 216). On 
+considering the [100] FM state, the symmetry of the lattice reduces to fourfold rotary reflection  
+𝑆̅4, which combines the fourfold rotational and mirror symmetries. The associated band structure 
+is semimetallic as shown in Fig. 3d without spin-orbit coupling. Importantly, the band structure 
+constitutes a band inversion between Γ8 (p-type) and  Γ6 (s-type) states reminiscent of half-Heulser 
+
+11 
+ 
+materials only in the spin-down states whereas spin-up states remain uninverted. Such a unique 
+band structure with single spin-channel band inversion can be closely associated with the half-
+metallic state of magnets and thus termed as HTS here. This HTS state can constitute multiband 
+anticrossing points comprised of opposite spin channels. In the presence of SOC, the band 
+anticrossing points are gapped, inducing a non-zero Berry curvature field and leading to large AHE 
+[see Fig. 3e]. Figure 3f shows the band structure of the A-type AFM state of TbPtBi (ground state). 
+The associated crystalline symmetries are threefold rotational symmetry (C3) about [111] direction 
+and spacetime symmetry 𝑇̃ = {𝑇|
+1
+2
+1
+2
+1
+2}, which combines both time-reversal symmetry and half-
+translation along [111] direction. The band structure is semimetallic with various band crossings 
+along the Γ − 𝑍 line and constitutes band inversion in both the spin-channels. This nontrivial band 
+inversion AFM state evolves to an HTS state with a single-spin channel band inversion in presence 
+of the magnetic field.  Such a band structure evolution with the magnetic field can lead to an 
+intermediate spin-canted state with various band anticrossing points, enhancing the Berry 
+curvature field and associated AHE.  
+We want to emphasize that the distinct anomalous Hall resistivity peak of TbPdBi occurs 
+near 5T (Fig.1a) where the system is in a spin-canted AFM state. We can therefore expect an 
+additional contribution to the AHE stemming from the canted AFM state. As noted in the 
+introduction, the non-collinear spin texture could induce large non-vanishing Berry curvature. 
+Since our TbPdBi possess the same magnetic structure as that of GdPtBi/TbPtBi [46], i.e., the 
+magnetic moments of Tb order ferromagnetically on the (111) planes, but are 
+antiferromagnetically coupled along the [111] direction. When the magnetic field is applied long 
+[100] direction, the magnetic moments will gradually tilt toward this direction, resulting in a spin-
+canted state.  A recent theoretical study indicates that, in AFM nodal line materials AMnBi2 (A=Ca, 
+
+12 
+ 
+and Yb), a strong AHE could be induced by a weak spin canting, and the anomalous Hall 
+conductivity keeps growing as the canting angle increases [47]. The remarkable anomalous Hall 
+resistivity peak near 5T in TbPdBi probably is likely accredited to the canted spin state formed by 
+Tb+ ions. Such an interpretation is supported by our Berry curvature calculation at spin canted state, 
+as discussed below. 
+To calculate the band structure and anomalous Hall conductivity (AHC) in the spin-canted 
+states of TbPdBi under an external magnetic, we employ the virtual crystal approximation (VCA) 
+with AFM and FM states model Hamiltonians as end members. The VCA Hamiltonian Hvca of the 
+spin canted state is defined as follows, 
+𝐻vca = (1 − 𝑥)𝐻AFM + 𝑥𝐻FM, 
+where 𝐻AFM (𝐻FM) is the Hamiltonian for AFM (FM) phase and x is the tuning parameter from 0 
+to 1. x = 0 (1) denotes purely AFM (FM) phase while 0 < x <1 describes the spin-canted states. In 
+the ground state (x = 0), the dominant Berry curvature Ωyz resides on anticrossing points in the ΓZ 
+direction as shown in Fig. 4a. With an increase of x to 0.1 (low external magnetic field), the strong 
+Berry curvature Ωyz although remains on the ΓZ path, the finite FM coupling splits the oppositely 
+spin-polarized bands, shifting the Berry curvature hot-spots to distinct energies [Fig. 4b]. In the 
+fully polarized FM phase, the Berry curvature Ωyz around Γ becomes significantly small [see Fig. 
+4c]. Calculated AHC for x = 0 ~ 1 as a function of energy is shown in Fig. 4d. The AHC is nearly 
+zero at x = 0 due to Berry curvature cancellation induced by effective  𝑇̃ symmetry. A dramatical 
+peak emerges near 0.1 eV above the Fermi level at x = 0.1 because the band splitting caused by 
+breaking both C3 and 𝑇̃ reduces the Berry curvature cancellation. The AHC peak gradually flattens 
+and shifts to a higher energy with increasing x. The negative AHC at the Fermi level grows 
+
+13 
+ 
+monotonically from x = 0 to 1, which implies that there are non-zero Berry curvatures away from 
+Γ point in the spin-canted AFM states.  
+      
+We consider the doping effect on the AHC behavior in the magnetic transition. For each 
+doping concentration, the chemical potential is determined by fixing the number of occupied states. 
+Positive (negative) doping concentration Δn indicates the number of occupied states above (below) 
+the original occupied particle number, i.e., Δn = n-n0, where n (n0) is the current (origin) occupied 
+particle number per formula unit cell. When Δn = 0.0085/f.u., σyz reaches a maximum near x = 0.1, 
+as shown in Fig. 4e, qualitatively consistent with our experimental observation (Fig.1a). Note that 
+the VCA cannot fully capture the behavior of magnetic state evolution under an external magnetic 
+field due to the lack of actual correlated-electronic corrections effects. In this sense, x is not 
+quantitatively comparable to the external field applied during the experiment. However, this 
+approach can give a reasonable physical insight into the band structure effects in spin-canted states. 
+On comparing to the experimental AHC (Fig. S4b), our calculated AHC are smaller than the 
+experimental value. Such a discrepancy further suggests that effects other than those not captured 
+in our VCA calculations are also at play. 
+Next, we first discuss the origin of the large Berry curvature in our system. Fig. 3d-e show 
+the band structure without and with SOC when the magnetic field is applied along [100] direction. 
+The band degeneracy lifting by exchange interaction is commonly seen in other Heusler 
+compounds, such as GdPtBi/TbPtBi [9,11,12]. However, the weaker exchange interaction of Tb 
+ions and nearly zero band inversion strength lead to the band splitting in TbPdBi smaller than that 
+in (Gd/Tb)PtBi. As a result, the spin-split bands create more crossing points near the Fermi-level. 
+A small gap is opened at the anti-crossing points with SOC, and the large non-vanished berry 
+curvature is accumulated here.  
+
+14 
+ 
+We note that in REPtBi (RE = Gd, Nd), the non-vanishing Berry curvature is induced by 
+the field-induced Weyl state. The location and number of Weyl nodes depend on the direction of 
+applied magnetic field B. For TbPdBi, our calculation indeed reveals a Weyl state. We found that 
+two pairs of Weyl nodes appear around 𝛤 point in the FM phase driven by the magnetic field 
+applied along [100] axis; the positions of these Weyl nodes are summarized in supplementary table 
+I. However, these Weyl nodes are far away from the Fermi level, which cannot induce large AHE. 
+Furthermore, the absence of chiral anomaly revealed in our magnetotransport measurements 
+indicates that the Weyl nodes do not contribute to the transport properties of TbPdBi. Hence, we 
+believe the giant AHC in TbPdBi cannot be attributed to the Weyl state.  
+The second origin of the large Berry curvature is the spin-canted state. As revealed by our 
+calculations, increasing the canted angle widens the band splitting and enlarges the gap at the anti-
+crossing points, which results in the Berry curvature along yz direction (Ωyz) continually increasing. 
+However, the Berry curvature arises from the anti-crossing points and the spin-canted state is not 
+the unique property for TbPdBi. It also has been observed in other half-Heusler compounds such 
+as GdPtBi [9,11], TbPtBi [12], and DyPtBi [39]; their anomalous Hall conductivity and Hall angle 
+are much smaller than that in TbPdBi. One may wonder what makes TbPdBi so unique. Our above 
+band structure calculations reveal TbPdBi shows a distinct topological state: TbPdBi is at the 
+critical point at the AFM ground state since its band inversion strength is nearly zero. At the field-
+driven FM state, it evolves to a HTS state: its spin-down bands maintain band inversion, while its 
+spin-up bands lose their band inversion and become topological trivial. We note a similar state has 
+been predicted in EuB6 [48], where a large AHE and large negative MR have been recently 
+reported [45]. Such findings imply that the HTS state probably plays a crucial role in enhancing 
+the AHE in topological materials.  As we stated before, the net Berry curvature requires the sum-
+
+15 
+ 
+up of the Berry curvature among all the occupied states. Thus, the number of the bands crossing 
+the Fermi level could affect the Berry curvature cancellation. In general, the compounds showing 
+good-metal behavior usually exhibit a large Berry curvature cancellation, resulting in a small net 
+Berry curvature. However, in some compounds with bad-metal or semimetal behavior, only a few 
+bands cross through the Fermi level, which could reduce the possibility of Berry curvature 
+cancellation. In TbPdBi, the spin-up and spin-down bands are pushed to opposite directions, with 
+a bandgap opening in the spin-up bands, which substantially reduces the number of the bands 
+which cross the Fermi level, thus minimizing the Berry curvature cancellation.   
+The exotic HTS state in TbPdBi leads to not only a large AHE but also a giant isotropic 
+negative MR. As described above, the spins are polarized near the Fermi level (spin-polarization 
+reaches 40%) when applying an external magnetic field, such that the carriers from spin-down 
+bands dominate the transport properties of TbPdBi.  In this case, spin-scattering is significantly 
+suppressed across the spin-flop transition, which leads to steep resistivity drop with the increase 
+of magnetic field. When the magnetic field reaches above 15T, the spins are fully polarized (Fig. 
+1b), so the MR remains constant. Since the large negative MR is attributed to such an exotic 3D 
+band structure, it is expected to be independent of the field orientation, which is exactly what we 
+observed in experiments (Fig. 2a & 2b). The temperature independence of MR(B) between 
+TN(=5.2K) and 20K (Fig. 2a&2b) can also well understood in terms of the field-induced FM HTS. 
+Although there is not a long-range spin-flop transition above TN, short-range AFM should exist in 
+a temperature range above TN at zero field so that a crossover-like transition from PM to FM should 
+occur under high magnetic fields, which is indeed manifested in the magnetization data measured 
+at 20K as shown in Fig. 2b. This indicates a FM HTS can extend to temperatures above TN under 
+high magnetic fields. However, when the temperature is increased above 50K, thermal excitation 
+
+16 
+ 
+would bring out the electrons from the spin-up bands, thus increasing the spin-scattering and 
+leading to the decrease of the magnitude of negative MR as shown in Fig. 2c.  
+In summary, we have observed an unusual AHE effect with a surprisingly large anomalous 
+Hall angle (tanH  2) and a giant negative MR in TbPdBi. Our theoretical analysis indicates that 
+these exotic transport properties originate from the unique HTS state of TbPdBi, dominated by the 
+contribution of spin-down bands. The greatly enhanced AHE results from a significantly reduced 
+Berry curvature cancellation between the spin-down and spin-up bands. Spin canting of the AFM 
+state under magnetic field increases the number of band anticrossing points and widens the gaps, 
+which in turn increases the Berry curvature hot spots near the Fermi level and accounts for the 
+distinct anomalous Hall resistivity peak at low fields. We also find that the HTS state enhances the 
+spin-polarization for the spin-down band in the FM phase, which explains the giant negative MR.  
+Our study unveils a new pathway for generating an extremely large AHE by tuning the electronic 
+structure and the magnetic state in half-Heusler materials. It thus advances the understanding of 
+the interplay between topological state and magnetism and provides clues for materials design for 
+spintronics and applications. 
+Methods 
+The single crystals of TbPdBi were synthesized using the Bi-flux method  [49]. The starting 
+materials of Tb, Pd, and Bi powders were mixed with a molar ratio of Tb: Pd: Bi=1:1:20, loaded 
+into Al2O3 crucibles, and sealed in quartz tubes under high vacuum. The mixtures of source 
+materials were heated to 1050°C in a crucible furnace and held at this temperature for 48 hours for 
+homogeneously melting, then followed by slow cooling down to 700°C at a rate of 3° C /h. Cubic-
+shape single crystals of TbPdBi were obtained by removing the excess Bi flux through centrifuging. 
+The structures and compositions of the grown crystals are confirmed by X-ray diffraction 
+
+17 
+ 
+measurements and Energy-dispersive X-ray spectroscopy (EDS). Magnetoresistivity and Hall 
+resistivity measurements were performed using a standard six-probe method in a Physical Property 
+Measurement System (PPMS, Quantum Design). The high-field transport measurements were 
+carried out at the National High Magnetic Field Laboratory (NHMFL) in Tallahassee. The 
+magnetoresistivity and Hall resistivity data presented in this paper are obtained through 
+symmetrizing and antisymmetrizing the longitudinal and transverse Hall resistivity data measured 
+at positive and negative magnetic fields respectively. All the samples used for transport 
+measurements were polished to rectangular shapes with dimensions of ~0.8mm × 0.7mm × 0.2mm, 
+and the polished surfaces were parallel to the (001) plane. The current was applied to the [100] or 
+[001] directions for both longitudinal resistivity and Hall resistivity measurements. Magnetization 
+measurements were measured using the National High Magnetic Field Pulsed Field Facility at Los 
+Alamos National Laboratory and the SQUID magnetometer (Quantum Design).  
+Electronic structure calculations were performed within the density functional theory 
+framework using the Vienna ab-initio simulation package (VASP) based on the projector-
+augmented wave method [50-52].The generalized gradient approximation (GGA) was employed 
+to include the exchange-correlation effects [53] and an on-site Coulomb interaction was added for 
+Tb f-electrons within the GGA+U scheme with Ueff = 10 eV. A plane wave cut-off energy of 500 
+eV was used and a 11 × 11 × 11 Γ centered k-point mesh to sample the first Brillouin zones of 
+cubic and trigonal unit cells associated with various magnetic ordering. Spin-orbit coupling effects 
+were included self-consistently. Transport properties were calculated using a first-principles 
+material-specific, 
+effective 
+tight-binding 
+model 
+Hamiltonian 
+generated 
+using 
+the 
+VASP2WANNIER90 interface [54]. Tb d and f orbitals, Pd s, p, and d orbitals, and Bi p orbitals 
+were included in constructing the wannier functions.  
+
+18 
+ 
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+Acknowledgment 
+The experimental part of this work is based upon research conducted at The Pennsylvania 
+State University Two-Dimensional Crystal Consortium–Materials Innovation Platform (2DCC-
+MIP), which is supported by NSF Cooperative Agreement No. DMR-2039351. Z.Q.M. also 
+acknowledges the support from NSF under Grant No. DMR 2211327. The work at Northeastern 
+University was supported by the Air Force Office of Scientific Research under award number 
+FA9550-20-1-0322 and benefited from the computational resources of Northeastern University's 
+Advanced Scientific Computation Center (ASCC) and the Discovery Cluster. The work at the 
+National High Magnetic Field Laboratory is supported by the NSF Cooperative Agreement 
+No.DMR-1644779 and No. DMR-1157490 and the State of Florida. The work at TIFR Mumbai is 
+supported by the Department of Atomic Energy of the Government of India under Project No. 12-
+R&D-TFR-5.10-0100.  
+Author contributions 
+The crystal growth and transport measurements were carried out and analyzed by Y.L.Z, Y.W., & Z.Q.M. 
+The high magnetic field measurements were carried out by Y.L.Z., D.G. S.H.L., L.J.M., J.C.P. & J. S. The 
+theoretical work was done by C.Y.H., H.L., B.S. & A.B. This work is supervised by Z.Q.M. (experiment) 
+and B.S. and A.B. (theory).  
+ 
+ 
+ 
+ 
+ 
+
+22 
+ 
+            
+ 
+Figure 1. (a) Hall resistivity of TbPdBi as a function of the magnetic field B up to 31T at various 
+temperatures. The Hall data have been anti-symmetrized to remove the minor component of 𝜌𝑥𝑥. 
+Inset: the schematic of the experimental setup for Hall measurements. (b) Left axis: longitudinal 
+resistivity 𝜌𝑥𝑥 and Hall resistivity 𝜌𝑥𝑦  at 4K for TbPdBi (marked as hollow circles). Right axis: 
+magnetization measured at various temperatures (solid line). (c) The tanΘ𝐻  of TbPdBi as a 
+function of magnetic field B at 4K (below Neel temperature),7K and 10K (above Neel temperature). 
+(d) The comparison of 𝜎𝑦𝑥
+𝐴𝐻 and tan Θ𝐻  between TbPdBi and other magnetic conductors 
+[4,6,8,9,12,14,17,18,39,45,55-57].  
+ 
+ 
+
+(a)
+1.2
+(b)
+10
+1.7K
+TbPdBi
+M(0.57K)
+4K
+0.8
+7K
+1.6
+T= 4K
+8
+cm
+10K
+5K
+15K
+u)
+20K
+( u)d
+1.2
+20K
+6
+(HB)
+0.0
+TbPtBi
+TbPdBi
+1.7K
+0.8
+4
+/f.u.)
+-0.4
+B
+Pxy
+2
+0.4
+-0.8
+(100)
+-1.2
+0.0
+-30
+-20
+-10
+0
+10
+20
+30
+0
+5
+10
+15
+20
+25
+30
+35
+B (T)
+B (T)
+3.0
+(d)
+(c)
+4K
+1500
+4 DyPtBi
+TbPdBi
+7K
+★
+1200
+Co,MnAl
+10K
+Fe,Sn
+Co,Sn,S
+EuB
+Co,MnGa
+1.5
+r.0)
+006
+?
+TbPtBi
+600
+Mn,Ge
+TbPdBi
+300
+MnSi
+GdPtBi
+EuCd,As,
+0.0
+oMn,Sn
+0
+10
+20
+30
+0.0
+0.5
+1.0
+1.5
+2.0
+B (T)
+tan@?23 
+ 
+ 
+Figure 2. Magnetoresistivity (MR) of TbPdBi: (a-b) Transverse (a) and longitudinal (b) MR = Δρ 
+/ρ0 = [ρ(B)–ρ(B = 0)]/ρ(B = 0) at 4K (below Neel temperature), 10K and 20K (above Neel 
+temperature). The MR at 4K and 10K were measured up to31T, while the MR for 20K was 
+measured up to 18T. The Insets show the schematic of the experimental setup for transverse and 
+longitudinal MR measurements. (c) Longitudinal MR versus magnetic field measured in PPMS at 
+various temperatures.  
+ 
+ 
+
+a)
+b
+TbPdBi B I I
+300 K
+B
+20
+-20
+0
+200 K
+TbPdBi
+TbPdBi
+-40
+(100)
+(100)
+-30
+MR
+20K
+MR
+20K
+MR
+-60
+-60
+100 K
+10K
+10K
+-60
+50 K
+-80
+4K
+-80
+4K
+4 K
+-90
+-100
+-100
+10 K
+-30
+-20
+-10
+0
+10
+20
+30
+-30 -20 -10
+10
+20
+30
+-10
+-5
+0
+5
+10
+B (T)
+B (T)
+B (T)24 
+ 
+Figure 3.  (a) Crystal structure of half-Heusler TbPdBi. [111] and [100] vectors denote the principal 
+magnetic axes for AFM and FM states, respectively. Spin structures of (b) [100] FM and (c) [111] 
+AFM magnetic structure in FM. Spins are aligned ferromagnetically in a (111) plane and 
+antiferromagnetically between the (111) planes in the AFM state shown in (c). (d) Calculated spin-
+resolved bulk band structure of FM TbPdBi without SOC considering the primitive unit cell. (e) 
+Spin-polarized band structure with SOC in (e) FM and (f) AFM states. Trigonal supercells are 
+considered in (e) and (f). Insets in (d)-(e) show the first Brillouin zones associated with the unit 
+cells employed in the calculations.  
+ 
+ 
+ 
+ 
+ 
+
+(a)
+(b)
+(c)
+[111]
+Tb
+Pd
+Bi
+[100]
+(d)
+(e)
+(f)
+S(111]
+1.2
+60
+60
+(%) I
+0.6
+0.6
+Spin-polarization 
+0.6
+e
+0.0
+0.0
+0
+0.0.
+0
+E
+0.6
+-0.6
+0.6
+1.2.
+1.2
+-60
+1.21
+-60
+M
+L
+r
+X
+M
+K
+Z
+F
+L
+r
+Z
+F
+L25 
+ 
+ 
+Figure 4.  Berry curvature Ωyz distribution of TbPdBi band structures for (a) x = 0 (AFM), (b) x = 
+0.1 (spin canted states) and (c) x =1 (FM) along selected high-symmetry directions. (d) Calculated 
+anomalous Hall conductivity σyz as function of energy for varying x = 0 (dark blue curve) to x = 1 
+(red curve). (e) Anomalous Hall conductivity as a function of x with Δn = 0.0085/f.u. See the text 
+for more details.  
+ 
+ 
+(a)
+(b)
+(c)
+σyz(Ω-1cm-1)
+E-EFM/AFM (eV)
+x = 0
+x = 0.1
+x = 1
+Ωyz(nm2)
+E-EFM\AFM (eV)
+Z
+Γ
+F
+Z
+Γ
+F
+E-EF,AFM (eV)
+Ωyz(nm2)
+Ωyz(nm2)
+E-EF,AFM (eV)
+Z
+Γ
+F
+x = 0 (AFM)
+x = 0.1 (canted state)
+x = 1 (FM)
+(d)
+(e)
+x
+σxy
+σyz
+σzx
+σ (Ω-1cm-1)
+
+0.2
+100
+0.1
+50
+0
+0
+-0.1
+-50
+-0.2
+1000.2
+100
+0.1
+50
+11
+0
+0
+-0.1
+-50
+-0.2
+-1000.2
+100
+0.1
+50
+0
+0
+-0.1
+-50
+11
+-0.2
+-100100
+50
+-50
+-100
+-150
+-0.5-0.4-0.3-0.2-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5120
+100
+80
+60
+40
+20
+0
+0
+0.05
+0.1
+0.15
+0.2
\ No newline at end of file
diff --git a/RtAzT4oBgHgl3EQfJPsM/content/tmp_files/load_file.txt b/RtAzT4oBgHgl3EQfJPsM/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..9ad20193c8edb79b1eb5ff6802cab291b0b3349a
--- /dev/null
+++ b/RtAzT4oBgHgl3EQfJPsM/content/tmp_files/load_file.txt
@@ -0,0 +1,1080 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf,len=1079
+page_content='1  Surprisingly large anomalous Hall effect and giant negative magnetoresistance in half-  topological semimetals  Yanglin Zhu1+, Cheng-Yi Huang2+, Yu Wang1, David Graf 3, Hsin Lin4, Seng Huat Lee1, John Singleton5,  Lujin Min1, Johanna C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Palmstrom5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Arun Bansil2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Bahadur Singh6*,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' and Zhiqiang Mao1*  1 Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' PA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 16802  2 Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Northeastern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' USA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 02115  3National High Magnetic Field Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Tallahassee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' FL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 32310  4 Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Academia Sinica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Taipei 11529,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Taiwan  5 National High Magnetic Field Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Pulse Field Facility,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Los Alamos National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Los Alamos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' NM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 87545  6 Department of Condensed Matter Physics and Materials Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Tata Institute of Fundamental  Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Mumbai 400005,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' India  Abstract  Large intrinsic anomalous Hall effect (AHE) due to the Berry curvature in magnetic  topological semimetals is attracting enormous interest due to its fundamental importance and  technological relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Mechanisms resulting in large intrinsic AHE include diverging Berry  curvature in Weyl semimetals, anticrossing nodal rings or points of non-trivial bands, and  noncollinear spin structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Here we show that a half-topological semimetal (HTS) state near a  topological critical point can provide a new mechanism for driving an exceptionally large AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  We reveal this through a systematic experimental and theoretical study of the antiferromagnetic  (AFM) half-Heusler compound TbPdBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' We not only observed an unusual AHE with a  surprisingly large anomalous Hall angle \uf051H (tan\uf051H \uf0bb 2, the largest among the antiferromagnets)  in its field-driven ferromagnetic (FM) phase, but also found a distinct Hall resistivity peak in the  canted AFM phase within a low field range, where its isothermal magnetization is nearly linearly  2  dependent on the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Moreover, we observed a nearly isotropic, giant negative  magnetoresistance with a magnitude of ~98%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Our in-depth theoretical modelling demonstrates  that these exotic transport properties originate from the HTS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' A minimal Berry curvature  cancellation between the trivial spin-up and nontrivial spin-down bands results not only in an  extremely large AHE, but it also enhances the spin polarization of the spin-down bands  substantially and thus leads to a giant negative magnetoresistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Our study advances the  understanding of the interplay between band topology and magnetism and offers new clues for  materials design for spintronics and other applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  +These two authors equally contribute to this work  emails: bahadur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='singh@tifr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='in, zim1@psu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='edu  3  Large intrinsic anomalous Hall effect (AHE) in topological semimetals has been a subject  under intensive studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Unlike the conventional Hall effect, which is caused by the Lorentz force  under an external magnetic field, the intrinsic AHE stems from the net Berry curvature \uf057 (k) of  the band structure [1-3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' \uf057 (k), which describes the geometry of the Bloch wavefunctions, is  determined by the topology of the band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' When a longitudinal electric field is applied,  \uf057 (k) imparts transverse velocity [\uf0b5 E \uf0b4 \uf057 (k)] to Bloch electrons and thus results in the AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' To  observe intrinsic AHE, time-reversal symmetry (TRS) is required to be broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Therefore, large  intrinsic AHE is usually expected in magnetic materials with Berry curvature hot spots near the  Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Prior studies have found that several types of magnetic topological materials can  exhibit large intrinsic AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' These include ferromagnetic (FM) Weyl semimetals such as Co3Sn2S2  [4,5] and Co2MnGa [6,7], which host Weyl nodal crossings that lie close to the Fermi level and  carry a diverging Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' With the TRS broken by ferromagnetism, the Berry curvature  contribution from the Weyl nodes with opposite chirality will not cancel out, resulting in a large  AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Strength of the AHE is usually characterized by the intrinsic anomalous Hall angle (AHA)  \uf051H, where \uf051H = tan-1( 𝜎𝑦𝑥  𝐴𝐻/𝜎𝑥𝑥), and 𝜎𝑦𝑥  𝐴𝐻 and 𝜎𝑥𝑥 are the anomalous Hall conductivity and  longitudinal conductivity, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Co3Sn2S2 and Co2MnGa both exhibit large AHA with  tan\uf051H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='2 (Co3Sn2S2) [4] and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='12 (Co2MnGa) [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Antiferromagnetic (AFM) half-Heusler  materials such as GdPtBi [9] and TbPtBi [10] also display a large intrinsic AHE with tan\uf051H values  as large as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='16-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='76 [9-12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Although these materials harbor Weyl nodes induced by magnetic  fields, their large AHE does not originate from Weyl nodes but arises from the large net Berry  curvature produced by the anticrossing of spin-split bands near the Fermi level  [9,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Besides the  large AHE caused by the presence of anticrossing Weyl nodes or bands, recent studies show that  4  gapped nodal rings can also generate extremely large AHE [13,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' This was experimentally  demonstrated in FM Heusler Co2MnAl [14], whose AHA reaches a record value with tan\uf051H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='21  at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' In addition, the non-collinear AFM structure can lead to a large Berry  curvature, resulting in a large intrinsic AHE [15,16];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' this has been seen in a range of AFM  topological materials with broken TRS, such as Mn3Sn [17] and Mn3Ge [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  In this paper, we report a surprisingly large AHE in the Pd-based half-Heusler compound  TbPdBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Its tan\uf051H value reaches ~ 2, which is the largest among all known magnetic topological  materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Our detailed theoretical analysis suggests that such an extremely large AHE originates  from a half-topological semimetal (HTS) state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' HTS is a long-sought topological state in materials  and can be viewed as a topological version of the half-metallic state in which electrons conduct in  only one spin channel, while the other spin channel is insulating [19-21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Since such a state could  generate low power consuming spin current, it holds great promise for applications in topological  quantum spintronic devices [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Moreover, it has been theoretically shown that a gap opening at  the non-trivial band crossing points transforms the HTS into a quantum anomalous Hall insulator  [23,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' These exotic properties have inspired extensive interest and a variety of material systems  have been predicted to be HTS, such as two-dimensional(2D) MnN [25], PrOBr [26], and PtCl3  [24];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' quasi-1D X2RhF6 (X=K, Rb, Cs) [27] and XYZ3 (X=Cs, Rb, Y=Cr, Cu, Z=Cl, I) [28];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 3D  MF3 (M=Pd, Mn) [29], LiV2F6 [30], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The Dirac/Weyl points or nodal rings in all these materials  are comprised of spin-polarized bands [21,30,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' However, these theoretical predictions are still  awaiting experimental verifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Our work here demonstrates that TbPdBi hosts a unique HTS  in proximity to a topological critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Such a peculiar HTS in TbPdBi results in not only an  unusually large AHE but also leads to a giant negative magnetoresistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Additionally, we find  that the Hall resistivity of TbPdBi exhibits a distinct anomalous peak in the low field range, where  5  its isothermal magnetization nearly linearly depends on field and this behavior can be understood  in terms of the Berry curvature enhancement induced by spin canting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Single crystals of TbPdBi were grown using the Bi-flux method (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' They  exhibit semi-metallic behavior, manifested by the broad peak in the temperature dependence of  resistivity \uf072xx around 50K (see supplemental Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' S1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The magnetic susceptibility (\uf063 )  measurements (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' S1a) show its AFM state with TN = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='2 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' \uf072xx exhibits a steeper drop below  TN, indicating electronic transport coupled with magnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' We performed a Curie-Weiss (CW)  fit for its temperature dependence of susceptibility \uf063(T);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' the best fit was obtained in the  temperature range of 100-300K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' S1b) which yields the effective magnetic moment (\uf06deff) of  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='4 \uf06dB/Tb, consistent with a prior report [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  From Hall resistivity \uf072xy measurements under high magnetic fields, we observed  exceptionally strong AHE in TbPdBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Figure 1a shows the \uf072xy data measured up to 31T at various  temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The most significant feature of the \uf072xy data of TbPdBi is that it exhibits a striking  peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' This peak not only occurs below TN (= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='2K) but also extends to temperatures above TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The  peak field is ~ 5T below TN, and slightly shifts to high field with increasing temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' \uf072xy(B)  gradually evolves into a linear field dependence after exhibiting a peak and this occurs above 15T  for T < TN, at higher temperatures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 15K & 20K), due to the shift of the peak to higher field,  the linear trend develops at higher fields, with the linear slope remaining similar to those of low  temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Such a distinct peak in \uf072xy has never been observed in any other half-Heusler  compounds, full-Heusler antiferromagnets, or conventional ferromagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' As noted above, prior  work has shown isostructural half Heusler compounds (Gd/Tb)PtBi also exhibits large AHE [9-  12];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' their \uf072xy data also display anomalous peaks near 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='5T where their AHAs achieve maxima with  6  tan\uf051H = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='16-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='76  [9-12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' However, the \uf072xy anomalous peak of (Tb/Dy)PtBi is far weaker than  that of TbPdBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' For comparison, we have added the \uf072xy data of TbPtBi at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='7K to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' While  its weak \uf072xy peak near 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='5T can be clearly resolved when the data is zoomed into the field range  of 0-9T (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 1(c) in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' [12]), it is hardly discernible when this data is plotted together with  the data of TbPdBi in the field range up to 31T as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 1a, suggesting that the replacement  of Pt by Pd leads to essential changes to the band structure, as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' In the field regime  where \uf072xy exhibits an anomalous peak, the longitudinal resistivity \uf072xx displays a drastic decrease,  followed by a saturation trend in the high field range where \uf072xy evolves into a linear field  dependence, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 2b which presents representative \uf072xx and \uf072xy data at 4K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Note that  the anomalous \uf072xy peak near 5T is not caused by the \uf072xx component which might not be removed  from the \uf072xy data’s anti-symmetrizing process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' this can be seen clearly from the raw data of \uf072xy and  \uf072xx shown in supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' S2 which shows the \uf072xx component in \uf072xy is negligibly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' To  find whether the unusual field dependences of \uf072xy and \uf072xy arise from a magnetic transition, we  measured the magnetization of TbPdBi up to 35T at various temperatures (see Methods) and added  the data from these measurements to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' From these data, we find a spin-flop transition near  15T and the saturated magnetic moment of Tb3+ in the FM phase is ~8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='5\uf06dB/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='57K,  comparable with the effective magnetic moment extracted from the CW fit (\uf06deff ~ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='4 \uf06dB/Tb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The  unusual field dependences of \uf072xx and \uf072xy are indeed coupled with such a magnetic spin-flop  transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The \uf072xy peak as well as the sharp drop of \uf072xx are present in the canted AFM state, while  the linear field dependence of \uf072xy and the \uf072xx saturation behavior occur in the polarized FM phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  These observations imply that the spin-flop transition leads to an electronic structure transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  7  As to be shown below, our theoretical calculations shows that such a spin-flop transition gives rise  to a HTS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Given that the remarkable ρxy peak is present in the CAFM state, its origin is most likely  associated with Berry curvature induced by noncolinear spin structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' In general, ρxy of magnetic  systems can be described by  ρxy= R0B + RSM +ρxy  T  , where the first term is normal Hall contribution  (R0, the normal Hall coefficient), the second term represents the anomalous Hall resistivity linearly  coupled with magnetization M (RS, the anomalous Hall coefficient), and the last term ρxy  T  is the  anomalous Hall resistivity arising from Berry curvature effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The remarkable ρxy peak of  TbPdBi present in the canted AFM state suggests it involves a significant component of anomalous  Hall resistivity Δρxy  AH(=ρxy- R0B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' To find how the magnetization contributes to Δρxy  AH, we have  plotted Δρxy  AH as a function of magnetization in supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' S3 (Note that Δρxy  AH is obtained  by subtracting the normal Hall contribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' the dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 1a which is inferred from  the high-field linear field dependence of  ρxy(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' We find Δρxy  AH strongly deviates from linear  dependence on magnetization and exhibits a striking peak, indicating the anomalous Hall  resistivity of TbPdBi involves a significant contribution of ρxy  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The maximal Δρxy  AH  near 5T is  ~  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='67 mΩ cm (supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' S4a), almost 5-15 times larger than that in (Gd/Tb)PtBi [9,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  From Δρxy  AH, we derive anomalous Hall conductivity σyx  AH using σyx  AH = Δρxy  AH/(ρxy2 +ρxx2) and AHA  as shown in supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' S4b and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The maximal value of tan\uf051H is ~ 2 at about 20T  and 10K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='1c), which is surprisingly large as compared to other magnetic topological materials,  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 1d, which plots tan\uf051H versus σyx  AH for TbPdBi and other magnetic topological  materials showing large AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Such an extremely large AHE seen in TbPdBi is unlikely induced  8  by extrinsic mechanisms such as skew scattering or side jump, since their induced Hall angle is  usually a few percent [33,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The intrinsic mechanisms due to Berry curvature should play a key  role in generating such an unusually large AHE in TbPdBi as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Previous studies  have shown Skyrmion magnetic lattices could give rise to an anomalous peak in ρxy  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  topological Hall effect [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Compared to the topological Hall effect of a prototype Skyrmion  system MnSi where anomalous Hall resistivity jump (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' ρxy  T ) induced by the spin-texture is less  than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='04 µΩ cm [36], our observed maximal Δρxy  AH value of ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='67 mΩ cm in TbPdBi is four orders  of magnitude larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' This implies that the extremely large AHE in TbPdBi should have a unique  mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' As to be shown below, it is indeed associated with a peculiar HTS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  The steep decrease of \uf072xx in field regions where \uf072xy exhibits an anomalous peak (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 1b)  suggests that TbPdBi exhibits large negative magnetoresistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Although we observed large  negative magnetoresistance in our earlier low-field (\uf0a3 9T)  measurements on TbPdBi [37], its  origin remained mysterious;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' it was also unclear whether its magnetoresistance saturates in high  field range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' With the advantage of high-field measurements and theoretical analyses, we have an  opportunity to address these issues in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Figures 2a-2b present the transverse and  longitudinal magnetoresistivity MR (=  \uf072𝑥𝑥(𝐵)− \uf072𝑥𝑥(0)  \uf072𝑥𝑥(0)  ) of TbPdBi, measured with the current  applied perpendicular and parallel to the magnetic field respectively at various temperatures (see  the insets to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 2a & 2b) (Note that these data were measured on the identical sample which was  used for the Hall resistivity measurements shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Both transverse and longitudinal MR  decreases steeply with increasing magnetic field, and then tend to saturate above 15T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' the  magnitude of the MR’s drop reaches ~98% as the field rises to 15T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Such a giant negative MR is  observed at both T \uf0a3 TN and T >TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The magnitude of MR remains nearly temperature independent  9  below 20K, but gradually decreases with increasing temperature above 20K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Even as the  temperature rises to 200K, the negative MR remains significant, with its magnitude being ~ 20%  at 9T (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The MR changes from negative to positive only at room temperature, with the  magnitude of positive MR being much smaller than that of negative MR at lower temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Since TbPdBi is a superconductor with Tc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='7K, its MR data measured at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='7K (the base  temperature of the measurement system) first shows a steep increase as the superconducting state  is suppressed by magnetic field, then followed by the steep decrease as discussed above (see  supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' We chose to not include this data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 2a-2b to avoid complications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' It  is worth pointing out that giant negative MR observed in TbPdBi does not occur to isostructural  compounds (Gd/Tb/Dy)PtBi [10-12,38,39], which again suggests TbPdBi has a distinct electronic  state from (Gd/Tb/Dy)PtBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Negative MR has been observed in various material systems and originates from several  different mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' In topological Weyl semimetals, the topological current induced by the  chiral anomaly could lead to large negative MR when the magnetic field is parallel to the current,  as observed in GdPtBi [11,38] and TbPtBi [10,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' When the magnetic field is rotated away from  the current, the chiral anomaly is gradually suppressed so that the sign of MR could change from  negative to positive above a certain rotation angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Prior studies have also shown materials with  macroscopic disorders may also cause negative longitudinal MR, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' polycrystalline Ag2+δSe [40],  gallium arsenide Quantum wells [41], and disordered topological insulator TlBi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='15Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='85Te2 [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Another mechanism is the spin-scattering suppression driven by spin flip/flop transition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' for  instance, the colossal MR in Ca3Ru2O7 can be attributed to this mechanism [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Apparently, none  of the above mechanisms is relevant to our observation of giant negative MR in TbPdBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' This is  because that (i) our observed negative MR is nearly independent of field orientation and tends to  10  saturate in the high field regime, which excludes the chiral anomaly effect, (ii) our samples are  high-quality single crystals and do not involve macroscopic disorders, and (iii) the giant negative  MR of TbPdBi occurs at both T \uf0a3 TN and T >TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' If its giant negative MR was due to the spin-  scattering suppression in the spin flop transition, we would expect its MR’s magnitude should  significantly drop as the spin-flop transition is suppressed at T >TN, which is inconsistent with the  observation of the temperature independence of MR between TN and 20K and the survival of large  negative MR in a wide temperature region above 20K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Additionally, we note that large  negative  transverse/longitudinal MR was recently reported in EuMnSb2 [44] and EuB6 [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The  origin of large negative MR in EuMnSb2 is ascribed to the field-induced metal-insulator transition,  while the large negative MR of EuB6 is suggested to be from the charge carriers’ high spin  polarization arising from a half Weyl semimetal [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' As to be shown below, our band structure  calculations for TbPdBi also reveals a high spin-polarization state, which is caused by a unique  FM HTS state close to the topological critical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Our observed giant negative MR in TbPdBi  provides strong support for such a HTS state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  To understand the unusual transport properties of TbPdBi discussed above, we now present  our detailed theoretical analysis of the electronic and transport properties of TbPdBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Figures 3a-c  show the arrangement of atoms in nonmagnetic, ferromagnetic (FM) with ordering vector [100],  and A-type AFM state with propagation vector [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The high-symmetry nonmagnetic state of  TbPdBi is described by the inversion asymmetric space group 𝑇𝑑  2 (𝐹4̅3𝑚 , No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 216).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' On  considering the [100] FM state, the symmetry of the lattice reduces to fourfold rotary reflection  𝑆̅4, which combines the fourfold rotational and mirror symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The associated band structure  is semimetallic as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 3d without spin-orbit coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Importantly, the band structure  constitutes a band inversion between Γ8 (p-type) and  Γ6 (s-type) states reminiscent of half-Heulser  11  materials only in the spin-down states whereas spin-up states remain uninverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Such a unique  band structure with single spin-channel band inversion can be closely associated with the half-  metallic state of magnets and thus termed as HTS here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' This HTS state can constitute multiband  anticrossing points comprised of opposite spin channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' In the presence of SOC, the band  anticrossing points are gapped, inducing a non-zero Berry curvature field and leading to large AHE  [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 3e].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Figure 3f shows the band structure of the A-type AFM state of TbPtBi (ground state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  The associated crystalline symmetries are threefold rotational symmetry (C3) about [111] direction  and spacetime symmetry 𝑇̃ = {𝑇|  1  2  1  2  1  2}, which combines both time-reversal symmetry and half-  translation along [111] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The band structure is semimetallic with various band crossings  along the Γ − 𝑍 line and constitutes band inversion in both the spin-channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' This nontrivial band  inversion AFM state evolves to an HTS state with a single-spin channel band inversion in presence  of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Such a band structure evolution with the magnetic field can lead to an  intermediate spin-canted state with various band anticrossing points, enhancing the Berry  curvature field and associated AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  We want to emphasize that the distinct anomalous Hall resistivity peak of TbPdBi occurs  near 5T (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='1a) where the system is in a spin-canted AFM state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' We can therefore expect an  additional contribution to the AHE stemming from the canted AFM state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' As noted in the  introduction, the non-collinear spin texture could induce large non-vanishing Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Since our TbPdBi possess the same magnetic structure as that of GdPtBi/TbPtBi [46], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=', the  magnetic moments of Tb order ferromagnetically on the (111) planes, but are  antiferromagnetically coupled along the [111] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' When the magnetic field is applied long  [100] direction, the magnetic moments will gradually tilt toward this direction, resulting in a spin-  canted state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' A recent theoretical study indicates that, in AFM nodal line materials AMnBi2 (A=Ca,  12  and Yb), a strong AHE could be induced by a weak spin canting, and the anomalous Hall  conductivity keeps growing as the canting angle increases [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The remarkable anomalous Hall  resistivity peak near 5T in TbPdBi probably is likely accredited to the canted spin state formed by  Tb+ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Such an interpretation is supported by our Berry curvature calculation at spin canted state,  as discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  To calculate the band structure and anomalous Hall conductivity (AHC) in the spin-canted  states of TbPdBi under an external magnetic, we employ the virtual crystal approximation (VCA)  with AFM and FM states model Hamiltonians as end members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The VCA Hamiltonian Hvca of the  spin canted state is defined as follows,  𝐻vca = (1 − 𝑥)𝐻AFM + 𝑥𝐻FM,  where 𝐻AFM (𝐻FM) is the Hamiltonian for AFM (FM) phase and x is the tuning parameter from 0  to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' x = 0 (1) denotes purely AFM (FM) phase while 0 < x <1 describes the spin-canted states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' In  the ground state (x = 0), the dominant Berry curvature Ωyz resides on anticrossing points in the ΓZ  direction as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' With an increase of x to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='1 (low external magnetic field), the strong  Berry curvature Ωyz although remains on the ΓZ path, the finite FM coupling splits the oppositely  spin-polarized bands, shifting the Berry curvature hot-spots to distinct energies [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 4b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' In the  fully polarized FM phase, the Berry curvature Ωyz around Γ becomes significantly small [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  4c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Calculated AHC for x = 0 ~ 1 as a function of energy is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The AHC is nearly  zero at x = 0 due to Berry curvature cancellation induced by effective  𝑇̃ symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' A dramatical  peak emerges near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='1 eV above the Fermi level at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='1 because the band splitting caused by  breaking both C3 and 𝑇̃ reduces the Berry curvature cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The AHC peak gradually flattens  and shifts to a higher energy with increasing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The negative AHC at the Fermi level grows  13  monotonically from x = 0 to 1, which implies that there are non-zero Berry curvatures away from  Γ point in the spin-canted AFM states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  We consider the doping effect on the AHC behavior in the magnetic transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' For each  doping concentration, the chemical potential is determined by fixing the number of occupied states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Positive (negative) doping concentration Δn indicates the number of occupied states above (below)  the original occupied particle number, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=', Δn = n-n0, where n (n0) is the current (origin) occupied  particle number per formula unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' When Δn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='0085/f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=', σyz reaches a maximum near x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='1,  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 4e, qualitatively consistent with our experimental observation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Note that  the VCA cannot fully capture the behavior of magnetic state evolution under an external magnetic  field due to the lack of actual correlated-electronic corrections effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' In this sense, x is not  quantitatively comparable to the external field applied during the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' However, this  approach can give a reasonable physical insight into the band structure effects in spin-canted states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  On comparing to the experimental AHC (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' S4b), our calculated AHC are smaller than the  experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Such a discrepancy further suggests that effects other than those not captured  in our VCA calculations are also at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Next, we first discuss the origin of the large Berry curvature in our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 3d-e show  the band structure without and with SOC when the magnetic field is applied along [100] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  The band degeneracy lifting by exchange interaction is commonly seen in other Heusler  compounds, such as GdPtBi/TbPtBi [9,11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' However, the weaker exchange interaction of Tb  ions and nearly zero band inversion strength lead to the band splitting in TbPdBi smaller than that  in (Gd/Tb)PtBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' As a result, the spin-split bands create more crossing points near the Fermi-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  A small gap is opened at the anti-crossing points with SOC, and the large non-vanished berry  curvature is accumulated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  14  We note that in REPtBi (RE = Gd, Nd), the non-vanishing Berry curvature is induced by  the field-induced Weyl state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The location and number of Weyl nodes depend on the direction of  applied magnetic field B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' For TbPdBi, our calculation indeed reveals a Weyl state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' We found that  two pairs of Weyl nodes appear around 𝛤 point in the FM phase driven by the magnetic field  applied along [100] axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' the positions of these Weyl nodes are summarized in supplementary table  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' However, these Weyl nodes are far away from the Fermi level, which cannot induce large AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Furthermore, the absence of chiral anomaly revealed in our magnetotransport measurements  indicates that the Weyl nodes do not contribute to the transport properties of TbPdBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Hence, we  believe the giant AHC in TbPdBi cannot be attributed to the Weyl state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  The second origin of the large Berry curvature is the spin-canted state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' As revealed by our  calculations, increasing the canted angle widens the band splitting and enlarges the gap at the anti-  crossing points, which results in the Berry curvature along yz direction (Ωyz) continually increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  However, the Berry curvature arises from the anti-crossing points and the spin-canted state is not  the unique property for TbPdBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' It also has been observed in other half-Heusler compounds such  as GdPtBi [9,11], TbPtBi [12], and DyPtBi [39];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' their anomalous Hall conductivity and Hall angle  are much smaller than that in TbPdBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' One may wonder what makes TbPdBi so unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Our above  band structure calculations reveal TbPdBi shows a distinct topological state: TbPdBi is at the  critical point at the AFM ground state since its band inversion strength is nearly zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' At the field-  driven FM state, it evolves to a HTS state: its spin-down bands maintain band inversion, while its  spin-up bands lose their band inversion and become topological trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' We note a similar state has  been predicted in EuB6 [48], where a large AHE and large negative MR have been recently  reported [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Such findings imply that the HTS state probably plays a crucial role in enhancing  the AHE in topological materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' As we stated before, the net Berry curvature requires the sum-  15  up of the Berry curvature among all the occupied states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Thus, the number of the bands crossing  the Fermi level could affect the Berry curvature cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' In general, the compounds showing  good-metal behavior usually exhibit a large Berry curvature cancellation, resulting in a small net  Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' However, in some compounds with bad-metal or semimetal behavior, only a few  bands cross through the Fermi level, which could reduce the possibility of Berry curvature  cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' In TbPdBi, the spin-up and spin-down bands are pushed to opposite directions, with  a bandgap opening in the spin-up bands, which substantially reduces the number of the bands  which cross the Fermi level, thus minimizing the Berry curvature cancellation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  The exotic HTS state in TbPdBi leads to not only a large AHE but also a giant isotropic  negative MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' As described above, the spins are polarized near the Fermi level (spin-polarization  reaches 40%) when applying an external magnetic field, such that the carriers from spin-down  bands dominate the transport properties of TbPdBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' In this case, spin-scattering is significantly  suppressed across the spin-flop transition, which leads to steep resistivity drop with the increase  of magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' When the magnetic field reaches above 15T, the spins are fully polarized (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  1b), so the MR remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Since the large negative MR is attributed to such an exotic 3D  band structure, it is expected to be independent of the field orientation, which is exactly what we  observed in experiments (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 2a & 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The temperature independence of MR(B) between  TN(=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='2K) and 20K (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 2a&2b) can also well understood in terms of the field-induced FM HTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Although there is not a long-range spin-flop transition above TN, short-range AFM should exist in  a temperature range above TN at zero field so that a crossover-like transition from PM to FM should  occur under high magnetic fields, which is indeed manifested in the magnetization data measured  at 20K as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' This indicates a FM HTS can extend to temperatures above TN under  high magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' However, when the temperature is increased above 50K, thermal excitation  16  would bring out the electrons from the spin-up bands, thus increasing the spin-scattering and  leading to the decrease of the magnitude of negative MR as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  In summary, we have observed an unusual AHE effect with a surprisingly large anomalous  Hall angle (tan\uf051H \uf0bb 2) and a giant negative MR in TbPdBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Our theoretical analysis indicates that  these exotic transport properties originate from the unique HTS state of TbPdBi, dominated by the  contribution of spin-down bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The greatly enhanced AHE results from a significantly reduced  Berry curvature cancellation between the spin-down and spin-up bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Spin canting of the AFM  state under magnetic field increases the number of band anticrossing points and widens the gaps,  which in turn increases the Berry curvature hot spots near the Fermi level and accounts for the  distinct anomalous Hall resistivity peak at low fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' We also find that the HTS state enhances the  spin-polarization for the spin-down band in the FM phase, which explains the giant negative MR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Our study unveils a new pathway for generating an extremely large AHE by tuning the electronic  structure and the magnetic state in half-Heusler materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' It thus advances the understanding of  the interplay between topological state and magnetism and provides clues for materials design for  spintronics and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Methods  The single crystals of TbPdBi were synthesized using the Bi-flux method  [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The starting  materials of Tb, Pd, and Bi powders were mixed with a molar ratio of Tb: Pd: Bi=1:1:20, loaded  into Al2O3 crucibles, and sealed in quartz tubes under high vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The mixtures of source  materials were heated to 1050°C in a crucible furnace and held at this temperature for 48 hours for  homogeneously melting, then followed by slow cooling down to 700°C at a rate of 3° C /h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Cubic-  shape single crystals of TbPdBi were obtained by removing the excess Bi flux through centrifuging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  The structures and compositions of the grown crystals are confirmed by X-ray diffraction  17  measurements and Energy-dispersive X-ray spectroscopy (EDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Magnetoresistivity and Hall  resistivity measurements were performed using a standard six-probe method in a Physical Property  Measurement System (PPMS, Quantum Design).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The high-field transport measurements were  carried out at the National High Magnetic Field Laboratory (NHMFL) in Tallahassee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The  magnetoresistivity and Hall resistivity data presented in this paper are obtained through  symmetrizing and antisymmetrizing the longitudinal and transverse Hall resistivity data measured  at positive and negative magnetic fields respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' All the samples used for transport  measurements were polished to rectangular shapes with dimensions of ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='8mm × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='7mm × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='2mm,  and the polished surfaces were parallel to the (001) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The current was applied to the [100] or  [001] directions for both longitudinal resistivity and Hall resistivity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Magnetization  measurements were measured using the National High Magnetic Field Pulsed Field Facility at Los  Alamos National Laboratory and the SQUID magnetometer (Quantum Design).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Electronic structure calculations were performed within the density functional theory  framework using the Vienna ab-initio simulation package (VASP) based on the projector-  augmented wave method [50-52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='The generalized gradient approximation (GGA) was employed  to include the exchange-correlation effects [53] and an on-site Coulomb interaction was added for  Tb f-electrons within the GGA+U scheme with Ueff = 10 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' A plane wave cut-off energy of 500  eV was used and a 11 × 11 × 11 Γ centered k-point mesh to sample the first Brillouin zones of  cubic and trigonal unit cells associated with various magnetic ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Spin-orbit coupling effects  were included self-consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Transport properties were calculated using a first-principles  material-specific,  effective  tight-binding  model  Hamiltonian  generated  using  the  VASP2WANNIER90 interface [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Tb d and f orbitals, Pd s, p, and d orbitals, and Bi p orbitals  were included in constructing the wannier functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  18  References:  [1]  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Onoda and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Nagaosa, Topological Nature of Anomalous Hall Effect in Ferromagnets, Journal  of the Physical Society of Japan 71 1, 19 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  [2]  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Haldane, Berry Curvature on the Fermi Surface: Anomalous Hall Effect as a Topological  Fermi-Liquid Property, Physical Review Letters 93 20, 206602 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  [3]  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Nagaosa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
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+page_content='  21  Acknowledgment  The experimental part of this work is based upon research conducted at The Pennsylvania  State University Two-Dimensional Crystal Consortium–Materials Innovation Platform (2DCC-  MIP), which is supported by NSF Cooperative Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
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+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' also  acknowledges the support from NSF under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' DMR 2211327.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=" The work at Northeastern  University was supported by the Air Force Office of Scientific Research under award number  FA9550-20-1-0322 and benefited from the computational resources of Northeastern University's  Advanced Scientific Computation Center (ASCC) and the Discovery Cluster." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The work at the  National High Magnetic Field Laboratory is supported by the NSF Cooperative Agreement  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='DMR-1644779 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' DMR-1157490 and the State of Florida.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The work at TIFR Mumbai is  supported by the Department of Atomic Energy of the Government of India under Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' 12-  R&D-TFR-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='10-0100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Author contributions  The crystal growth and transport measurements were carried out and analyzed by Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='Z, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
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+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  The high magnetic field measurements were carried out by Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=', D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
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+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
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+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
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+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The  theoretical work was done by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
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+page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' & A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' This work is supervised by Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' (experiment)  and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' (theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  22  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' (a) Hall resistivity of TbPdBi as a function of the magnetic field B up to 31T at various  temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The Hall data have been anti-symmetrized to remove the minor component of 𝜌𝑥𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  Inset: the schematic of the experimental setup for Hall measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' (b) Left axis: longitudinal  resistivity 𝜌𝑥𝑥 and Hall resistivity 𝜌𝑥𝑦  at 4K for TbPdBi (marked as hollow circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Right axis:  magnetization measured at various temperatures (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' (c) The tanΘ𝐻  of TbPdBi as a  function of magnetic field B at 4K (below Neel temperature),7K and 10K (above Neel temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  (d) The comparison of 𝜎𝑦𝑥  𝐴𝐻 and tan Θ𝐻  between TbPdBi and other magnetic conductors  [4,6,8,9,12,14,17,18,39,45,55-57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  (a)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='2  (b)  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='7K  TbPdBi  M(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='57K)  4K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='8  7K  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='6  T= 4K  8  cm  10K  5K  15K  u)  20K  ( u)d  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='2  20K  6  (HB)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='0  TbPtBi  TbPdBi  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='7K  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='8  4  /f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=')  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='4  B  Pxy  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='8  (100)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='0  30  20  10  0  10  20  30  0  5  10  15  20  25  30  35  B (T)  B (T)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='0  (d)  (c)  4K  1500  4 DyPtBi  TbPdBi  7K  ★  1200  Co,MnAl  10K  Fe,Sn  Co,Sn,S  EuB  Co,MnGa  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='5  r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='0)  006  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  TbPtBi  600  Mn,Ge  TbPdBi  300  MnSi  GdPtBi  EuCd,As,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='0  oMn,Sn  0  10  20  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='0  B (T)  tan@?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='23  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Magnetoresistivity (MR) of TbPdBi: (a-b) Transverse (a) and longitudinal (b) MR = Δρ  /ρ0 = [ρ(B)–ρ(B = 0)]/ρ(B = 0) at 4K (below Neel temperature), 10K and 20K (above Neel  temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The MR at 4K and 10K were measured up to31T, while the MR for 20K was  measured up to 18T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' The Insets show the schematic of the experimental setup for transverse and  longitudinal MR measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' (c) Longitudinal MR versus magnetic field measured in PPMS at  various temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  a)  b  TbPdBi B I I  300 K  B  20  20  0  200 K  TbPdBi  TbPdBi  40  (100)  (100)  30  MR  20K  MR  20K  MR  60  60  100 K  10K  10K  60  50 K  80  4K  80  4K  4 K  90  100  100  10 K  30  20  10  0  10  20  30  30 -20 -10  10  20  30  10  5  0  5  10  B (T)  B (T)  B (T)24  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' (a) Crystal structure of half-Heusler TbPdBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' [111] and [100] vectors denote the principal  magnetic axes for AFM and FM states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Spin structures of (b) [100] FM and (c) [111]  AFM magnetic structure in FM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Spins are aligned ferromagnetically in a (111) plane and  antiferromagnetically between the (111) planes in the AFM state shown in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' (d) Calculated spin-  resolved bulk band structure of FM TbPdBi without SOC considering the primitive unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' (e)  Spin-polarized band structure with SOC in (e) FM and (f) AFM states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Trigonal supercells are  considered in (e) and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Insets in (d)-(e) show the first Brillouin zones associated with the unit  cells employed in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  (a)  (b)  (c)  [111]  Tb  Pd  Bi  [100]  (d)  (e)  (f)  S(111]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='2  60  60  (%) I  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
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+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='2  60  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='21  60  M  L  r  X  M  K  Z  F  L  r  Z  F  L25  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content=' Berry curvature Ωyz distribution of TbPdBi band structures for (a) x = 0 (AFM), (b) x =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
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+page_content=' (d) Calculated  anomalous Hall conductivity σyz as function of energy for varying x = 0 (dark blue curve) to x = 1  (red curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
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+page_content=' See the text  for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
+page_content='  (a)  (b)  (c)  σyz(Ω-1cm-1)  E-EFM/AFM (eV)  x = 0  x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtAzT4oBgHgl3EQfJPsM/content/2301.01074v1.pdf'}
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+Signed Directed Graph Contrastive Learning with Structure
+and Laplacian Augmentation
+Taewook Ko
+taewook.ko@snu.ac.kr
+Seoul National University
+Republic of Korea
+Yoonhyuk Choi
+younhyuk95@snu.ac.kr
+Seoul National University
+Republic of Korea
+Chong-Kwon Kim
+ckim@kentech.ac.kr
+Korea Institute of Energy Technology
+Republic of Korea
+ABSTRACT
+Graph contrastive learning has become a powerful technique for
+several graph mining tasks. It learns discriminative representation
+from different perspectives of augmented graphs. Ubiquitous in our
+daily life, singed-directed graphs are the most complex and tricky
+to analyze among various graph types. That is why singed-directed
+graph contrastive learning has not been studied much yet, while there
+are many contrastive studies for unsigned and undirected. Thus, this
+paper proposes a novel signed-directed graph contrastive learning,
+SDGCL. It makes two different structurally perturbed graph views
+and gets node representations via magnetic Laplacian perturbation.
+We use a node-level contrastive loss to maximize the mutual infor-
+mation between the two graph views. The model is jointly learned
+with contrastive and supervised objectives. The graph encoder of
+SDGCL does not depend on social theories or predefined assump-
+tions. Therefore it does not require finding triads or selecting neigh-
+bors to aggregate. It leverages only the edge signs and directions
+via magnetic Laplacian. To the best of our knowledge, it is the first
+to introduce magnetic Laplacian perturbation and signed spectral
+graph contrastive learning. The superiority of the proposed model
+is demonstrated through exhaustive experiments on four real-world
+datasets. SDGCL shows better performance than other state-of-the-
+art on four evaluation metrics.
+KEYWORDS
+graph contrastive learning, magnetic Laplacian, signed directed
+graph, link sign prediction
+1
+INTRODUCTION
+Various types of online social graphs are developing as the use of
+the internet increases. Social graph research attracts attention from
+researchers with its usefulness for various downstream tasks such
+as prediction, classification, and recommendation[9, 11, 17, 53].
+Signed-directed graphs, which express the relationship between
+users as like/dislike or trust/distrust, have the most information and
+are the most valuable among several graph types. However, it has
+difficulties to analyze them caused of severe noise and complexity.
+Many network embedding and graph convolution studies for signed-
+directed graphs have been proposed based on the two well-known
+social theories, balance and status.[18, 19] Balance theory is the
+concept that "A friend of my friend is my friend, and an enemy of
+my friend is my enemy." Furthermore, status theory sets the relative
+user ranks with the relation of positive and negative edges. The ideas
+of most existing studies, including SiNE[47], SGCN[8], SNEA[31],
+and SDGNN[21], are derived from the two social theories. Recently,
+some studies that do not use them have been proposed. ROSE[45]
+proposed a user-role-based methodology by transforming a graph
+into an unsigned bipartite. SDGCN[26] proposed a spectral convolu-
+tion via a signed magnetic Laplacian matrix.
+Contrastive learning[1, 16] is a successful learning mechanism in
+computer vision research with fewer labels or a self-supervised envi-
+ronment. It learns discriminative representations through contrastive
+loss from the data itself. They maximize the mutual information of
+representations of augmented images. Recently, many graph stud-
+ies utilizing contrastive learning have been suggested. Graph con-
+trastive learning models generate graph views in their own ways,
+such as edge or node deletion[51], addition[52], random-walk[38],
+or attribute masking[56]. JOAO[50] and AD-GCL[43] learn optimal
+perturbing distribution according to the dataset and SimGRACE[49]
+adds gaussian noise on the trained parameters for encoder perturba-
+tion. They define contrastive loss with the augmented representations
+of nodes or graphs. Graph contrastive learning also aims to maxi-
+mize the mutual information of different graph views. However, not
+much research has been conducted on signed-directed graphs.
+SGCL[39] is the first study to apply contrastive learning on signed-
+directed. They make two graph views by giving random perturba-
+tions on edge signs and directions. Then the views are divided into
+positive graph views and negative graph views. The positive graph
+view is constructed with only the positive edges of a graph view, and
+the same for the negative graph view. Then run graph convolution
+on the different graph views to get augmented node representations.
+Though SGCL is innovative, there are some limitations. First, it
+loses the context of the original graph if we separate the edges into
+different graphs by the edge types. The mutual influence between
+edge types cannot be trained. For example, valuable patterns like the
+balance theory cannot be reflected. Second, most real-world signed
+graphs have a nearly 90% of positive edge ratio[27]. Therefore, the
+negative views are unsuitable for graph convolutions due to the
+sparsity issue. In this study, we propose a Signed Directed Graph
+Contrastive Learning(SDGCL), which improves these limitations.
+SDGCL utilizes two levels of perturbation. We apply random edge
+perturbation similar to SGCN. It changes edge signs and direc-
+tions to generate augmented graph views. Then we introduce mag-
+netic Laplacian perturbation. It changes a phase parameter 𝑞 of the
+magnetic Laplacian matrix. 𝑞 controls the phase angle between the
+real- and imaginary axis[26]. Magnetic Laplacian was studied in
+the field of quantum physics[5, 35, 40]. Thanks to its Hermitian
+properties, it has been used in graph convolutions for directed and
+signed-directed[41, 54]. Signed magnetic Laplacian encodes graph
+sturucture, including edge signs and directions. It reflects the graph
+information and allows the model to learn the patterns of nodes and
+edges. We define a spectral convolution layer with the perturbed
+Laplacian by the idea of graph signal processing[7, 15, 25]. SDGCL
+gets the two augmented node representations with the graph encoder
+arXiv:2301.05163v1  [cs.LG]  12 Jan 2023
+
+Taewook Ko, et al.
+Figure 1: Example of a signed-directed graph and its structure
+perturbation. Dashed edges indicate perturbed edges.
+and defines node-level contrast loss[39, 56] for contrastive learning.
+The performance of the proposed model was evaluated with the link
+sign prediction task, which is widely selected to evaluate the signed-
+directed graph embedding. It was tested with four real-world graphs
+and showed excellent performance compared to various baselines
+for signed-directed graph embedding and graph contrast learning.
+The followings are the contributions of this paper
+• This paper proposes a novel signed-directed graph contrastive
+learning model, SDGCL.
+• It augments node representations via two stages, graph struc-
+ture perturbation and magnetic Laplacian perturbation.
+• To the best of our knowledge, it is the first to introduce mag-
+netic Laplacian perturbation and is the first spectral signed
+graph contrastive model.
+• The proposed model has the best link sign prediction perfor-
+mance in real-world graph experiments.
+2
+RELATED WORKS
+2.1
+Signed-Directed Graph Embedding
+Most signed-directed graph studies utilize sociological stories.[18,
+19] SiNE[47] is an undirected model that uses balance theory, defin-
+ing an objective function: friend nodes increase their similarity.
+BESIDE[3] focused on the bridge edges to overcome the limitation
+of triads and utilized both social theories. SGCN[8] defined a novel
+balanced and unbalanced path for neighbor aggregation. SNEA[31]
+and SiGAT[20] used attention mechanism. SDGNN defined four
+different weight matrices to distinguish the edge signs and directions.
+And they proposed triad loss based on the theories. Balance and sta-
+tus theories are developed with the patterns of the user triads. About
+60-70% of triads on real-world graphs satisfy the theories[24, 29].
+Though the theories are crucial paradigms in signed-directed graph
+research, not all the triads satisfy them. Moreover, it is not a rule
+that all users should follow. Models that depend on the theories are
+not working well with users with little or no triads. On top of that,
+such models require much computational cost to find triads or paths.
+Recently, ROSE[23] proposed a methodology using user-role by
+transforming graphs into unsigned bipartite. SDGCN[26] proposed
+a spectral convolution model by proving the positive semidefinite of
+signed magnetic Laplacian.
+2.2
+Magnetic Laplacian
+Magnetic Laplacian was first introduced in the field of quan-
+tum mechanics to analyze the charged particle under magnetic
+flux.[5, 32, 35, 40] But thanks to its Hermitian properties, the mag-
+netic Laplacian has been utilized in directed graph studies.[14, 34]
+Some application studies such as graph clustering[4, 6] node[54]
+and graph representation learning[12] are delivered. MagNet[54]
+proved the positive semidefinite property of the magnetic Laplacian
+and proposed a spectral graph convolution. Recently, some studies
+extended the magnetic Laplacian to the signed graph. SDGCN[26]
+and SigMagNet[10] define signed magnetic Laplacian. The signed
+magnetic Laplacian matrix uniquely encodes edge types of signed-
+directed graphs with real- and imaginary numbers. They proposed
+spectral graph convolution with the signed magnetic Laplacian.
+2.3
+Graph Contrastive Learning
+SimCLR[1] and MoCo[16] proposed contrastive learning and
+achieved great success in image classification. Many follow-up stud-
+ies were proposed thanks to the increasing learning efficiency via
+self-supervised. DGI[46], GMI[37] adopted the contrastive learning
+to graph studies. They measure mutual information of input and node
+representations. InfoGraph[42] proposed patch-level representation.
+GCC employed random-walk sampling to make positive and nega-
+tive samples. Since then, several graph augmentations[51, 52, 56]
+and graph encoders[39, 49] have been proposed for contrastive learn-
+ing. There are contrastive studies for graph clustering[36, 55], node
+embedding[56], DDI prediction[30, 48], and recommendation[33].
+However, signed-directed graph contrastive studies are not discussed
+much. To the best of our knowledge, SGCL[39] is the only one.
+SGCL makes graph views by randomly perturbing the edges. Then
+they construct positive- and negative-graph views by separating
+edges by the edge type. However, the edges of signed-directed graphs
+have meaningful relationships with each other. Therefore, the con-
+text of the data cannot be properly interpreted when we arbitrarily
+separate graph views by edge types. Moreover, they are inefficient
+in that they need to find positively or negatively linked nodes for
+every aggregation stage. To overcomes these limitations, this study
+proposes a novel graph contrastive learning model.
+3
+PROBLEM FORMULATION
+Let G = (𝑉, E+, E−) be a signed-directed graph where 𝑉 is a set
+of nodes like users in social graphs. E+ ⊆ 𝑉 × 𝑉 indicates positive
+edge matrix, and E− ⊆ 𝑉 × 𝑉 is of negative edges. Positive and
+negative edges represent user relationships, such as like/dislike or
+trust/distrust. For example E+𝑢,𝑣 equals 1 if there is a positive edge
+from node 𝑢 to node 𝑣; otherwise, 0. Similarly, if there is a negative
+edge from node 𝑣 to 𝑢, E−𝑣,𝑢 equals 1. Note that E+ ∩ E− = ∅.
+Because we do not accept users having two different edges to the
+other user simultaneously, such as "John loves Jane and also hates
+her." A user may be connected to other users by one of the three
+relations(none, positive, negative). Thus, there are nine-edge types
+between a user pair. The goal of this paper is to map the nodes 𝑢 ∈ 𝑉
+into the low-dimensional embedding vectors 𝑧𝑢 ∈ R𝑑 with a given
+graph G as:
+𝑓 (G) = 𝑍,
+𝑓 is a transformation function and 𝑍 ∈ R|𝑉 |×𝑑 is an embedding ma-
+trix. Each row of 𝑍 represents the node embedding with dimension
+size 𝑑.
+
+edge
+perturbation
+original graph
+augmented graph viewSigned Directed Graph Contrastive Learning with Structure and Laplacian Augmentation
+4
+MODEL FRAMEWORK
+4.1
+Graph Augmentation
+4.1.1
+Structure Perturbation. There are two methods for struc-
+ture perturbation, edge sign perturbing and edge direction perturbing.
+In edge sign perturbing, we change the edge signs randomly from
+the given graph. For example, we sample 𝑝% of positive edges and
+change their signs to negative. And the same for the negative edges.
+Similarly, we sample 𝑟% of edges and change their directions to in-
+verse. If an edge is bidirectional, we change it to directional arbitrary.
+With this random edge perturbation, we get structurally perturbed
+graph views, �G = (𝑉, ˜E+, ˜E+). Figure 1 shows an example of struc-
+ture perturbation. As already known by balance and status theories,
+there is important information in the edge sign and directions. We
+may harm the vital triad patterns via random perturbation. However,
+we expect some amount of perturbation would be helpful to learn
+robust representations from noisy real-world data. Moreover, the
+perturbing can discover and exploit some relationships that might ex-
+ist. The model can improve the generalization performance through
+random perturbations.
+4.1.2
+Signed-Directed Magnetic Laplacian. Before introduc-
+ing Laplacian Perturbation, we define the signed-directed magnetic
+Laplacian. Graph Laplacian is useful to encode graph sturucture
+with degree and adjacency matrices, L = D − A. They are not only
+positive semidefinite but also have non-negative eigenvalues and as-
+sociate orthonormal eigenvectors. With that properties, [25] and [15]
+proposed the idea of spectral graph convolution. However, graph
+Laplacian is asymmetric when a graph is directed or signed-directed.
+They have complex eigenvalues and do not satisfy the properties
+for spectral convolution. Thus, [10, 26, 41] proposed a novel mag-
+netic Laplacian matrix representing the structure of signed-directed
+graphs and satisfying positive semidefinite. First of all, we define a
+complex Hermitian adjacency matrix as follow,
+H𝑞 = A𝑠 ⊙ P𝑞.
+A𝑠 := 1
+2 (A + A⊺) ⊆ 𝑉 × 𝑉 is a symmetrized adjacency matrix,
+and P𝑞 ⊆ 𝑉 × 𝑉 is a phase matrix with complex numbers. ⊙ is an
+element-wise multiplication operation. The definition of the phase
+matrix is,
+P𝑞(𝑢, 𝑣) :=
+exp(𝑖Θ𝑞
+𝑢𝑣)A𝑢𝑣 + exp(𝑖Θ𝑞
+𝑢𝑣)A𝑣𝑢
+|exp(𝑖Θ𝑞
+𝑢𝑣)A𝑢𝑣 + exp(𝑖Θ𝑞
+𝑢𝑣)A𝑣𝑢| + 𝜖
+.
+Θ𝑞
+𝑢𝑣 = 𝑞E+𝑢𝑣 +(𝜋 +𝑞)E−𝑢𝑣 and Θ𝑢𝑣
+𝑞 = −𝑞E+𝑣𝑢 +(𝜋 −𝑞)E−𝑣𝑢 Be careful
+with the subscripts orders. 𝑞 is a hyperparameter lies in [0,𝜋/2]. The
+symmetrized adjacency matrix encodes the node connectivity, and
+the phase matrix encodes link directions and signs with different
+phase values. Figure 2(b) shows the edge encodings of the defined
+Hermitian adjacency matrix H𝑞. They uniquely encode the nine-
+edge types of signed-directed graphs. It tells node connections and
+edge types as well. We can see that H𝑞 is a complex numbered skew-
+symmetric form, a complex Hermitian matrix. Then, we define the
+signed-directed magnetic Laplacians with this Hermitian adjacency
+by,
+L𝑞
+𝑈 := D𝑠 − H𝑞 = D𝑠 − A𝑠 ⊙ P𝑞
+L𝑞
+𝑁 := I − (D− 1
+2
+𝑠
+A𝑠D− 1
+2
+𝑠
+) ⊙ P𝑞,
+Figure 2: Edge encoding and meaning of 𝑞. It shows edge encod-
+ing values of a complex Hermitian adjacency matrix. 𝑞 controls
+the phase angle.
+D𝑠 is a symmetric degree matrix, similar to A𝑠. L𝑞
+𝑈 and L𝑞
+𝑁 are un-
+normalized and normalized signed-directed magnetic Laplacians. It
+is well known that the skew-symmetric, complex Hermitian matrix
+is positive semidefinite[10, 26, 41]. Thus, the Laplacians are posi-
+tive semidefinite and diagonalizable by spectral decomposition. For
+example, the normalized Laplacian is diagonalized by,
+L𝑞
+𝑁 = UΛU†.
+Each column of U is eigenvector u𝑘 and U† is a conjugate transpose
+of U. Λ is a diagonal matrix where the elements are 𝑘-th eigenvalues
+Λ𝑘,𝑘 = 𝜆𝑘. The eigenvalues and eigenvectors contain the structural
+information of the signed-directed graph. We leverage this matrix to
+define spectral graph convolution.
+4.1.3
+Laplacian Perturbation. Figure 2(a) describes the meaning
+of 𝑞 in magnetic Laplacian. 𝑞 controls the phase angle of encoded
+values between the real- and imaginary axes. If 𝑞 is small, the Lapla-
+cian puts less attention on directional information. It becomes an
+undirected model when the 𝑞 is 0. However, the high 𝑞 value also
+harms the encoding validity. 𝑞 influences the encoding effectiveness
+of the signed-directed magnetic Laplacian. In this subsection, we
+introduce Laplacian perturbation via 𝑞 variation. We randomly se-
+lect 𝑞 value from 0 to 0.4𝜋. The Laplacian matrix varies according
+to the 𝑞 variations, though the meaning of the graph structure that
+the Laplacian represents is still the same. This kind of Laplacian
+perturbation is a useful technique that makes graph augmentation
+without distortion of the original data[44].
+In short, we make two structurally perturbed graph views �G1 and
+�G2, through random edge changes. Then we get perturbed signed-
+directed magnetic Laplacian �L
+𝑞1 and �L
+𝑞2 from the graph views with
+randomly chosen 𝑞 values. With these Laplacian matrices, we define
+graph encoders.
+4.2
+Graph Encoder
+4.2.1
+Spectral Convolution via Magnetic Laplacian. The
+signed-directed magnetic Laplacian L, is diagonalizable with eigen-
+vector matrix U, and diagonal eigenvalues Λ. Several graph convolu-
+tion studies [7, 15] utilize the eigenvectors as discrete Fourier modes
+
+Edge type
+H (u,)
+imag-axis
+u→v
+(cosq + isinq)
+u←v
+uv
+(cosq - isinq)
+u←v
+(cosq - ising)
+u←v
+u-→v
+u→v
+ (cosq + isinq)
+uv
+uv
+uv
+real-axis
+cosq
+u-→v
+u←v
+uv
+-cosq
+uv
+u与v
+isinq
+u乌v
+-isinq
+u
+V
+0
+(a)
+(b)Taewook Ko, et al.
+Figure 3: Model overview
+of graph signal processing. The graph signals are transformed into
+the Fourier domain through graph Fourier transform x : 𝑉 → C by
+ˆx = U†x. The inverse Fourier transform formula is defined as follow
+thanks to the unitarity of U,
+x = Uˆx =
+𝑁
+∑︁
+𝑘=1
+ˆx(𝑘)u𝑘.
+The spectral convolution operation of graph signal processing is
+described as
+g𝜃 ∗ x = Ug𝜃U†x,
+where g𝜃 = 𝑑𝑖𝑎𝑔(𝜃) is a trainable filter. For efficient calculation,
+[15] proposed a truncated Chebyshev polynomial expansion of the
+filter by,
+g𝜃′(Λ) ≈
+𝐾
+∑︁
+𝑘=0
+𝜃 ′
+𝑘𝑇𝑘 ( ˜Λ).
+𝑇0(𝑥) = 1,𝑇1(𝑥) = 𝑥, and 𝑇𝑘 = 2𝑥𝑇𝑘−1(𝑥) +𝑇𝑘−2(𝑥) for 𝑘 ≥ 2. 𝑘 is
+an expansion order. ˜Λ =
+2
+𝜆𝑚𝑎𝑥 Λ−I is a normalized eigenvalue matrix,
+and 𝜆𝑚𝑎𝑥 is the largest eigenvalue. 𝜃 ′
+𝑘 are Chebyshev coefficients.
+We have a simplified form of spectral graph convolution by,
+g𝜃′ ∗ x =
+𝐾
+∑︁
+𝑘=0
+𝜃 ′
+𝑘𝑇𝑘 ( ˜L)𝑥,
+where ˜L =
+2
+𝜆𝑚𝑎𝑥 L − I analogous to ˜Λ.
+4.2.2
+Spectral Convolution Layer. We define the spectral con-
+volution layer with the approximated signed-directed spectral convo-
+lution operation. We set 𝑘 as 1, the maximum polynomial order, and
+𝜆𝑚𝑎𝑥 is assumed 2 to make it practical. Similar to GCN[25], we set
+𝜃 = 𝜃 ′
+0 = −𝜃 ′
+1. Then we have
+g𝜃′ ∗ x ≈ 𝜃 (I + (D− 1
+2
+𝑠
+A𝑠D− 1
+2
+𝑠
+) ⊙ P𝑞)x.
+The spectral convolution layer is defined as
+H = ( ˜D− 1
+2
+𝑠
+˜A𝑠 ˜D− 1
+2
+𝑠
+⊙ P𝑞)XW.
+H ∈ R𝑁 ×𝐹 is the convoluted graph signals or representations. X ∈
+R𝑁 ×𝐶 is the input signal.𝐶 and 𝐹 are the numbers of input and output
+channels. W ∈ R𝐶×𝐹 is a learnable matrix. It is a renormalization
+trick that I + (D− 1
+2
+𝑠
+A𝑠D− 1
+2
+𝑠
+) ⊙ P𝑞 → ˜D− 1
+2
+𝑠
+˜A𝑠 ˜D− 1
+2
+𝑠
+⊙ P𝑞 where ˜As =
+A𝑠 + I and ˜Ds(𝑖,𝑖) = �
+𝑗 ˜As(𝑖, 𝑗). It prevents gradient vanishing and
+exploding problems.
+4.2.3
+Signed-Directed Graph Encoder. The graph encoder
+stacks 𝐿 layers of proposed spectral convolution layer. The 𝑙-th
+layer feature vector x(𝑙) is defined as,
+x(𝑙)
+𝑗
+= 𝜎(
+𝐹𝑙−1
+∑︁
+𝑖=1
+Y(𝑙)
+𝑖𝑗 x(𝑙−1)
+𝑖
++ b(𝑙)
+𝑗 ).
+We use a novel activation function for the complex elements. The
+activation function 𝜎 is an complex version of ReLU, 𝜎(𝑧) = 𝑧, if
+−𝜋/2 ≤ arg(𝑧) ≤ 𝜋/2, otherwise, 𝜎(𝑧) = 0. The feature matrix X(𝐿)
+has both real and imaginary values. Here we introduce unwinding
+operation. It converts real and imaginary features into the same
+domain.
+X(𝐿)
+unwind = [real(X(𝐿))||imag(X(𝐿)) ⊗ (−𝑖)].
+We add a fully connected layer after unwinding to get the node
+representation.
+Z = 𝜎(X(𝐿)
+unwindW𝐿+1 + B(𝐿+1))
+Z ∈ R𝑁 ×𝐷 is the output node representation. The augmented two
+graph views are fed into the spectral graph encoder and makes the
+augmented node representations.
+
+Structure Perturb
+Laplacian Perturb
+Projection &
+graph view
+I graph encoder
+Contrastive Objectives
+inter-negative
+iCGN with L
+: Projection
+Embedding
+Representation 1
+intra-negative
+CGN with L
+ Projection
+q2
+inter-positive
+Embedding 2:
+Representation 2Signed Directed Graph Contrastive Learning with Structure and Laplacian Augmentation
+4.3
+Contrastive Objective
+There are two graph views after perturbations, and the graph encoder
+makes node representations of them Z1 and Z2. The goal of the
+contrastive objective is that the embeddings of the same nodes agree
+with each other. At the same time, they distinguish from the other
+nodes. We apply a MLP projection layer to improve the discrimina-
+tive power.[2, 22] We define the contrastive losses with the projected
+representation M.
+4.3.1
+Inter-view Loss. It is considered that the two identical
+nodes from different views are inter-positive pairs. On the other
+hand, the rest of the nodes are considered inter-negative pairs. For
+example, a node 𝑢 from graph view 1 and another node 𝑢 from graph
+view 2 are inter-positive pairs. And others nodes 𝑣 ∈ V and 𝑣 ≠ 𝑢
+from graph view 2 are inter-negative pairs with a node 𝑢 from graph
+view 1. Even though the augmented representations from different
+graph views are different, they represent the same nodes. We want
+to maximize the agreement of the positive pair representations m𝑣1
+𝑢
+and m𝑣2
+𝑢 . While minimize the agreements of the negative pairs m𝑣1
+𝑢
+and m𝑣2𝑣 . The goal of the inter-view objective is to maximize the
+similarity of positive pairs and minimize the negative pairs.
+L𝑖𝑛𝑡𝑒𝑟 = − 1
+𝑁
+𝑁
+∑︁
+𝑖=𝑖
+log
+exp(𝑠𝑖𝑚(m𝑣1
+𝑖 , m𝑣2
+𝑖 )/𝜏)
+�𝑁
+𝑗=1,𝑗≠𝑖 exp(𝑠𝑖𝑚(m𝑣1
+𝑖 , m𝑣2
+𝑗 )/𝜏)
+𝑠𝑖𝑚 is a cosine similarity operation, and 𝜏 is a temperature parameter.
+4.3.2
+Intra-view Loss. We define intra-view loss as similar to
+inter-view loss. The inter-view loss compares the projected node
+representations between the two different graph views. In contrast,
+inter-view loss calculates the discriminative loss within a graph
+view. Each node in a graph view is a unique user. All nodes have
+their representations, and we need to distinguish them from others.
+Therefore, we define inter-view loss as,
+L𝑖𝑛𝑡𝑟𝑎 = − 1
+𝑁
+𝑁
+∑︁
+𝑖=𝑖
+log
+1
+�𝑁
+𝑗=1,𝑗≠𝑖 exp(𝑠𝑖𝑚(m𝑣
+𝑖 , m𝑣
+𝑗 )/𝜏)
+.
+4.3.3
+Contrastive Loss. The whole contrastive loss is the sum of
+the inter- and intra-view loss functions. The goal of the contrastive
+loss is to let the model learn discriminative power from the projected
+node representations and maximize the positive agreements/mutual
+information.
+L𝑐𝑜𝑛𝑡𝑟𝑎𝑠𝑡𝑖𝑣𝑒 = L𝑖𝑛𝑡𝑒𝑟 + L𝑖𝑛𝑡𝑟𝑎
+4.4
+Prediction and Label Loss
+For model training, we not only use contrastive loss but also uti-
+lize label loss. The augmented two graph views are inputs of graph
+encoder and makes two node representations Z1 and Z2. The rep-
+resentations are concatenated and fed into output layer. The output
+layer makes the final node representation.
+R = 𝜎([Z1 ∥ Z2]W𝑜𝑢𝑡 + B𝑜𝑢𝑡)
+R ⊆ 𝑉 ×𝑑 and W𝑜𝑢𝑡 ⊆ 2𝑑 ×𝑑 where 𝑑 is embedding dimension. We
+make prediction results with this output node representation. The
+prediction layer is defined as,
+ˆ𝑦𝑢,𝑣 = 𝜎([r1
+𝑢||r2
+𝑣]W𝑝𝑟𝑒𝑑 + B𝑝𝑟𝑒𝑑)
+Prediction value estimates the link sign when there is an edge from
+node 𝑢 to 𝑣. We define the label loss with the prediction error.
+L𝑙𝑎𝑏𝑒𝑙 = −
+|E+ |
+∑︁
+𝑢,𝑣∈E+
+𝑦𝑢,𝑣logˆ𝑦𝑢,𝑣 −
+|E− |
+∑︁
+𝑢,𝑣∈E−
+(1 − 𝑦𝑢,𝑣)log(1 − ˆ𝑦𝑢,𝑣)
+SDGCL is trained with the following objective function,
+L = 𝛼 × L𝑐𝑜𝑛𝑡𝑟𝑎𝑠𝑡𝑖𝑣𝑒 + L𝑙𝑎𝑏𝑒𝑙.
+𝛼 is weight of contrastive loss. Note that, unlike other graph con-
+trastive studies, it leverages supervised labels.
+5
+EXPERIMENTS
+5.1
+Datasets and Metrics
+We evaluate the model with four real-world signed directed graph
+datasets widely used in the link sign prediction task. Bitcoin-Alpha1
+and Bitcoin-OTC2[27] are extracted from Bitcoin trading platforms.
+Nodes are users, and edges are user relationships. Users can score
+the others on a scale of -10 to +10. Edges higher than 0 are treated
+as positive edges, otherwise negative edges. Epinions3[13] is a who-
+trust-whom network crawled from a consumer review site. Users
+can notate trust or distrust to reviews of other users. Slashdot4[28] is
+a social network of user community site. Especially they share new
+information. Users tag others as friends or foes, and we can construct
+positive and negative edges with this information. Moreover, we
+adopt four metrics, AUC, macro-F1, micro-F1, and binary-F1, for
+unbiased evaluation.
+Dataset
+# nodes
+# pos links
+# neg links
+positive ratio
+Bitcoin-Alpha
+3,783
+22,650
+1,536
+0.937
+Bitcoin-OTC
+5,881
+32,029
+3,563
+0.900
+Epinions
+131,828
+717,667
+123,705
+0.853
+Slashdot
+82,144
+425,072
+124,130
+0.774
+Table 1: Dataset statistics.
+Table 1 shows the statistics of the datasets. BitCoin datasets have
+relatively small size of nodes and edges. The positive and negative
+ratios are highly imbalanced. Three of the datasets have nearly 90%
+of positives among the edges. Otherwise, Slashdot has a 23% share
+of negative edges. The four datasets have various graph sizes and
+positive ratios. It is suitable for unbiased evaluation.
+5.2
+Baselines
+We implemented seven baselines to compare the model performance.
+There are three signed graph convolutions, three constative learnings,
+and one signed graph contrastive model.
+• SGCN[8] defines (un)balanced path based on the balanced
+theory for neighbor aggregation. It is the first signed convolu-
+tion model but does not consider the edge directions.
+1http://www.btc-alpha.com
+2http://www.bitcoin-otc.com
+3http://www.epinions.com
+4http://www.slashdot.com
+
+Taewook Ko, et al.
+Signed Convolution
+Contrastive Learning
+Signed Contrastive Learning
+Dataset
+Metric
+SGCN
+SDGNN
+SDGCN
+GraphCL
+GCA
+SimGRACE
+SGCL
+SDGCL-s
+SDGCL-l
+SDGCL
+Bitcoin-Alpha
+AUC
+0.782
+0.835
+0.858
+0.814
+0.838
+0.823
+0.849
+0.896
+0.883
+0.886
+Macro-F1
+0.668
+0.683
+0.723
+0.653
+0.671
+0.657
+0.712
+0.740
+0.744
+0.754
+Micro-F1
+0.899
+0.909
+0.923
+0.907
+0.913
+0.919
+0.923
+0.947
+0.942
+0.949
+Binary-F1
+0.941
+0.947
+0.958
+0.950
+0.953
+0.957
+0.959
+0.973
+0.969
+0.971
+Bitcoin-OTC
+AUC
+0.832
+0.879
+0.887
+0.852
+0.868
+0.859
+0.893
+0.914
+0.902
+0.910
+Macro-F1
+0.710
+0.751
+0.773
+0.725
+0.743
+0.725
+0.781
+0.803
+0.796
+0.802
+Micro-F1
+0.886
+0.902
+0.911
+0.904
+0.907
+0.906
+0.920
+0.935
+0.930
+0.937
+Binary-F1
+0.924
+0.938
+0.950
+0.948
+0.948
+0.948
+0.956
+0.964
+0.962
+0.965
+Epinions
+AUC
+0.848
+0.914
+0.939
+0.839
+0.911
+0.913
+0.876
+0.941
+0.943
+0.942
+Macro-F1
+0.741
+0.831
+0.850
+0.726
+0.814
+0.812
+0.798
+0.861
+0.865
+0.863
+Micro-F1
+0.893
+0.912
+0.925
+0.887
+0.913
+0.915
+0.909
+0.934
+0.936
+0.936
+Binary-F1
+0.937
+0.944
+0.956
+0.936
+0.950
+0.951
+0.948
+0.962
+0.963
+0.963
+Slashdot
+AUC
+0.740
+0.849
+0.886
+0.813
+0.870
+0.865
+0.783
+0.900
+0.891
+0.902
+Macro-F1
+0.688
+0.729
+0.780
+0.667
+0.750
+0.745
+0.683
+0.792
+0.785
+0.789
+Micro-F1
+0.786
+0.823
+0.855
+0.813
+0.842
+0.833
+0.811
+0.864
+0.859
+0.863
+Binary-F1
+0.869
+0.889
+0.908
+0.887
+0.902
+0.895
+0.884
+0.915
+0.911
+0.914
+Table 2: Link sign prediction performance. Bold and underline indicate the best and the second performance respectively. The per-
+formances are the average score of 10 experiments with different seed sets.
+• SDGNN[21] proposed four weight matrices to aggregate the
+neighbor features according to the edge types adaptively.
+• SDGCN[26] is the first spectral convolution for signed-
+directed graphs. They proposed a signed magnetic Laplacian
+to overcome the disadvantage of traditional graph Laplacian.
+• GraphCL[51] is a graph contrastive model with node and
+edge augmentations. They randomly perturbed graph struc-
+tures by dropping or adding edges and nodes.
+• GCA[56] proposed score-based graph augmentation methods
+and introduced novel node-level contrastive objectives.
+• SimGRACE[49] tried to overcome the cumbersome augmen-
+tation search. They introduced a novel graph encoder pertur-
+bation rather than graph augmentation.
+• SGCL[39] is a graph contrastive model for signed-directed
+graphs. It perturbs the edge sign and directions to get positive
+and negative graph views. Each view makes node representa-
+tions and defines a node-level contrastive loss.
+For a fair comparison, all the baselines were implemented under the
+same environments, such as embedding dimension, learning epochs,
+and convolution layers. Though the graph contrastive baselines are
+designed for self-supervised learning, we train them with the same
+supervised loss. Moreover, we removed the read-out process of
+GraphCL and SimGRACE, which are designed for graph embedding.
+5.3
+Implementation Details
+We follow the hyperparameter settings of the original papers of
+each model. The node embedding dimension is set to 64 for all the
+baselines to make the same learning capacity. The edges are split
+into 60:20:20 for training, validation, and test sets. However, we did
+not use all the positive edges as training instances during the training
+stage. We sampled positive edges at the ratio of 3:1. It is because
+the ratio of the positive edge is too high. If we use all positive
+edges for training, the model would be easy to make positives only.
+Nevertheless, we utilize all edge instances for the validation and test
+phase. The structure perturbing ratio 𝑝 and 𝑟 are set to 0.1 for all
+datasets. And the magnetic Laplacian phase 𝑞 is randomly selected
+from [0, 0.1𝜋, 0.2𝜋, 0.3𝜋, 0.4𝜋] for every iterations. The influence
+of the perturbing ratio and 𝑞 variation is analyzed in Figure 5. The
+contrastive loss weight 𝛼 is set to 0.2. Graph encoder stacks two
+signed-directed spectral convolution layers. We use Adam optimizer
+with learning rate = 0.001, weight decay = 0.001. All experiments
+are run 10 times with different seed sets to avoid randomness and get
+the average score. The experiments are conducted on Xeon E5-2660
+v4 and accelerated via Nvidia Titan XP 12G GPU. The software is
+implemented via Ubuntu v16.4 with python v3.6 and Pytorch v1.8.0.
+6
+RESULTS
+6.1
+Link Sign Prediction
+We summarize the prediction results in Table 2. There are three
+proposed models. SDGCL-s is a model only with structure aug-
+mentation, and SDGCL-l is a model with Laplacian augmentation.
+SDGCL is the one that adopts both augmentations. Consequently, the
+proposed SDGCL and its variants always show the best performance
+in all datasets and all metrics. For Bitcoin-Alpha, Bitcoin-OTC, and
+Slashdot datasets, SDGCL-composite and SDGCL-structure have
+the highest score alternatively. SDGCL-composite and SDGCL-
+Laplacian get the highest score in the Epinions dataset.
+SGCL or SDGCN shows the best performance among the baselines.
+SDGCN could achieve better performance than others thanks to the
+signed-directed spectral convolution. They could fully enjoy the sign
+and direction information. SGCL is the only model for signed graph
+contrastive. They could improve the performance via a contrastive
+learning mechanism than other baselines. However, SGCL shows
+poor performance on Epinions and Slashdot. We think there are
+two reasons. First, the advantage of contrastive learning is maxi-
+mized with scarce labeled datasets. That is why the model can learn
+more from the contrastive loss. However, if the label is rich enough,
+
+Signed Directed Graph Contrastive Learning with Structure and Laplacian Augmentation
+Figure 4: Structure perturbing analysis. x-axis indicates perturbing ratio.
+Figure 5: Laplacian perturbing analysis. x-axis indicates noise variance.
+the utility of contrastive learning is lessened. Second, we ruin the
+original dataset context by random perturbation of SGCL. Epinions
+and Slashdot have larger sizes of edges. The hidden patterns and
+joint distributions of edges are far more complicated. We harm the
+original data information by perturbing many edges and splitting
+them into different graph views. Unlike SGCL, our proposed model
+minimizes the structure perturbation by proposing Laplacian aug-
+mentation. Furthermore, we do not split positive and negative edges
+in different graph views.
+6.2
+Perturbing Analysis
+Here we analyze the effect of structure and Laplacian augmentation.
+Figure 4 shows that the performance varies according to the edge
+perturbing ratio. Laplacian perturbation is not adopted in this ex-
+periment to get the effect of edge perturbing. When the perturbing
+ratios are zero, AUC and Macro-F1 scores are low. We can see the
+importance of structure perturbing. The performances go up with
+a small perturbing ratio and go down with a large value. Figure 5
+shows the performances of magnetic Laplacian phase 𝑞 variation.
+We set default 𝑞 as 0.1𝜋 and add some Gaussian noise with standard
+deviation. At the same time, the edge perturbing ratios are set to
+zeros. The x-axis shows the standard deviation values. We cap the
+maximum value of 𝑞 to 0.5𝜋. Similar to structure perturbation, zero
+standard deviations show low performances. Zero standard deviation
+means there is no Laplacian perturbation. They have similar trends
+that go up and go down. However, Laplacian perturbation is not as
+sensitive as structure perturbation. Though graph augmentations are
+compulsory in contrastive learning, much perturbation is detrimental
+to training. Structure perturbation gives direct contrastive informa-
+tion while it may lose the data context. Laplacian perturbation gives
+indirect perturbation of graph data but is still effective. SDGCL
+combines these two augmentations for better and more stable graph
+augmentation.
+6.3
+Ablation Study
+Table 3 shows the AUC scores of ablation studies. We check the ef-
+fects of SDGCL components. w/o Laplacian, w/o structure, and w/o
+augmentation are the variation of graph augmentations. Especially,
+w/o augmentation shows the lowest performance. As we expected,
+the model leverages the advantage of contrastive learning. It shows
+that augmentations are important in our model. w/o contrastive loss
+is a model with 𝛼 = 0 and the model uses label loss only. Note
+that it does not mean that the model does not utilize the benefits of
+contrastive learning. Even though the contrastive loss weight is zero,
+label loss is calculated with augmented representations of graph
+views. Thus, the model is still leveraging the contrastive effects. w/o
+projection is a model without a projection layer. It is well known that
+the projection layer is useful for robust contrastive learning[2, 22].
+Bitcoin-Alpha
+Bitcoin-OTC
+Epinions
+Slashdot
+SDGCL
+0.886
+0.910
+0.942
+0.902
+w/o structure aug
+0.883
+0.902
+0.943
+0.891
+w/o Laplacian aug
+0.896
+0.914
+0.941
+0.900
+w/o augmentation
+0.846
+0.889
+0.936
+0.884
+w/o contrastive loss
+0.878
+0.901
+0.941
+0.891
+w/o projection
+0.872
+0.904
+0.939
+0.897
+Table 3: Ablation Study
+
+0.89
+AUC(left)
+0.79
+0.82
+Macro F1(right)
+0.83
+0.89
+0.94
+0.90
+0.77
+0.91
+0.87
+0.81
+0.87
+0.75
+0.90
+0.800.93
+0.79
+0.88
++0.85
+0.85
+0.77
+0.89
+0.92
+0.78
+0.86
+0.75
+0.83
+0.71
+0.83
+0.05 0.075 0.1
+0.15
+0.2
+0
+0.05 0.075 0.1
+0.15
+0.2
+0.91
+0.0250.050.0750.1
+0.15
+0.0250.050.0750.1
+0.15 0.730.87
+ 0.790
+AUC(left)
+0.76
++0.81
+0.89
+Macro F1(right)
+0.90
+0.87
+0.86
+0.785
+0.940
+0.85
+0.88
+0.74
+0.88
+0.85
+0.780
+0.83
+0.935
+0.87
+0.72
+0.86
+LL0+
+0
+0.1
+0.2
+0.3
+0.4
+0.1
+0.2
+0.3
+0.4
+0
+0.1
+0.2
+0.3
+0.40.84
+0
+0.1
+0.2
+0.3
+0.4Taewook Ko, et al.
+7
+CONCLUSION
+This paper proposed SDGCL, a signed-directed graph contrastive
+learning model. To make augmented node representations, we intro-
+duced two levels of the perturbation process. Structure perturbation
+randomly changes the signs and directions of edges. Even though
+it may lose some vital information from the original graphs, we
+check that it makes the model noise-robust. Laplacian perturbation
+changes the phase parameter 𝑞 for every training iteration to give
+perturbation. It does not directly influence the graph information
+but perturbs the Laplacian matrix. Then we define the inter- and
+intra-view contrastive objectives with the augmented node repre-
+sentations. Unlike other signed graph studies, our model does not
+depend on social theories or assumptions. Also, there are no arbitrary
+graph-splitting processes. Our graph encoder is based on the Lapla-
+cian matrix; it only enjoys the structural information. Therefore it
+could reduce computational costs. We evaluate the proposed model
+with four real-world datasets and seven baselines. The experimental
+results show that our proposed model has superior performance to
+others. SDGCL and their variants ranked at the top for every dataset
+and metric.
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+
diff --git a/StE4T4oBgHgl3EQflw2y/content/tmp_files/load_file.txt b/StE4T4oBgHgl3EQflw2y/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf,len=990
+page_content='Signed Directed Graph Contrastive Learning with Structure  and Laplacian Augmentation  Taewook Ko  taewook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='ko@snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='kr  Seoul National University  Republic of Korea  Yoonhyuk Choi  younhyuk95@snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='kr  Seoul National University  Republic of Korea  Chong-Kwon Kim  ckim@kentech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='kr  Korea Institute of Energy Technology  Republic of Korea  ABSTRACT  Graph contrastive learning has become a powerful technique for  several graph mining tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It learns discriminative representation  from different perspectives of augmented graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Ubiquitous in our  daily life, singed-directed graphs are the most complex and tricky  to analyze among various graph types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' That is why singed-directed  graph contrastive learning has not been studied much yet, while there  are many contrastive studies for unsigned and undirected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Thus, this  paper proposes a novel signed-directed graph contrastive learning,  SDGCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It makes two different structurally perturbed graph views  and gets node representations via magnetic Laplacian perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  We use a node-level contrastive loss to maximize the mutual infor-  mation between the two graph views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The model is jointly learned  with contrastive and supervised objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The graph encoder of  SDGCL does not depend on social theories or predefined assump-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Therefore it does not require finding triads or selecting neigh-  bors to aggregate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It leverages only the edge signs and directions  via magnetic Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' To the best of our knowledge, it is the first  to introduce magnetic Laplacian perturbation and signed spectral  graph contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The superiority of the proposed model  is demonstrated through exhaustive experiments on four real-world  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' SDGCL shows better performance than other state-of-the-  art on four evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  KEYWORDS  graph contrastive learning, magnetic Laplacian, signed directed  graph, link sign prediction  1  INTRODUCTION  Various types of online social graphs are developing as the use of  the internet increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Social graph research attracts attention from  researchers with its usefulness for various downstream tasks such  as prediction, classification, and recommendation[9, 11, 17, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Signed-directed graphs, which express the relationship between  users as like/dislike or trust/distrust, have the most information and  are the most valuable among several graph types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' However, it has  difficulties to analyze them caused of severe noise and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Many network embedding and graph convolution studies for signed-  directed graphs have been proposed based on the two well-known  social theories, balance and status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' [18, 19] Balance theory is the  concept that "A friend of my friend is my friend, and an enemy of  my friend is my enemy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='" Furthermore, status theory sets the relative  user ranks with the relation of positive and negative edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The ideas  of most existing studies, including SiNE[47], SGCN[8], SNEA[31],  and SDGNN[21], are derived from the two social theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Recently,  some studies that do not use them have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' ROSE[45]  proposed a user-role-based methodology by transforming a graph  into an unsigned bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' SDGCN[26] proposed a spectral convolu-  tion via a signed magnetic Laplacian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Contrastive learning[1, 16] is a successful learning mechanism in  computer vision research with fewer labels or a self-supervised envi-  ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It learns discriminative representations through contrastive  loss from the data itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' They maximize the mutual information of  representations of augmented images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Recently, many graph stud-  ies utilizing contrastive learning have been suggested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Graph con-  trastive learning models generate graph views in their own ways,  such as edge or node deletion[51], addition[52], random-walk[38],  or attribute masking[56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' JOAO[50] and AD-GCL[43] learn optimal  perturbing distribution according to the dataset and SimGRACE[49]  adds gaussian noise on the trained parameters for encoder perturba-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' They define contrastive loss with the augmented representations  of nodes or graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Graph contrastive learning also aims to maxi-  mize the mutual information of different graph views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' However, not  much research has been conducted on signed-directed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  SGCL[39] is the first study to apply contrastive learning on signed-  directed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' They make two graph views by giving random perturba-  tions on edge signs and directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Then the views are divided into  positive graph views and negative graph views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The positive graph  view is constructed with only the positive edges of a graph view, and  the same for the negative graph view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Then run graph convolution  on the different graph views to get augmented node representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Though SGCL is innovative, there are some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' First, it  loses the context of the original graph if we separate the edges into  different graphs by the edge types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The mutual influence between  edge types cannot be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' For example, valuable patterns like the  balance theory cannot be reflected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Second, most real-world signed  graphs have a nearly 90% of positive edge ratio[27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Therefore, the  negative views are unsuitable for graph convolutions due to the  sparsity issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' In this study, we propose a Signed Directed Graph  Contrastive Learning(SDGCL), which improves these limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  SDGCL utilizes two levels of perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We apply random edge  perturbation similar to SGCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It changes edge signs and direc-  tions to generate augmented graph views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Then we introduce mag-  netic Laplacian perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It changes a phase parameter 𝑞 of the  magnetic Laplacian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' 𝑞 controls the phase angle between the  real- and imaginary axis[26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Magnetic Laplacian was studied in  the field of quantum physics[5, 35, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Thanks to its Hermitian  properties, it has been used in graph convolutions for directed and  signed-directed[41, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Signed magnetic Laplacian encodes graph  sturucture, including edge signs and directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It reflects the graph  information and allows the model to learn the patterns of nodes and  edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We define a spectral convolution layer with the perturbed  Laplacian by the idea of graph signal processing[7, 15, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' SDGCL  gets the two augmented node representations with the graph encoder  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='05163v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='LG]  12 Jan 2023  Taewook Ko, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Figure 1: Example of a signed-directed graph and its structure  perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Dashed edges indicate perturbed edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  and defines node-level contrast loss[39, 56] for contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  The performance of the proposed model was evaluated with the link  sign prediction task, which is widely selected to evaluate the signed-  directed graph embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It was tested with four real-world graphs  and showed excellent performance compared to various baselines  for signed-directed graph embedding and graph contrast learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  The followings are the contributions of this paper  This paper proposes a novel signed-directed graph contrastive  learning model, SDGCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  It augments node representations via two stages, graph struc-  ture perturbation and magnetic Laplacian perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  To the best of our knowledge, it is the first to introduce mag-  netic Laplacian perturbation and is the first spectral signed  graph contrastive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  The proposed model has the best link sign prediction perfor-  mance in real-world graph experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  2  RELATED WORKS  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='1  Signed-Directed Graph Embedding  Most signed-directed graph studies utilize sociological stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' [18,  19] SiNE[47] is an undirected model that uses balance theory, defin-  ing an objective function: friend nodes increase their similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  BESIDE[3] focused on the bridge edges to overcome the limitation  of triads and utilized both social theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' SGCN[8] defined a novel  balanced and unbalanced path for neighbor aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' SNEA[31]  and SiGAT[20] used attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' SDGNN defined four  different weight matrices to distinguish the edge signs and directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  And they proposed triad loss based on the theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Balance and sta-  tus theories are developed with the patterns of the user triads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' About  60-70% of triads on real-world graphs satisfy the theories[24, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Though the theories are crucial paradigms in signed-directed graph  research, not all the triads satisfy them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Moreover, it is not a rule  that all users should follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Models that depend on the theories are  not working well with users with little or no triads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' On top of that,  such models require much computational cost to find triads or paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Recently, ROSE[23] proposed a methodology using user-role by  transforming graphs into unsigned bipartite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' SDGCN[26] proposed  a spectral convolution model by proving the positive semidefinite of  signed magnetic Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='2  Magnetic Laplacian  Magnetic Laplacian was first introduced in the field of quan-  tum mechanics to analyze the charged particle under magnetic  flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' [5, 32, 35, 40] But thanks to its Hermitian properties, the mag-  netic Laplacian has been utilized in directed graph studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' [14, 34]  Some application studies such as graph clustering[4, 6] node[54]  and graph representation learning[12] are delivered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' MagNet[54]  proved the positive semidefinite property of the magnetic Laplacian  and proposed a spectral graph convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Recently, some studies  extended the magnetic Laplacian to the signed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' SDGCN[26]  and SigMagNet[10] define signed magnetic Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The signed  magnetic Laplacian matrix uniquely encodes edge types of signed-  directed graphs with real- and imaginary numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' They proposed  spectral graph convolution with the signed magnetic Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='3  Graph Contrastive Learning  SimCLR[1] and MoCo[16] proposed contrastive learning and  achieved great success in image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Many follow-up stud-  ies were proposed thanks to the increasing learning efficiency via  self-supervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' DGI[46], GMI[37] adopted the contrastive learning  to graph studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' They measure mutual information of input and node  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' InfoGraph[42] proposed patch-level representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  GCC employed random-walk sampling to make positive and nega-  tive samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Since then, several graph augmentations[51, 52, 56]  and graph encoders[39, 49] have been proposed for contrastive learn-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' There are contrastive studies for graph clustering[36, 55], node  embedding[56], DDI prediction[30, 48], and recommendation[33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  However, signed-directed graph contrastive studies are not discussed  much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' To the best of our knowledge, SGCL[39] is the only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  SGCL makes graph views by randomly perturbing the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Then  they construct positive- and negative-graph views by separating  edges by the edge type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' However, the edges of signed-directed graphs  have meaningful relationships with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Therefore, the con-  text of the data cannot be properly interpreted when we arbitrarily  separate graph views by edge types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Moreover, they are inefficient  in that they need to find positively or negatively linked nodes for  every aggregation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' To overcomes these limitations, this study  proposes a novel graph contrastive learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  3  PROBLEM FORMULATION  Let G = (𝑉, E+, E−) be a signed-directed graph where 𝑉 is a set  of nodes like users in social graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' E+ ⊆ 𝑉 × 𝑉 indicates positive  edge matrix, and E− ⊆ 𝑉 × 𝑉 is of negative edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Positive and  negative edges represent user relationships, such as like/dislike or  trust/distrust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' For example E+𝑢,𝑣 equals 1 if there is a positive edge  from node 𝑢 to node 𝑣;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' otherwise, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Similarly, if there is a negative  edge from node 𝑣 to 𝑢, E−𝑣,𝑢 equals 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Note that E+ ∩ E− = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Because we do not accept users having two different edges to the  other user simultaneously, such as "John loves Jane and also hates  her.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='" A user may be connected to other users by one of the three  relations(none, positive, negative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Thus, there are nine-edge types  between a user pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The goal of this paper is to map the nodes 𝑢 ∈ 𝑉  into the low-dimensional embedding vectors 𝑧𝑢 ∈ R𝑑 with a given  graph G as:  𝑓 (G) = 𝑍,  𝑓 is a transformation function and 𝑍 ∈ R|𝑉 |×𝑑 is an embedding ma-  trix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Each row of 𝑍 represents the node embedding with dimension  size 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  edge  perturbation  original graph  augmented graph viewSigned Directed Graph Contrastive Learning with Structure and Laplacian Augmentation  4  MODEL FRAMEWORK  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='1  Graph Augmentation  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='1  Structure Perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' There are two methods for struc-  ture perturbation, edge sign perturbing and edge direction perturbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  In edge sign perturbing, we change the edge signs randomly from  the given graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' For example, we sample 𝑝% of positive edges and  change their signs to negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' And the same for the negative edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Similarly, we sample 𝑟% of edges and change their directions to in-  verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' If an edge is bidirectional, we change it to directional arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  With this random edge perturbation, we get structurally perturbed  graph views, �G = (𝑉, ˜E+, ˜E+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Figure 1 shows an example of struc-  ture perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' As already known by balance and status theories,  there is important information in the edge sign and directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We  may harm the vital triad patterns via random perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' However,  we expect some amount of perturbation would be helpful to learn  robust representations from noisy real-world data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Moreover, the  perturbing can discover and exploit some relationships that might ex-  ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The model can improve the generalization performance through  random perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='2  Signed-Directed Magnetic Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Before introduc-  ing Laplacian Perturbation, we define the signed-directed magnetic  Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Graph Laplacian is useful to encode graph sturucture  with degree and adjacency matrices, L = D − A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' They are not only  positive semidefinite but also have non-negative eigenvalues and as-  sociate orthonormal eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' With that properties, [25] and [15]  proposed the idea of spectral graph convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' However, graph  Laplacian is asymmetric when a graph is directed or signed-directed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  They have complex eigenvalues and do not satisfy the properties  for spectral convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Thus, [10, 26, 41] proposed a novel mag-  netic Laplacian matrix representing the structure of signed-directed  graphs and satisfying positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' First of all, we define a  complex Hermitian adjacency matrix as follow,  H𝑞 = A𝑠 ⊙ P𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  A𝑠 := 1  2 (A + A⊺) ⊆ 𝑉 × 𝑉 is a symmetrized adjacency matrix,  and P𝑞 ⊆ 𝑉 × 𝑉 is a phase matrix with complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' ⊙ is an  element-wise multiplication operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The definition of the phase  matrix is,  P𝑞(𝑢, 𝑣) :=  exp(𝑖Θ𝑞  𝑢𝑣)A𝑢𝑣 + exp(𝑖Θ𝑞  𝑢𝑣)A𝑣𝑢  |exp(𝑖Θ𝑞  𝑢𝑣)A𝑢𝑣 + exp(𝑖Θ𝑞  𝑢𝑣)A𝑣𝑢| + 𝜖  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Θ𝑞  𝑢𝑣 = 𝑞E+𝑢𝑣 +(𝜋 +𝑞)E−𝑢𝑣 and Θ𝑢𝑣  𝑞 = −𝑞E+𝑣𝑢 +(𝜋 −𝑞)E−𝑣𝑢 Be careful  with the subscripts orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' 𝑞 is a hyperparameter lies in [0,𝜋/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The  symmetrized adjacency matrix encodes the node connectivity, and  the phase matrix encodes link directions and signs with different  phase values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Figure 2(b) shows the edge encodings of the defined  Hermitian adjacency matrix H𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' They uniquely encode the nine-  edge types of signed-directed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It tells node connections and  edge types as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We can see that H𝑞 is a complex numbered skew-  symmetric form, a complex Hermitian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Then, we define the  signed-directed magnetic Laplacians with this Hermitian adjacency  by,  L𝑞  𝑈 := D𝑠 − H𝑞 = D𝑠 − A𝑠 ⊙ P𝑞  L𝑞  𝑁 := I − (D− 1  2  𝑠  A𝑠D− 1  2  𝑠  ) ⊙ P𝑞,  Figure 2: Edge encoding and meaning of 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It shows edge encod-  ing values of a complex Hermitian adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' 𝑞 controls  the phase angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  D𝑠 is a symmetric degree matrix, similar to A𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' L𝑞  𝑈 and L𝑞  𝑁 are un-  normalized and normalized signed-directed magnetic Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It  is well known that the skew-symmetric, complex Hermitian matrix  is positive semidefinite[10, 26, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Thus, the Laplacians are posi-  tive semidefinite and diagonalizable by spectral decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' For  example, the normalized Laplacian is diagonalized by,  L𝑞  𝑁 = UΛU†.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Each column of U is eigenvector u𝑘 and U† is a conjugate transpose  of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Λ is a diagonal matrix where the elements are 𝑘-th eigenvalues  Λ𝑘,𝑘 = 𝜆𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The eigenvalues and eigenvectors contain the structural  information of the signed-directed graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We leverage this matrix to  define spectral graph convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='3  Laplacian Perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Figure 2(a) describes the meaning  of 𝑞 in magnetic Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' 𝑞 controls the phase angle of encoded  values between the real- and imaginary axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' If 𝑞 is small, the Lapla-  cian puts less attention on directional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It becomes an  undirected model when the 𝑞 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' However, the high 𝑞 value also  harms the encoding validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' 𝑞 influences the encoding effectiveness  of the signed-directed magnetic Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' In this subsection, we  introduce Laplacian perturbation via 𝑞 variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We randomly se-  lect 𝑞 value from 0 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='4𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The Laplacian matrix varies according  to the 𝑞 variations, though the meaning of the graph structure that  the Laplacian represents is still the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' This kind of Laplacian  perturbation is a useful technique that makes graph augmentation  without distortion of the original data[44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  In short, we make two structurally perturbed graph views �G1 and  �G2, through random edge changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Then we get perturbed signed-  directed magnetic Laplacian �L  𝑞1 and �L  𝑞2 from the graph views with  randomly chosen 𝑞 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' With these Laplacian matrices, we define  graph encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='2  Graph Encoder  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='1  Spectral Convolution via Magnetic Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The  signed-directed magnetic Laplacian L, is diagonalizable with eigen-  vector matrix U, and diagonal eigenvalues Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Several graph convolu-  tion studies [7, 15] utilize the eigenvectors as discrete Fourier modes  Edge type  H (u,)  imag-axis  u→v  (cosq + isinq)  u←v  uv  (cosq - isinq)  u←v  (cosq - ising)  u←v  u-→v  u→v  (cosq + isinq)  uv  uv  uv  real-axis  cosq  u-→v  u←v  uv  cosq  uv  u与v  isinq  u乌v  isinq  u  V  0  (a)  (b)Taewook Ko, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Figure 3: Model overview  of graph signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The graph signals are transformed into  the Fourier domain through graph Fourier transform x : 𝑉 → C by  ˆx = U†x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The inverse Fourier transform formula is defined as follow  thanks to the unitarity of U,  x = Uˆx =  𝑁  ∑︁  𝑘=1  ˆx(𝑘)u𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  The spectral convolution operation of graph signal processing is  described as  g𝜃 ∗ x = Ug𝜃U†x,  where g𝜃 = 𝑑𝑖𝑎𝑔(𝜃) is a trainable filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' For efficient calculation,  [15] proposed a truncated Chebyshev polynomial expansion of the  filter by,  g𝜃′(Λ) ≈  𝐾  ∑︁  𝑘=0  𝜃 ′  𝑘𝑇𝑘 ( ˜Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  𝑇0(𝑥) = 1,𝑇1(𝑥) = 𝑥, and 𝑇𝑘 = 2𝑥𝑇𝑘−1(𝑥) +𝑇𝑘−2(𝑥) for 𝑘 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' 𝑘 is  an expansion order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' ˜Λ =  2  𝜆𝑚𝑎𝑥 Λ−I is a normalized eigenvalue matrix,  and 𝜆𝑚𝑎𝑥 is the largest eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' 𝜃 ′  𝑘 are Chebyshev coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  We have a simplified form of spectral graph convolution by,  g𝜃′ ∗ x =  𝐾  ∑︁  𝑘=0  𝜃 ′  𝑘𝑇𝑘 ( ˜L)𝑥,  where ˜L =  2  𝜆𝑚𝑎𝑥 L − I analogous to ˜Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='2  Spectral Convolution Layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We define the spectral con-  volution layer with the approximated signed-directed spectral convo-  lution operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We set 𝑘 as 1, the maximum polynomial order, and  𝜆𝑚𝑎𝑥 is assumed 2 to make it practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Similar to GCN[25], we set  𝜃 = 𝜃 ′  0 = −𝜃 ′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Then we have  g𝜃′ ∗ x ≈ 𝜃 (I + (D− 1  2  𝑠  A𝑠D− 1  2  𝑠  ) ⊙ P𝑞)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  The spectral convolution layer is defined as  H = ( ˜D− 1  2  𝑠  ˜A𝑠 ˜D− 1  2  𝑠  ⊙ P𝑞)XW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  H ∈ R𝑁 ×𝐹 is the convoluted graph signals or representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' X ∈  R𝑁 ×𝐶 is the input signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='𝐶 and 𝐹 are the numbers of input and output  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' W ∈ R𝐶×𝐹 is a learnable matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It is a renormalization  trick that I + (D− 1  2  𝑠  A𝑠D− 1  2  𝑠  ) ⊙ P𝑞 → ˜D− 1  2  𝑠  ˜A𝑠 ˜D− 1  2  𝑠  ⊙ P𝑞 where ˜As =  A𝑠 + I and ˜Ds(𝑖,𝑖) = �  𝑗 ˜As(𝑖, 𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It prevents gradient vanishing and  exploding problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='3  Signed-Directed Graph Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The graph encoder  stacks 𝐿 layers of proposed spectral convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The 𝑙-th  layer feature vector x(𝑙) is defined as,  x(𝑙)  𝑗  = 𝜎(  𝐹𝑙−1  ∑︁  𝑖=1  Y(𝑙)  𝑖𝑗 x(𝑙−1)  𝑖  + b(𝑙)  𝑗 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  We use a novel activation function for the complex elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The  activation function 𝜎 is an complex version of ReLU, 𝜎(𝑧) = 𝑧, if  −𝜋/2 ≤ arg(𝑧) ≤ 𝜋/2, otherwise, 𝜎(𝑧) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The feature matrix X(𝐿)  has both real and imaginary values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Here we introduce unwinding  operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It converts real and imaginary features into the same  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  X(𝐿)  unwind = [real(X(𝐿))||imag(X(𝐿)) ⊗ (−𝑖)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  We add a fully connected layer after unwinding to get the node  representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Z = 𝜎(X(𝐿)  unwindW𝐿+1 + B(𝐿+1))  Z ∈ R𝑁 ×𝐷 is the output node representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The augmented two  graph views are fed into the spectral graph encoder and makes the  augmented node representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Structure Perturb  Laplacian Perturb  Projection &  graph view  I graph encoder  Contrastive Objectives  inter-negative  iCGN with L  : Projection  Embedding  Representation 1  intra-negative  CGN with L  Projection  q2  inter-positive  Embedding 2:  Representation 2Signed Directed Graph Contrastive Learning with Structure and Laplacian Augmentation  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='3  Contrastive Objective  There are two graph views after perturbations, and the graph encoder  makes node representations of them Z1 and Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The goal of the  contrastive objective is that the embeddings of the same nodes agree  with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' At the same time, they distinguish from the other  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We apply a MLP projection layer to improve the discrimina-  tive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' [2, 22] We define the contrastive losses with the projected  representation M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='1  Inter-view Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It is considered that the two identical  nodes from different views are inter-positive pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' On the other  hand, the rest of the nodes are considered inter-negative pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' For  example, a node 𝑢 from graph view 1 and another node 𝑢 from graph  view 2 are inter-positive pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' And others nodes 𝑣 ∈ V and 𝑣 ≠ 𝑢  from graph view 2 are inter-negative pairs with a node 𝑢 from graph  view 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Even though the augmented representations from different  graph views are different, they represent the same nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We want  to maximize the agreement of the positive pair representations m𝑣1  𝑢  and m𝑣2  𝑢 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' While minimize the agreements of the negative pairs m𝑣1  𝑢  and m𝑣2𝑣 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The goal of the inter-view objective is to maximize the  similarity of positive pairs and minimize the negative pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  L𝑖𝑛𝑡𝑒𝑟 = − 1  𝑁  𝑁  ∑︁  𝑖=𝑖  log  exp(𝑠𝑖𝑚(m𝑣1  𝑖 , m𝑣2  𝑖 )/𝜏)  �𝑁  𝑗=1,𝑗≠𝑖 exp(𝑠𝑖𝑚(m𝑣1  𝑖 , m𝑣2  𝑗 )/𝜏)  𝑠𝑖𝑚 is a cosine similarity operation, and 𝜏 is a temperature parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='2  Intra-view Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We define intra-view loss as similar to  inter-view loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The inter-view loss compares the projected node  representations between the two different graph views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' In contrast,  inter-view loss calculates the discriminative loss within a graph  view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Each node in a graph view is a unique user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' All nodes have  their representations, and we need to distinguish them from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Therefore, we define inter-view loss as,  L𝑖𝑛𝑡𝑟𝑎 = − 1  𝑁  𝑁  ∑︁  𝑖=𝑖  log  1  �𝑁  𝑗=1,𝑗≠𝑖 exp(𝑠𝑖𝑚(m𝑣  𝑖 , m𝑣  𝑗 )/𝜏)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='3  Contrastive Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The whole contrastive loss is the sum of  the inter- and intra-view loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The goal of the contrastive  loss is to let the model learn discriminative power from the projected  node representations and maximize the positive agreements/mutual  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  L𝑐𝑜𝑛𝑡𝑟𝑎𝑠𝑡𝑖𝑣𝑒 = L𝑖𝑛𝑡𝑒𝑟 + L𝑖𝑛𝑡𝑟𝑎  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='4  Prediction and Label Loss  For model training, we not only use contrastive loss but also uti-  lize label loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The augmented two graph views are inputs of graph  encoder and makes two node representations Z1 and Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The rep-  resentations are concatenated and fed into output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The output  layer makes the final node representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  R = 𝜎([Z1 ∥ Z2]W𝑜𝑢𝑡 + B𝑜𝑢𝑡)  R ⊆ 𝑉 ×𝑑 and W𝑜𝑢𝑡 ⊆ 2𝑑 ×𝑑 where 𝑑 is embedding dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We  make prediction results with this output node representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The  prediction layer is defined as,  ˆ𝑦𝑢,𝑣 = 𝜎([r1  𝑢||r2  𝑣]W𝑝𝑟𝑒𝑑 + B𝑝𝑟𝑒𝑑)  Prediction value estimates the link sign when there is an edge from  node 𝑢 to 𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We define the label loss with the prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  L𝑙𝑎𝑏𝑒𝑙 = −  |E+ |  ∑︁  𝑢,𝑣∈E+  𝑦𝑢,𝑣logˆ𝑦𝑢,𝑣 −  |E− |  ∑︁  𝑢,𝑣∈E−  (1 − 𝑦𝑢,𝑣)log(1 − ˆ𝑦𝑢,𝑣)  SDGCL is trained with the following objective function,  L = 𝛼 × L𝑐𝑜𝑛𝑡𝑟𝑎𝑠𝑡𝑖𝑣𝑒 + L𝑙𝑎𝑏𝑒𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  𝛼 is weight of contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Note that, unlike other graph con-  trastive studies, it leverages supervised labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  5  EXPERIMENTS  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='1  Datasets and Metrics  We evaluate the model with four real-world signed directed graph  datasets widely used in the link sign prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Bitcoin-Alpha1  and Bitcoin-OTC2[27] are extracted from Bitcoin trading platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Nodes are users, and edges are user relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Users can score  the others on a scale of -10 to +10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Edges higher than 0 are treated  as positive edges, otherwise negative edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Epinions3[13] is a who-  trust-whom network crawled from a consumer review site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Users  can notate trust or distrust to reviews of other users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Slashdot4[28] is  a social network of user community site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Especially they share new  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Users tag others as friends or foes, and we can construct  positive and negative edges with this information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Moreover, we  adopt four metrics, AUC, macro-F1, micro-F1, and binary-F1, for  unbiased evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Dataset  # nodes  # pos links  # neg links  positive ratio  Bitcoin-Alpha  3,783  22,650  1,536  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='937  Bitcoin-OTC  5,881  32,029  3,563  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='900  Epinions  131,828  717,667  123,705  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='853  Slashdot  82,144  425,072  124,130  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='774  Table 1: Dataset statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Table 1 shows the statistics of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' BitCoin datasets have  relatively small size of nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The positive and negative  ratios are highly imbalanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Three of the datasets have nearly 90%  of positives among the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Otherwise, Slashdot has a 23% share  of negative edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The four datasets have various graph sizes and  positive ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It is suitable for unbiased evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='2  Baselines  We implemented seven baselines to compare the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  There are three signed graph convolutions, three constative learnings,  and one signed graph contrastive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  SGCN[8] defines (un)balanced path based on the balanced  theory for neighbor aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It is the first signed convolu-  tion model but does not consider the edge directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  1http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='btc-alpha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='com  2http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='bitcoin-otc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='com  3http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
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+page_content='914  Table 2: Link sign prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Bold and underline indicate the best and the second performance respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The per-  formances are the average score of 10 experiments with different seed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  SDGNN[21] proposed four weight matrices to aggregate the  neighbor features according to the edge types adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  SDGCN[26] is the first spectral convolution for signed-  directed graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' They proposed a signed magnetic Laplacian  to overcome the disadvantage of traditional graph Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  GraphCL[51] is a graph contrastive model with node and  edge augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' They randomly perturbed graph struc-  tures by dropping or adding edges and nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  GCA[56] proposed score-based graph augmentation methods  and introduced novel node-level contrastive objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  SimGRACE[49] tried to overcome the cumbersome augmen-  tation search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' They introduced a novel graph encoder pertur-  bation rather than graph augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  SGCL[39] is a graph contrastive model for signed-directed  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It perturbs the edge sign and directions to get positive  and negative graph views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Each view makes node representa-  tions and defines a node-level contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  For a fair comparison, all the baselines were implemented under the  same environments, such as embedding dimension, learning epochs,  and convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Though the graph contrastive baselines are  designed for self-supervised learning, we train them with the same  supervised loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Moreover, we removed the read-out process of  GraphCL and SimGRACE, which are designed for graph embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='3  Implementation Details  We follow the hyperparameter settings of the original papers of  each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The node embedding dimension is set to 64 for all the  baselines to make the same learning capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The edges are split  into 60:20:20 for training, validation, and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' However, we did  not use all the positive edges as training instances during the training  stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We sampled positive edges at the ratio of 3:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It is because  the ratio of the positive edge is too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' If we use all positive  edges for training, the model would be easy to make positives only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Nevertheless, we utilize all edge instances for the validation and test  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The structure perturbing ratio 𝑝 and 𝑟 are set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='1 for all  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' And the magnetic Laplacian phase 𝑞 is randomly selected  from [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='1𝜋, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='2𝜋, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='3𝜋, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='4𝜋] for every iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The influence  of the perturbing ratio and 𝑞 variation is analyzed in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The  contrastive loss weight 𝛼 is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Graph encoder stacks two  signed-directed spectral convolution layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We use Adam optimizer  with learning rate = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='001, weight decay = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' All experiments  are run 10 times with different seed sets to avoid randomness and get  the average score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The experiments are conducted on Xeon E5-2660  v4 and accelerated via Nvidia Titan XP 12G GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The software is  implemented via Ubuntu v16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='4 with python v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='6 and Pytorch v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  6  RESULTS  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='1  Link Sign Prediction  We summarize the prediction results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' There are three  proposed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' SDGCL-s is a model only with structure aug-  mentation, and SDGCL-l is a model with Laplacian augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  SDGCL is the one that adopts both augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Consequently, the  proposed SDGCL and its variants always show the best performance  in all datasets and all metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' For Bitcoin-Alpha, Bitcoin-OTC, and  Slashdot datasets, SDGCL-composite and SDGCL-structure have  the highest score alternatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' SDGCL-composite and SDGCL-  Laplacian get the highest score in the Epinions dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  SGCL or SDGCN shows the best performance among the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  SDGCN could achieve better performance than others thanks to the  signed-directed spectral convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' They could fully enjoy the sign  and direction information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' SGCL is the only model for signed graph  contrastive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' They could improve the performance via a contrastive  learning mechanism than other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' However, SGCL shows  poor performance on Epinions and Slashdot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We think there are  two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' First, the advantage of contrastive learning is maxi-  mized with scarce labeled datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' That is why the model can learn  more from the contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' However, if the label is rich enough,  Signed Directed Graph Contrastive Learning with Structure and Laplacian Augmentation  Figure 4: Structure perturbing analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' x-axis indicates perturbing ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Figure 5: Laplacian perturbing analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' x-axis indicates noise variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  the utility of contrastive learning is lessened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Second, we ruin the  original dataset context by random perturbation of SGCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Epinions  and Slashdot have larger sizes of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The hidden patterns and  joint distributions of edges are far more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We harm the  original data information by perturbing many edges and splitting  them into different graph views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Unlike SGCL, our proposed model  minimizes the structure perturbation by proposing Laplacian aug-  mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Furthermore, we do not split positive and negative edges  in different graph views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='2  Perturbing Analysis  Here we analyze the effect of structure and Laplacian augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Figure 4 shows that the performance varies according to the edge  perturbing ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Laplacian perturbation is not adopted in this ex-  periment to get the effect of edge perturbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' When the perturbing  ratios are zero, AUC and Macro-F1 scores are low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We can see the  importance of structure perturbing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The performances go up with  a small perturbing ratio and go down with a large value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Figure 5  shows the performances of magnetic Laplacian phase 𝑞 variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  We set default 𝑞 as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='1𝜋 and add some Gaussian noise with standard  deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' At the same time, the edge perturbing ratios are set to  zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The x-axis shows the standard deviation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We cap the  maximum value of 𝑞 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='5𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Similar to structure perturbation, zero  standard deviations show low performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Zero standard deviation  means there is no Laplacian perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' They have similar trends  that go up and go down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' However, Laplacian perturbation is not as  sensitive as structure perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Though graph augmentations are  compulsory in contrastive learning, much perturbation is detrimental  to training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Structure perturbation gives direct contrastive informa-  tion while it may lose the data context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Laplacian perturbation gives  indirect perturbation of graph data but is still effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' SDGCL  combines these two augmentations for better and more stable graph  augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='3  Ablation Study  Table 3 shows the AUC scores of ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We check the ef-  fects of SDGCL components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' w/o Laplacian, w/o structure, and w/o  augmentation are the variation of graph augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Especially,  w/o augmentation shows the lowest performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' As we expected,  the model leverages the advantage of contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It shows  that augmentations are important in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' w/o contrastive loss  is a model with 𝛼 = 0 and the model uses label loss only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Note  that it does not mean that the model does not utilize the benefits of  contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Even though the contrastive loss weight is zero,  label loss is calculated with augmented representations of graph  views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Thus, the model is still leveraging the contrastive effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' w/o  projection is a model without a projection layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It is well known that  the projection layer is useful for robust contrastive learning[2, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Bitcoin-Alpha  Bitcoin-OTC  Epinions  Slashdot  SDGCL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
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+page_content='4Taewook Ko, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  7  CONCLUSION  This paper proposed SDGCL, a signed-directed graph contrastive  learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' To make augmented node representations, we intro-  duced two levels of the perturbation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Structure perturbation  randomly changes the signs and directions of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Even though  it may lose some vital information from the original graphs, we  check that it makes the model noise-robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Laplacian perturbation  changes the phase parameter 𝑞 for every training iteration to give  perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' It does not directly influence the graph information  but perturbs the Laplacian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Then we define the inter- and  intra-view contrastive objectives with the augmented node repre-  sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Unlike other signed graph studies, our model does not  depend on social theories or assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Also, there are no arbitrary  graph-splitting processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Our graph encoder is based on the Lapla-  cian matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' it only enjoys the structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Therefore it  could reduce computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' We evaluate the proposed model  with four real-world datasets and seven baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' The experimental  results show that our proposed model has superior performance to  others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' SDGCL and their variants ranked at the top for every dataset  and metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  REFERENCES  [1] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' A  simple framework for contrastive learning of visual representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' In Interna-  tional conference on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' PMLR, 1597–1607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  [2] Ting Chen, Simon Kornblith, Kevin Swersky, Mohammad Norouzi, and Geoffrey E  Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Big self-supervised models are strong semi-supervised learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
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+page_content='  [3] Yiqi Chen, Tieyun Qian, Huan Liu, and Ke Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
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+page_content=' In Proceedings of the 27th acm international  conference on information and knowledge management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
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+page_content=' A note on Markov normalized magnetic eigenmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  Applied and Computational Harmonic Analysis 43, 2 (2017), 370–380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  [5] Yves Colin de Verdière.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Magnetic interpretation of the nodal defect on  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Analysis & PDE 6, 5 (2013), 1235–1242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  [6] Mihai Cucuringu, Huan Li, He Sun, and Luca Zanetti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Hermitian ma-  trices for clustering directed graphs: insights and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' In International  Conference on Artificial Intelligence and Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' PMLR, 983–992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  [7] Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Convolu-  tional neural networks on graphs with fast localized spectral filtering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Advances in  neural information processing systems 29 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  [8] Tyler Derr, Yao Ma, and Jiliang Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Signed graph convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  In 2018 IEEE International Conference on Data Mining (ICDM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' IEEE, 929–934.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  [9] Wenqi Fan, Yao Ma, Qing Li, Yuan He, Eric Zhao, Jiliang Tang, and Dawei Yin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Graph neural networks for social recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' In The world wide web  conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
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+page_content='  [10] Stefano Fiorini, Stefano Coniglio, Michele Ciavotta, and Enza Messina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  SigMaNet: One Laplacian to Rule Them All.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
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+page_content=' Protein interface  prediction using graph convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
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+page_content=' In Joint European Conference on Machine Learning and Knowledge  Discovery in Databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' Springer, 447–463.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content='  [13] Ramanthan Guha, Ravi Kumar, Prabhakar Raghavan, and Andrew Tomkins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
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+page_content='  Propagation of trust and distrust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
+page_content=' In Proceedings of the 13th international confer-  ence on World Wide Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StE4T4oBgHgl3EQflw2y/content/2301.05163v1.pdf'}
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diff --git a/TdE5T4oBgHgl3EQfaw-V/content/tmp_files/2301.05591v1.pdf.txt b/TdE5T4oBgHgl3EQfaw-V/content/tmp_files/2301.05591v1.pdf.txt
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+Draft version January 16, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+Exploring magnetic loops and serpentine fields in the quiet Sun with the GRIS-IFU
+Ryan J. Campbell
+,1 Ricardo Gafeira
+,2, 3 Mihalis Mathioudakis
+,1 Carlos Quintero Noda
+,4, 5 and
+Manuel Collados
+4, 5
+1Astrophysics Research Centre, Queen’s University of Belfast, Northern Ireland, BT7 1NN, UK
+2Instituto de Astrof´ısica de Andaluc´ıa, Apartado de Correos 3004, 18080 Granada, Spain
+3Instituto de Astrof´ısica e Ciˆencias do Espa¸co, Departamento de F´ısica da Universidade de Coimbra,
+Rua do Observat´orio s/n, 3040-004 Coimbra, Portuga
+4Instituto de Astrof´ısica de Canarias, V´ıa L´actea s/n, E-38205 La Laguna, Tenerife, Spain
+5Departamento de Astrof´ısica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain
+ABSTRACT
+Synthetic observations produced from radiative magnetohydronamic simulations have predicted that higher
+polarization fractions in the quiet solar photosphere would be revealed by increasing the total integration time
+of observations at GREGOR resolutions. We present recently acquired disk centre observations of the Fe I
+15648.5 ˚A line obtained with the GREGOR telescope equipped with the GRIS-IFU during excellent seeing
+conditions, showing exceptionally high polarization fractions. Our observation reveal an internetwork region
+with a majority (> 60%) of magnetised pixels displaying a clear transverse component of the magnetic field.
+This result is in stark contrast to previous disk-centre GRIS-IFU observations in this spectral line, which had
+predominantly vertical magnetic fields in the deep photosphere. At the same time, the median magnetic field
+strength is weaker than previous GRIS-IFU observations, indicating that the larger fraction of polarization
+signals cannot be explained by a more active target. We use the Stokes Inversion based on Response functions
+(SIR) code to analyse the data, performing over 45 million inversions, and interrogate the impact of two
+conflicting approaches to the treatment of noise on the retrieval of the magnetic inclination and azimuth.
+We present several case studies of the zoo of magnetic features present in these data, including small-scale
+magnetic loops that seem to be embedded in a sea of magnetism, and serpentine fields, focusing on regions
+where full-vector spectropolarimetry has been achieved. We also present a new open-source Python 3 analysis
+tool, SIR Explorer (SIRE), that we use to examine the dynamics of these small-scale magnetic features.
+Keywords: Sun: photosphere — Sun: magnetic fields — Sun: infrared — Sun: granulation
+1. INTRODUCTION
+Lites et al. (2008) revealed the quiet Sun (QS) inter-
+network (IN) as dominated by horizontal (i.e. transverse
+with respect to the solar normal) magnetic fields at an
+effective spatial resolution of 0.3′′. However, this result
+is not undisputed and remains subject to contradiction
+by other studies, as reviewed by Steiner & Rezaei (2012).
+For instance, the lack of variation in the degree of linear
+and circular polarisation recorded in near infrared (NIR)
+observations by Mart´ınez Gonz´alez et al. (2008) at 0.8′′
+as a function of different heliocentric angles points to
+a QS magnetic field which has no preferential bias in
+orientation. If the distribution of magnetic field incli-
+nations is isotropic, its probability density function is
+given as,
+P(γ) = sin γ
+2
+,
+(1)
+where γ is the magnetic inclination angle, defined as
+the angle between the magnetic vector and solar nor-
+mal, such that P(γ) has a maximum at 90◦. It is per-
+haps counter-intuitive, but this would mean most of the
+fields are transverse, because for the magnetic field to be
+aligned along the line-of-sight (LOS) it has to point in
+one of two possible directions, but to be transverse there
+are many more possible directions (S´anchez Almeida &
+Mart´ınez Gonz´alez 2011).
+The greatest difficulty in constraining the γ from in-
+versions of IN observations in a consistent way results
+from the differing treatments of varying levels of noise.
+In the weak field regime, a vertical field produces a larger
+amplitude circular polarization profile than the ampli-
+tude of a linear polarization profile produced by a hori-
+zontal field of equal strength at disk centre. This creates
+an intrinsic bias against being able to confidently detect
+arXiv:2301.05591v1  [astro-ph.SR]  13 Jan 2023
+
+ID2
+Campbell et al.
+0
+3.3
+6.6
+9.9
+X [arcsec]
+0
+3.75
+7.5
+Y [arcsec]
+Stokes I
+0.9
+1.0
+1.1
+[Ic]
+0
+3.3
+6.6
+9.9
+X [arcsec]
+0
+3.75
+7.5
+Y [arcsec]
+Stokes Q
+−0.005
+0.000
+0.005
+[Ic]
+0
+3.3
+6.6
+9.9
+X [arcsec]
+0
+3.75
+7.5
+Y [arcsec]
+Stokes U
+−0.005
+0.000
+0.005
+[Ic]
+0
+3.3
+6.6
+9.9
+X [arcsec]
+0
+3.75
+7.5
+Y [arcsec]
+Stokes V
+−0.01
+0.00
+0.01
+[Ic]
+0
+3.3
+6.6
+9.9
+X [arcsec]
+0
+3.75
+7.5
+Y [arcsec]
+T
+7000
+7500
+8000
+[K]
+0
+3.3
+6.6
+9.9
+X [arcsec]
+0
+3.75
+7.5
+Y [arcsec]
+vLOS
+−2
+0
+2
+[km/s]
+0
+3.3
+6.6
+9.9
+X [arcsec]
+0
+3.75
+7.5
+Y [arcsec]
+γ
+0
+90
+180
+[∘]
+0
+3.3
+6.6
+9.9
+X [arcsec]
+0
+3.75
+7.5
+Y [arcsec]
+αmB
+0
+25
+50
+[G]
+Figure 1. Sample frame from a GRIS-IFU scan of a quiet Sun region on the 24 August 2021 (scan D) showing in the left
+column from top to bottom the continuum-normalized Stokes I, Q, U, and V , and in the right column from top to bottom
+the T, vLOS, αmB, and γ as derived from SIR inversions (scenario 3). Stokes I is shown at a wavelength of 15650.59 ˚A in the
+continuum and the polarization profiles are shown at 15648.36 ˚A in the blue lobes of the geff = 3 line. Any pixel which does
+not have a Stokes Q, U, or V profile with maximum unsigned amplitude across the 15648.5 ˚A line below the 5σn threshold has
+been set to zero and masked from the plots of Stokes Q, Stokes U, Stokes V , αmB, and γ. T is shown at logτ5000˚
+A = 0.5, while
+the other model parameters are shown at constant in depth.
+horizontal fields at disk centre when a given noise thresh-
+old is equally applied to Stokes Q, U, and V . By pro-
+ducing models with deliberately purely vertical fields,
+synthesizing the Stokes vector, and adding noise before
+inverting again, Borrero & Kobel (2011) show that an
+inversion code will return an overabundance of horizon-
+tal inclinations. As a result, even when one inverts only
+those pixels with at least one Stokes Q, U, or V pro-
+file above a noise threshold, and most of the pixels have
+only Stokes V above the threshold, this arguably results
+in a possible bias in favour of horizontal fields as the in-
+version code interprets noise in Stokes Q and U as real
+signals.
+The deepest Hinode/SP integrations show circular
+and linear polarisation in 88% and 53% of the field of
+view (FOV), respectively, but this comes at the expense
+of spatio-temporal resolution which distorts the polar-
+ization signals (Bellot Rubio & Orozco Su´arez 2012).
+Most recently, observations of the IN with visible lines
+at the ground-based Swedish Solar Telescope (SST;
+
+Exploring magnetic loops and serpentine fields in the quiet Sun with the GRIS-IFU
+3
+Goˇsi´c et al. (2021, 2022); Ledvina et al. (2022)) and
+balloon-borne Sunrise experiment (Danilovic et al. 2010;
+Mart´ınez Gonz´alez et al. 2012; Kianfar et al. 2018) have
+provided good statistics in terms of the longitudinal field
+and even of cancellations, but visible photospheric lines
+still struggle to confidently detect the horizontal fields
+that should be present in magnetic loops along the polar-
+ity inversion line (PIL) without significant spatial, spec-
+tral, or temporal binning. Observations with the NIR
+line pair at GREGOR with the GREGOR Infrared Spec-
+trograph (GRIS; Lagg et al. (2016); Mart´ınez Gonz´alez
+et al. (2016)) have demonstrated higher efficacy at mea-
+suring linear polarization at similar spatial resolutions.
+This is despite the fact that modelling with simulations
+by Steiner et al. (2008) shows the vertical field being
+prominent in the deep photosphere, where the NIR line
+pair is sensitive (Quintero Noda et al. 2021), with the
+horizontal component of the magnetic field becoming
+increasingly important in the upper photosphere, where
+the visible line pair is sensitive, with convective over-
+shooting responsible for expelling horizontal fields to
+greater heights. Campbell et al. (2021a) have used the
+Integral Field Unit (IFU) mode of GRIS to examine
+the dynamics of small-scale magnetic features, including
+magnetic loops. Two-dimensional spectrographs like the
+GRIS-IFU are uniquely capable of providing time-series
+imaging with high polarimetric sensitivity and spectral
+resolution simultaneously, with the size of the FOV and
+cadence acting as competing factors.
+Campbell et al. (2021b) have predicted, using syn-
+thetic observations produced from MURaM simulations
+(V¨ogler et al. 2005) degraded to GREGOR/GRIS-IFU
+resolutions, that increasing the total integration time
+of the observations will yield higher fractions of polar-
+ization and allow the magnetic inclination to be better
+constrained in a larger fraction of the FOV. We report
+observations calibrated based on these predictions.
+2. OBSERVATIONS
+Repeated observations of disk centre QS regions
+were taken in late August 2021 using the GRIS-IFU
+(Dominguez-Tagle et al. 2022) at GREGOR (Schmidt
+et al. 2012; Kleint et al. 2020). The GRIS-IFU was op-
+erated in double sampling mode, with a 3 (vertical) by
+2 (horizontal) mosaic pattern, with an exposure time of
+60 ms per polarimetric state, and 20 accumulations (for
+a total integration time per pixel of 4.8 seconds). This
+resulted in a cadence of 103 seconds between frames.
+Compared to similar 2019 scans, the exposure time and
+accumulations were increased to target a higher S/N,
+but the number of steps in the mosaic pattern was re-
+duced to constrain the cadence between frames.
+The
+FOV of the scans is therefore 12.15′′ by 9.024′′ with
+a spatial sampling of 0.135′′ by 0.188′′ in the x- and
+y−directions, respectively.
+The GRIS-IFU observed a 40 ˚A spectral window that
+includes the highly magnetically sensitive photospheric
+Fe I line at 15648.5 ˚A. The spectral dispersion was de-
+termined to be 39.83 m˚A/pixel by comparison with a
+degraded Fourier Transform Spectrometer (FTS) atlas
+(Livingston & Wallace 1991).
+During the observing campaign we benefitted from
+very good seeing conditions. Table 1 lists five datasets
+which are candidates for analysis as they were taken
+with good to excellent seeing, as quantified by the lo-
+cally measured Fried parameter, r0, and the root-mean-
+square continuum intensity contrast, δIrms. The GRIS
+reduction pipeline (Collados et al. 2012) was employed
+for dark current removal, flat fielding, polarimetric cal-
+ibration, and cross-talk corrections.
+3. RESULTS
+3.1. Polarization analysis
+In order to quantify the fraction of pixels exhibiting
+a polarization profile confidently above the noise level
+in the datasets listed in Table 1, we adopt the method
+used by Lagg et al. (2016); Campbell et al. (2021a). The
+purpose of this analysis is to enable a comparison be-
+tween the new GRIS-IFU datasets, previous GRIS-IFU
+datasets, and datasets from other facilities. For a given
+Stokes parameter, we determine the noise level, σn, by
+calculating the standard deviation in the continuum in
+each frame. The explicit assumption is made that the
+continuum is unpolarized and the primary type of noise
+is photon noise. We then determine whether a pixel has
+a Stokes Q, U, or V profile with maximum amplitude
+greater than a threshold set at 5σn across the 15648.5 ˚A
+line. If a given pixel has a maximum amplitude in Stokes
+V greater than the threshold, the pixel is said to have a
+confidently measured circular polarization (CP) signal.
+Similarly, if a given pixel has a maximum amplitude in
+Stokes Q or U greater than the threshold, the pixel is
+said to have a confidently measured linear polarization
+(LP) signal. If a Stokes vector has neither LP or CP, it
+is said to have no polarization (NP).
+Table 1 shows the time-averaged results of this anal-
+ysis.
+The exceptionally high polarization fractions of
+scans D and E stand out as being much larger than
+scans A, B, and C. Figure 1 shows a sample frame from
+scan D. This frame had a δIrms of 3.4%, a maximal r0
+of 22 cm, and shows an abundance of polarization. In
+terms of understanding why the polarization fractions of
+scan D and E are higher than the other scans, the first
+factor to consider is that the GRIS-IFU FOV is very
+
+4
+Campbell et al.
+Table 1. Time-averaged percentage of linear (LP) and cir-
+cular (CP) polarization profiles with maximum amplitudes
+above 5σn for level 1 GRIS-IFU/GREGOR data. The per-
+centages are calculated relative to the full FOV. The 1σn
+noise level is determined by the calculation of the standard
+deviation in the relevant Stokes parameter at continuum
+wavelengths in each frame. All scans were taken in August
+2021 and the times are given in universal time.
+Label
+Day, time
+%LP
+%CP
+%LP & CP
+%NP
+A
+19, 08:14
+12.7
+43.4
+8.6
+52.5
+B
+20, 08:56
+12.4
+42.2
+7.4
+52.8
+C
+23, 08:00
+6.7
+28.7
+3.7
+68.2
+D
+24, 07:50
+29.3
+64.7
+22.7
+28.6
+E
+24, 08:55
+30.7
+66.5
+24.6
+27.4
+Table 2.
+As for Table 1, but for the PCA-RVM recon-
+structed GRIS-IFU data. The noise threshold is set at 5σn
+as measured in the continuum of the original level 1 data.
+Label
+%LP
+%CP
+%LP & CP
+%NP
+D
+22.0
+59.0
+16.2
+35.3
+E
+23.2
+60.7
+17.6
+33.8
+small. There is therefore always a risk when observing
+with the GRIS-IFU that one could point to a so-called
+‘void’, where there is little detectable magnetic flux, or
+indeed the opposite. The second factor to consider is
+the quality and stability of the seeing conditions. From
+the r0 and δIrms values for each of the scans, it is clear
+that the seeing conditions for scans D and E were supe-
+rior in both quality and stability. For instance, scan A
+had a peak δIrms of only 3% and a low of 1.8%, while
+scan D had a peak of 3.4% and low of 2.7%. This is im-
+portant as atmospheric stability is essential to achieve a
+maximal effective spatial resolution, and impacts heavily
+on the polarization fractions recorded because opposite
+polarity profiles cancel out within the spatio-temporal
+resolution element with insufficient spatial resolution.
+Principal component analysis (PCA) with 15 retained
+eigenvectors is applied to scans D and E for the purpose
+of removing noise. As eludicated by Borrero & Kobel
+(2011), while the probability of photon noise producing
+a signal greater than 3σn is only 0.3%, the probabil-
+ity increases dramatically when more and more wave-
+length points are considered. By applying PCA to re-
+move noise and using a stringent 5σn noise threshold,
+we are reducing the probability that a false signal will
+survive and enter the analysis. As interference fringes
+are present in the wavelength domain of the polariza-
+tion profiles in some pixels, a relevance vector machine
+(RVM) is employed to remove these defects and produce
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+Amplitude [Ic]
+0
+10
+20
+30
+40
+50
+60
+N [%]
+LP
+CP
+Scan D
+Scan E
+6 May 19
+Figure 2. Area occupied by pixels with a LP (red lines)
+or CP (blue lines) signal for scan D (dotted lines), scan E
+(dashed lines), and the scan taken on the 6 May 2019 (solid
+lines) for given amplitudes. The data has undergone PCA-
+RVM reconstruction and had any Stokes profiles with maxi-
+mum unsigned amplitude across the 15648.5 ˚A line below the
+5σn threshold in the 2021 data, and 3σn in the 2019 data,
+set to zero.
+apparently noiseless polarization profiles. The full re-
+construction process is the same as employed by Camp-
+bell et al. (2021a) on GRIS-IFU/GREGOR data and is
+described in detail therein. By applying the PCA-RVM
+reconstruction process to scans D and E, this enables a
+comparison with the PCA-RVM reconstructed statistics
+reported by Campbell et al. (2021a). For this purpose
+in this paper we will use the scan taken on 6 May 2019,
+henceforth referenced simply as the 2019 data.
+Table 2 shows the time-averaged polarization fractions
+for scan D and E after application of the PCA-RVM re-
+construction process. Figure 2 shows the time-averaged
+polarization fractions of scans D and E as a function of
+amplitude alongside the scan taken on 6 May 2019. In
+this case, any Stokes profile whose maximum unsigned
+amplitude across the 15648.5 ˚A line does not exceed the
+5σn threshold for the two 2021 datasets, and 3σn for the
+2019 dataset, has been set to zero. From Fig. 2 it is clear
+that the 5σn threshold is on average at a slightly lower
+amplitude in scans D and E than the 3σn threshold in
+the 2019 scan. Seeing-induced cross-talk becomes more
+likely with longer exposures, as with the longer total in-
+tegration time it is more likely that the modulations of
+the Stokes parameters will not be conducted fast enough
+for Earth’s atmosphere to be considered ‘frozen’ during
+measurements (Collados 1999). We choose a stringent
+5σn threshold for the August 2021 data that still al-
+lows us to access weaker polarization signals than in the
+2019 data, because the former has a lower noise level
+than the latter, but is sufficiently large to allow us to
+assume any seeing-induced cross-talk produced is un-
+likely to have an amplitude of this magnitude. There is
+
+Exploring magnetic loops and serpentine fields in the quiet Sun with the GRIS-IFU
+5
+a significantly higher fraction of CP and LP in the new
+scans than the old scans. Perhaps the most significant
+difference between the old and new scans is the much
+higher percentage of the FOV which has both LP and
+CP. Previously, between 3 − 5% of the FOV had both
+LP and CP, but in the new scans this number has signif-
+icantly increased to 16−18% at 5σn. This means that γ
+should be retrievable from inversions in a much higher
+fraction of the FOV and it would also be anticipated
+that a larger fraction of pixels would harbour fields with
+an intermediate (as opposed to highly vertical or highly
+inclined) inclination.
+3.2. Inversion strategies
+We use the Stokes Inversion based on Response func-
+tions (SIR) (Ruiz Cobo & del Toro Iniesta 1992) code to
+invert scans D and E. We adapt the parallelized wrap-
+per to SIR written by Gafeira et al. (2021) to enable
+the kinematic and magnetic parameters of the input
+model(s) to be randomized within physically sensible
+upper and lower boundaries at every iteration.
+This
+is an essential step to maximise the statistical chances
+of achieving the global χ2 minimum solution for each
+Stokes vector.
+For the inversion, we adopt the same
+scheme as tested on real observations of the NIR line
+pair by Mart´ınez Gonz´alez et al. (2016); Campbell et al.
+(2021a) and on synthetic observations by Campbell et al.
+(2021b).
+This is a two-component inversion with one
+non-magnetic (i.e. the magnetic field strength, B, was
+set to zero and was not allowed to vary) and one mag-
+netic model, which are combined by SIR in accordance
+to their respective filling factors, α, which is another free
+parameter. For clarity, we refer to the filling factor of the
+magnetic model as αm. SIR allows the user to determine
+the optical depths at which perturbations will be made
+between iterations by selecting a number of nodes in
+each variable parameter, with the full parameter strati-
+fication in optical depth given by cubic splines or linear
+interpolation. Apart from temperature, T, for which 4
+nodes were selected, each parameter, including B, was
+forced to be constant in optical depth. The macrotur-
+bulent velocity, vmac, was included as a free parameter
+in the non-magnetic model and forced to be the same
+in the magnetic model. The T was also determined by
+the non-magnetic model and forced to be the same in
+the magnetic model. The line of sight velocity, vLOS,
+and microturbulent velocity, vmic, were free parameters
+in both models. Full PCA-RVM reconstruction was ap-
+plied to the datasets before inversion. We used the em-
+pirically determined atomic parameters made available
+by Trelles Arjona et al. (2021) and abundances from
+Asplund et al. (2009). Included in the inversion were
+seven spectral lines, including five Fe I lines with rest
+wavelengths of 15645.02 ˚A, 15645.303 ˚A, 15648.514 ˚A,
+15652.873 ˚A, 15662.017 ˚A, 15665.241 ˚A, one blended
+Fe II line with rest wavelength 15648.514 ˚A, and one
+blended O I line with rest wavelength 15665.098 ˚A.
+We interrogate the impact of noise on the retrieval of
+B, γ, and αm statistically by taking more than one ap-
+proach to data treatment. In order to accurately deter-
+mine γ, one must measure both linear polarization and
+circular polarization confidently above the noise level.
+However, it can be argued that the noise level itself
+places limits on the amplitude of the weakest polariza-
+tion profiles. We divide the approaches into three sce-
+narios. In the first scenario (S1), the full datasets were
+inverted with all Stokes parameters provided to SIR for
+each pixel without any consideration for signal ampli-
+tudes relative to the σn level. In scenario 2 (S2), any
+individual Stokes Q, U, or V profile in a given pixel
+whose maximum amplitude across the 15648.5 ˚A line
+does not reach the 5σn threshold was set to zero before
+inversion. In scenario 3 (S3), any individual Stokes Q,
+U, or V profile that does not reach the 5σn threshold was
+set to zero before inversion as in S2, with the exception
+that if one linear polarization parameter was confidently
+measured then the weaker linear polarization parameter
+was not set to zero despite being below the threshold.
+This third scenario is included to investigate whether
+only one linear polarization parameter (i.e. cases where
+Stokes Q is > 5σn but Stokes U is not, or vice versa) is
+sufficient to constrain γ and if there are any impacts on
+other magnetic parameters when the weaker linear po-
+larization parameter is set to zero and the stronger one is
+not. Each inversion was repeated 50 times with random-
+ized initial models, resulting in over 45 million inversions
+in total for all three scenarios and both datasets.
+3.3. Magnetic filling factors and field strengths
+Figure 3 shows the distributions of B, γ, αmB, αm,
+and |cos φ|, where φ is the azimuthal angle of the mag-
+netic field vector in the plane perpendicular to the ob-
+server’s LOS, for each scenario and scans D and E. First,
+the αm distribution looks very similar in all scenarios.
+However, the B distribution is significantly different be-
+tween S1 on the one hand and S2 and S3 on the other
+hand below 600 G for both scans. In S1, the B distri-
+bution peaks at around 100 G, but for S2 and S3 there
+is a much larger number of pixels with B values below
+100 G. Since B is determined by the longitudinal and
+transverse components, setting linear and circular polar-
+ization signals to zero naturally results in lower values
+of B overall. As a consequence, the αmB distributions
+for S1 on one hand and S2 and S3 on the other hand
+
+6
+Campbell et al.
+0
+20
+40
+60
+80
+100 120 140 160 180
+γ [deg]
+0.0%
+1.0%
+2.0%
+3.0%
+4.0%
+5.0%
+6.0%
+N [%]
+0
+200
+400
+600
+800
+1000 1200 1400
+B [G]
+0.0%
+2.0%
+4.0%
+6.0%
+8.0%
+10.0%
+12.0%
+14.0%
+16.0%
+N [%]
+Scenario 3
+Scenario 2
+Scenario 1
+0
+20
+40
+60
+80
+100
+120
+αmB [G]
+0.0%
+2.0%
+4.0%
+6.0%
+8.0%
+10.0%
+12.0%
+14.0%
+N [%]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+αm [G]
+0.0%
+2.0%
+4.0%
+6.0%
+8.0%
+10.0%
+12.0%
+14.0%
+N [%]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+|cos ϕ| [deg]
+0.0%
+2.0%
+4.0%
+6.0%
+8.0%
+10.0%
+N [%]
+0
+20
+40
+60
+80
+100 120 140 160 180
+γ [deg]
+0.0%
+1.0%
+2.0%
+3.0%
+4.0%
+5.0%
+6.0%
+7.0%
+N [%]
+0
+200
+400
+600
+800
+1000 1200 1400
+B [G]
+0.0%
+2.5%
+5.0%
+7.5%
+10.0%
+12.5%
+15.0%
+17.5%
+N [%]
+Scenario 3
+Scenario 2
+Scenario 1
+0
+20
+40
+60
+80
+100
+120
+αmB [G]
+0.0%
+2.0%
+4.0%
+6.0%
+8.0%
+10.0%
+12.0%
+14.0%
+N [%]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+αm [G]
+0.0%
+2.5%
+5.0%
+7.5%
+10.0%
+12.5%
+15.0%
+17.5%
+N [%]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+|cos ϕ| [deg]
+0.0%
+2.0%
+4.0%
+6.0%
+8.0%
+N [%]
+Figure 3. Histograms of B (top row), γ (second row), αmB (third row), αm (fourth row), and |cos φ| (bottom row) returned
+from the inversions. The histograms are shown for three inversion scenarios (S1, S2, and S3) as described in the text. The left
+and right columns show the distributions for scan D and E, respectively. Histograms are weighted with respect to the total
+number of pixels, i.e. including those with no polarization. Pixels which had no measured polarization in any Stokes parameter
+are excluded from the histogram (i.e. only pixels with maximum absolute amplitude across the 15648.5 ˚A line greater than the
+5σn threshold in at least one Stokes parameter are included).
+
+Exploring magnetic loops and serpentine fields in the quiet Sun with the GRIS-IFU
+7
+also differ in a similar way.
+Tables 3 and 4 show for
+scans D and E, respectively, the median αmB and B for
+each scenario. In addition, the median transverse and
+longitudinal components, B⊥ and B∥, as well as αmB⊥,
+and αmB∥, are also shown. First of all, the median B,
+B⊥, and B∥ values under S2 are all significantly lower
+than the equivalent values reported by Campbell et al.
+(2021a). These weak average values underline that the
+target region is IN in nature. Further, as these fields can
+be understood as firmly in the weak field regime, where
+the circular polarization amplitudes are proportional to
+B (and linear polarization amplitudes are proportional
+to B2), recording lower median values in these parame-
+ters is sensible on the inclusion of weaker Stokes profiles
+that are accessed in the approximate range 2−2.4×10−3
+Ic that could not be confidently measured in 2019. The
+median αm values are very small, typically at 0.04. How-
+ever, this is a consequence of the shape of the distribu-
+tion - the mean value is an order of magnitude bigger
+typically at 0.4 − 0.5. The ratio B⊥/B∥ has been re-
+ported in other studies (Orozco Su´arez & Bellot Rubio
+2012; Campbell et al. 2021a) and is in fact slightly larger
+than 2019, indicating that as we increase the S/N, and
+access weaker fields, the magnetic field is getting more
+horizontal on average. Since the ratio B⊥/B∥ would be
+π/2 for an isotropic distribution, this also indicates that
+accessing weaker fields has returned a distribution that
+is further from the isotropic case. There is also a clear
+difference between scans D and E, with stronger B, B∥,
+and B⊥ values returned by SIR in all scenarios in scan
+E relative to scan D. It is also clear from the ratios that
+scan E is slightly more longitudinal on average than scan
+D. Finally, there is a clear difference between S2 and S3
+for both scans, with the median B values being about
+20 G stronger when the weakest linear polarization pa-
+rameters are not eliminated. Further, the B⊥/B∥ ratio
+is slightly larger in S3 compared to S2 in both scans,
+and the median αm values are unchanged.
+3.4. Inclinations and azimuths
+The γ distributions in Fig. 3 differ significantly be-
+tween S1 on one hand and S2 and S3 on the other. The
+shape of the γ distribution in S1 resembles the distri-
+butions derived from semi-empirical models produced
+by Borrero & Kobel (2011) with large noise thresholds
+(> 4.5σn) specifically in terms of the small dip at 90◦.
+As the authors explain, this dip is a consequence of as-
+serting a large noise threshold in the weak field regime.
+Thus, by setting the threshold at 5σn, the dip in the γ
+distribution is created by excluding the weakest fields.
+This dip is also observed in the distribution from MU-
+RaM simulations by Campbell et al. (2021b). For S2
+Table 3. Unsigned median magnetic field strengths and flux
+densities, and its horizontal (transverse) and vertical (longi-
+tudinal) components, for scan D. The median B is measured
+across all pixels with at least one polarization parameter with
+maximum amplitude > 5σn across the 15648.5 ˚A line, but
+when computing the corresponding B⊥ and B∥ component
+only profiles that had Stokes Q or U > 5σn in the former
+case, and Stokes V > 5σn in the latter case, are included.
+Scenario 1
+Scenario 2
+Scenario 3
+B
+206.4 G
+119.3 G
+140.6 G
+B∥
+92.6 G
+71.4 G
+77.9 G
+B⊥
+200.9 G
+153.2 G
+188.4 G
+αm
+0.04
+0.04
+0.04
+αmB
+7.3 G
+4.7 G
+5.6 G
+αmB∥
+3.3 G
+2.8 G
+3.1 G
+αmB⊥
+7.1 G
+6.0 G
+7.5 G
+B⊥/B∥
+2.2
+2.1
+2.4
+Table 4. As in Table 3, but for scan E.
+Scenario 1
+Scenario 2
+Scenario 3
+B
+233.3 G
+163.6 G
+182.8 G
+B∥
+107.5 G
+98.2 G
+101.8 G
+B⊥
+221.1 G
+182.3 G
+220.1 G
+αm
+0.04
+0.04
+0.04
+αmB
+9.7 G
+6.0 G
+8.2 G
+αmB∥
+4.5 G
+3.6 G
+4.6 G
+αmB⊥
+9.2 G
+6.7 G
+9.9 G
+B⊥/B∥
+2.1
+1.9
+2.2
+Table 5.
+Percentage of pixels with inclinations in given
+ranges as determined by SIR under the three scenarios for
+scan D. The percentages are computed relative to the total
+number of pixels with at least one Stokes profile with maxi-
+mum amplitude > 5σn across the 15648.5 ˚A line.
+Scenario
+classification
+range [◦]
+1 [%]
+2 [%]
+3 [%]
+highly vertical
+γ < 16
+3.9
+16.6
+16.0
+highly vertical
+γ > 164
+4.5
+17.5
+17.2
+intermediate
+15 < γ < 75
+33.5
+25.1
+24.6
+intermediate
+105 < γ < 165
+35.7
+25.2
+25.5
+highly inclined
+74 < γ < 106
+22.4
+15.6
+16.7
+and S3, the distribution is similar to Campbell et al.
+(2021a) in that large peaks are recorded at 0, 90, and
+180◦, which is a consequence of setting noisy Stokes Q,
+U, and V profiles to zero. For instance, a Stokes vec-
+tor with only Stokes Q above the noise threshold and
+Stokes V set to zero would be expected to return a γ
+
+8
+Campbell et al.
+Table 6. As in Table 6, but for scan E.
+Scenario
+classification
+range [◦]
+1 [%]
+2 [%]
+3 [%]
+highly vertical
+γ < 16
+5.0
+19.0
+18.3
+highly vertical
+γ > 164
+5.3
+19.5
+19.2
+intermediate
+15 < γ < 75
+33.9
+23.1
+23.0
+intermediate
+105 < γ < 165
+35.7
+24.2
+24.0
+highly inclined
+74 < γ < 106
+20.2
+14.1
+15.5
+value close to 90◦ by SIR when a good fit is achieved,
+which is the reason for the central peak in the γ distri-
+bution. However, if this profile had a noisy Stokes V
+included when inverted, the γ would be more likely to
+show a more intermediate inclination.
+Tables 5 and 6 classify the inclinations in terms of
+highly vertical, highly inclined, and intermediately in-
+clined fields. The difference compared to 2019 is in the
+intermediately inclined fields, which have a much higher
+population. Campbell et al. (2021a) report over 70% of
+pixels had a highly vertical classification, with a minor-
+ity of pixels having a significant transverse component
+(i.e. classified as either intermediate or highly inclined).
+In scans D and E, remarkably, the situation is reversed -
+only 34.1% in scan D and 38.5% in scan E under S2 have
+a highly vertical classification, with the majority having
+a significant transverse component. The explanation for
+this has already been discussed in Sect. 3.1: there are a
+much larger number of pixels with both LP and CP in
+scans D and E. The very close similarity between S2 and
+S3 indicates that eliminating the weaker linear polariza-
+tion parameter does not result in an altered distribution
+of γ values, i.e. only the stronger linear polarization pa-
+rameter is necessary to retrieve γ (in addition to Stokes
+V , if a value that differs from close to 90◦ is to be re-
+turned). Of course, there is a difference in φ. The reality
+is that regardless of the scenario, since both Stokes Q
+and U must be measured confidently to constrain φ, and
+so few pixels have both, retrieving φ (even without dis-
+ambiguation) in a statistical sense is not feasible in this
+data. Nevertheless, the peaks in the | cos φ| distribution
+at 0.7 and 1 is a consequence of setting one of Stokes Q
+or U to zero while the other exceeds the 5σ threshold,
+forcing φ to be such that cos φ is returned as −1, −0.7,
+0.7, or 1 for φ values of 0, 45, 135, or 180◦ (or indeed
+these values shifted by 180◦ due to a lack of disambigua-
+tion). Finally, the clear difference between scans D and
+E is elucidated again, with slightly larger numbers of
+highly vertical classifications in scan E relative to scan
+D, and indeed slightly larger numbers of intermediate
+and highly inclined classifications in scan D relative to
+scan E.
+3.5. Case studies
+A key strength of the GRIS-IFU lies in its ability
+to image the dynamics of small-scale magnetic fea-
+tures with high S/N, spatial resolution, and spectral
+resolution simultaneously.
+We now present two case
+studies and in order to make the analysis more effi-
+cient, we have developed an open-source application,
+SIR Explorer (SIRE), that enables users to navigate the
+multi-dimensional input and output files of SIR inver-
+sions. This tool is described in Appendix A and is used
+throughout this section. Figure 4 shows a magnetic loop
+visible in scan D. Figure 5 shows a close-up of the po-
+larization in the boxed region in Fig. 4. Here we define
+the total linear polarization as,
+Ltot =
+� λr
+λb [Q2(λ) + U 2(λ)]
+1
+2 dλ
+Ic
+� λr
+λb dλ
+.
+(2)
+Figure 4 shows a magnetic feature whose structure re-
+sembles a magnetic loop. There is a large patch of linear
+polarization, which bridges two patches (or foot-points)
+of opposite polarity vertical fields.
+As seen from the
+Dopplergram, the linear polarization appears at the cen-
+tre of the granule while the circular polarization is lo-
+cated in the intergranular lanes (IGLs). Over time, the
+circular polarization remains cemented in the lanes as
+the granule evolves. As the linear polarization vanishes
+below our detection capabilities, the two patches of cir-
+cular polarization no longer appear to be in close con-
+tact. This magnetic feature has a lifetime of less than 10
+minutes. This shows how the evolution of the magnetic
+loop is influenced by the granular evolution. There are
+many pixels in this structure which have all three polar-
+ization profiles above the 5σn threshold. Figure 6 shows
+three example Stokes vectors, whose spatio-temporal lo-
+cations are outlined in Fig. 5, across the PIL of the mag-
+netic loop. The upper-most vector, vector 1.a, has a pos-
+itive polarity but the field is highly inclined (γ = 95◦)
+and stronger than the median (B = 330 G, αmB = 198
+G). The middle vector, vector 1.b, has no confidently
+measured Stokes V profile. The pixel is located in the
+PIL and thus the physical reason for the lack of a circu-
+lar polarization profile could be either due to mixing of
+opposite polarities within the spatio-temporal resolution
+element or the magnetic vector could be purely transver-
+sal. The field is still strong (B = 381 G), but the fill-
+ing factor is halved compared to profile 1.a (αm = 0.3)
+and thus the αmB is significantly lower (αmB = 114
+G). Due to the lack of a confidently measured Stokes
+V signal, we cannot access the longitudinal component
+
+Exploring magnetic loops and serpentine fields in the quiet Sun with the GRIS-IFU
+9
+0
+20
+40
+αmB [G]
+0
+3.75
+7.5
+Y [arcsec]
+8:33:24 UT
+0.0000
+0.0005
+0.0010
+0.0015
+0.0020
+Ltot [Ic]
+−0.004
+−0.002
+0.000
+0.002
+0.004
+Stokes V [Ic]
+−2
+−1
+0
+1
+2
+vLOS [km/s]
+0
+90
+180
+γ [∘]
+0
+3.75
+7.5
+Y [arcsec]
+8:36:51 UT
+0
+3.75
+7.5
+Y [arcsec]
+8:38:34 UT
+0
+3.75
+7.5
+Y [arcsec]
+8:40:17 UT
+0
+3.3
+6.6
+X [arcsec]
+0
+3.3
+6.6
+X [arcsec]
+0
+3.75
+7.5
+Y [arcsec]
+8:43:44 UT
+0
+3.3
+6.6
+X [arcsec]
+0
+3.3
+6.6
+X [arcsec]
+0
+3.3
+6.6
+X [arcsec]
+Figure 4. Case study of a magnetic loop in scan D. Shown from left to right is the Ltot, Stokes V at 15648.36 ˚A in the blue
+lobes of the geff = 3 line, vLOS, γ, and αmB. The last three parameters are derived from SIR inversions under S2. The rows
+show subsequent frames with the time-stamp in the upper-right corner of the left-most plot. The box (solid line) highlights the
+spatio-temporal location of the distinct magnetic loop.
+8:36:51 UT
+Ltot [Ic]
+1.a
+1.b
+1.c
+Stokes V [Ic]
+Figure 5. Close-up of the polarization in the boxed region
+in Fig. 4, showing a distinct magnetic loop. The total linear
+polarization is shown in the left panel and Stokes V is shown
+in the right panel. The full Stokes vectors of the three out-
+lined pixels are shown in Fig. 6. The pixels are separated by
+0.188′′ each.
+of the field, and thus the αmB is likely under-estimated
+in vector 1.b. Finally, in the pixel below the PIL, vec-
+tor 1.c, the polarity of the field is negative but the field
+is still strong (B = 397 G, αmB = 159 G) and highly
+inclined (γ = 83◦). All three Stokes vectors have very
+strong Stokes Q and U signals, and thus there is infor-
+mation about φ available in each profile. The φ values
+are very consistent, differing by only 1 − 2◦. Of course,
+the ambiguity in the φ remains, so although φ values of
+123 − 124◦ are returned by SIR, values of 303 − 304◦
+would also be equally acceptable solutions. Of course,
+as all three pixels are located in the granule their vLOS
+values are consistently negative.
+Figure 7 shows the second case study of a series of
+magnetic loops present in scan E. In this region, several
+magnetic loops, whose locations are highlighted with
+boxes in Fig. 7, are present and appear to form a contin-
+uous “serpentine” structure. Alternatively, these could
+be independent loops formed by the action of a small-
+
+10
+Campbell et al.
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+Stokes I/Ic
+SIR
+Observed profile
+−0.004
+−0.002
+0.000
+0.002
+0.004
+Stokes Q/Ic
+15647.4
+15651.4
+15655.4
+15659.4
+15663.3
+wavelength [Å]
+−0.015
+−0.010
+−0.005
+0.000
+0.005
+0.010
+Stokes U/Ic
+15647.4
+15651.4
+15655.4
+15659.4
+15663.3
+wavelength [Å]
+−0.004
+−0.002
+0.000
+0.002
+0.004
+0.006
+Stokes V/Ic
+VECTOR 1.a - B = 330.0 [G], α = 0.6, γ = 94.7 [ ∘ ], ϕ = 123.9 [ ∘ ], vLOS = -1.2 [km/s]
+0.6
+0.7
+0.8
+0.9
+1.0
+1.1
+Stokes I/Ic
+SIR
+Observed profile
+−0.006
+−0.004
+−0.002
+0.000
+0.002
+0.004
+0.006
+Stokes Q/Ic
+15647.4
+15651.4
+15655.4
+15659.4
+15663.3
+wavelength [Å]
+−0.015
+−0.010
+−0.005
+0.000
+0.005
+0.010
+0.015
+Stokes U/Ic
+15647.4
+15651.4
+15655.4
+15659.4
+15663.3
+wavelength [Å]
+−0.002
+−0.001
+0.000
+0.001
+0.002
+Stokes V/Ic
+VECTOR 1.b - B = 381.3 [G], α = 0.3, γ = 90.1 [ ∘ ], ϕ = 124.0 [ ∘ ], vLOS = -1.1 [km/s]
+0.7
+0.8
+0.9
+1.0
+1.1
+Stokes I/Ic
+SIR
+Observed profile
+−0.004
+−0.002
+0.000
+0.002
+0.004
+0.006
+0.008
+Stokes Q/Ic
+15647.4
+15651.4
+15655.4
+15659.4
+15663.3
+wavelength [Å]
+−0.015
+−0.010
+−0.005
+0.000
+0.005
+0.010
+0.015
+0.020
+Stokes U/Ic
+15647.4
+15651.4
+15655.4
+15659.4
+15663.3
+wavelength [Å]
+−0.006
+−0.004
+−0.002
+0.000
+0.002
+0.004
+0.006
+Stokes V/Ic
+VECTOR 1.c - B = 397.3 [G], α = 0.4, γ = 82.9 [ ∘ ], ϕ = 122.7 [ ∘ ], vLOS = -1.4 [km/s]
+Figure 6. Full observed Stokes vectors 1.a, 1.b, and 1.c (black, dashed lines), along with the corresponding synthetic vectors
+derived from SIR inversions (red, solid lines). Stokes Q, U, and V have been PCA-RVM reconstructed. The horizontal (dot-
+dashed) lines show the 5σn noise thresholds. The locations of vectors 1.a, 1.b, and 1.c are shown in Fig. 5.
+scale dynamo. Most of the magnetic flux is located in
+IGLs, with linear polarization present in the PIL which
+is most commonly located at the granule-IGL boundary
+or in the granule. A close-up of the magnetic loop high-
+lighted by the solid box in Fig. 7 is shown in Fig. 8 for
+two frames. Since opposite polarity Stokes V profiles
+cancel in the PIL, the most common location to find
+pixels with all three polarization parameters above the
+noise threshold is one pixel adjacent to the PIL. For in-
+stance, vectors 2.a and 2.b, shown in Fig. 9 with their
+spatio-temporal location shown in Fig 8, are located on
+either side of the PIL and thus have opposing polarities
+(γ = 83◦ versus γ = 102◦). The B and αmB values
+of profile 2.a are smaller (B = 243 G, αmB = 97 G)
+than profile 2.b (B = 311 G, αmB = 155 G) as the
+magnetic loop increases in strength as it emerges in the
+frame from which profile 2.a is selected and reaches its
+peak in the frame from which profile 2.b is found. Both
+of these pixels are located in granular upflows.
+Considering the wider context of the second case
+study, the series of magnetic loops are connected to a
+much stronger longitudinal magnetic element shown in
+
+Exploring magnetic loops and serpentine fields in the quiet Sun with the GRIS-IFU
+11
+0
+20
+40
+αmB [G]
+3.75
+7.5
+Y [arcsec]
+9:22:44 UT
+0.0000
+0.0005
+0.0010
+0.0015
+0.0020
+Ltot [Ic]
+−0.004
+−0.002
+0.000
+0.002
+0.004
+Stokes V [Ic]
+−2
+−1
+0
+1
+2
+vLOS [km/s]
+0
+90
+180
+γ [∘]
+3.75
+7.5
+Y [arcsec]
+9:24:27 UT
+3.75
+7.5
+Y [arcsec]
+9:26:11 UT
+3.3
+6.6
+9.9
+X [arcsec]
+3.3
+6.6
+9.9
+X [arcsec]
+3.75
+7.5
+Y [arcsec]
+9:27:54 UT
+3.3
+6.6
+9.9
+X [arcsec]
+3.3
+6.6
+9.9
+X [arcsec]
+3.3
+6.6
+9.9
+X [arcsec]
+Figure 7. Case study of a series of magnetic loops in scan E. The plots follow the same layout as in Fig. 4. The solid, dashed,
+dotted and dot-dashed red boxes highlight the locations of magnetic loops. The solid, magenta ellipse highlights the location of
+a strong longitudinal patch of magnetic flux.
+the bottom-right of Fig. 7 in the ellipse. The tempo-
+ral evolution of this patch of longitudinal flux can be
+traced throughout the full 1 hour time series and there-
+fore persists even as the much shorter-lived magnetic
+loops appear and disappear. This magnetic element is
+first visible at the start of the time series further to the
+right, and is migrated and coalesced by granular evolu-
+tion until it covers a smaller surface area by the end of
+the scan. The presence of this strong longitudinal flux
+element is reminiscent of the magnetic loop reported by
+Campbell et al. (2021a), which was also located next to
+a strong longitudinal magnetic element. For normal Zee-
+man triplets, with an effective Land´e g-factor, geff, char-
+acterized by two split σ components and an unshifted π
+component, the two σ components are separated from
+the rest wavelength, λ0, by
+∆λ = ±
+e
+4πmecλ2
+0geffB ≈ ±4.67 × 10−13λ2
+0geffB,
+(3)
+where ∆λ and λ0 are in units of ˚A and B is in units of G,
+c is the speed of light, me is the mass of an electron, and
+e is the electron charge. In what is known as the strong
+field approximation (SFA), by measuring ∆λ it is there-
+fore possible to measure B directly from the splitting of
+the lobes of Stokes V when the field is strong enough
+(Khomenko et al. 2003; Nelson et al. 2021). SIRE pro-
+vides a simple calculator which allows users to quickly
+calculate an estimate of B based on the separation of
+the lobes, which can also be measured using the wave-
+length slider. Figure 10 shows a sample Stokes V profile
+from the structure. The Stokes V profile is asymmetric,
+particularly in the red σ lobe, but SIR can still produce
+an estimate of B even without introducing the gradients
+in vLOS or B that would be necessary to better fit this
+profile. The pixel had no linear polarization. The val-
+ues returned from SIR were: B = 854 G, αm = 0.14,
+γ = 0◦, vLOS = 2.2 km/s. The SFA estimate is returned
+as BSFA = 812 G. The wavelength positions from which
+this estimate was measured is indicated in Fig. 10. SIR
+must determine B such that a good fit is found for all
+lines, and as such it is not abnormal that there is a small
+difference in the measurements, especially given that the
+wavelength position of the synthetic and observed blue
+σ lobe differ very slightly. In any case, a shift of a single
+increment in wavelength is equivalent to 58 G, and so the
+values are within an acceptable uncertainty interval of
+each other, and serves as a sanity check demonstrating
+that the inversions are well calibrated.
+4. DISCUSSION
+The quest to reveal higher fractions of the FOV as
+displaying confidently measured Zeeman-induced polar-
+ization signal continues. The fact that 65−67% of the IN
+FOV has confidently measured polarization in at least
+
+12
+Campbell et al.
+9:22:44 UT
+Ltot [Ic]
+2.a
+Stokes V [Ic]
+9:24:27 UT
+2.b
+Figure 8. As in Fig 5, but for a magnetic loop outlined by
+the solid box in Fig. 7. The full Stokes vectors of the two
+outlined pixels are shown in Fig. 9.
+one polarization parameter is a remarkable feat for the
+GREGOR/GRIS-IFU, however linear and circular po-
+larization is only confidently detected simultaneously in
+16 − 18%. The increase in δIrms could be the result of
+having achieved closer to diffraction limited resolution.
+Indeed, the exceptional seeing conditions during scans D
+and E could be responsible for this. However, it is cer-
+tainly the case that these observations have not achieved
+the diffraction limited spatial resolution that GREGOR
+is capable of, because the spatial sampling of the GRIS-
+IFU in the y−direction, even in double sampling mode,
+is insufficient. The increase in δIrms could also be the
+result of a small reduction in stray light. Another possi-
+bility for the large polarization fractions recorded is that
+we were lucky with the target, and these explanations
+are not mutually exclusive.
+Lites et al. (2008) report a mean vertical apparent
+flux density of 11 G. This value, determined from Hin-
+ode/SP observations of the Fe I 6301/2 ˚A line pair, was
+found assuming αm = 1. However, the authors them-
+selves recommend that this value should be reduced by
+30% to 7.7 G, before comparison with other works, as
+the assumption that αm = 1 is almost certainly un-
+true. The very large fraction of the FOV in scans D and
+E which have a CP signal makes a comparison worth-
+while. The corresponding average value from this work
+is the median αmB⊥, listed in Table 3 and
+4 as be-
+tween 2.8 G and 4.6 G. As their mean is calculated over
+the full FOV of the Hinode/SP scan, and their FOV is
+much larger and includes network patches, it is sensible
+that the value for the GREGOR/GRIS-IFU is smaller.
+Further as the NIR Fe I line pair is more magnetically
+sensitive than the visible line pair, it is expected that we
+record a smaller average value in our maps. However,
+the value is a function of the effective spatial resolu-
+tion and S/N - indeed, in deep mode observations Lites
+et al. (2008) return a distribution for the vertical appar-
+ent longitudinal flux density that peaks at a much lower
+value of 1.2 G. Our value therefore rests comfortably
+between these two values. Aside from a comparison, it
+must be emphasised that all of these values could be
+under-estimates if there remains a substantial amount
+of unresolved mixed-polarities.
+The much larger number of pixels which have a LP
+signal in scans D and E, relative to 2019, allowed us
+to infer the inclination angle in a larger fraction of the
+FOV. Using SIR, we were then able to determine that a
+large majority (> 60%) of the pixels which had at least
+one Stokes parameter with a maximum signal greater
+than 5σn across the 15648.5 ˚A line were classifiable as ei-
+ther highly inclined or intermediately inclined. In other
+words, a majority of magnetised pixels displayed a sig-
+nificant transverse magnetic component of the magnetic
+field. This is because the number of intermediately in-
+clined fields outnumbers both the highly inclined and
+highly vertical populations. The ratio B⊥/B∥ is larger
+than 2019, which indicates that as the S/N increased,
+and thus we were able to access weaker fields, the mag-
+netic field is revealed as more transverse on average.
+Determining this ratio from observations is important
+because it is useful for optimisation of the initial and
+boundary conditions of radiative magnetohydrodynamic
+simulations (Steiner et al. 2008). The question which
+naturally follows is whether this is an uncommon quirk
+of the specific observational target in scans D and E, or
+whether this is a glimpse of what we can expect to see
+when larger aperture telescopes like the Daniel K. In-
+ouye Solar Telescope (DKIST: Rimmele et al. (2020))
+and European Solar Telescope (EST: Quintero Noda
+et al. (2022)) take similar observations in the NIR lines
+with significantly higher effective spatial resolution. Im-
+portantly, this result was obtained even with a very con-
+servative approach to noise tolerance that favours longi-
+tudinal inclinations. In scenario 2 (and 3), we set Stokes
+profiles which did not satisfy the 5σn threshold to zero,
+which means in many pixels the γ is likely to be either
+0◦, 90◦, or 180◦, and this is reflected in the large peaks at
+these values in the γ distributions in Fig. 3. Of course,
+Stokes Q and U are set to zero in a much larger number
+of pixels than Stokes V . While our approach favours
+longitudinal inclinations, that does not mean our deter-
+mination of B⊥/B∥ is biased in favour of B∥. It must
+be stressed that the opposite is true - our determina-
+
+Exploring magnetic loops and serpentine fields in the quiet Sun with the GRIS-IFU
+13
+0.6
+0.7
+0.8
+0.9
+1.0
+Stokes I/Ic
+SIR
+Observed profile
+−0.006
+−0.004
+−0.002
+0.000
+0.002
+0.004
+Stokes Q/Ic
+15647.4
+15651.4
+15655.4
+15659.4
+15663.3
+wavelength [Å]
+−0.004
+−0.002
+0.000
+0.002
+0.004
+Stokes U/Ic
+15647.4
+15651.4
+15655.4
+15659.4
+15663.3
+wavelength [Å]
+−0.006
+−0.004
+−0.002
+0.000
+0.002
+0.004
+0.006
+Stokes V/Ic
+VECTOR 2.a - B = 243.2 [G], α = 0.4, γ = 83.1 [ ∘ ], ϕ = 112.5 [ ∘ ], vLOS = -1.0 [km/s]
+0.6
+0.7
+0.8
+0.9
+1.0
+Stokes I/Ic
+SIR
+Observed profile
+−0.008
+−0.006
+−0.004
+−0.002
+0.000
+0.002
+0.004
+Stokes Q/Ic
+15647.4
+15651.4
+15655.4
+15659.4
+15663.3
+wavelength [Å]
+−0.015
+−0.010
+−0.005
+0.000
+0.005
+0.010
+Stokes U/Ic
+15647.4
+15651.4
+15655.4
+15659.4
+15663.3
+wavelength [Å]
+−0.010
+−0.005
+0.000
+0.005
+0.010
+Stokes V/Ic
+VECTOR 2.b - B = 310.6 [G], α = 0.5, γ = 102.2 [ ∘ ], ϕ = -33.8 [ ∘ ], vLOS = -1.3 [km/s]
+Figure 9. As in Fig. 6, but for vectors 2.a and 2.b. The spatio-temporal locations of the pixels are shown in Fig. 8.
+15647.32
+15648.52
+15649.71
+wavelength [Å]
+−0.010
+−0.005
+0.000
+0.005
+0.010
+0.015
+Stokes V [Ic]
+2Δλ
+Observed profile
+SIR
+Figure 10. Sample Stokes V profile for a pixel located in
+the strong magnetic element in case study 2. The dotted,
+blue lines show the wavelength positions of the blue and red
+lobes for the most magnetically sensitive line, from which
+the strong field approximation is used to estimate B∥ = 812
+G. The synthetic profile from SIR is also shown, which has
+model parameters of B = 854 G, αm = 0.14, γ = 0◦, vLOS =
+2.2 km/s.
+tion of B⊥/B∥ is biased in favour of horizontal fields
+because B⊥ is determined over a much smaller popula-
+tion of pixels with significantly stronger fields (Steiner
+& Rezaei 2012).
+To determine an unbiased value for
+B⊥/B∥ it may be necessary to approach this problem
+by applying a threshold to each Stokes parameter sep-
+arately whose amplitude is determined by the magnetic
+field strength, rather than applying the same threshold
+to Stokes Q, U, and V as determined purely noise (i.e.
+by the amplitude of the 5σn threshold).
+The magnetic loops shown in both case studies are
+clear demonstrations of how the photospheric magnetic
+field is organized in the QS IN. Of particular signifi-
+cance is the apparently serpentine nature of this mag-
+netism.
+In terms of magnetic loops in the IN, there
+are a large number of statistical studies that report on
+the phenomenon, especially from the Imaging Magne-
+tograph (IMaX)/Sunrise experiment (Danilovic et al.
+2010; Mart´ınez Gonz´alez et al. 2012) and the SST (Goˇsi´c
+et al. 2021, 2022; Ledvina et al. 2022). Most of these
+studies focus on analysing bi-polar patches of circular
+polarization where cancellation could take place and
+where horizontal fields are expected to be found along
+the PIL. For instance, the SST observations by Ledvina
+et al. (2022) with the Fe I 5576 ˚A and 6301 ˚A lines, and
+with a FOV 30 times greater by area than the GRIS-
+IFU scans in this paper, found 38 magnetic loops in a
+
+14
+Campbell et al.
+42.5 minute time series, but find no evidence of hori-
+zontal fields at the PIL in any of them.
+In terms of
+serpentine magnetism, Harra et al. (2010) reported on
+observations of a serpentine magnetic field between two
+large bi-polar patches of magnetic flux in an emerging
+flux region.
+However, their target was not an IN re-
+gion. We have revealed a serpentine structure in the QS
+IN for the first time with unambiguous linear polariza-
+tion along the PIL. However, there is still a clear need
+for higher resolution observations. If one examines the
+circular polarization at the PIL, the mixing of opposite
+polarity signals within the spatio-temporal resolution el-
+ement means full-vector spectropolarimetry remains elu-
+sive (see vector 1.b in Fig. 6). Nevertheless, full-vector
+spectropolarimetry is achieved just one pixel adjacent in
+both case scenarios examined. Further there is a need
+for multi-wavelength facilities with spectral diagnostics
+sensitive to the upper photosphere and lower chromo-
+sphere, such as will be available at the DKIST (Rast
+et al. 2021), the EST (Quintero Noda et al. 2022), and
+also the Sunrise III balloon experiment when launched,
+which will allow us to assess whether these small-scale
+cancellation sites, which could be pervasive across the
+quiet solar surface, are capable of contributing to the
+heating of the lower chromosphere.
+APPENDIX
+A. SIR EXPLORER
+As spectropolarimetric solar datasets become ever larger and more complex, the multi-dimensional data cubes
+produced from observatories become increasingly non-trivial to analyse. Inversions of this multi-dimensional data
+increase in complexity too as the number of spectral lines that are inverted continues to increase irrespective of the
+data volume. To meet the challenge of inverting such large data cubes, the parallelized Python wrapper (Gafeira
+et al. 2021) to both the SIR (Ruiz Cobo & del Toro Iniesta 1992) and the Departure coefficient aided Stokes Inversion
+based on Response functions (DeSIRe; Ruiz Cobo et al. (2022)) codes enables users to easily spread the computational
+problem across many CPU nodes on high performance computing facilities. However, even when the data is inverted,
+the data products still need to be analysed and browsing the data products of inversions, of which there are many,
+becomes increasingly cumbersome. SIR Explorer (SIRE), is a Python 3.9 graphical user interface (GUI) application
+that aims to make the analysis and exploration of inversion inputs and outputs associated with SIR/DeSIRe a little
+faster and a little easier (Campbell 2023).
+Figure 11 shows the S2 inversions of scan D loaded into SIRE. The user interface (UI) of SIRE is split into three
+main regions:
+1. The control panel, located on the left of the main window, which allows the user to load datasets,
+2. The three canvases, upon which datasets are displayed, and
+3. The widget bar, underneath the canvases, which has three sliders to control the frame (FR), wavelength (WL),
+and optical depth (OD) indices.
+SIRE is designed to pick up where the parallelized Python wrapper to SIR/DeSIRe leaves off. SIRE’s required input
+file structure, detailed in Table 7, is therefore dictated by the output file structure of the wrapper. There are three
+mandatory files that are always required. The first mandatory file is the primary models output by SIR/DeSIRe,
+which are an array of the one-dimensional atmospheric files with a user-defined number of optical depth points, n(τ).
+The other two mandatory files are the observed profiles, which are the input Stokes vectors provided to SIR/DeSIRe,
+and the synthetic profiles, which are the final output Stokes vectors produced by SIR/DeSIRe when solving radiative
+transfer, which must both have the same number of wavelength points, n(λ).
+The observed profiles do not need
+to be real observations. If inverting synthetic vectors produced from simulation snapshots, these may synthetic in
+origin. There are a number of optional files that can be provided. These include: secondary models, if SIR/DeSIRe is
+employed with two models, primary (and secondary) macroturbulence files which contains the macroturbulence of the
+model and also the α and stray light fraction values, and binary map(s), which are a user-defined array that may be
+provided to remove selected pixels from the maps of the magnetic parameters.
+There are three canvas objects: the maps where images of the Stokes profiles and model parameters may be displayed,
+the plots on the upper right of the interface of the observed and synthetic Stokes profiles for the selected pixel, and
+the plots on the bottom right of the interface of the model parameters as a function of optical depth for the selected
+
+Exploring magnetic loops and serpentine fields in the quiet Sun with the GRIS-IFU
+15
+Figure 11. Main window of SIRE with scan D loaded into the application.
+Table 7.
+SIRE’s input files and the required array structure.
+The ⋆ symbol indicates that the file is mandatory in all
+circumstances. The ⋄ symbol indicates that the file is mandatory when two models are provided.
+Input file
+Array shape (t = 1)
+Array shape (t > 1)
+Primary models ⋆ ⋄
+[11, n(τ), y, x]
+[t, 11, n(τ), y, x]
+Observed profiles ⋆ ⋄
+[4, n(λ), y, x]
+[t, 4, n(λ), y, x]
+Synthetic profiles ⋆ ⋄
+[4, n(λ), y, x]
+[t, 4, n(λ), y, x]
+Secondary models ⋄
+[11, n(τ), y, x]
+[t, 11, n(τ), y, x]
+Primary macroturbulence files ⋄
+[3, y, x]
+[t, 3, y, x]
+Secondary macroturbulence files
+[3, y, x]
+[t, 3, y, x]
+Binary map(s)
+[y, x]
+[t, y, x]
+pixel. Each of the canvases may be resized in proportion to one another, and each of them may be collapsed entirely.
+The control panel provides the user with the ability to determine exactly which parameters should be shown in each
+canvas. For instance, in Fig 11 only maps of Stokes V , αmB, vLOS, and γ are selected and shown.
+The most important functionality of SIRE concerns the way in which the user is able to navigate the dataset. The
+controls have been designed to be as simple and fast as possible. When a dataset is first loaded, the pixel with zeroth
+co-ordinates in each dimension (t = 0, n(λ) = 0, y = 0, x = 0, n(τ) = 0) will be plot by default. The user can left-click
+any of the maps and the synthetic Stokes vector, observed Stokes vector, and model parameters for the corresponding
+pixel co-ordinates will be plotted. In addition, the user may adjust the selected pixel by using key-pressing the ‘UP’,
+‘DOWN’, ‘LEFT’, and ‘RIGHT’ arrow keys on their keyboard. The location of the selected pixel is denoted by vertical
+and horizontal lines on the maps. The sliders in the widget bar allow the user to adjust the frame (FR), wavelength
+(WL), and optical depth (OD) indices at which the data is retrieved from the appropriate files. The user can left-click
+and drag the sliders. The code that updates the maps and plots will not be executed until the slider is released. The
+
+Files|
+Maps
+Information
+Stokes V[lc]
+Stokes / [lc]
+Update profiles axes
+1.00 
+0.004
+V
+V
+Obs 
+7.52
+0.75 
+V
+Syn
+Update models axes
+ 0.002
+5.64
+15643.42
+15647.4
+15651.39
+15655.37
+15659.35
+15663.34
+15667.32
+ 0.000
+ Show Stokes I
+wavelength [A]
+3.76
+ Show Stokes Q
+-0.002
+Stokes Q [lc]
+ Show Stokes U
+0.004
+0.005 -
+0.0 
+0.000
+ Show Stokes V
+0.0
+2.7
+5.4
+8.1
+10.8
+15643.42
+15647.4
+15651.39
+15655.37
+15659.35
+15663.34
+WL min:0
+WL max:600
+X [arcsec.]
+15667.32
+wavelength [A]
+α B [G]
+Set wavelength range
+40
+Stokes U [lc]
+0.01 -
+7.52
+Reset wavelength range
+0.00
+30
+V
+W
+-0.01 -
+15643.42
+15647.4
+15651.39
+15655.37
+15659.35
+15663.34
+15667.32
+√ Show temperature
+20
+3.76
+wavelength [A]
+ Show magnetic field strength/flux
+ Show velocity
+1.88
+Stokes V [lc]
+ Show inclination
+0.01 
+0.0
+ Show azimuth
+0.0 
+2.7 
+5.4
+8.1
+10.8
+-0.01
+:V
+X [arcsec.]
+15643.42
+15647.4
+15651.39
+15655.37
+15659.35
+15663.34
+15667.32
+OD min:0
+OD max:54
+wavelength [A]
+VLos [km/s]
+Set optical depth range
+7.52
+Reset optical depth range
+T [K] 
+B [G]
+F 0
+SFA calculator
+3.76
+mod 1
+400
+mod 2
+Change wavelength scale
+.88
+6000
+200 -
+0.0 
+2.7
+5.4
+8.1
+10.8
+4000
+X [arcsec.]
+-3 
+y[deg.]
+log(T500nm)
+log(T500nm)
+005
+7.52
+VLos [km/s]
+Y[°]
+60 
+100
+3.76
+-1 :
+40
+20 -
+F 0
+0.0
+Display
+0.0
+2.7
+5.4
+8.1
+10.8
+-2
+-3
+-2 
+X [arcsec.]
+log(T50onm)
+log(T50onm)
+Reload
+Global Preferences
+FR: 27
+WL: 121
+OD: 1016
+Campbell et al.
+user can also key-press ‘Q’ and ‘E’ to decrease or increase, respectively, the frame index by 1. Similarly, the user can
+key-press ‘A’ and ‘D’ for wavelength, or ‘Z’ and ‘C’ for optical depth.
+The authors would like to thank the anonymous referee whose feedback helped to significantly improve this manuscript.
+We express our appreciation also to Carlos Dominguez-Tagle, whose work with the GRIS-IFU made these observations
+possible, and to all the engineering, operating, and technical staff at GREGOR for their assistance during the observing
+campaign, including Miguel Esteves Perez, Saida Milena D´ıaz Castillo, Karin Gerber, and Oliver Wiloth. Gratitude
+is extended to Juan Manuel Borrero and Lucia Kleint for their advice and insightful discussions. RJC thanks Robert
+Ryans for IT support and assistance in utilizing QUB’s high performance computing (HPC) facilities. We thank Carsten
+Denker and Christoph Kuckein for assistance with operating the High-resolution Fast Imager (HiFI) instrument and
+associated data reduction.
+This research has received financial support from the European Union’s Horizon 2020
+research and innovation program under grant agreement No.
+824135 (SOLARNET). RJC acknowledges support
+from the Northern Ireland Department for the Economy (DfE) for the award of a PhD studentship. RJC and MM
+acknowledge support from the Science and Technology Facilities Council (STFC) under grant No. ST/P000304/1 &
+ST/T00021X/1. The 1.5−meter GREGOR solar telescope was built by a German consortium under the leadership
+of the Leibniz-Institute for Solar Physics (KIS) in Freiburg with the Leibniz Institute for Astrophysics Potsdam,
+the Institute for Astrophysics G¨ottingen, and the Max Planck Institute for Solar System Research in G¨ottingen as
+partners, and with contributions by the Instituto de Astrof´ısica de Canarias and the Astronomical Institute of the
+Academy of Sciences of the Czech Republic. The redesign of the GREGOR AO and instrument distribution optics
+was carried out by KIS whose technical staff is gratefully acknowledged. Helioseismic and Magnetic Imager (HMI)
+magnetograms, courtesy of NASA/SDO and the AIA, EVE, and HMI science teams, were used during observations
+for target selection. This work was supported by Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) through the research
+grants UIDB/04434/2020 and UIDP/04434/2020.
+Facilities: GREGOR solar telescope (Schmidt et al. 2012; Kleint et al. 2020).
+Software: SIR explorer, (Campbell 2023), Astropy (Astropy Collaboration et al. 2022), Matplotlib (Hunter 2007),
+Numpy (Harris et al. 2020).
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+
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new file mode 100644
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+page_content='1 Ricardo Gafeira  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content=' 3 Mihalis Mathioudakis  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1 Carlos Quintero Noda  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 5 and  Manuel Collados  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 5  1Astrophysics Research Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Queen’s University of Belfast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Northern Ireland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' BT7 1NN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' UK  2Instituto de Astrof´ısica de Andaluc´ıa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Apartado de Correos 3004,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 18080 Granada,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Spain  3Instituto de Astrof´ısica e Ciˆencias do Espa¸co,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Departamento de F´ısica da Universidade de Coimbra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Rua do Observat´orio s/n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 3040-004 Coimbra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Portuga  4Instituto de Astrof´ısica de Canarias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' V´ıa L´actea s/n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' E-38205 La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Tenerife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Spain  5Departamento de Astrof´ısica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Universidad de La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' E-38206 La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Tenerife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Spain  ABSTRACT  Synthetic observations produced from radiative magnetohydronamic simulations have predicted that higher  polarization fractions in the quiet solar photosphere would be revealed by increasing the total integration time  of observations at GREGOR resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' We present recently acquired disk centre observations of the Fe I  15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 ˚A line obtained with the GREGOR telescope equipped with the GRIS-IFU during excellent seeing  conditions, showing exceptionally high polarization fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Our observation reveal an internetwork region  with a majority (> 60%) of magnetised pixels displaying a clear transverse component of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  This result is in stark contrast to previous disk-centre GRIS-IFU observations in this spectral line, which had  predominantly vertical magnetic fields in the deep photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' At the same time, the median magnetic field  strength is weaker than previous GRIS-IFU observations, indicating that the larger fraction of polarization  signals cannot be explained by a more active target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' We use the Stokes Inversion based on Response functions  (SIR) code to analyse the data, performing over 45 million inversions, and interrogate the impact of two  conflicting approaches to the treatment of noise on the retrieval of the magnetic inclination and azimuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  We present several case studies of the zoo of magnetic features present in these data, including small-scale  magnetic loops that seem to be embedded in a sea of magnetism, and serpentine fields, focusing on regions  where full-vector spectropolarimetry has been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' We also present a new open-source Python 3 analysis  tool, SIR Explorer (SIRE), that we use to examine the dynamics of these small-scale magnetic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Keywords: Sun: photosphere — Sun: magnetic fields — Sun: infrared — Sun: granulation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' INTRODUCTION  Lites et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2008) revealed the quiet Sun (QS) inter-  network (IN) as dominated by horizontal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' transverse  with respect to the solar normal) magnetic fields at an  effective spatial resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' However, this result  is not undisputed and remains subject to contradiction  by other studies, as reviewed by Steiner & Rezaei (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  For instance, the lack of variation in the degree of linear  and circular polarisation recorded in near infrared (NIR)  observations by Mart´ınez Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2008) at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='8′′  as a function of different heliocentric angles points to  a QS magnetic field which has no preferential bias in  orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' If the distribution of magnetic field incli-  nations is isotropic, its probability density function is  given as,  P(γ) = sin γ  2  ,  (1)  where γ is the magnetic inclination angle, defined as  the angle between the magnetic vector and solar nor-  mal, such that P(γ) has a maximum at 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' It is per-  haps counter-intuitive, but this would mean most of the  fields are transverse, because for the magnetic field to be  aligned along the line-of-sight (LOS) it has to point in  one of two possible directions, but to be transverse there  are many more possible directions (S´anchez Almeida &  Mart´ınez Gonz´alez 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  The greatest difficulty in constraining the γ from in-  versions of IN observations in a consistent way results  from the differing treatments of varying levels of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  In the weak field regime, a vertical field produces a larger  amplitude circular polarization profile than the ampli-  tude of a linear polarization profile produced by a hori-  zontal field of equal strength at disk centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' This creates  an intrinsic bias against being able to confidently detect  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='05591v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='SR]  13 Jan 2023  ID2  Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='5  Y [arcsec]  αmB  0  25  50  [G]  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Sample frame from a GRIS-IFU scan of a quiet Sun region on the 24 August 2021 (scan D) showing in the left  column from top to bottom the continuum-normalized Stokes I, Q, U, and V , and in the right column from top to bottom  the T, vLOS, αmB, and γ as derived from SIR inversions (scenario 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Stokes I is shown at a wavelength of 15650.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='59 ˚A in the  continuum and the polarization profiles are shown at 15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='36 ˚A in the blue lobes of the geff = 3 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Any pixel which does  not have a Stokes Q, U, or V profile with maximum unsigned amplitude across the 15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 ˚A line below the 5σn threshold has  been set to zero and masked from the plots of Stokes Q, Stokes U, Stokes V , αmB, and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' T is shown at logτ5000˚  A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5, while  the other model parameters are shown at constant in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  horizontal fields at disk centre when a given noise thresh-  old is equally applied to Stokes Q, U, and V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' By pro-  ducing models with deliberately purely vertical fields,  synthesizing the Stokes vector, and adding noise before  inverting again, Borrero & Kobel (2011) show that an  inversion code will return an overabundance of horizon-  tal inclinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' As a result, even when one inverts only  those pixels with at least one Stokes Q, U, or V pro-  file above a noise threshold, and most of the pixels have  only Stokes V above the threshold, this arguably results  in a possible bias in favour of horizontal fields as the in-  version code interprets noise in Stokes Q and U as real  signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  The deepest Hinode/SP integrations show circular  and linear polarisation in 88% and 53% of the field of  view (FOV), respectively, but this comes at the expense  of spatio-temporal resolution which distorts the polar-  ization signals (Bellot Rubio & Orozco Su´arez 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Most recently, observations of the IN with visible lines  at the ground-based Swedish Solar Telescope (SST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Exploring magnetic loops and serpentine fields in the quiet Sun with the GRIS-IFU  3  Goˇsi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2021, 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Ledvina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2022)) and  balloon-borne Sunrise experiment (Danilovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Mart´ınez Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Kianfar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2018) have  provided good statistics in terms of the longitudinal field  and even of cancellations, but visible photospheric lines  still struggle to confidently detect the horizontal fields  that should be present in magnetic loops along the polar-  ity inversion line (PIL) without significant spatial, spec-  tral, or temporal binning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Observations with the NIR  line pair at GREGOR with the GREGOR Infrared Spec-  trograph (GRIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Lagg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Mart´ınez Gonz´alez  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2016)) have demonstrated higher efficacy at mea-  suring linear polarization at similar spatial resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  This is despite the fact that modelling with simulations  by Steiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2008) shows the vertical field being  prominent in the deep photosphere, where the NIR line  pair is sensitive (Quintero Noda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2021), with the  horizontal component of the magnetic field becoming  increasingly important in the upper photosphere, where  the visible line pair is sensitive, with convective over-  shooting responsible for expelling horizontal fields to  greater heights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2021a) have used the  Integral Field Unit (IFU) mode of GRIS to examine  the dynamics of small-scale magnetic features, including  magnetic loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Two-dimensional spectrographs like the  GRIS-IFU are uniquely capable of providing time-series  imaging with high polarimetric sensitivity and spectral  resolution simultaneously, with the size of the FOV and  cadence acting as competing factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2021b) have predicted, using syn-  thetic observations produced from MURaM simulations  (V¨ogler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2005) degraded to GREGOR/GRIS-IFU  resolutions, that increasing the total integration time  of the observations will yield higher fractions of polar-  ization and allow the magnetic inclination to be better  constrained in a larger fraction of the FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' We report  observations calibrated based on these predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' OBSERVATIONS  Repeated observations of disk centre QS regions  were taken in late August 2021 using the GRIS-IFU  (Dominguez-Tagle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2022) at GREGOR (Schmidt  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Kleint et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The GRIS-IFU was op-  erated in double sampling mode, with a 3 (vertical) by  2 (horizontal) mosaic pattern, with an exposure time of  60 ms per polarimetric state, and 20 accumulations (for  a total integration time per pixel of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='8 seconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' This  resulted in a cadence of 103 seconds between frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Compared to similar 2019 scans, the exposure time and  accumulations were increased to target a higher S/N,  but the number of steps in the mosaic pattern was re-  duced to constrain the cadence between frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  The  FOV of the scans is therefore 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='15′′ by 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='024′′ with  a spatial sampling of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='135′′ by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='188′′ in the x- and  y−directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  The GRIS-IFU observed a 40 ˚A spectral window that  includes the highly magnetically sensitive photospheric  Fe I line at 15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The spectral dispersion was de-  termined to be 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='83 m˚A/pixel by comparison with a  degraded Fourier Transform Spectrometer (FTS) atlas  (Livingston & Wallace 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  During the observing campaign we benefitted from  very good seeing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Table 1 lists five datasets  which are candidates for analysis as they were taken  with good to excellent seeing, as quantified by the lo-  cally measured Fried parameter, r0, and the root-mean-  square continuum intensity contrast, δIrms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The GRIS  reduction pipeline (Collados et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2012) was employed  for dark current removal, flat fielding, polarimetric cal-  ibration, and cross-talk corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' RESULTS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Polarization analysis  In order to quantify the fraction of pixels exhibiting  a polarization profile confidently above the noise level  in the datasets listed in Table 1, we adopt the method  used by Lagg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The  purpose of this analysis is to enable a comparison be-  tween the new GRIS-IFU datasets, previous GRIS-IFU  datasets, and datasets from other facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' For a given  Stokes parameter, we determine the noise level, σn, by  calculating the standard deviation in the continuum in  each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The explicit assumption is made that the  continuum is unpolarized and the primary type of noise  is photon noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' We then determine whether a pixel has  a Stokes Q, U, or V profile with maximum amplitude  greater than a threshold set at 5σn across the 15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 ˚A  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' If a given pixel has a maximum amplitude in Stokes  V greater than the threshold, the pixel is said to have a  confidently measured circular polarization (CP) signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Similarly, if a given pixel has a maximum amplitude in  Stokes Q or U greater than the threshold, the pixel is  said to have a confidently measured linear polarization  (LP) signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' If a Stokes vector has neither LP or CP, it  is said to have no polarization (NP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Table 1 shows the time-averaged results of this anal-  ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  The exceptionally high polarization fractions of  scans D and E stand out as being much larger than  scans A, B, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Figure 1 shows a sample frame from  scan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' This frame had a δIrms of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4%, a maximal r0  of 22 cm, and shows an abundance of polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' In  terms of understanding why the polarization fractions of  scan D and E are higher than the other scans, the first  factor to consider is that the GRIS-IFU FOV is very  4  Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Time-averaged percentage of linear (LP) and cir-  cular (CP) polarization profiles with maximum amplitudes  above 5σn for level 1 GRIS-IFU/GREGOR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The per-  centages are calculated relative to the full FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The 1σn  noise level is determined by the calculation of the standard  deviation in the relevant Stokes parameter at continuum  wavelengths in each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' All scans were taken in August  2021 and the times are given in universal time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Label  Day, time  %LP  %CP  %LP & CP  %NP  A  19, 08:14  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  B  20, 08:56  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='8  C  23, 08:00  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2  D  24, 07:50  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  E  24, 08:55  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  As for Table 1, but for the PCA-RVM recon-  structed GRIS-IFU data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The noise threshold is set at 5σn  as measured in the continuum of the original level 1 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Label  %LP  %CP  %LP & CP  %NP  D  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  E  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='8  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' There is therefore always a risk when observing  with the GRIS-IFU that one could point to a so-called  ‘void’, where there is little detectable magnetic flux, or  indeed the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The second factor to consider is  the quality and stability of the seeing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' From  the r0 and δIrms values for each of the scans, it is clear  that the seeing conditions for scans D and E were supe-  rior in both quality and stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' For instance, scan A  had a peak δIrms of only 3% and a low of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='8%, while  scan D had a peak of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4% and low of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' This is im-  portant as atmospheric stability is essential to achieve a  maximal effective spatial resolution, and impacts heavily  on the polarization fractions recorded because opposite  polarity profiles cancel out within the spatio-temporal  resolution element with insufficient spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Principal component analysis (PCA) with 15 retained  eigenvectors is applied to scans D and E for the purpose  of removing noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' As eludicated by Borrero & Kobel  (2011), while the probability of photon noise producing  a signal greater than 3σn is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3%, the probabil-  ity increases dramatically when more and more wave-  length points are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' By applying PCA to re-  move noise and using a stringent 5σn noise threshold,  we are reducing the probability that a false signal will  survive and enter the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' As interference fringes  are present in the wavelength domain of the polariza-  tion profiles in some pixels, a relevance vector machine  (RVM) is employed to remove these defects and produce  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='008  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='010  Amplitude [Ic]  0  10  20  30  40  50  60  N [%]  LP  CP  Scan D  Scan E  6 May 19  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Area occupied by pixels with a LP (red lines)  or CP (blue lines) signal for scan D (dotted lines), scan E  (dashed lines), and the scan taken on the 6 May 2019 (solid  lines) for given amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The data has undergone PCA-  RVM reconstruction and had any Stokes profiles with maxi-  mum unsigned amplitude across the 15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 ˚A line below the  5σn threshold in the 2021 data, and 3σn in the 2019 data,  set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  apparently noiseless polarization profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The full re-  construction process is the same as employed by Camp-  bell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2021a) on GRIS-IFU/GREGOR data and is  described in detail therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' By applying the PCA-RVM  reconstruction process to scans D and E, this enables a  comparison with the PCA-RVM reconstructed statistics  reported by Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' For this purpose  in this paper we will use the scan taken on 6 May 2019,  henceforth referenced simply as the 2019 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Table 2 shows the time-averaged polarization fractions  for scan D and E after application of the PCA-RVM re-  construction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Figure 2 shows the time-averaged  polarization fractions of scans D and E as a function of  amplitude alongside the scan taken on 6 May 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' In  this case, any Stokes profile whose maximum unsigned  amplitude across the 15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 ˚A line does not exceed the  5σn threshold for the two 2021 datasets, and 3σn for the  2019 dataset, has been set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2 it is clear  that the 5σn threshold is on average at a slightly lower  amplitude in scans D and E than the 3σn threshold in  the 2019 scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Seeing-induced cross-talk becomes more  likely with longer exposures, as with the longer total in-  tegration time it is more likely that the modulations of  the Stokes parameters will not be conducted fast enough  for Earth’s atmosphere to be considered ‘frozen’ during  measurements (Collados 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' We choose a stringent  5σn threshold for the August 2021 data that still al-  lows us to access weaker polarization signals than in the  2019 data, because the former has a lower noise level  than the latter, but is sufficiently large to allow us to  assume any seeing-induced cross-talk produced is un-  likely to have an amplitude of this magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' There is  Exploring magnetic loops and serpentine fields in the quiet Sun with the GRIS-IFU  5  a significantly higher fraction of CP and LP in the new  scans than the old scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Perhaps the most significant  difference between the old and new scans is the much  higher percentage of the FOV which has both LP and  CP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Previously, between 3 − 5% of the FOV had both  LP and CP, but in the new scans this number has signif-  icantly increased to 16−18% at 5σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' This means that γ  should be retrievable from inversions in a much higher  fraction of the FOV and it would also be anticipated  that a larger fraction of pixels would harbour fields with  an intermediate (as opposed to highly vertical or highly  inclined) inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Inversion strategies  We use the Stokes Inversion based on Response func-  tions (SIR) (Ruiz Cobo & del Toro Iniesta 1992) code to  invert scans D and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' We adapt the parallelized wrap-  per to SIR written by Gafeira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2021) to enable  the kinematic and magnetic parameters of the input  model(s) to be randomized within physically sensible  upper and lower boundaries at every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  This  is an essential step to maximise the statistical chances  of achieving the global χ2 minimum solution for each  Stokes vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  For the inversion, we adopt the same  scheme as tested on real observations of the NIR line  pair by Mart´ınez Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  (2021a) and on synthetic observations by Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  This is a two-component inversion with one  non-magnetic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' the magnetic field strength, B, was  set to zero and was not allowed to vary) and one mag-  netic model, which are combined by SIR in accordance  to their respective filling factors, α, which is another free  parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' For clarity, we refer to the filling factor of the  magnetic model as αm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' SIR allows the user to determine  the optical depths at which perturbations will be made  between iterations by selecting a number of nodes in  each variable parameter, with the full parameter strati-  fication in optical depth given by cubic splines or linear  interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Apart from temperature, T, for which 4  nodes were selected, each parameter, including B, was  forced to be constant in optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The macrotur-  bulent velocity, vmac, was included as a free parameter  in the non-magnetic model and forced to be the same  in the magnetic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The T was also determined by  the non-magnetic model and forced to be the same in  the magnetic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The line of sight velocity, vLOS,  and microturbulent velocity, vmic, were free parameters  in both models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Full PCA-RVM reconstruction was ap-  plied to the datasets before inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' We used the em-  pirically determined atomic parameters made available  by Trelles Arjona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2021) and abundances from  Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Included in the inversion were  seven spectral lines, including five Fe I lines with rest  wavelengths of 15645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='02 ˚A, 15645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='303 ˚A, 15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='514 ˚A,  15652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='873 ˚A, 15662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='017 ˚A, 15665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='241 ˚A, one blended  Fe II line with rest wavelength 15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='514 ˚A, and one  blended O I line with rest wavelength 15665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='098 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  We interrogate the impact of noise on the retrieval of  B, γ, and αm statistically by taking more than one ap-  proach to data treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' In order to accurately deter-  mine γ, one must measure both linear polarization and  circular polarization confidently above the noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  However, it can be argued that the noise level itself  places limits on the amplitude of the weakest polariza-  tion profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' We divide the approaches into three sce-  narios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' In the first scenario (S1), the full datasets were  inverted with all Stokes parameters provided to SIR for  each pixel without any consideration for signal ampli-  tudes relative to the σn level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' In scenario 2 (S2), any  individual Stokes Q, U, or V profile in a given pixel  whose maximum amplitude across the 15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 ˚A line  does not reach the 5σn threshold was set to zero before  inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' In scenario 3 (S3), any individual Stokes Q,  U, or V profile that does not reach the 5σn threshold was  set to zero before inversion as in S2, with the exception  that if one linear polarization parameter was confidently  measured then the weaker linear polarization parameter  was not set to zero despite being below the threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  This third scenario is included to investigate whether  only one linear polarization parameter (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' cases where  Stokes Q is > 5σn but Stokes U is not, or vice versa) is  sufficient to constrain γ and if there are any impacts on  other magnetic parameters when the weaker linear po-  larization parameter is set to zero and the stronger one is  not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Each inversion was repeated 50 times with random-  ized initial models, resulting in over 45 million inversions  in total for all three scenarios and both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Magnetic filling factors and field strengths  Figure 3 shows the distributions of B, γ, αmB, αm,  and |cos φ|, where φ is the azimuthal angle of the mag-  netic field vector in the plane perpendicular to the ob-  server’s LOS, for each scenario and scans D and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' First,  the αm distribution looks very similar in all scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  However, the B distribution is significantly different be-  tween S1 on the one hand and S2 and S3 on the other  hand below 600 G for both scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' In S1, the B distri-  bution peaks at around 100 G, but for S2 and S3 there  is a much larger number of pixels with B values below  100 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Since B is determined by the longitudinal and  transverse components, setting linear and circular polar-  ization signals to zero naturally results in lower values  of B overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' As a consequence, the αmB distributions  for S1 on one hand and S2 and S3 on the other hand  6  Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='0%  N [%]  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Histograms of B (top row), γ (second row), αmB (third row), αm (fourth row), and |cos φ| (bottom row) returned  from the inversions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The histograms are shown for three inversion scenarios (S1, S2, and S3) as described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The left  and right columns show the distributions for scan D and E, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Histograms are weighted with respect to the total  number of pixels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' including those with no polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Pixels which had no measured polarization in any Stokes parameter  are excluded from the histogram (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' only pixels with maximum absolute amplitude across the 15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 ˚A line greater than the  5σn threshold in at least one Stokes parameter are included).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Exploring magnetic loops and serpentine fields in the quiet Sun with the GRIS-IFU  7  also differ in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Tables 3 and 4 show for  scans D and E, respectively, the median αmB and B for  each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' In addition, the median transverse and  longitudinal components, B⊥ and B∥, as well as αmB⊥,  and αmB∥, are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' First of all, the median B,  B⊥, and B∥ values under S2 are all significantly lower  than the equivalent values reported by Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  (2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' These weak average values underline that the  target region is IN in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Further, as these fields can  be understood as firmly in the weak field regime, where  the circular polarization amplitudes are proportional to  B (and linear polarization amplitudes are proportional  to B2), recording lower median values in these parame-  ters is sensible on the inclusion of weaker Stokes profiles  that are accessed in the approximate range 2−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4×10−3  Ic that could not be confidently measured in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The  median αm values are very small, typically at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' How-  ever, this is a consequence of the shape of the distribu-  tion - the mean value is an order of magnitude bigger  typically at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The ratio B⊥/B∥ has been re-  ported in other studies (Orozco Su´arez & Bellot Rubio  2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2021a) and is in fact slightly larger  than 2019, indicating that as we increase the S/N, and  access weaker fields, the magnetic field is getting more  horizontal on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Since the ratio B⊥/B∥ would be  π/2 for an isotropic distribution, this also indicates that  accessing weaker fields has returned a distribution that  is further from the isotropic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' There is also a clear  difference between scans D and E, with stronger B, B∥,  and B⊥ values returned by SIR in all scenarios in scan  E relative to scan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' It is also clear from the ratios that  scan E is slightly more longitudinal on average than scan  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Finally, there is a clear difference between S2 and S3  for both scans, with the median B values being about  20 G stronger when the weakest linear polarization pa-  rameters are not eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Further, the B⊥/B∥ ratio  is slightly larger in S3 compared to S2 in both scans,  and the median αm values are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Inclinations and azimuths  The γ distributions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 3 differ significantly be-  tween S1 on one hand and S2 and S3 on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The  shape of the γ distribution in S1 resembles the distri-  butions derived from semi-empirical models produced  by Borrero & Kobel (2011) with large noise thresholds  (> 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5σn) specifically in terms of the small dip at 90◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  As the authors explain, this dip is a consequence of as-  serting a large noise threshold in the weak field regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Thus, by setting the threshold at 5σn, the dip in the γ  distribution is created by excluding the weakest fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  This dip is also observed in the distribution from MU-  RaM simulations by Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' For S2  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Unsigned median magnetic field strengths and flux  densities, and its horizontal (transverse) and vertical (longi-  tudinal) components, for scan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The median B is measured  across all pixels with at least one polarization parameter with  maximum amplitude > 5σn across the 15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 ˚A line, but  when computing the corresponding B⊥ and B∥ component  only profiles that had Stokes Q or U > 5σn in the former  case, and Stokes V > 5σn in the latter case, are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Scenario 1  Scenario 2  Scenario 3  B  206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4 G  119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3 G  140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6 G  B∥  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6 G  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4 G  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9 G  B⊥  200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9 G  153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2 G  188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4 G  αm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='04  αmB  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3 G  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7 G  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6 G  αmB∥  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3 G  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='8 G  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1 G  αmB⊥  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1 G  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0 G  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 G  B⊥/B∥  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' As in Table 3, but for scan E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Scenario 1  Scenario 2  Scenario 3  B  233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3 G  163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6 G  182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='8 G  B∥  107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 G  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2 G  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='8 G  B⊥  221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1 G  182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3 G  220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1 G  αm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='04  αmB  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7 G  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0 G  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2 G  αmB∥  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 G  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6 G  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6 G  αmB⊥  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2 G  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7 G  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9 G  B⊥/B∥  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Percentage of pixels with inclinations in given  ranges as determined by SIR under the three scenarios for  scan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The percentages are computed relative to the total  number of pixels with at least one Stokes profile with maxi-  mum amplitude > 5σn across the 15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 ˚A line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Scenario  classification  range [◦]  1 [%]  2 [%]  3 [%]  highly vertical  γ < 16  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0  highly vertical  γ > 164  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2  intermediate  15 < γ < 75  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  intermediate  105 < γ < 165  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  highly inclined  74 < γ < 106  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7  and S3, the distribution is similar to Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  (2021a) in that large peaks are recorded at 0, 90, and  180◦, which is a consequence of setting noisy Stokes Q,  U, and V profiles to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' For instance, a Stokes vec-  tor with only Stokes Q above the noise threshold and  Stokes V set to zero would be expected to return a γ  8  Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' As in Table 6, but for scan E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Scenario  classification  range [◦]  1 [%]  2 [%]  3 [%]  highly vertical  γ < 16  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  highly vertical  γ > 164  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2  intermediate  15 < γ < 75  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0  intermediate  105 < γ < 165  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0  highly inclined  74 < γ < 106  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  value close to 90◦ by SIR when a good fit is achieved,  which is the reason for the central peak in the γ distri-  bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' However, if this profile had a noisy Stokes V  included when inverted, the γ would be more likely to  show a more intermediate inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Tables 5 and 6 classify the inclinations in terms of  highly vertical, highly inclined, and intermediately in-  clined fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The difference compared to 2019 is in the  intermediately inclined fields, which have a much higher  population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2021a) report over 70% of  pixels had a highly vertical classification, with a minor-  ity of pixels having a significant transverse component  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' classified as either intermediate or highly inclined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  In scans D and E, remarkably, the situation is reversed -  only 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1% in scan D and 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5% in scan E under S2 have  a highly vertical classification, with the majority having  a significant transverse component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The explanation for  this has already been discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1: there are a  much larger number of pixels with both LP and CP in  scans D and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The very close similarity between S2 and  S3 indicates that eliminating the weaker linear polariza-  tion parameter does not result in an altered distribution  of γ values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' only the stronger linear polarization pa-  rameter is necessary to retrieve γ (in addition to Stokes  V , if a value that differs from close to 90◦ is to be re-  turned).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Of course, there is a difference in φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The reality  is that regardless of the scenario, since both Stokes Q  and U must be measured confidently to constrain φ, and  so few pixels have both, retrieving φ (even without dis-  ambiguation) in a statistical sense is not feasible in this  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Nevertheless, the peaks in the | cos φ| distribution  at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7 and 1 is a consequence of setting one of Stokes Q  or U to zero while the other exceeds the 5σ threshold,  forcing φ to be such that cos φ is returned as −1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7, or 1 for φ values of 0, 45, 135, or 180◦ (or indeed  these values shifted by 180◦ due to a lack of disambigua-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Finally, the clear difference between scans D and  E is elucidated again, with slightly larger numbers of  highly vertical classifications in scan E relative to scan  D, and indeed slightly larger numbers of intermediate  and highly inclined classifications in scan D relative to  scan E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Case studies  A key strength of the GRIS-IFU lies in its ability  to image the dynamics of small-scale magnetic fea-  tures with high S/N, spatial resolution, and spectral  resolution simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  We now present two case  studies and in order to make the analysis more effi-  cient, we have developed an open-source application,  SIR Explorer (SIRE), that enables users to navigate the  multi-dimensional input and output files of SIR inver-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' This tool is described in Appendix A and is used  throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Figure 4 shows a magnetic loop  visible in scan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Figure 5 shows a close-up of the po-  larization in the boxed region in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Here we define  the total linear polarization as,  Ltot =  � λr  λb [Q2(λ) + U 2(λ)]  1  2 dλ  Ic  � λr  λb dλ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  (2)  Figure 4 shows a magnetic feature whose structure re-  sembles a magnetic loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' There is a large patch of linear  polarization, which bridges two patches (or foot-points)  of opposite polarity vertical fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  As seen from the  Dopplergram, the linear polarization appears at the cen-  tre of the granule while the circular polarization is lo-  cated in the intergranular lanes (IGLs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Over time, the  circular polarization remains cemented in the lanes as  the granule evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' As the linear polarization vanishes  below our detection capabilities, the two patches of cir-  cular polarization no longer appear to be in close con-  tact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' This magnetic feature has a lifetime of less than 10  minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' This shows how the evolution of the magnetic  loop is influenced by the granular evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' There are  many pixels in this structure which have all three polar-  ization profiles above the 5σn threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Figure 6 shows  three example Stokes vectors, whose spatio-temporal lo-  cations are outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 5, across the PIL of the mag-  netic loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The upper-most vector, vector 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='a, has a pos-  itive polarity but the field is highly inclined (γ = 95◦)  and stronger than the median (B = 330 G, αmB = 198  G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The middle vector, vector 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='b, has no confidently  measured Stokes V profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The pixel is located in the  PIL and thus the physical reason for the lack of a circu-  lar polarization profile could be either due to mixing of  opposite polarities within the spatio-temporal resolution  element or the magnetic vector could be purely transver-  sal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The field is still strong (B = 381 G), but the fill-  ing factor is halved compared to profile 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='a (αm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3)  and thus the αmB is significantly lower (αmB = 114  G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Due to the lack of a confidently measured Stokes  V signal, we cannot access the longitudinal component  Exploring magnetic loops and serpentine fields in the quiet Sun with the GRIS-IFU  9  0  20  40  αmB [G]  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='75  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  Y [arcsec]  8:33:24 UT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0020  Ltot [Ic]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  Stokes V [Ic]  −2  −1  0  1  2  vLOS [km/s]  0  90  180  γ [∘]  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='75  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  Y [arcsec]  8:36:51 UT  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='75  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  Y [arcsec]  8:38:34 UT  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='75  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  Y [arcsec]  8:40:17 UT  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  X [arcsec]  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  X [arcsec]  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='75  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  Y [arcsec]  8:43:44 UT  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  X [arcsec]  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  X [arcsec]  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  X [arcsec]  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Case study of a magnetic loop in scan D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Shown from left to right is the Ltot, Stokes V at 15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='36 ˚A in the blue  lobes of the geff = 3 line, vLOS, γ, and αmB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The last three parameters are derived from SIR inversions under S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The rows  show subsequent frames with the time-stamp in the upper-right corner of the left-most plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The box (solid line) highlights the  spatio-temporal location of the distinct magnetic loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  8:36:51 UT  Ltot [Ic]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='a  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='c  Stokes V [Ic]  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Close-up of the polarization in the boxed region  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 4, showing a distinct magnetic loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The total linear  polarization is shown in the left panel and Stokes V is shown  in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The full Stokes vectors of the three out-  lined pixels are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The pixels are separated by  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='188′′ each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  of the field, and thus the αmB is likely under-estimated  in vector 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Finally, in the pixel below the PIL, vec-  tor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='c, the polarity of the field is negative but the field  is still strong (B = 397 G, αmB = 159 G) and highly  inclined (γ = 83◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' All three Stokes vectors have very  strong Stokes Q and U signals, and thus there is infor-  mation about φ available in each profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The φ values  are very consistent, differing by only 1 − 2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Of course,  the ambiguity in the φ remains, so although φ values of  123 − 124◦ are returned by SIR, values of 303 − 304◦  would also be equally acceptable solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Of course,  as all three pixels are located in the granule their vLOS  values are consistently negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Figure 7 shows the second case study of a series of  magnetic loops present in scan E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' In this region, several  magnetic loops, whose locations are highlighted with  boxes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 7, are present and appear to form a contin-  uous “serpentine” structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Alternatively, these could  be independent loops formed by the action of a small-  10  Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1  Stokes I/Ic  SIR  Observed profile  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='004  Stokes Q/Ic  15647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='3  wavelength [Å]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='015  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='010  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='010  Stokes U/Ic  15647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  wavelength [Å]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='006  Stokes V/Ic  VECTOR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='a - B = 330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0 [G], α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6, γ = 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7 [ ∘ ], ϕ = 123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9 [ ∘ ], vLOS = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2 [km/s]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1  Stokes I/Ic  SIR  Observed profile  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='006  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='006  Stokes Q/Ic  15647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='3  wavelength [Å]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='015  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='010  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='015  Stokes U/Ic  15647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  wavelength [Å]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='001  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  Stokes V/Ic  VECTOR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='b - B = 381.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3 [G], α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3, γ = 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1 [ ∘ ], ϕ = 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0 [ ∘ ], vLOS = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1 [km/s]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1  Stokes I/Ic  SIR  Observed profile  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='008  Stokes Q/Ic  15647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  wavelength [Å]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='015  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='010  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='020  Stokes U/Ic  15647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  wavelength [Å]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='006  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='006  Stokes V/Ic  VECTOR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='c - B = 397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3 [G], α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4, γ = 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9 [ ∘ ], ϕ = 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7 [ ∘ ], vLOS = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4 [km/s]  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Full observed Stokes vectors 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='a, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='b, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='c (black, dashed lines), along with the corresponding synthetic vectors  derived from SIR inversions (red, solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Stokes Q, U, and V have been PCA-RVM reconstructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The horizontal (dot-  dashed) lines show the 5σn noise thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The locations of vectors 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='a, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='b, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='c are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  scale dynamo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Most of the magnetic flux is located in  IGLs, with linear polarization present in the PIL which  is most commonly located at the granule-IGL boundary  or in the granule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' A close-up of the magnetic loop high-  lighted by the solid box in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 7 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 8 for  two frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Since opposite polarity Stokes V profiles  cancel in the PIL, the most common location to find  pixels with all three polarization parameters above the  noise threshold is one pixel adjacent to the PIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' For in-  stance, vectors 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='a and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='b, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 9 with their  spatio-temporal location shown in Fig 8, are located on  either side of the PIL and thus have opposing polarities  (γ = 83◦ versus γ = 102◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The B and αmB values  of profile 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='a are smaller (B = 243 G, αmB = 97 G)  than profile 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='b (B = 311 G, αmB = 155 G) as the  magnetic loop increases in strength as it emerges in the  frame from which profile 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='a is selected and reaches its  peak in the frame from which profile 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='b is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Both  of these pixels are located in granular upflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Considering the wider context of the second case  study, the series of magnetic loops are connected to a  much stronger longitudinal magnetic element shown in  Exploring magnetic loops and serpentine fields in the quiet Sun with the GRIS-IFU  11  0  20  40  αmB [G]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='75  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  Y [arcsec]  9:22:44 UT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0020  Ltot [Ic]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='004  Stokes V [Ic]  −2  −1  0  1  2  vLOS [km/s]  0  90  180  γ [∘]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='75  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  Y [arcsec]  9:24:27 UT  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='75  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  Y [arcsec]  9:26:11 UT  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9  X [arcsec]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9  X [arcsec]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='75  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5  Y [arcsec]  9:27:54 UT  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9  X [arcsec]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9  X [arcsec]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9  X [arcsec]  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Case study of a series of magnetic loops in scan E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The plots follow the same layout as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The solid, dashed,  dotted and dot-dashed red boxes highlight the locations of magnetic loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The solid, magenta ellipse highlights the location of  a strong longitudinal patch of magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  the bottom-right of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 7 in the ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The tempo-  ral evolution of this patch of longitudinal flux can be  traced throughout the full 1 hour time series and there-  fore persists even as the much shorter-lived magnetic  loops appear and disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' This magnetic element is  first visible at the start of the time series further to the  right, and is migrated and coalesced by granular evolu-  tion until it covers a smaller surface area by the end of  the scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The presence of this strong longitudinal flux  element is reminiscent of the magnetic loop reported by  Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2021a), which was also located next to  a strong longitudinal magnetic element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' For normal Zee-  man triplets, with an effective Land´e g-factor, geff, char-  acterized by two split σ components and an unshifted π  component, the two σ components are separated from  the rest wavelength, λ0, by  ∆λ = ±  e  4πmecλ2  0geffB ≈ ±4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='67 × 10−13λ2  0geffB,  (3)  where ∆λ and λ0 are in units of ˚A and B is in units of G,  c is the speed of light, me is the mass of an electron, and  e is the electron charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' In what is known as the strong  field approximation (SFA), by measuring ∆λ it is there-  fore possible to measure B directly from the splitting of  the lobes of Stokes V when the field is strong enough  (Khomenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' SIRE pro-  vides a simple calculator which allows users to quickly  calculate an estimate of B based on the separation of  the lobes, which can also be measured using the wave-  length slider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Figure 10 shows a sample Stokes V profile  from the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The Stokes V profile is asymmetric,  particularly in the red σ lobe, but SIR can still produce  an estimate of B even without introducing the gradients  in vLOS or B that would be necessary to better fit this  profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The pixel had no linear polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The val-  ues returned from SIR were: B = 854 G, αm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='14,  γ = 0◦, vLOS = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The SFA estimate is returned  as BSFA = 812 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The wavelength positions from which  this estimate was measured is indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' SIR  must determine B such that a good fit is found for all  lines, and as such it is not abnormal that there is a small  difference in the measurements, especially given that the  wavelength position of the synthetic and observed blue  σ lobe differ very slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' In any case, a shift of a single  increment in wavelength is equivalent to 58 G, and so the  values are within an acceptable uncertainty interval of  each other, and serves as a sanity check demonstrating  that the inversions are well calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' DISCUSSION  The quest to reveal higher fractions of the FOV as  displaying confidently measured Zeeman-induced polar-  ization signal continues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The fact that 65−67% of the IN  FOV has confidently measured polarization in at least  12  Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  9:22:44 UT  Ltot [Ic]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='a  Stokes V [Ic]  9:24:27 UT  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='b  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' As in Fig 5, but for a magnetic loop outlined by  the solid box in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The full Stokes vectors of the two  outlined pixels are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  one polarization parameter is a remarkable feat for the  GREGOR/GRIS-IFU, however linear and circular po-  larization is only confidently detected simultaneously in  16 − 18%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The increase in δIrms could be the result of  having achieved closer to diffraction limited resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Indeed, the exceptional seeing conditions during scans D  and E could be responsible for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' However, it is cer-  tainly the case that these observations have not achieved  the diffraction limited spatial resolution that GREGOR  is capable of, because the spatial sampling of the GRIS-  IFU in the y−direction, even in double sampling mode,  is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The increase in δIrms could also be the  result of a small reduction in stray light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Another possi-  bility for the large polarization fractions recorded is that  we were lucky with the target, and these explanations  are not mutually exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Lites et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2008) report a mean vertical apparent  flux density of 11 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' This value, determined from Hin-  ode/SP observations of the Fe I 6301/2 ˚A line pair, was  found assuming αm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' However, the authors them-  selves recommend that this value should be reduced by  30% to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7 G, before comparison with other works, as  the assumption that αm = 1 is almost certainly un-  true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The very large fraction of the FOV in scans D and  E which have a CP signal makes a comparison worth-  while.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The corresponding average value from this work  is the median αmB⊥, listed in Table 3 and  4 as be-  tween 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='8 G and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' As their mean is calculated over  the full FOV of the Hinode/SP scan, and their FOV is  much larger and includes network patches, it is sensible  that the value for the GREGOR/GRIS-IFU is smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Further as the NIR Fe I line pair is more magnetically  sensitive than the visible line pair, it is expected that we  record a smaller average value in our maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' However,  the value is a function of the effective spatial resolu-  tion and S/N - indeed, in deep mode observations Lites  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2008) return a distribution for the vertical appar-  ent longitudinal flux density that peaks at a much lower  value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Our value therefore rests comfortably  between these two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Aside from a comparison, it  must be emphasised that all of these values could be  under-estimates if there remains a substantial amount  of unresolved mixed-polarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  The much larger number of pixels which have a LP  signal in scans D and E, relative to 2019, allowed us  to infer the inclination angle in a larger fraction of the  FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Using SIR, we were then able to determine that a  large majority (> 60%) of the pixels which had at least  one Stokes parameter with a maximum signal greater  than 5σn across the 15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 ˚A line were classifiable as ei-  ther highly inclined or intermediately inclined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' In other  words, a majority of magnetised pixels displayed a sig-  nificant transverse magnetic component of the magnetic  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' This is because the number of intermediately in-  clined fields outnumbers both the highly inclined and  highly vertical populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The ratio B⊥/B∥ is larger  than 2019, which indicates that as the S/N increased,  and thus we were able to access weaker fields, the mag-  netic field is revealed as more transverse on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Determining this ratio from observations is important  because it is useful for optimisation of the initial and  boundary conditions of radiative magnetohydrodynamic  simulations (Steiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The question which  naturally follows is whether this is an uncommon quirk  of the specific observational target in scans D and E, or  whether this is a glimpse of what we can expect to see  when larger aperture telescopes like the Daniel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' In-  ouye Solar Telescope (DKIST: Rimmele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2020))  and European Solar Telescope (EST: Quintero Noda  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2022)) take similar observations in the NIR lines  with significantly higher effective spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Im-  portantly, this result was obtained even with a very con-  servative approach to noise tolerance that favours longi-  tudinal inclinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' In scenario 2 (and 3), we set Stokes  profiles which did not satisfy the 5σn threshold to zero,  which means in many pixels the γ is likely to be either  0◦, 90◦, or 180◦, and this is reflected in the large peaks at  these values in the γ distributions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Of course,  Stokes Q and U are set to zero in a much larger number  of pixels than Stokes V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' While our approach favours  longitudinal inclinations, that does not mean our deter-  mination of B⊥/B∥ is biased in favour of B∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' It must  be stressed that the opposite is true - our determina-  Exploring magnetic loops and serpentine fields in the quiet Sun with the GRIS-IFU  13  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0  Stokes I/Ic  SIR  Observed profile  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='006  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  Stokes Q/Ic  15647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15659.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  wavelength [Å]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='004  Stokes U/Ic  15647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='3  wavelength [Å]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='006  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='006  Stokes V/Ic  VECTOR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='a - B = 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2 [G], α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4, γ = 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='1 [ ∘ ], ϕ = 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 [ ∘ ], vLOS = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0 [km/s]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='0  Stokes I/Ic  SIR  Observed profile  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='008  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='004  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='004  Stokes Q/Ic  15647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='3  wavelength [Å]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='015  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='010  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='010  Stokes U/Ic  15647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='4  15655.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='4  15663.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3  wavelength [Å]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='010  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='010  Stokes V/Ic  VECTOR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='b - B = 310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='6 [G], α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5, γ = 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2 [ ∘ ], ϕ = -33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='8 [ ∘ ], vLOS = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='3 [km/s]  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 6, but for vectors 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='a and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The spatio-temporal locations of the pixels are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  15647.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='32  15648.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='52  15649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='71  wavelength [Å]  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='010  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='015  Stokes V [Ic]  2Δλ  Observed profile  SIR  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Sample Stokes V profile for a pixel located in  the strong magnetic element in case study 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The dotted,  blue lines show the wavelength positions of the blue and red  lobes for the most magnetically sensitive line, from which  the strong field approximation is used to estimate B∥ = 812  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The synthetic profile from SIR is also shown, which has  model parameters of B = 854 G, αm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='14, γ = 0◦, vLOS =  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='2 km/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  tion of B⊥/B∥ is biased in favour of horizontal fields  because B⊥ is determined over a much smaller popula-  tion of pixels with significantly stronger fields (Steiner  & Rezaei 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  To determine an unbiased value for  B⊥/B∥ it may be necessary to approach this problem  by applying a threshold to each Stokes parameter sep-  arately whose amplitude is determined by the magnetic  field strength, rather than applying the same threshold  to Stokes Q, U, and V as determined purely noise (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  by the amplitude of the 5σn threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  The magnetic loops shown in both case studies are  clear demonstrations of how the photospheric magnetic  field is organized in the QS IN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Of particular signifi-  cance is the apparently serpentine nature of this mag-  netism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  In terms of magnetic loops in the IN, there  are a large number of statistical studies that report on  the phenomenon, especially from the Imaging Magne-  tograph (IMaX)/Sunrise experiment (Danilovic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Mart´ınez Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2012) and the SST (Goˇsi´c  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Ledvina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Most of these  studies focus on analysing bi-polar patches of circular  polarization where cancellation could take place and  where horizontal fields are expected to be found along  the PIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' For instance, the SST observations by Ledvina  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2022) with the Fe I 5576 ˚A and 6301 ˚A lines, and  with a FOV 30 times greater by area than the GRIS-  IFU scans in this paper, found 38 magnetic loops in a  14  Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5 minute time series, but find no evidence of hori-  zontal fields at the PIL in any of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  In terms of  serpentine magnetism, Harra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2010) reported on  observations of a serpentine magnetic field between two  large bi-polar patches of magnetic flux in an emerging  flux region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  However, their target was not an IN re-  gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' We have revealed a serpentine structure in the QS  IN for the first time with unambiguous linear polariza-  tion along the PIL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' However, there is still a clear need  for higher resolution observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' If one examines the  circular polarization at the PIL, the mixing of opposite  polarity signals within the spatio-temporal resolution el-  ement means full-vector spectropolarimetry remains elu-  sive (see vector 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='b in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Nevertheless, full-vector  spectropolarimetry is achieved just one pixel adjacent in  both case scenarios examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Further there is a need  for multi-wavelength facilities with spectral diagnostics  sensitive to the upper photosphere and lower chromo-  sphere, such as will be available at the DKIST (Rast  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2021), the EST (Quintero Noda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2022), and  also the Sunrise III balloon experiment when launched,  which will allow us to assess whether these small-scale  cancellation sites, which could be pervasive across the  quiet solar surface, are capable of contributing to the  heating of the lower chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  APPENDIX  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' SIR EXPLORER  As spectropolarimetric solar datasets become ever larger and more complex, the multi-dimensional data cubes  produced from observatories become increasingly non-trivial to analyse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Inversions of this multi-dimensional data  increase in complexity too as the number of spectral lines that are inverted continues to increase irrespective of the  data volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' To meet the challenge of inverting such large data cubes, the parallelized Python wrapper (Gafeira  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' 2021) to both the SIR (Ruiz Cobo & del Toro Iniesta 1992) and the Departure coefficient aided Stokes Inversion  based on Response functions (DeSIRe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Ruiz Cobo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' (2022)) codes enables users to easily spread the computational  problem across many CPU nodes on high performance computing facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' However, even when the data is inverted,  the data products still need to be analysed and browsing the data products of inversions, of which there are many,  becomes increasingly cumbersome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' SIR Explorer (SIRE), is a Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='9 graphical user interface (GUI) application  that aims to make the analysis and exploration of inversion inputs and outputs associated with SIR/DeSIRe a little  faster and a little easier (Campbell 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Figure 11 shows the S2 inversions of scan D loaded into SIRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The user interface (UI) of SIRE is split into three  main regions:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The control panel, located on the left of the main window, which allows the user to load datasets,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The three canvases, upon which datasets are displayed, and  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The widget bar, underneath the canvases, which has three sliders to control the frame (FR), wavelength (WL),  and optical depth (OD) indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  SIRE is designed to pick up where the parallelized Python wrapper to SIR/DeSIRe leaves off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' SIRE’s required input  file structure, detailed in Table 7, is therefore dictated by the output file structure of the wrapper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' There are three  mandatory files that are always required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The first mandatory file is the primary models output by SIR/DeSIRe,  which are an array of the one-dimensional atmospheric files with a user-defined number of optical depth points, n(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  The other two mandatory files are the observed profiles, which are the input Stokes vectors provided to SIR/DeSIRe,  and the synthetic profiles, which are the final output Stokes vectors produced by SIR/DeSIRe when solving radiative  transfer, which must both have the same number of wavelength points, n(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  The observed profiles do not need  to be real observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' If inverting synthetic vectors produced from simulation snapshots, these may synthetic in  origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' There are a number of optional files that can be provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' These include: secondary models, if SIR/DeSIRe is  employed with two models, primary (and secondary) macroturbulence files which contains the macroturbulence of the  model and also the α and stray light fraction values, and binary map(s), which are a user-defined array that may be  provided to remove selected pixels from the maps of the magnetic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  There are three canvas objects: the maps where images of the Stokes profiles and model parameters may be displayed,  the plots on the upper right of the interface of the observed and synthetic Stokes profiles for the selected pixel, and  the plots on the bottom right of the interface of the model parameters as a function of optical depth for the selected  Exploring magnetic loops and serpentine fields in the quiet Sun with the GRIS-IFU  15  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Main window of SIRE with scan D loaded into the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  SIRE’s input files and the required array structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  The ⋆ symbol indicates that the file is mandatory in all  circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The ⋄ symbol indicates that the file is mandatory when two models are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Input file  Array shape (t = 1)  Array shape (t > 1)  Primary models ⋆ ⋄  [11, n(τ), y, x]  [t, 11, n(τ), y, x]  Observed profiles ⋆ ⋄  [4, n(λ), y, x]  [t, 4, n(λ), y, x]  Synthetic profiles ⋆ ⋄  [4, n(λ), y, x]  [t, 4, n(λ), y, x]  Secondary models ⋄  [11, n(τ), y, x]  [t, 11, n(τ), y, x]  Primary macroturbulence files ⋄  [3, y, x]  [t, 3, y, x]  Secondary macroturbulence files  [3, y, x]  [t, 3, y, x]  Binary map(s)  [y, x]  [t, y, x]  pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Each of the canvases may be resized in proportion to one another, and each of them may be collapsed entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  The control panel provides the user with the ability to determine exactly which parameters should be shown in each  canvas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' For instance, in Fig 11 only maps of Stokes V , αmB, vLOS, and γ are selected and shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  The most important functionality of SIRE concerns the way in which the user is able to navigate the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The  controls have been designed to be as simple and fast as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' When a dataset is first loaded, the pixel with zeroth  co-ordinates in each dimension (t = 0, n(λ) = 0, y = 0, x = 0, n(τ) = 0) will be plot by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The user can left-click  any of the maps and the synthetic Stokes vector, observed Stokes vector, and model parameters for the corresponding  pixel co-ordinates will be plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' In addition, the user may adjust the selected pixel by using key-pressing the ‘UP’,  ‘DOWN’, ‘LEFT’, and ‘RIGHT’ arrow keys on their keyboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The location of the selected pixel is denoted by vertical  and horizontal lines on the maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The sliders in the widget bar allow the user to adjust the frame (FR), wavelength  (WL), and optical depth (OD) indices at which the data is retrieved from the appropriate files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The user can left-click  and drag the sliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The code that updates the maps and plots will not be executed until the slider is released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The  Files|  Maps  Information  Stokes V[lc]  Stokes / [lc]  Update profiles axes  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='  user can also key-press ‘Q’ and ‘E’ to decrease or increase, respectively, the frame index by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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+page_content='  The authors would like to thank the anonymous referee whose feedback helped to significantly improve this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  We express our appreciation also to Carlos Dominguez-Tagle, whose work with the GRIS-IFU made these observations  possible, and to all the engineering, operating, and technical staff at GREGOR for their assistance during the observing  campaign, including Miguel Esteves Perez, Saida Milena D´ıaz Castillo, Karin Gerber, and Oliver Wiloth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Gratitude  is extended to Juan Manuel Borrero and Lucia Kleint for their advice and insightful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' RJC thanks Robert  Ryans for IT support and assistance in utilizing QUB’s high performance computing (HPC) facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' We thank Carsten  Denker and Christoph Kuckein for assistance with operating the High-resolution Fast Imager (HiFI) instrument and  associated data reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  This research has received financial support from the European Union’s Horizon 2020  research and innovation program under grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  824135 (SOLARNET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' RJC acknowledges support  from the Northern Ireland Department for the Economy (DfE) for the award of a PhD studentship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' RJC and MM  acknowledge support from the Science and Technology Facilities Council (STFC) under grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' ST/P000304/1 &  ST/T00021X/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='5−meter GREGOR solar telescope was built by a German consortium under the leadership  of the Leibniz-Institute for Solar Physics (KIS) in Freiburg with the Leibniz Institute for Astrophysics Potsdam,  the Institute for Astrophysics G¨ottingen, and the Max Planck Institute for Solar System Research in G¨ottingen as  partners, and with contributions by the Instituto de Astrof´ısica de Canarias and the Astronomical Institute of the  Academy of Sciences of the Czech Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' The redesign of the GREGOR AO and instrument distribution optics  was carried out by KIS whose technical staff is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' Helioseismic and Magnetic Imager (HMI)  magnetograms, courtesy of NASA/SDO and the AIA, EVE, and HMI science teams, were used during observations  for target selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content=' This work was supported by Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) through the research  grants UIDB/04434/2020 and UIDP/04434/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
+page_content='  Facilities: GREGOR solar telescope (Schmidt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TdE5T4oBgHgl3EQfaw-V/content/2301.05591v1.pdf'}
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diff --git a/TtE5T4oBgHgl3EQfAg5m/content/tmp_files/2301.05379v1.pdf.txt b/TtE5T4oBgHgl3EQfAg5m/content/tmp_files/2301.05379v1.pdf.txt
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+Benefits and Limitations of Remote Work to
+LGBTQIA+ Software Professionals
+Ronnie E. de Souza Santos
+Cape Breton University
+Sydney, NS, Canada
+Email: ronnie desouza@cbu.ca
+Cleyton V. C. de Magalh˜aes
+CESAR School
+Recife, Brazil
+Email: cvcm@cesar.school
+Paul Ralph
+Dalhousie University
+Halifax, Canada
+Email: paulralph@dal.ca
+Abstract—Background. The mass transition to remote work
+amid the COVID-19 pandemic profoundly affected software
+professionals, who abruptly shifted into ostensibly temporary
+home offices. The effects of this transition on these professionals
+are complex, depending on the particularities of the context and
+individuals. Recent studies advocate for remote structures to
+create opportunities for many equity-deserving groups; however,
+remote work can also be challenging for some individuals, such
+as women and individuals with disabilities. As the discussions on
+equity, diversity, and inclusion increase in software engineering, it
+is important to explore the realities and perspectives of different
+equity-deserving groups to develop strategies that can support
+them post-pandemic. Objective. This study aims to investigate the
+effects of remote work on LGBTQIA+ software professionals.
+Method. A grounded theory approach was applied, based on
+information collected from two main sources: a survey question-
+naire with a sample of 57 LGBTQIA+ software professionals and
+nine follow-up interviews with individuals from this sample. This
+sample included professionals of different genders, ethnicities,
+sexual orientations, and levels of experience. Consistent with
+grounded theory methodology, the process of data collection
+and analysis was conducted iteratively using three stages of
+coding: line-by-line, focused, and theoretical. Member checking
+was used to validate the findings obtained from interpreting the
+experiences commented on by LGBTQIA+ software profession-
+als. Findings. Our findings demonstrate that (1) remote work
+benefits LGBTQIA+ people by increasing security and visibil-
+ity; (2) remote work harms LGBTQIA+ software professionals
+through isolation and invisibility; (3) the benefits outweigh the
+drawbacks; (4) the drawbacks can be mitigated by supportive
+measures developed by software companies. Conclusion. This
+paper investigated how remote work can affect LGBTQIA+
+software professionals and presented a set of recommendations on
+how software companies can address the benefits and limitations
+associated with this work model. In summary, we concluded
+that remote work has a crucial role in increasing diversity and
+inclusion in the software industry.
+Abstract—General Abstract. Remote work is here to stay. There
+is no denying it, as some software professionals would rather
+quit their jobs than return to the office full-time. Therefore,
+software companies want to understand how the remote working
+model can be successfully used without causing major issues.
+The problem is that the effects of remote work are complex
+because they depend on individual and group characteristics that
+require careful evaluation. In this scenario, one thing has been
+extremely positive: remote work is helping to increase diversity in
+software engineering by fostering new opportunities and better
+work conditions for individuals from equity-deserving groups,
+for instance, LGBTQIA+ software professionals. The software
+industry is overly homogeneous, most of the professionals who
+work in this area are heterosexual men (a reflection of the uni-
+versity courses on computer science and software engineering),
+but diversity can only be good for an area that strongly depends
+on creativity and innovation. What better way to innovate than
+putting several individuals from different backgrounds and with
+various experiences to work together? Remote work plays an
+important role in improving equity, diversity, and inclusion in
+the software industry. In this paper, we discuss how remote
+work is affecting software professionals from the LGBTQIA+
+community and provide a list of recommendations to support
+software companies in dealing with this work model.
+Index Terms—EDI, equity, diversity, inclusion, software pro-
+fessionals, LGBTQIA+.
+I. INTRODUCTION
+Equity, diversity, and inclusion (EDI) are central topics
+being discussed in society nowadays. EDI is a complex
+phenomenon centered on developing approaches for providing
+equal opportunity for individuals (equity) while recognizing
+their personal, social, and cultural differences (diversity) and
+encouraging them to participate in environments and debates
+(inclusion) [1], [2]. Discussions on EDI are gradually increas-
+ing in Computer Science as well [3]. Since technology has
+a direct impact on our society, and our society is plural, the
+lack of diversity among those who develop software creates
+limitations in the use of technologies as software solutions
+might exclude groups of individuals [4]–[6].
+Diversity is essential in developing technologies to sup-
+port our multifaceted society (Albusays, 2021). However,
+the software industry is experiencing a diversity crisis, as
+software teams are mainly composed of heterosexual men [4].
+Currently, there is a lack of knowledge on how EDI should
+be addressed and integrated into software engineering [6]–[8].
+The main consensus among researchers is that diversity needs
+to be addressed from the academy to the software industry
+by increasing opportunities, improving debates, and fostering
+safe spaces for equity-deserving groups.
+Recently, the COVID-19 pandemic intensified discussions
+about EDI in software engineering. As the lockdowns forced
+software professionals to work from home, remote work struc-
+tures created better work-life balance, more flexibility, and
+less commuting, which resulted in new opportunities for many
+underrepresented groups in the software industry: caregivers
+(especially mothers), LGBTQIA+ individuals (especially trans
+people), and people with disabilities [4], [9], [10]. Now,
+arXiv:2301.05379v1  [cs.SE]  13 Jan 2023
+
+as the pandemic seems to be under control, many software
+companies are permanently transitioning to remote and hybrid
+work allowing software professionals to decide where they will
+work, i.e., in the office or any other place [11], [12]. Thus, the
+post-pandemic scenario tends to keep increasing diversity and
+inclusion in the software industry as more job opportunities
+will be available for professionals that cannot afford to work
+in an office every day.
+Since remote work fosters new job opportunities, we need
+to understand the impacts of remote work arrangements on
+individuals from equity-deserving groups because (1) EDI
+groups experience remote work differently, as reported in
+previous studies [9], [13], and (2) EDI is essential for having
+a more just and inclusive society. In the context of software
+engineering, understanding the impacts of remote work on
+software professionals from EDI groups is an important step
+toward sharpening practices, strategies, and tools required
+to improve equity, diversity, and inclusion in the software
+industry. In this study, we explore the impacts of remote work
+on LGBTQIA+ software professionals aiming to answer the
+following research question:
+Research
+Question: How does remote work affect
+LGBTQIA+ software professionals?
+In particular, we explore the experience of software profes-
+sionals from the LGBTQIA+ community because the research
+on EDI in software engineering that focuses on LGBTQIA+
+software professionals is very scarce [3], [5]. From this
+introduction, the rest of this study is organized as follows.
+In Section II, we discuss existing studies on EDI in software
+engineering. In Section III, we present the method applied in
+this study. In Section IV, we present our findings which are
+discussed in Section V. Finally, we summarize the contribu-
+tions of this study and opportunities for future research.
+II. BACKGROUND
+Remote work has received attention from researchers from
+several areas since this is expected to be the norm in many
+industries across the globe [14]. In software engineering,
+studies are being developed on remote work for at least
+two reasons. First, software professionals are more likely to
+continue working remotely since they are already used to
+this type of work structure (e.g., global software development
+and distributed teams) [15]. Second, software professionals
+are responsible for creating and adapting technologies to
+support employees from other industries, which might also
+face adaptations in the post-pandemic [16].
+Considering the first reason, as software engineering follows
+the world’s transformations and their impacts on society,
+understanding the long-term implications of the post-pandemic
+in the software industry and practitioners is crucial. Over
+the past two years, studies revealed that remote work can
+produce several benefits and positively affect those who work
+in software development. However, some limitations have also
+been observed in the same period [17]. Remote work was
+demonstrated to benefit software professionals by providing
+them with more flexibility, better work-life balance, and in-
+creased satisfaction [9], [10]. On the other hand, remote work
+can foster problems at the team level, creating communication
+issues, increasing coordination challenges, and reducing team
+cohesion [18], [19].
+The main challenge associated with remote work is the
+fact that its effects are intimately dependent on individual
+characteristics and in the particularities of some groups of
+professionals [9]. For instance, remote work can be worse
+for women and caregivers (e.g., parents), as they might have
+to deal with many distractions while performing software
+development activities that demand problem-solving skills
+and high levels of concentration [9]. Remote work might
+promote more opportunities for people with disabilities [20];
+however, it can also force this group of individuals to deal
+with poor ergonomics [9]. Remote structures allow transgender
+software developers to have control over their identities [13];
+however, in other fields, researchers are concerned that remote
+interactions can undermine diversity [21].
+III. METHOD
+Aiming to address the above-cited problem and the lack of
+discussions on the impacts of remote work in some equity-
+deserving groups, we developed a qualitative study using con-
+structivist grounded theory [22]. Grounded Theory is a family
+of methods for inductively generating theories based on the
+data collected from real-life contexts to identify and describe
+concepts, behaviors, and experiences [23]. It is widely applied
+in social sciences and is well-suited for exploring social,
+cultural, and human aspects of software development [24].
+Grounded theory is appropriate for this study because it
+helped us understand the research problem using the experi-
+ence of a small group of individuals [25]. in this process, due
+to the characteristics of our targeted population, we could not
+expect reaching a significant number of professionals to partic-
+ipate in the study. Therefore, the grounded theory effectively
+supported us in constructing concepts that can be transferred
+to other contexts through theoretical generalization [26].
+Thus, following the guidelines to conduct grounded theory,
+we used well-defined rounds of data collection and analysis,
+including inductive coding, memoing, constant comparison,
+and theoretical sampling. This process allowed us to identify
+concepts emerging from the field rather than fitting data into
+preconceived theories [23].
+A. Participants
+The participants of this study are individuals from a very
+specific population: LGBTQIA+ software professionals, e.g.,
+developers, QAs, analysts, designers, and managers, among
+others directly participating in the software development pro-
+cess. These professionals belong to a hidden population—
+a population that cannot be easily defined or enumerated
+based on existing knowledge—which makes selecting indi-
+viduals to participate in studies a challenging activity [27].
+In this study, we followed guidelines that suggest treating
+software professionals as a hidden population [28]. Moreover,
+2
+
+individuals from the LGBTQIA+ community are commonly
+treated as a hidden population in studies from other fields, in
+particular, because many of its members are not comfortable
+with discussing aspects of their sexuality due to the risk of
+being exposed to structural and social discrimination [29].
+B. Instrumentation
+We applied two approaches to data collection, aiming to
+reach as many individuals as possible and carefully access the
+experience of LGBTQIA+ professionals without having them
+reveal their identities. The approaches applied were a survey
+questionnaire and semi-structured interviews. Due to the hid-
+den nature of the population targeted in this study, we could
+not expect a statistically significant number of respondents
+in a survey (e.g., 1000 LGBTQIA+ software professionals)
+at the same time that a significant number of volunteers for
+interviews was also challenging to reach. Therefore, we used
+the best data collection methods (survey and interviews) to
+obtain data and the best of the grounded theory method to
+work with a small sample of participants.
+We designed the questionnaire so that we could anony-
+mously obtain basic information about the participants. The
+questionnaire was mostly open-ended as we were targeting
+to collect qualitative data about the professionals’ experience.
+We started by asking individuals to provide their age, gender,
+sexual orientation, ethnicity, and country. Then, we added
+questions about their work and their company, including their
+level of education, level of experience working in the software
+industry, role (e.g., developer, tester, designer, manager, etc.),
+and the number of employees working in their company. Fi-
+nally, we asked the following questions about their experience
+working remotely:
+• Would
+you
+say
+that
+remote/hybrid
+work
+benefits
+LGBTQIA+ individuals working in the software indus-
+try? Please, describe in what ways.
+• Would you say that remote/hybrid work creates chal-
+lenges for LGBTQIA+ individuals in the software indus-
+try? In what ways?
+• Do you know any software professionals from the
+LGBTQIA+ community that benefited from working
+from home on a regular basis? Tell us a bit about this.
+• Do you know any professional from the LGBTQIA+
+community that faced disadvantages caused by working
+from home on a regular basis? Tell us a bit about this.
+• Is remote work more inclusive for LGBTQIA+ software
+professionals? In which ways?
+• Does
+remote/work
+create
+more
+opportunities
+for
+LGBTQIA+ software professionals? How?
+All questions were optional, so individuals would be free to
+answer only the ones that they wanted to answer. At the end
+of the questionnaire, we asked those who would be interested
+in discussing more aspects of remote work and how it affects
+the software professionals from the LGBTQIA+ community to
+provide their email to be contacted for a 30-minute interview.
+The questionnaire was developed in two languages English
+and Portuguese.
+C. Data Collection
+We started the data collection with the questionnaire, which
+was applied using the following sampling approaches [28]:
+• Convenience sampling: We invited participants based on
+their availability. The first author was a member of the
+target population (i.e., an LGBTQIA+ professional who
+worked with software development for six years) and
+invited members of the population from his contacts list.
+• Purposive sampling: We used the communication chan-
+nels of a large software company in South America to
+advertise our questionnaire to over 1,000 software pro-
+fessionals and expected that individuals from the targeted
+population would participate in the study. This company
+was founded in 1996, specialized in on-demand software
+development, and creates technologies for clients from
+various sectors, including finance, telecommunication,
+manufacturing, and services. In addition to this company,
+we advertised the questionnaire in online communities
+from social media websites.
+• Snowballing sampling: We requested that individuals
+from the convenience and the purposive sampling invite
+other LGBTQIA+ software professionals to participate in
+the study.
+We started with the questionnaire because individuals from
+hidden populations usually avoid outsiders [30]. However,
+members of such populations often know each other [28].
+Therefore, by applying the questionnaire first, we were able to
+apply sampling techniques commonly used in surveys. Once
+we reached the population, we could identify individuals to be
+interviewed, providing us with additional details beyond those
+collected with the open-ended questions in the survey.
+Data collection using the questionnaire happened between
+June 10th and July 20th, 2022. Following the nature of the
+grounded theory method and the approach to collecting and
+analyzing data in rounds, we considered the application of the
+questionnaire as the first round of the study. Next, we invited
+the participants who volunteered for interviews and conducted
+three rounds of interviews. These interviews were conducted
+online with nine participants (three per round) using Google
+Meet. In these three rounds, we focused on obtaining more
+details about the answers already provided in the question-
+naire. In addition, as the rounds evolved, participants were
+asked about the role of diversity on software teams and the
+effects of remote work on software teams. The nine interviews
+ranged from 20 to 43 minutes. This produced four hours and
+18 minutes of audio and 97 pages of transcripts.
+D. Data Analysis
+We started data analysis by applying descriptive statistics
+[31] to summarize the information about our sample of
+participants. Descriptive statistics allowed us to present the
+distribution of participants’ answers regarding their personal
+and professional profiles.
+Following this, we analyzed the answers to open-ended
+questions in the questionnaire (first round). Then, we contin-
+ued using qualitative data analysis in the interviews (following
+3
+
+three rounds). Qualitative analysis was conducted based on
+three coding stages: line-by-line coding was focused on iden-
+tifying initial codes and building concepts within our data;
+focused coding supported the establishment of connection
+among the concepts and the definition of high-level categories
+of concepts; theoretical coding was used to emphasize the core
+category and explain the main story arising from the data [23],
+[25].
+In this process, we began with line-by-line open coding,
+which allowed us to identify and refine emerging concepts
+from the answers to the open-ended questions in the survey
+[23]. We repeated this strategy in the first round of interviews
+through a systematic process supported by audio recordings,
+which helped identify the properties of codes and their mean-
+ing. Following the second round of interviews, we started cat-
+egorizing these codes and establishing connections using fo-
+cused coding. Focused coding was performed through repeated
+analyses and comparisons among emerging categories [25].
+This process continued during the third and final round of
+interviews. When these transcripts were integrated into the
+analysis, we used theoretical coding [25] to rearrange the
+categories and highlight a central story about the experience
+of our participants.
+E. Ethics
+Before answering questions about their background and
+their experience with remote work in the questionnaire, each
+professional read about the goal of the research and the
+study’s relevance to the software industry. Also, they needed
+to explicitly agree to provide data to the study by selecting
+the option at the beginning of the questionnaire. Following
+this, volunteers agreed to be interviewed by providing their
+email address and then again, orally, at the beginning of their
+interview. Before each interview, the interviewer explained
+the goal of the research one more time and the need to
+collect more data in addition to the questionnaire. Participants
+were guaranteed data confidentiality and de-identification of
+their quotes. They were also informed about the voluntary
+nature of their participation and the right to stop the interview
+and withdraw from the research at any time. No participants
+withdrew. A university research ethics board approved this
+research.
+IV. FINDINGS
+This section presents the demographics of our sample,
+followed by the categories that emerged from the analysis,
+which are the benefits and limitations of remote work for
+LGBTQIA+ software professionals. Our findings are illus-
+trated with quotations extracted from the questionnaire and the
+interviews. Some quotations have been translated into English
+and may read awkwardly as we have transcribed and translated
+them as accurately as possible.
+A. Demographics
+Our survey reached 81 individuals. However, some partici-
+pants were excluded from the survey for not being part of the
+targeted population:
+• one woman and one man identified themselves as het-
+erosexual, despite stating that they were software profes-
+sionals from the LGBTQIA+ community in the screening
+questions.
+• 22 individuals only completed the screening questions;
+they answered neither demographic nor open-ended ques-
+tions.
+However, our sample of 57 individuals varies as not all
+participants answered the entire questionnaire. For instance:
+• 57 (100%) individuals provided their level of education.
+• 56 (98%) individuals provided their role in the software
+team.
+• 53 (93%) individuals provided their age, gender, and
+ethnicity.
+• 52 (91%) individuals provided their country and their
+level of experience.
+• 45 (79%) individuals provided information about the
+number of employees in their companies.
+• 25 (44%) individuals answered the question about their
+sexual orientation.
+• 13 (23%) individuals answered at least one question
+about how remote work affects LGBTQIA+ software
+professionals.
+• 10 (17%) individuals provided us with an email for an
+interview.
+Participants’ ages varied from 21 to 50 years old. Partic-
+ipants had an average of 5.7 years of work experience in
+software development; the most experienced professional in
+the sample has worked in the software industry for 25 years,
+and the least for one year. Table I presents more details
+about our sample and its variations. Finally, we were able
+to interview 16% of the survey (9/57), who provided more
+details on the topics addressed in the research.
+B. Benefits of Remote Work to LGBTQIA+ Software Profes-
+sionals
+Participants pointed out benefits that were observed in the
+general professional context (e.g., the increase in job oppor-
+tunities for the community) as well as benefits observed at
+the individual and team level when professionals are working
+in the software industry. Table II summarizes the benefits
+identified in this study and the evidence obtained from par-
+ticipants’ experiences. These benefits are Job Opportunities,
+Engagement, Identity Disclosure Control, Physical Safety,
+and Toxicity Avoidance. All of these benefits are extremely
+valuable for LGBTQIA+ software professionals because the
+software industry is still an area dominated by heterosexual
+men.
+Remote work increases access to job opportunities in the
+software industry, and this benefits LGBTQIA+ individuals
+greatly. In general, this could be seen as a simple cor-
+relation; that is, more remote jobs available means more
+jobs available for LGBTQIA+ professionals. However, for
+LGBTQIA+ individuals, this aspect goes beyond the increase
+in the number of positions. These are professionals that have
+4
+
+TABLE I
+DEMOGRAPHICS
+Gender
+Men
+34
+Women
+14
+Non-binary
+3
+Not informed
+4
+Ethnicity
+White
+34
+Mixed ethnic
+13
+Black
+2
+Arab
+2
+Asian
+1
+Latino
+1
+Not informed
+4
+Sexual Orientation
+Gay
+13
+Bisexual
+7
+Lesbian
+3
+Pansexual
+1
+Asexual
+1
+Not informed
+32
+Education
+High-School
+8
+Bachelor
+30
+Post-baccalaureate
+12
+Master
+6
+Ph.D.
+1
+Roles and Activitiesa
+Testing
+28
+Programming
+25
+Requirements
+17
+Design
+14
+Architecture
+10
+Management
+3
+Not informed
+1
+Location
+Brazil
+44
+Canada
+2
+Portugal
+2
+Sweden
+2
+Netherlands
+1
+US
+1
+Not informed
+5
+Company Size
+0 - 9 employees
+1
+10 - 99 employees
+3
+100 - 999 employees
+5
+1,000 - 9,999 employees
+35
+More than 10,000 employees
+1
+Not informed
+12
+Notes: asome professionals reported performing multiple activities or having
+more than one roles in the team
+suffered discrimination for years. Our participants explained
+that online recruitment tends to be slightly less biased by
+physical characteristics. Moreover, LGBTQIA+ professionals
+can apply for jobs that before were primarily available to those
+living in major urban centers. This means that they can choose
+to work where they always lived instead of adapting to new
+places or having to move away to unfriendly locales.
+For many LGBTQIA+ individuals, disclosing their identity
+is never easy, especially when entering a new environment.
+Trans people and non-binary are apprehensive about precon-
+ceived judgments about their physical appearances. Gender-
+affirming care (e.g., medical treatments and therapy) is an
+additional concern for software professionals who are trans-
+gender. Other individuals from the community who do not
+fit into the gender binary (e.g., male or female stereotype)
+suffer from fear and uncertainty in relation to how they will
+be seen and treated by co-workers. Remote environments
+established a place where LGBTQIA+ software professionals
+feel comfortable as they can control their identity disclosure,
+for instance, by deciding when they want to use their cameras.
+Remote work structure allows LGBTQIA+ software pro-
+fessionals to regulate the level of interaction and contact
+with their teammates to the point that they feel comfortable
+around them. Our participants revealed that their engagement
+with other team members increased while working remotely
+because they could gradually get used to their co-workers
+and engage accordingly. Joining a software team before (i.e.,
+in person) was challenging for LGBTQIA+ professionals
+because they were worried about acceptance in the software
+industry work environment since it is essentially heterosexual.
+In contrast, in remote environments, they will first become
+comfortable with their team, then engage, and once they feel
+included by the team, they can finally interact in person. This is
+a know me before you judge me effect that cannot be afforded
+in primarily in-person environments.
+Many LGBTQIA+ individuals still feel unsafe and afraid
+of physical attacks. This reality is not different from those
+working in the software industry. Transgender professionals
+who participated in this study reported the risks that they are
+exposed to every day, in particular, those who live in what,
+according to them, is the country where more trans women are
+killed in the world. Working from home reduces the exposure
+of these professionals to unsafe environments, such as public
+transportation during peak hours in countries that are more
+dangerous to them. By not having to go to the office daily,
+these professionals can use safer alternative commuting, such
+as taxis and private rides, whenever they need to be in the
+office.
+Office environments can be extremely uncomfortable for
+LGBTQIA+ professionals who are exposed to coworkers’
+negative comments, attitudes, and actions. These toxic behav-
+iors vary from the act of being constantly observed to being
+a target for constant commentaries in the office. Although
+this is not an act of physical violence, such conduct affects
+individuals’ well-being and their sense of belonging in relation
+to the company. Hybrid work allows LGBTQIA+ software
+professionals to decide the best moment to be in the office
+and avoid toxicity from others when necessary.
+The
+above-cited
+benefits
+are
+extremely
+valuable
+for
+LGBTQIA+ software professionals because the software in-
+dustry is still an area dominated by heterosexual males, which
+ends up undermining the inclusion of other groups of individ-
+uals, even indirectly. Remote work and its benefits, therefore,
+create more opportunities for diversity in software engineering
+while also helping software teams to be more inclusive.
+C. Limitations of Remote Work to LGBTQIA+ Software Pro-
+fessionals
+Although remote work produced benefits that improve the
+experience of LGBTQIA+ individuals working in the software
+industry, there are also observable limitations reported by our
+5
+
+TABLE II
+BENEFITS OF REMOTE WORK FOR LGBTQIA+ SOFTWARE PROFESSIONALS
+Construct
+Definition
+Example Evidence
+Job
+Opportunities
+The availability of
+employment that
+LGBTQIA+ software
+professionals can easily
+access.
+“it gives professionals the chance to work in more inclusive companies that are distant from their
+cities.” (P10)
+“my girlfriend is from another state, and now I am living with her and working remotely.” (P15)
+“software companies are in need of talented professionals and this [remote work] can be a good opportunity
+for people from the LGBTQIA+ community.” (P16)
+Engagement
+The level of interaction
+of LGBTQIA+ software
+professionals with their
+teammates.
+“it increased the engagement of those who were afraid of participating before [in-person]” (P12)
+“remote work allowed me to engage with new people, cultures, experiences and different realities [in the
+team].” (P16)
+Identity
+Disclosure
+Control
+The individual decision
+of concealing or
+communicating and
+expressing their gender
+and sexual orientation.
+“virtual tools allow people to communicate without having to expose themselves either for fear or as an
+option.” (P12)
+“I can choose how I show up on camera, to exclude gender identifiers that make me dysphoric or
+communicate a gender that is incorrect.” (P07)
+“relying on slack has allowed our tabs/NB colleagues to state their pronouns in their bio for easy
+reference.” (P11)
+Physical Safety
+The individual perception
+of being safe and
+protected from any risk
+to their physical integrity.
+“we don’t have to be exposed to public transportation and streets where trans people are killed as if this
+was a sport.” (P03)
+“For those who don’t belong in the heteronormative context, working from home is safer.” (P07)
+Toxicity
+Avoidance
+The sentiment of being
+distant from other
+people’s negative
+comments, attitudes, and
+actions.
+“Small types of violence that can happen in-person [such as comments and staring] are less common in
+virtual environments on Slack or email.” (P03)
+“Remote or hybrid work reduces situations where people keep staring at you or making you feel
+uncomfortable.” (P04)
+“In my experience, it reduces the chance of me suffering discrimination because everything is recorded
+(videos, screenshots, etc.)“ (P10)
+TABLE III
+LIMITATIONS OF REMOTE WORK FOR LGBTQIA+ SOFTWARE PROFESSIONALS
+Construct
+Definition
+Example Evidence
+Self-isolation
+The strategy of increasing
+isolation seeking to feel
+safe from external
+attacks.
+“it is not all positive as it allows us [by option] to reduce our contact with people from other eth-
+nicities, sexual orientation, social classes, therefore, reducing our adaptation capability.” (P01)
+“if you are working from home because in-person you need to hide any of your characteristics,
+then, I don’t see it as inclusion.” (P04)
+Invisibility
+Feeling
+The impression of being
+alone, with little or no
+contact with other
+members of the
+LGBTQIA+ community
+in the company.
+“it reduces our ability to adapt (. . . ) and experience beyond the ‘bubble’ that we live in.” (P01)
+“sometimes it feels very lonely and difficult to be just around a family that is not very
+supportive.” (P08)
+“workplaces to talk about these themes are scarce and talking about it in remote environments
+is even rarer.” (P12)
+participants that cannot be overlooked. Most of the benefits
+described above are related to LGBTQIA+ software profes-
+sionals distancing themselves from their coworkers and orga-
+nizations for their protection. This aspect creates limitations
+that need to be managed at the team and organizational levels,
+namely, feelings of invisibility and self-isolation.
+By self-isolating, individuals might shield themselves from
+unsafe and uncomfortable situations. However, one participant
+reported this could be experienced as a fake protection because
+the elements of violence against LGBTQIA+ software profes-
+sionals will not disappear; they would just be obscured. There-
+fore, it is essential that software companies design strategies
+to increase inclusion so that professionals will feel welcome
+in the workspace; such that remote work is a choice, not an
+escape.
+In addition, not all LGBTQIA+ software professionals have
+supportive families. Some of them encounter, in their com-
+panies and their teams, people that they can rely on. Remote
+structures create barriers for LGBTQIA+ software developers
+to meet with other members of the community that also work
+in software development, which creates the feeling of invisi-
+bility. Again, individuals reported that software companies and
+not themselves should be responsible for designing strategies
+to increase the visibility of this group.
+6
+
+D. How does remote work affect LGBTQIA+ software profes-
+sionals?
+Working from home affects LGBTQIA+ software profes-
+sionals in different ways. Primarily, our data reveals these
+professionals can benefit greatly from the flexibility provided
+by this work structure, and the opportunity of choosing where
+to work supports these individuals in dealing with several
+struggles faced for years by the community in general. How-
+ever, remote work also can be associated with advantages
+that can be observed at the individual level and that require
+organizational actions to smooth such problems.
+In general, remote work increased the access of LGBTQIA+
+software professionals to work opportunities. In particular,
+those who live in suburban areas and had difficulty joining
+the software industry before are now finding jobs in software
+companies without leaving the safety of their communities.
+In addition, remote work has allowed LGBTQIA+ software
+professionals to avoid acts of violence, both physical and
+emotional, resulting from discrimination in the workspace and
+the commuting between home and work. These are advantages
+of remote work that organizations might explore to develop
+strategies that will increase diversity in the software industry
+by bringing talented LGBTQIA+ professionals to work in the
+area.
+Only facilitating the access of LGBTQIA+ professionals
+to jobs in the software industry is not enough to guarantee
+fairness in this environment that for years has been extremely
+non-diverse and even unfriendly to equity-deserving groups.
+Considering this aspect, remote work turned out to be more
+inclusive as it facilitates LGBTQIA+ individuals to control
+their identities and regulate their interaction with their teams
+based on how accepted they feel. However, inclusion in the
+software industry depends on the organizations’ attitudes to
+create strategies that reinforce the importance of embracing
+diversity in the workspace. Such strategies should prioritize
+communication and networking, allowing LGBTQIA+ soft-
+ware professionals to develop connections within the organi-
+zation, therefore, avoiding isolation.
+Two out of the three EDI principles were observed in this
+research, namely, diversity and inclusion. Although no aspects
+of equity were revealed in this study, our findings suggest
+that diversity and inclusion are critical elements to increasing
+LGBTQIA+ visibility in the software industry, which is funda-
+mental to ensuring fair treatment and opportunities despite of
+professionals’ gender and sexuality. Visibility is an essential
+aspect of reducing intolerance, unfairness, and inequity against
+LGBTQIA+ people, especially in environments predominantly
+heterosexual, such as the software industry. Remote work plays
+a vital role in this matter.
+V. DISCUSSIONS
+Our investigation demonstrated that remote work could
+benefit LGBTQIA+ software professionals greatly. In general,
+remote work supports a structure that improves LGBTQIA+
+visibility in the software industry through increasing diversity
+and inclusion, supported by the benefits observed in this study
+as summarized in Fig 1. and discussed below.
+A. Enfolding the Literature
+Diversity and inclusion are crucial factors for organizations
+[32]. Diversity is a core element for developing new ideas,
+which is the key to innovation [33], while inclusion supports
+productivity, talent retention, and engagement [34].
+Diversity is fundamental for innovative environments as
+innovation is the enabler of business transformation in an ever-
+changing world where technology (e.g., products, processes,
+or services) is constantly evolving [35]. Diverse teams improve
+innovation because they are more creative, more effective, and
+better coordinated [5], [36]. Remote work creates opportu-
+nities for software teams to be more diverse by adding to
+their portfolio the experiences of LGBTQIA+ professionals.
+LGBTQIA+ individuals are reported to be more creative and
+cope with high levels of autonomy, which are two essential
+aspects of software development nowadays, in particular in
+agile environments [37].
+Remote work supports the inclusion of LGBTQIA+ profes-
+sionals in software teams, as these professionals can slowly
+adjust to their team (e.g., controlling the camera, engaging via
+chat or call, etc.) to the point that they feel comfortable enough
+to interact regularly, different from in-person interaction which
+allows no adjustments. Inclusion is important to software
+companies because it enhances organizational commitment,
+consequently increasing productivity, job satisfaction, and re-
+taining talented professionals in software teams. Although in
+this study, we are explicitly discussing inclusion from the
+perspective of LGBTQIA+ software professionals, perceived
+inclusion is a factor that can affect all individuals in an
+organization [38].
+As
+remote
+work
+fosters
+diversity
+and
+inclusion
+of
+LGBTQIA+
+software
+professionals,
+we
+expect
+that
+LGBTQIA+ visibility will grow gradually in the software
+industry.
+On
+the
+one
+hand,
+visibility
+is
+essential
+in
+strengthening the sense of individual belonging and security
+for LGBTQ+ people, therefore improving several aspects
+of software development, such as teamwork and team
+resilience [39]. On the other hand, this visibility is expected
+to
+transcend
+the
+boundaries
+of
+software
+development
+environments and become more frequent in the software
+products and technologies that nowadays impact several
+aspects of our society (e.g., work, education, and leisure
+[4]), producing diverse and inclusive solutions achieving
+individuals from a variety of profiles.
+B. Implications
+Our findings have implications for academia and research.
+Studies focused on LGBTQIA+ software professionals are ex-
+tremely rare. We contribute to the theme with a comprehensive
+investigation focused on the effects of remote work on these
+professionals by exploring the experience of several groups of
+individuals that compose this community. To the best of our
+7
+
+Fig. 1. The effects of remote work on LGBTQIA+ software professionals
+knowledge, this is the first study to have such a diverse popula-
+tion of LGBTQIA+ individuals in software engineering since
+we were able to reach lesbians, gays, bisexuals, transgenders,
+asexuals, and pansexuals, among others.
+Our study calls the attention of researchers to the need for
+developing studies regarding equity in software engineering
+for LGBTQIA+ professionals. Recently, studies have been
+addressing equity in the context of others underrepresented
+groups. We highlight the importance of increasing research
+efforts on addressing equity to reach the LGBTQIA+ com-
+munity, while our findings are expected to initiate further
+discussion on this theme in academia.
+From a methodological perspective, our study is an applied
+example of how to deal with hidden populations in software
+engineering, in particular when addressing a group of individ-
+uals who are sensitive in relation to data collection strategies.
+Our data collection strategy can be used to guide other studies
+that deal with complex hidden populations.
+As for industrial practice, our findings demonstrate several
+aspects of the work the software organizations can conduct
+to improve the experience of LGBTQIA+ software profes-
+sionals. Software companies can benefit significantly from
+using remote work to compose highly diverse teams that
+reflect our multifaceted society. Our findings also highlight
+the importance of actions to improve inclusion in the software
+industry.
+Based on our findings and regarding LGBTQIA+ software
+professionals, we recommend that software companies:
+• Apply unbiased recruitment and hiring process to avoid
+discrimination.
+• Create inclusive onboarding processes, including EDI
+training for software teams.
+• Develop democratic remote work structures, allowing
+professionals to choose their workspace, which will help
+LGBTQIA+ software professionals to better deal with
+violence, toxicity, and other problems related to in-person
+work.
+• Understand the specific needs of LGBTQIA+ profession-
+als and provide them with the appropriate support, e.g.,
+regarding cameras and video calls, and gender-affirming
+care, among others.
+• Foster a culture of diversity and inclusion that embraces
+and welcomes LGBTQIA+ professionals.
+• Support the creation of channels and committees to help
+LGBTQIA+ professionals within the company to connect
+among themselves and others, thus, avoiding isolation.
+• Celebrate diversity and inclusion, improving the visibility
+of LGBTQIA+ professionals in the company.
+• Acknowledge the role of diversity in developing innova-
+tive technologies in modern society, thus, increasing the
+interest of equity-deserving groups in software engineer-
+ing, e.g., students.
+We understand that most of these recommendations only
+apply to organizations that are already LGBTQIA+ friendly
+and many software companies still discriminate against
+LGBTQIA+ individuals. We expect that the findings presented
+in this research, in particular, the discussions on how diversity
+is essential to the development of software for our modern
+8
+
+Job
+Opportunities
+Diversity in
+Physical
+Safety
+Software
+Engineering
+Toxicity
+LGBTQIA+ Visibility
+Avoidance
+in Software
+Engineering
++
+Engagement
+Remote Work
+Inclusion in
+Software
+Identity
+Engineering
+Disclosure
+Control
+Self-distance
+Isolation
+EDI Organizational
+Strategiessociety help policymakers to build strategies to change this
+reality.
+C. Limitations
+The main limitation of this study is the number of partici-
+pants from whom we were able to collect data. Investigating
+a very particular hidden population, such as LGBTQIA+ soft-
+ware professionals, is challenging because many individuals
+avoid exposure, mainly because software engineering is a
+predominantly heterosexual environment. This limitation was
+demonstrated when 22 individuals quit the study right after
+completing the screening part of our questionnaire. We tried
+to work around this limitation by following the all is data
+principle of grounded theory [23], opening our minds to collect
+data both completely anonymously using a questionnaire and
+subsequently interviewing those who provided their contact
+for further discussions. This allowed us to have a sample of
+57 individuals; however, we acknowledge that our findings
+are based on the experience of a limited number of software
+professionals.
+Another limitation is related to regional aspects. As 77%
+of our sample comprises participants from Brazil, a great
+deal of the experience of these participants is associated with
+their local context. Although software teams usually apply
+similar processes across the globe, and even though most of
+our participants have international clients, which brings them
+close to other cultures, several aspects of equity, diversity, and
+inclusion are particularly linked to organizational behaviors
+that are more influenced by regional factors and less influ-
+enced by software development processes. Therefore, the local
+context (e.g., the participants’ country) remains a limitation
+when EDI in the software industry is under investigation.
+We tried to work around this limitation by having a version
+of the questionnaire built in English and advertised across
+social media and international forums of software profession-
+als. However, the participation of individuals from multiple
+countries remained low.
+Finally, regarding the quality aspects of our method and
+to avoid threats to the study’s validity in terms of credibility,
+originality, resonance, and usefulness, we followed Charmaz’s
+[25] criteria to evaluate grounded theory studies. To support
+the quality and credibility of our findings, we provided direct
+quotations from questionnaires and interviews to illustrate
+the interpretation of our participants’ experiences. This inter-
+pretation was accessed by conducting member-checking with
+three interviewees who agreed to participate by commenting
+about the obtained categories after a brief presentation and
+explanation of the findings.
+D. Future Work
+Equity, diversity, and inclusion in software engineering
+are topics with many opportunities for investigation. There
+are many gaps about this theme in the software industry.
+Additionally, it also requires investigations in the academic
+context, especially on how to increase diversity and inclusion
+in software engineering courses.
+Regarding our findings, the immediate future works related
+to them are:
+• Conduct an in-depth investigation of the factors revealed
+in this study and their effects on several aspects of
+software development, e.g., the relation of diversity and
+team conflicts, how diversity could improve team re-
+silience, and the relationship among diversity, inclusion,
+and software practices, among others.
+• Work on the generalization of the theory by conducting
+quantitative studies based on a worldwide survey.
+• Develop investigations on the perspective of heterosex-
+ual professionals about EDI and its impacts on soft-
+ware development since people who are not part of the
+LGBTQIA+ community demonstrated interest in partici-
+pating in the present research.
+• Investigate strategies to improve equity in the software
+industry aiming to increase the access of LGBTQIA+
+professionals to job opportunities in the area.
+Finally, we are interested in exploring EDI strategies that
+are being successfully applied in other industries to determine
+how they can be transferred to software companies by applying
+transformative research methods (i.e., action research and
+design science [40]).
+VI. CONCLUSION
+The present study investigated the effects of remote work
+structures on LGBTQIA+ software professionals. Using a
+grounded theory approach, we explored the experience of
+57 individuals from different groups within the community,
+e.g., lesbians, gays, bisexuals, and asexuals, including both
+cisgender and transgender people.
+In summary, we concluded that remote work has a crucial
+role in increasing diversity and inclusion in the software indus-
+try. Remote work allows LGBTQIA+ software professionals to
+access a variety of job opportunities. Remote work also allows
+these professionals to have more control over their identities
+and their interaction with other professionals.
+Regarding limitations, remote work might create barriers for
+LGBTQIA+ software professionals, as it can increase isolation
+and self-distancing among these individuals. Practices devel-
+oped by software companies to increase LGBTQIA+ visibility
+among employees are essential to reduce these problems.
+No aspect related to equity was observed in this study,
+which indicates a gap that requires immediate investigation.
+However, we generally concluded that remote work is positive
+for LGBTQIA+ software professionals. Our analysis also
+demonstrates that software development can benefit signifi-
+cantly from more diverse teams, including the improvement
+of aspects related to innovation, problem-solving, teamwork,
+and team resilience.
+DATA AVAILABILITY
+Supplementary material is available on Figshare: https:
+//figshare.com/account/home#/projects/157305
+9
+
+ACKNOWLEDGMENTS
+The authors would like to thank all of the participants who
+participated in this study, and for all LGBTQIA+ software
+professionals out there, we would like to say that we see you.
+You are not alone!
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+
diff --git a/TtE5T4oBgHgl3EQfAg5m/content/tmp_files/load_file.txt b/TtE5T4oBgHgl3EQfAg5m/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf,len=644
+page_content='Benefits and Limitations of Remote Work to  LGBTQIA+ Software Professionals  Ronnie E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' de Souza Santos  Cape Breton University  Sydney, NS, Canada  Email: ronnie desouza@cbu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='ca  Cleyton V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' de Magalh˜aes  CESAR School  Recife, Brazil  Email: cvcm@cesar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='school  Paul Ralph  Dalhousie University  Halifax, Canada  Email: paulralph@dal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='ca  Abstract—Background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' The mass transition to remote work  amid the COVID-19 pandemic profoundly affected software  professionals, who abruptly shifted into ostensibly temporary  home offices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' The effects of this transition on these professionals  are complex, depending on the particularities of the context and  individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Recent studies advocate for remote structures to  create opportunities for many equity-deserving groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' however,  remote work can also be challenging for some individuals, such  as women and individuals with disabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' As the discussions on  equity, diversity, and inclusion increase in software engineering, it  is important to explore the realities and perspectives of different  equity-deserving groups to develop strategies that can support  them post-pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' This study aims to investigate the  effects of remote work on LGBTQIA+ software professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' A grounded theory approach was applied, based on  information collected from two main sources: a survey question-  naire with a sample of 57 LGBTQIA+ software professionals and  nine follow-up interviews with individuals from this sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' This  sample included professionals of different genders, ethnicities,  sexual orientations, and levels of experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Consistent with  grounded theory methodology, the process of data collection  and analysis was conducted iteratively using three stages of  coding: line-by-line, focused, and theoretical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Member checking  was used to validate the findings obtained from interpreting the  experiences commented on by LGBTQIA+ software profession-  als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Our findings demonstrate that (1) remote work  benefits LGBTQIA+ people by increasing security and visibil-  ity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' (2) remote work harms LGBTQIA+ software professionals  through isolation and invisibility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' (3) the benefits outweigh the  drawbacks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' (4) the drawbacks can be mitigated by supportive  measures developed by software companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' This  paper investigated how remote work can affect LGBTQIA+  software professionals and presented a set of recommendations on  how software companies can address the benefits and limitations  associated with this work model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In summary, we concluded  that remote work has a crucial role in increasing diversity and  inclusion in the software industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Abstract—General Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Remote work is here to stay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' There  is no denying it, as some software professionals would rather  quit their jobs than return to the office full-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Therefore,  software companies want to understand how the remote working  model can be successfully used without causing major issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  The problem is that the effects of remote work are complex  because they depend on individual and group characteristics that  require careful evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In this scenario, one thing has been  extremely positive: remote work is helping to increase diversity in  software engineering by fostering new opportunities and better  work conditions for individuals from equity-deserving groups,  for instance, LGBTQIA+ software professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' The software  industry is overly homogeneous, most of the professionals who  work in this area are heterosexual men (a reflection of the uni-  versity courses on computer science and software engineering),  but diversity can only be good for an area that strongly depends  on creativity and innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' What better way to innovate than  putting several individuals from different backgrounds and with  various experiences to work together?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Remote work plays an  important role in improving equity, diversity, and inclusion in  the software industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In this paper, we discuss how remote  work is affecting software professionals from the LGBTQIA+  community and provide a list of recommendations to support  software companies in dealing with this work model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Index Terms—EDI, equity, diversity, inclusion, software pro-  fessionals, LGBTQIA+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' INTRODUCTION  Equity, diversity, and inclusion (EDI) are central topics  being discussed in society nowadays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' EDI is a complex  phenomenon centered on developing approaches for providing  equal opportunity for individuals (equity) while recognizing  their personal, social, and cultural differences (diversity) and  encouraging them to participate in environments and debates  (inclusion) [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Discussions on EDI are gradually increas-  ing in Computer Science as well [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Since technology has  a direct impact on our society, and our society is plural, the  lack of diversity among those who develop software creates  limitations in the use of technologies as software solutions  might exclude groups of individuals [4]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Diversity is essential in developing technologies to sup-  port our multifaceted society (Albusays, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' However,  the software industry is experiencing a diversity crisis, as  software teams are mainly composed of heterosexual men [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Currently, there is a lack of knowledge on how EDI should  be addressed and integrated into software engineering [6]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  The main consensus among researchers is that diversity needs  to be addressed from the academy to the software industry  by increasing opportunities, improving debates, and fostering  safe spaces for equity-deserving groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Recently, the COVID-19 pandemic intensified discussions  about EDI in software engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' As the lockdowns forced  software professionals to work from home, remote work struc-  tures created better work-life balance, more flexibility, and  less commuting, which resulted in new opportunities for many  underrepresented groups in the software industry: caregivers  (especially mothers), LGBTQIA+ individuals (especially trans  people), and people with disabilities [4], [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Now,  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='05379v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='SE]  13 Jan 2023  as the pandemic seems to be under control, many software  companies are permanently transitioning to remote and hybrid  work allowing software professionals to decide where they will  work, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', in the office or any other place [11], [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Thus, the  post-pandemic scenario tends to keep increasing diversity and  inclusion in the software industry as more job opportunities  will be available for professionals that cannot afford to work  in an office every day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Since remote work fosters new job opportunities, we need  to understand the impacts of remote work arrangements on  individuals from equity-deserving groups because (1) EDI  groups experience remote work differently, as reported in  previous studies [9], [13], and (2) EDI is essential for having  a more just and inclusive society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In the context of software  engineering, understanding the impacts of remote work on  software professionals from EDI groups is an important step  toward sharpening practices, strategies, and tools required  to improve equity, diversity, and inclusion in the software  industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In this study, we explore the impacts of remote work  on LGBTQIA+ software professionals aiming to answer the  following research question:  Research  Question: How does remote work affect  LGBTQIA+ software professionals?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  In particular, we explore the experience of software profes-  sionals from the LGBTQIA+ community because the research  on EDI in software engineering that focuses on LGBTQIA+  software professionals is very scarce [3], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' From this  introduction, the rest of this study is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  In Section II, we discuss existing studies on EDI in software  engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In Section III, we present the method applied in  this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In Section IV, we present our findings which are  discussed in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Finally, we summarize the contribu-  tions of this study and opportunities for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' BACKGROUND  Remote work has received attention from researchers from  several areas since this is expected to be the norm in many  industries across the globe [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In software engineering,  studies are being developed on remote work for at least  two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' First, software professionals are more likely to  continue working remotely since they are already used to  this type of work structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', global software development  and distributed teams) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Second, software professionals  are responsible for creating and adapting technologies to  support employees from other industries, which might also  face adaptations in the post-pandemic [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Considering the first reason, as software engineering follows  the world’s transformations and their impacts on society,  understanding the long-term implications of the post-pandemic  in the software industry and practitioners is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Over  the past two years, studies revealed that remote work can  produce several benefits and positively affect those who work  in software development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' However, some limitations have also  been observed in the same period [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Remote work was  demonstrated to benefit software professionals by providing  them with more flexibility, better work-life balance, and in-  creased satisfaction [9], [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' On the other hand, remote work  can foster problems at the team level, creating communication  issues, increasing coordination challenges, and reducing team  cohesion [18], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  The main challenge associated with remote work is the  fact that its effects are intimately dependent on individual  characteristics and in the particularities of some groups of  professionals [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' For instance, remote work can be worse  for women and caregivers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', parents), as they might have  to deal with many distractions while performing software  development activities that demand problem-solving skills  and high levels of concentration [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Remote work might  promote more opportunities for people with disabilities [20];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  however, it can also force this group of individuals to deal  with poor ergonomics [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Remote structures allow transgender  software developers to have control over their identities [13];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  however, in other fields, researchers are concerned that remote  interactions can undermine diversity [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' METHOD  Aiming to address the above-cited problem and the lack of  discussions on the impacts of remote work in some equity-  deserving groups, we developed a qualitative study using con-  structivist grounded theory [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Grounded Theory is a family  of methods for inductively generating theories based on the  data collected from real-life contexts to identify and describe  concepts, behaviors, and experiences [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' It is widely applied  in social sciences and is well-suited for exploring social,  cultural, and human aspects of software development [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Grounded theory is appropriate for this study because it  helped us understand the research problem using the experi-  ence of a small group of individuals [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' in this process, due  to the characteristics of our targeted population, we could not  expect reaching a significant number of professionals to partic-  ipate in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Therefore, the grounded theory effectively  supported us in constructing concepts that can be transferred  to other contexts through theoretical generalization [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Thus, following the guidelines to conduct grounded theory,  we used well-defined rounds of data collection and analysis,  including inductive coding, memoing, constant comparison,  and theoretical sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' This process allowed us to identify  concepts emerging from the field rather than fitting data into  preconceived theories [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Participants  The participants of this study are individuals from a very  specific population: LGBTQIA+ software professionals, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=',  developers, QAs, analysts, designers, and managers, among  others directly participating in the software development pro-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' These professionals belong to a hidden population—  a population that cannot be easily defined or enumerated  based on existing knowledge—which makes selecting indi-  viduals to participate in studies a challenging activity [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  In this study, we followed guidelines that suggest treating  software professionals as a hidden population [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Moreover,  2  individuals from the LGBTQIA+ community are commonly  treated as a hidden population in studies from other fields, in  particular, because many of its members are not comfortable  with discussing aspects of their sexuality due to the risk of  being exposed to structural and social discrimination [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Instrumentation  We applied two approaches to data collection, aiming to  reach as many individuals as possible and carefully access the  experience of LGBTQIA+ professionals without having them  reveal their identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' The approaches applied were a survey  questionnaire and semi-structured interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Due to the hid-  den nature of the population targeted in this study, we could  not expect a statistically significant number of respondents  in a survey (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', 1000 LGBTQIA+ software professionals)  at the same time that a significant number of volunteers for  interviews was also challenging to reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Therefore, we used  the best data collection methods (survey and interviews) to  obtain data and the best of the grounded theory method to  work with a small sample of participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  We designed the questionnaire so that we could anony-  mously obtain basic information about the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' The  questionnaire was mostly open-ended as we were targeting  to collect qualitative data about the professionals’ experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  We started by asking individuals to provide their age, gender,  sexual orientation, ethnicity, and country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Then, we added  questions about their work and their company, including their  level of education, level of experience working in the software  industry, role (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', developer, tester, designer, manager, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' ),  and the number of employees working in their company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Fi-  nally, we asked the following questions about their experience  working remotely:  Would  you  say  that  remote/hybrid  work  benefits  LGBTQIA+ individuals working in the software indus-  try?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Please, describe in what ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Would you say that remote/hybrid work creates chal-  lenges for LGBTQIA+ individuals in the software indus-  try?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In what ways?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Do you know any software professionals from the  LGBTQIA+ community that benefited from working  from home on a regular basis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Tell us a bit about this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Do you know any professional from the LGBTQIA+  community that faced disadvantages caused by working  from home on a regular basis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Tell us a bit about this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Is remote work more inclusive for LGBTQIA+ software  professionals?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In which ways?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Does  remote/work  create  more  opportunities  for  LGBTQIA+ software professionals?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' How?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  All questions were optional, so individuals would be free to  answer only the ones that they wanted to answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' At the end  of the questionnaire, we asked those who would be interested  in discussing more aspects of remote work and how it affects  the software professionals from the LGBTQIA+ community to  provide their email to be contacted for a 30-minute interview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  The questionnaire was developed in two languages English  and Portuguese.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Data Collection  We started the data collection with the questionnaire, which  was applied using the following sampling approaches [28]:  Convenience sampling: We invited participants based on  their availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' The first author was a member of the  target population (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', an LGBTQIA+ professional who  worked with software development for six years) and  invited members of the population from his contacts list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Purposive sampling: We used the communication chan-  nels of a large software company in South America to  advertise our questionnaire to over 1,000 software pro-  fessionals and expected that individuals from the targeted  population would participate in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' This company  was founded in 1996, specialized in on-demand software  development, and creates technologies for clients from  various sectors, including finance, telecommunication,  manufacturing, and services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In addition to this company,  we advertised the questionnaire in online communities  from social media websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Snowballing sampling: We requested that individuals  from the convenience and the purposive sampling invite  other LGBTQIA+ software professionals to participate in  the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  We started with the questionnaire because individuals from  hidden populations usually avoid outsiders [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' However,  members of such populations often know each other [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Therefore, by applying the questionnaire first, we were able to  apply sampling techniques commonly used in surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Once  we reached the population, we could identify individuals to be  interviewed, providing us with additional details beyond those  collected with the open-ended questions in the survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Data collection using the questionnaire happened between  June 10th and July 20th, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Following the nature of the  grounded theory method and the approach to collecting and  analyzing data in rounds, we considered the application of the  questionnaire as the first round of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Next, we invited  the participants who volunteered for interviews and conducted  three rounds of interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' These interviews were conducted  online with nine participants (three per round) using Google  Meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In these three rounds, we focused on obtaining more  details about the answers already provided in the question-  naire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In addition, as the rounds evolved, participants were  asked about the role of diversity on software teams and the  effects of remote work on software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' The nine interviews  ranged from 20 to 43 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' This produced four hours and  18 minutes of audio and 97 pages of transcripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Data Analysis  We started data analysis by applying descriptive statistics  [31] to summarize the information about our sample of  participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Descriptive statistics allowed us to present the  distribution of participants’ answers regarding their personal  and professional profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Following this, we analyzed the answers to open-ended  questions in the questionnaire (first round).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Then, we contin-  ued using qualitative data analysis in the interviews (following  3  three rounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Qualitative analysis was conducted based on  three coding stages: line-by-line coding was focused on iden-  tifying initial codes and building concepts within our data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  focused coding supported the establishment of connection  among the concepts and the definition of high-level categories  of concepts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' theoretical coding was used to emphasize the core  category and explain the main story arising from the data [23],  [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  In this process, we began with line-by-line open coding,  which allowed us to identify and refine emerging concepts  from the answers to the open-ended questions in the survey  [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' We repeated this strategy in the first round of interviews  through a systematic process supported by audio recordings,  which helped identify the properties of codes and their mean-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Following the second round of interviews, we started cat-  egorizing these codes and establishing connections using fo-  cused coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Focused coding was performed through repeated  analyses and comparisons among emerging categories [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  This process continued during the third and final round of  interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' When these transcripts were integrated into the  analysis, we used theoretical coding [25] to rearrange the  categories and highlight a central story about the experience  of our participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Ethics  Before answering questions about their background and  their experience with remote work in the questionnaire, each  professional read about the goal of the research and the  study’s relevance to the software industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Also, they needed  to explicitly agree to provide data to the study by selecting  the option at the beginning of the questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Following  this, volunteers agreed to be interviewed by providing their  email address and then again, orally, at the beginning of their  interview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Before each interview, the interviewer explained  the goal of the research one more time and the need to  collect more data in addition to the questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Participants  were guaranteed data confidentiality and de-identification of  their quotes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' They were also informed about the voluntary  nature of their participation and the right to stop the interview  and withdraw from the research at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' No participants  withdrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' A university research ethics board approved this  research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' FINDINGS  This section presents the demographics of our sample,  followed by the categories that emerged from the analysis,  which are the benefits and limitations of remote work for  LGBTQIA+ software professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Our findings are illus-  trated with quotations extracted from the questionnaire and the  interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Some quotations have been translated into English  and may read awkwardly as we have transcribed and translated  them as accurately as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Demographics  Our survey reached 81 individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' However, some partici-  pants were excluded from the survey for not being part of the  targeted population:  one woman and one man identified themselves as het-  erosexual, despite stating that they were software profes-  sionals from the LGBTQIA+ community in the screening  questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  22 individuals only completed the screening questions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  they answered neither demographic nor open-ended ques-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  However, our sample of 57 individuals varies as not all  participants answered the entire questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' For instance:  57 (100%) individuals provided their level of education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  56 (98%) individuals provided their role in the software  team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  53 (93%) individuals provided their age, gender, and  ethnicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  52 (91%) individuals provided their country and their  level of experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  45 (79%) individuals provided information about the  number of employees in their companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  25 (44%) individuals answered the question about their  sexual orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  13 (23%) individuals answered at least one question  about how remote work affects LGBTQIA+ software  professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  10 (17%) individuals provided us with an email for an  interview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Participants’ ages varied from 21 to 50 years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Partic-  ipants had an average of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='7 years of work experience in  software development;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' the most experienced professional in  the sample has worked in the software industry for 25 years,  and the least for one year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Table I presents more details  about our sample and its variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Finally, we were able  to interview 16% of the survey (9/57), who provided more  details on the topics addressed in the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Benefits of Remote Work to LGBTQIA+ Software Profes-  sionals  Participants pointed out benefits that were observed in the  general professional context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', the increase in job oppor-  tunities for the community) as well as benefits observed at  the individual and team level when professionals are working  in the software industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Table II summarizes the benefits  identified in this study and the evidence obtained from par-  ticipants’ experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' These benefits are Job Opportunities,  Engagement, Identity Disclosure Control, Physical Safety,  and Toxicity Avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' All of these benefits are extremely  valuable for LGBTQIA+ software professionals because the  software industry is still an area dominated by heterosexual  men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Remote work increases access to job opportunities in the  software industry, and this benefits LGBTQIA+ individuals  greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In general, this could be seen as a simple cor-  relation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' that is, more remote jobs available means more  jobs available for LGBTQIA+ professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' However, for  LGBTQIA+ individuals, this aspect goes beyond the increase  in the number of positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' These are professionals that have  4  TABLE I  DEMOGRAPHICS  Gender  Men  34  Women  14  Non-binary  3  Not informed  4  Ethnicity  White  34  Mixed ethnic  13  Black  2  Arab  2  Asian  1  Latino  1  Not informed  4  Sexual Orientation  Gay  13  Bisexual  7  Lesbian  3  Pansexual  1  Asexual  1  Not informed  32  Education  High-School  8  Bachelor  30  Post-baccalaureate  12  Master  6  Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  1  Roles and Activitiesa  Testing  28  Programming  25  Requirements  17  Design  14  Architecture  10  Management  3  Not informed  1  Location  Brazil  44  Canada  2  Portugal  2  Sweden  2  Netherlands  1  US  1  Not informed  5  Company Size  0 - 9 employees  1  10 - 99 employees  3  100 - 999 employees  5  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='000 - 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='999 employees  35  More than 10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='000 employees  1  Not informed  12  Notes: asome professionals reported performing multiple activities or having  more than one roles in the team  suffered discrimination for years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Our participants explained  that online recruitment tends to be slightly less biased by  physical characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Moreover, LGBTQIA+ professionals  can apply for jobs that before were primarily available to those  living in major urban centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' This means that they can choose  to work where they always lived instead of adapting to new  places or having to move away to unfriendly locales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  For many LGBTQIA+ individuals, disclosing their identity  is never easy, especially when entering a new environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Trans people and non-binary are apprehensive about precon-  ceived judgments about their physical appearances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Gender-  affirming care (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', medical treatments and therapy) is an  additional concern for software professionals who are trans-  gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Other individuals from the community who do not  fit into the gender binary (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', male or female stereotype)  suffer from fear and uncertainty in relation to how they will  be seen and treated by co-workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Remote environments  established a place where LGBTQIA+ software professionals  feel comfortable as they can control their identity disclosure,  for instance, by deciding when they want to use their cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Remote work structure allows LGBTQIA+ software pro-  fessionals to regulate the level of interaction and contact  with their teammates to the point that they feel comfortable  around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Our participants revealed that their engagement  with other team members increased while working remotely  because they could gradually get used to their co-workers  and engage accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Joining a software team before (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=',  in person) was challenging for LGBTQIA+ professionals  because they were worried about acceptance in the software  industry work environment since it is essentially heterosexual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  In contrast, in remote environments, they will first become  comfortable with their team, then engage, and once they feel  included by the team, they can finally interact in person.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' This is  a know me before you judge me effect that cannot be afforded  in primarily in-person environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Many LGBTQIA+ individuals still feel unsafe and afraid  of physical attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' This reality is not different from those  working in the software industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Transgender professionals  who participated in this study reported the risks that they are  exposed to every day, in particular, those who live in what,  according to them, is the country where more trans women are  killed in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Working from home reduces the exposure  of these professionals to unsafe environments, such as public  transportation during peak hours in countries that are more  dangerous to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' By not having to go to the office daily,  these professionals can use safer alternative commuting, such  as taxis and private rides, whenever they need to be in the  office.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Office environments can be extremely uncomfortable for  LGBTQIA+ professionals who are exposed to coworkers’  negative comments, attitudes, and actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' These toxic behav-  iors vary from the act of being constantly observed to being  a target for constant commentaries in the office.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Although  this is not an act of physical violence, such conduct affects  individuals’ well-being and their sense of belonging in relation  to the company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Hybrid work allows LGBTQIA+ software  professionals to decide the best moment to be in the office  and avoid toxicity from others when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  The  above-cited  benefits  are  extremely  valuable  for  LGBTQIA+ software professionals because the software in-  dustry is still an area dominated by heterosexual males, which  ends up undermining the inclusion of other groups of individ-  uals, even indirectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Remote work and its benefits, therefore,  create more opportunities for diversity in software engineering  while also helping software teams to be more inclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Limitations of Remote Work to LGBTQIA+ Software Pro-  fessionals  Although remote work produced benefits that improve the  experience of LGBTQIA+ individuals working in the software  industry, there are also observable limitations reported by our  5  TABLE II  BENEFITS OF REMOTE WORK FOR LGBTQIA+ SOFTWARE PROFESSIONALS  Construct  Definition  Example Evidence  Job  Opportunities  The availability of  employment that  LGBTQIA+ software  professionals can easily  access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  “it gives professionals the chance to work in more inclusive companies that are distant from their  cities.” (P10)  “my girlfriend is from another state, and now I am living with her and working remotely.” (P15)  “software companies are in need of talented professionals and this [remote work] can be a good opportunity  for people from the LGBTQIA+ community.” (P16)  Engagement  The level of interaction  of LGBTQIA+ software  professionals with their  teammates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  “it increased the engagement of those who were afraid of participating before [in-person]” (P12)  “remote work allowed me to engage with new people, cultures, experiences and different realities [in the  team].” (P16)  Identity  Disclosure  Control  The individual decision  of concealing or  communicating and  expressing their gender  and sexual orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  “virtual tools allow people to communicate without having to expose themselves either for fear or as an  option.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' (P12)  “I can choose how I show up on camera,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' to exclude gender identifiers that make me dysphoric or  communicate a gender that is incorrect.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' (P07)  “relying on slack has allowed our tabs/NB colleagues to state their pronouns in their bio for easy  reference.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' (P11)  Physical Safety  The individual perception  of being safe and  protected from any risk  to their physical integrity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  “we don’t have to be exposed to public transportation and streets where trans people are killed as if this  was a sport.” (P03)  “For those who don’t belong in the heteronormative context, working from home is safer.” (P07)  Toxicity  Avoidance  The sentiment of being  distant from other  people’s negative  comments, attitudes, and  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  “Small types of violence that can happen in-person [such as comments and staring] are less common in  virtual environments on Slack or email.” (P03)  “Remote or hybrid work reduces situations where people keep staring at you or making you feel  uncomfortable.” (P04)  “In my experience, it reduces the chance of me suffering discrimination because everything is recorded  (videos, screenshots, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' )“ (P10)  TABLE III  LIMITATIONS OF REMOTE WORK FOR LGBTQIA+ SOFTWARE PROFESSIONALS  Construct  Definition  Example Evidence  Self-isolation  The strategy of increasing  isolation seeking to feel  safe from external  attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  “it is not all positive as it allows us [by option] to reduce our contact with people from other eth-  nicities, sexual orientation, social classes, therefore, reducing our adaptation capability.” (P01)  “if you are working from home because in-person you need to hide any of your characteristics,  then, I don’t see it as inclusion.” (P04)  Invisibility  Feeling  The impression of being  alone, with little or no  contact with other  members of the  LGBTQIA+ community  in the company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  “it reduces our ability to adapt (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' ) and experience beyond the ‘bubble’ that we live in.” (P01)  “sometimes it feels very lonely and difficult to be just around a family that is not very  supportive.” (P08)  “workplaces to talk about these themes are scarce and talking about it in remote environments  is even rarer.” (P12)  participants that cannot be overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Most of the benefits  described above are related to LGBTQIA+ software profes-  sionals distancing themselves from their coworkers and orga-  nizations for their protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' This aspect creates limitations  that need to be managed at the team and organizational levels,  namely, feelings of invisibility and self-isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  By self-isolating, individuals might shield themselves from  unsafe and uncomfortable situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' However, one participant  reported this could be experienced as a fake protection because  the elements of violence against LGBTQIA+ software profes-  sionals will not disappear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' they would just be obscured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' There-  fore, it is essential that software companies design strategies  to increase inclusion so that professionals will feel welcome  in the workspace;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' such that remote work is a choice, not an  escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  In addition, not all LGBTQIA+ software professionals have  supportive families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Some of them encounter, in their com-  panies and their teams, people that they can rely on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Remote  structures create barriers for LGBTQIA+ software developers  to meet with other members of the community that also work  in software development, which creates the feeling of invisi-  bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Again, individuals reported that software companies and  not themselves should be responsible for designing strategies  to increase the visibility of this group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  6  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' How does remote work affect LGBTQIA+ software profes-  sionals?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Working from home affects LGBTQIA+ software profes-  sionals in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Primarily, our data reveals these  professionals can benefit greatly from the flexibility provided  by this work structure, and the opportunity of choosing where  to work supports these individuals in dealing with several  struggles faced for years by the community in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' How-  ever, remote work also can be associated with advantages  that can be observed at the individual level and that require  organizational actions to smooth such problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  In general, remote work increased the access of LGBTQIA+  software professionals to work opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In particular,  those who live in suburban areas and had difficulty joining  the software industry before are now finding jobs in software  companies without leaving the safety of their communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  In addition, remote work has allowed LGBTQIA+ software  professionals to avoid acts of violence, both physical and  emotional, resulting from discrimination in the workspace and  the commuting between home and work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' These are advantages  of remote work that organizations might explore to develop  strategies that will increase diversity in the software industry  by bringing talented LGBTQIA+ professionals to work in the  area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Only facilitating the access of LGBTQIA+ professionals  to jobs in the software industry is not enough to guarantee  fairness in this environment that for years has been extremely  non-diverse and even unfriendly to equity-deserving groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Considering this aspect, remote work turned out to be more  inclusive as it facilitates LGBTQIA+ individuals to control  their identities and regulate their interaction with their teams  based on how accepted they feel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' However, inclusion in the  software industry depends on the organizations’ attitudes to  create strategies that reinforce the importance of embracing  diversity in the workspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Such strategies should prioritize  communication and networking, allowing LGBTQIA+ soft-  ware professionals to develop connections within the organi-  zation, therefore, avoiding isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Two out of the three EDI principles were observed in this  research, namely, diversity and inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Although no aspects  of equity were revealed in this study, our findings suggest  that diversity and inclusion are critical elements to increasing  LGBTQIA+ visibility in the software industry, which is funda-  mental to ensuring fair treatment and opportunities despite of  professionals’ gender and sexuality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Visibility is an essential  aspect of reducing intolerance, unfairness, and inequity against  LGBTQIA+ people, especially in environments predominantly  heterosexual, such as the software industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Remote work plays  a vital role in this matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' DISCUSSIONS  Our investigation demonstrated that remote work could  benefit LGBTQIA+ software professionals greatly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' In general,  remote work supports a structure that improves LGBTQIA+  visibility in the software industry through increasing diversity  and inclusion, supported by the benefits observed in this study  as summarized in Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' and discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Enfolding the Literature  Diversity and inclusion are crucial factors for organizations  [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Diversity is a core element for developing new ideas,  which is the key to innovation [33], while inclusion supports  productivity, talent retention, and engagement [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Diversity is fundamental for innovative environments as  innovation is the enabler of business transformation in an ever-  changing world where technology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', products, processes,  or services) is constantly evolving [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Diverse teams improve  innovation because they are more creative, more effective, and  better coordinated [5], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Remote work creates opportu-  nities for software teams to be more diverse by adding to  their portfolio the experiences of LGBTQIA+ professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  LGBTQIA+ individuals are reported to be more creative and  cope with high levels of autonomy, which are two essential  aspects of software development nowadays, in particular in  agile environments [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Remote work supports the inclusion of LGBTQIA+ profes-  sionals in software teams, as these professionals can slowly  adjust to their team (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', controlling the camera, engaging via  chat or call, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=') to the point that they feel comfortable enough  to interact regularly, different from in-person interaction which  allows no adjustments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Inclusion is important to software  companies because it enhances organizational commitment,  consequently increasing productivity, job satisfaction, and re-  taining talented professionals in software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Although in  this study, we are explicitly discussing inclusion from the  perspective of LGBTQIA+ software professionals, perceived  inclusion is a factor that can affect all individuals in an  organization [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  As  remote  work  fosters  diversity  and  inclusion  of  LGBTQIA+  software  professionals,  we  expect  that  LGBTQIA+ visibility will grow gradually in the software  industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  On  the  one  hand,  visibility  is  essential  in  strengthening the sense of individual belonging and security  for LGBTQ+ people, therefore improving several aspects  of software development, such as teamwork and team  resilience [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' On the other hand, this visibility is expected  to  transcend  the  boundaries  of  software  development  environments and become more frequent in the software  products and technologies that nowadays impact several  aspects of our society (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', work, education, and leisure  [4]), producing diverse and inclusive solutions achieving  individuals from a variety of profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Implications  Our findings have implications for academia and research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Studies focused on LGBTQIA+ software professionals are ex-  tremely rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' We contribute to the theme with a comprehensive  investigation focused on the effects of remote work on these  professionals by exploring the experience of several groups of  individuals that compose this community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' To the best of our  7  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' The effects of remote work on LGBTQIA+ software professionals  knowledge, this is the first study to have such a diverse popula-  tion of LGBTQIA+ individuals in software engineering since  we were able to reach lesbians, gays, bisexuals, transgenders,  asexuals, and pansexuals, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Our study calls the attention of researchers to the need for  developing studies regarding equity in software engineering  for LGBTQIA+ professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Recently, studies have been  addressing equity in the context of others underrepresented  groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' We highlight the importance of increasing research  efforts on addressing equity to reach the LGBTQIA+ com-  munity, while our findings are expected to initiate further  discussion on this theme in academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  From a methodological perspective, our study is an applied  example of how to deal with hidden populations in software  engineering, in particular when addressing a group of individ-  uals who are sensitive in relation to data collection strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Our data collection strategy can be used to guide other studies  that deal with complex hidden populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  As for industrial practice, our findings demonstrate several  aspects of the work the software organizations can conduct  to improve the experience of LGBTQIA+ software profes-  sionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Software companies can benefit significantly from  using remote work to compose highly diverse teams that  reflect our multifaceted society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Our findings also highlight  the importance of actions to improve inclusion in the software  industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Based on our findings and regarding LGBTQIA+ software  professionals, we recommend that software companies:  Apply unbiased recruitment and hiring process to avoid  discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Create inclusive onboarding processes, including EDI  training for software teams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Develop democratic remote work structures, allowing  professionals to choose their workspace, which will help  LGBTQIA+ software professionals to better deal with  violence, toxicity, and other problems related to in-person  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Understand the specific needs of LGBTQIA+ profession-  als and provide them with the appropriate support, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=',  regarding cameras and video calls, and gender-affirming  care, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Foster a culture of diversity and inclusion that embraces  and welcomes LGBTQIA+ professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Support the creation of channels and committees to help  LGBTQIA+ professionals within the company to connect  among themselves and others, thus, avoiding isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Celebrate diversity and inclusion, improving the visibility  of LGBTQIA+ professionals in the company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Acknowledge the role of diversity in developing innova-  tive technologies in modern society, thus, increasing the  interest of equity-deserving groups in software engineer-  ing, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  We understand that most of these recommendations only  apply to organizations that are already LGBTQIA+ friendly  and many software companies still discriminate against  LGBTQIA+ individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' We expect that the findings presented  in this research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' in particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' the discussions on how diversity  is essential to the development of software for our modern  8  Job  Opportunities  Diversity in  Physical  Safety  Software  Engineering  Toxicity  LGBTQIA+ Visibility  Avoidance  in Software  Engineering  +  Engagement  Remote Work  Inclusion in  Software  Identity  Engineering  Disclosure  Control  Self-distance  Isolation  EDI Organizational  Strategiessociety help policymakers to build strategies to change this  reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Limitations  The main limitation of this study is the number of partici-  pants from whom we were able to collect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Investigating  a very particular hidden population, such as LGBTQIA+ soft-  ware professionals, is challenging because many individuals  avoid exposure, mainly because software engineering is a  predominantly heterosexual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' This limitation was  demonstrated when 22 individuals quit the study right after  completing the screening part of our questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' We tried  to work around this limitation by following the all is data  principle of grounded theory [23], opening our minds to collect  data both completely anonymously using a questionnaire and  subsequently interviewing those who provided their contact  for further discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' This allowed us to have a sample of  57 individuals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' however, we acknowledge that our findings  are based on the experience of a limited number of software  professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Another limitation is related to regional aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' As 77%  of our sample comprises participants from Brazil, a great  deal of the experience of these participants is associated with  their local context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Although software teams usually apply  similar processes across the globe, and even though most of  our participants have international clients, which brings them  close to other cultures, several aspects of equity, diversity, and  inclusion are particularly linked to organizational behaviors  that are more influenced by regional factors and less influ-  enced by software development processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Therefore, the local  context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', the participants’ country) remains a limitation  when EDI in the software industry is under investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  We tried to work around this limitation by having a version  of the questionnaire built in English and advertised across  social media and international forums of software profession-  als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' However, the participation of individuals from multiple  countries remained low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Finally, regarding the quality aspects of our method and  to avoid threats to the study’s validity in terms of credibility,  originality, resonance, and usefulness, we followed Charmaz’s  [25] criteria to evaluate grounded theory studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' To support  the quality and credibility of our findings, we provided direct  quotations from questionnaires and interviews to illustrate  the interpretation of our participants’ experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' This inter-  pretation was accessed by conducting member-checking with  three interviewees who agreed to participate by commenting  about the obtained categories after a brief presentation and  explanation of the findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Future Work  Equity, diversity, and inclusion in software engineering  are topics with many opportunities for investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' There  are many gaps about this theme in the software industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Additionally, it also requires investigations in the academic  context, especially on how to increase diversity and inclusion  in software engineering courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Regarding our findings, the immediate future works related  to them are:  Conduct an in-depth investigation of the factors revealed  in this study and their effects on several aspects of  software development, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', the relation of diversity and  team conflicts, how diversity could improve team re-  silience, and the relationship among diversity, inclusion,  and software practices, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Work on the generalization of the theory by conducting  quantitative studies based on a worldwide survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Develop investigations on the perspective of heterosex-  ual professionals about EDI and its impacts on soft-  ware development since people who are not part of the  LGBTQIA+ community demonstrated interest in partici-  pating in the present research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Investigate strategies to improve equity in the software  industry aiming to increase the access of LGBTQIA+  professionals to job opportunities in the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Finally, we are interested in exploring EDI strategies that  are being successfully applied in other industries to determine  how they can be transferred to software companies by applying  transformative research methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', action research and  design science [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' CONCLUSION  The present study investigated the effects of remote work  structures on LGBTQIA+ software professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Using a  grounded theory approach, we explored the experience of  57 individuals from different groups within the community,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=', lesbians, gays, bisexuals, and asexuals, including both  cisgender and transgender people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  In summary, we concluded that remote work has a crucial  role in increasing diversity and inclusion in the software indus-  try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Remote work allows LGBTQIA+ software professionals to  access a variety of job opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Remote work also allows  these professionals to have more control over their identities  and their interaction with other professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  Regarding limitations, remote work might create barriers for  LGBTQIA+ software professionals, as it can increase isolation  and self-distancing among these individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Practices devel-  oped by software companies to increase LGBTQIA+ visibility  among employees are essential to reduce these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  No aspect related to equity was observed in this study,  which indicates a gap that requires immediate investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  However, we generally concluded that remote work is positive  for LGBTQIA+ software professionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Our analysis also  demonstrates that software development can benefit signifi-  cantly from more diverse teams, including the improvement  of aspects related to innovation, problem-solving, teamwork,  and team resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  DATA AVAILABILITY  Supplementary material is available on Figshare: https:  //figshare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='com/account/home#/projects/157305  9  ACKNOWLEDGMENTS  The authors would like to thank all of the participants who  participated in this study, and for all LGBTQIA+ software  professionals out there, we would like to say that we see you.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  You are not alone!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  REFERENCES  [1] JIBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' (2022) Equity, diversity, inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Available: https:  //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='jibc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='ca/teaching-learning/equity-diversity-inclusion-edi  [2] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
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+page_content=' Journal of Managerial Psychology, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  [39] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Sinton, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Baines, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Thornalley, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Ilangovan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Kurt,  “Increasing the visibility of lgbtq+ researchers in stem,” The Lancet, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  397, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' 10269, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' 77–79, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content='  [40] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
+page_content=' Wohlin and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TtE5T4oBgHgl3EQfAg5m/content/2301.05379v1.pdf'}
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+DEEP RESIDUAL COMPENSATION CONVOLUTIONAL NETWORK
+WITHOUT BACKPROPAGATION
+A PREPRINT
+Mubarakah M. Alotaibi
+Department of Computer Science
+University of York, Taif University
+York, UK, Taif, Saudi Arabia
+mmma512@york.ac.uk
+Richard C. Wilson
+Department of Computer Science
+University of York
+York, UK
+richard.wilson@york.ac.uk
+January 30, 2023
+ABSTRACT
+PCANet and its variants provided good accuracy results for classification tasks. However, despite the
+importance of network depth in achieving good classification accuracy, these networks were trained
+with a maximum of nine layers. In this paper, we introduce a residual compensation convolutional
+network, which is the first PCANet-like network trained with hundreds of layers while improving
+classification accuracy. The design of the proposed network consists of several convolutional
+layers, each followed by post-processing steps and a classifier. To correct the classification errors
+and significantly increase the network’s depth, we train each layer with new labels derived from
+the residual information of all its preceding layers. This learning mechanism is accomplished
+by traversing the network’s layers in a single forward pass without backpropagation or gradient
+computations. Our experiments on four distinct classification benchmarks (MNIST, CIFAR-10,
+CIFAR-100, and TinyImageNet) show that our deep network outperforms all existing PCANet-like
+networks and is competitive with several traditional gradient-based models.
+Keywords PCANet · DCTNet · DCCNet · CCANet · OSNet · LDANet · Multi-layer PCANet · classification
+1
+INTRODUCTION
+1.1
+Background and Related Works
+Deep learning is a non-task-specific learning technique that uses hierarchical structures to automatically learn represen-
+tations from raw data [1]. Since Alexnet [14] won the 2012 ImageNet challenge, convolutional neural networks (CNNs)
+have been widely used for image classification with great success. VGG [19], ResNet [10] and Network in network
+[16] are a few examples of standard CNNs. Despite their success, CNNs are usually trained with a large number of
+parameters, requiring intensive parameter updates and a lot of data for training, which might increase the computational
+cost even with GPU-equipped computing resources [8].
+Chan et al. [3] presented PCANet as an alternative to deep CNNs for classification tasks on small datasets. The network
+structure comprises two cascaded principal component analysis (PCA) layers followed by binary hashing, block-wise
+histogram and a classifier. Unlike CNNs, PCANet learns its filter bank in a non-iterative layer-by-layer fashion through
+PCA applied to image-based patches. This training mechanism provides a faster training time advantage to PCANet
+over conventional CNNs. With its simple architecture, PCANet has achieved state-of-the-art performance across several
+datasets, including MNIST, extended Yale B, AR and FERET.
+The success of PCANet has inspired a family of related strategies. Generally speaking, PCANet-related research has
+fallen into one of two categories: those focusing on improving the features representation process or those attempting to
+increase network depth. The work in this paper fits within the second category.
+arXiv:2301.11663v1  [cs.CV]  27 Jan 2023
+
+arXiv Template
+A PREPRINT
+In an effort to enhance PCANet’s feature representation, several articles studied networks that maintained PCANet’s
+fundamental structure while producing different features using different convolutional filters. DCTNet [18], CCANet
+[20] and ICANet [22] are examples of PCANet-like networks that create their filter banks using unsupervised approaches.
+The LDANet [3], DCCNet [6] and OSNet [7] are a few examples of those that use supervised approaches to produce
+their filter banks. DFSNet [8] is a good example of a network that uses semi-supervised filters.
+In order to increase the network depth, several studies have attempted to address PCANet’s primary issue, the features
+explosion problem, which limits its depth to only two layers. The block-wise histogram is one factor that contributes to
+this problem, as the number of bins required to calculate the histogram features grows exponentially with the number of
+filters in the second stage. Fan et al. [5] significantly reduced PCANet’s features by replacing the histogram pooling
+with second-order pooling. The per-channel convolution mechanism is another factor that contributes to the increasing
+dimensionality problem in PCANet. PCANet+ [17] provided an alternative solution to this issue by proposing a filter
+ensemble mechanism that aggregates the feature maps over all channels, similar to CNNs. By adopting CNN-like
+filters, as in [17], replacing PCANet’s binarisation step with the z-score method and using second-order pooling and late
+fusion, Alotaibi and Wilson [2] were able to expand PCANet depth to nine layers and outperform the original design.
+In contrast to prior studies that aimed to increase network depth by tackling the dimensionality issue, our objective is to
+explore the design of a hundreds-layer PCANet-like network by optimising the classification process at each layer.
+1.2
+Proposed Network
+In this article, we present a residual compensation convolutional learning framework to achieve accuracy from a
+considerably increased depth while simultaneously correcting network errors as we traverse it. The proposed network
+inherits the simplicity of PCANet-like networks and is trained in a single forward-pass without gradient computations
+or backpropagation. A comprehensive description of the network architecture, along with its training procedures, is
+described in Section 2. The experimental section (3) consists of three subsections. In Section 3.2, we evaluate the
+performance of the proposed network on MNIST, CIFAR-10, CIFAR-100 and TinyImageNet against gradient-based
+and non-gradient architectures without data augmentation. Section 3.3 examines the influence of several network
+parameters, such as the number of filters (3.3.1) and the learning rate (3.3.2), on the accuracy of the proposed model.
+The final subsection (Section 3.4) addresses the use of data augmentation to improve the model’s accuracy. Finally, the
+conclusions and future works are discussed in Section 4.
+2
+NETWORK ARCHITECTURE
+2.1
+Problem Formulation
+Consider a classification problem with N training images X(1) ∈ Rm×n×d×N, where m and n are the images’ spatial
+dimensions, and d ∈ {1, 3} is the number of channels for greyscale or colour images. Let T represent the C classes to
+which the images originally belonged. The objective of our deep residual compensation convolution network (ResCNet)
+is to construct a PCANet-like network that achieves high accuracy from a considerable network depth and is trained
+without using gradient descent or backpropagation. Precisely, the network structure should be structured hierarchically
+such that, as network depth increases, succeeding layers compensate for the classification errors of previous layers. By
+combining the predicted probabilities of the network layers, the model should obtain a high accuracy.
+2.2
+Design Overview
+The network architecture, as shown in Figure 1, consists of multiple convolutional layers, each followed by post-
+processing steps and a linear discriminant analysis (LDA) classifier. Each convolutional layer, except the first layer,
+receives as input the concatenation of its previous layer’s outputs and the original features. After post-processing the
+feature maps of the first layer, an LDA classifier is trained to categorise the extracted features using the original classes.
+By contrast, the LDA classifiers of the subsequent layers, referred to as compensation layers, are trained using new
+classes learnt from the residual information of the preceding layers. In fact, each layer produces two outputs ( ˜Y (i) and
+T (i+1) in Figure 1, where i represents the ith layer). The first provides the predicted probabilities of the network at
+that layer, whereas the second represents the new classes produced for training the subsequent layer. To produce the
+network’s outputs in each layer ( ˜Y (i) in Figure 1), the predicted probabilities of that layer are added to or subtracted
+from those of its preceding layers. This combination mechanism maintains the network’s predicted probabilities of
+being in the range of between 0 and 1 and reduces the error of preceding layers. Section 2.3 describes in detail the
+network’s components, including the convolutional layers (Section 2.3.1), post-processing procedures (Section 2.3.2)
+and residual mechanism (Section 2.3.3).
+2
+
+arXiv Template
+A PREPRINT
+filters
+filters
++
+filters
++
+Post-processing
+Residual
+mechanism 
+Post-processing
+Post-processing
+Residual  
+mechanism 
+Residual  
+mechanism 
+Figure 1: The deep residual compensation convolutional network architecture. T (1) represents the original classes.
+2.3
+Network Components
+2.3.1
+Convolutional Layers
+Let X(1) ∈ Rm×n×d×N represent N training images, T is their C classes, and O(i) ∈ Rm×n×di×N represents the
+feature maps produced by the ith layer. The i + 1 layer receives as input the output of the ith layer concatenated with
+the original features; this input is represented as X(i+1) ∈ Rm×n×(d+di)×N. We then divide the input images (X(i+1))
+into patches of k × k size and a stride of 1 pixel, where k is the filter size and is a user-predefined parameter. The
+resulting matrix is P ∈ Rk2(di+d)× ˜m˜nN, where ˜m = (m − k) + 1, ˜n = (n − k) + 1 and m and n are the width and
+the height of the images, respectively. The filter learning process relies on applying any non-gradient-based method to
+the extracted local patches P and collecting their weights to represent the filters used in the i + 1 layer. The PCA filter
+bank by Low et al. [17] is an example of such filters. The authors first centralised the extracted patches to their mean to
+obtain ¯P ∈ Rk2(di+d)× ˜m˜nN. They then applied PCA to the centralised patches, where the principal components of
+¯P ¯P T can be computed by solving the following optimisation problem:
+min
+V ∈R(k2)×(di+d) || ¯P − V V T ¯P||2
+F ,
+s.t. V T V = I,
+(1)
+where I represents the identity matrix. The convolutional output of the i + 1 layer can then be expressed as follows:
+O(i+1) = Xi+1 ∗ W (i) ∈ Rm×n×di+1,
+(2)
+where di+1 represents the number of filters in layer i + 1, X(i+1) is the input images that are zero padded to obtain the
+same image size outputs, ∗ represents the convolution operation, and W (i) denotes the PCA filters expressed as follows:
+W (i) =
+mat
+k×k×(di+d)qs, s = 1, 2, . . . , di+1,
+(3)
+where qs is the sth principal eigenvector of ¯P ¯P T .
+The stacked-LDA filter bank is another example of filters generated without relying on gradient descent or backpropaga-
+tion. Appendix A discusses these filters in detail. These filters are computed using an iterative process that involves
+selecting a subset of the localised patches P and then applying an LDA classifier to train the selected patches with
+their classes. The algorithm searches for patches with separable classes and accumulates their weights, which are
+subsequently used as stacked-LDA filters.
+This article focuses mainly on the network architecture rather than investigating the filter type used. In our experiments
+(Section 3), we use semi-supervised stacked-LDA filters created by combining 50% of the supervised stacked-LDA
+filters with 50% of the unsupervised PCA filters. However, different filter types can be employed, such as those used by
+Ng and Teoh [18], Yang et al. [20] and Gatto et al. [8].
+3
+
+arXiv Template
+A PREPRINT
+2.3.2
+Post-processing Steps
+The feature maps of each convolutional layer are post-processed using a ReLU non-linear activation function, followed
+by second-order pooling and a multi-level spatial pyramid pooling. The ReLU function is applied to the feature maps
+of each layer but not between the layers. The feature maps are then pooled locally using the second-order pooling
+mechanism described by Alotaibi and Wilson [2]. Let X(i)
+j
+∈ Rm×n×di denote the jth activations map in the ith
+layer, where m and n are the spatial dimensions of the images, and di represents the number of filters in layer i. The
+calculation of the second-order pooling starts by dividing the tensors of X(i)
+j
+into patches of the same size, which could
+be overlapped, e.g. (r × c). Each of these patches is then normalised using the z-score method, defined as z = x−µ
+σ ,
+where µ and σ represent the mean and the standard deviation of the data. Next, the channel-wise covariance matrix of
+each patch, after reshaping each of them to rc × di, is computed. Because of the symmetry property of the covariance
+matrix, the number of second-order features is calculated as the number of patches×( di×di
+2
++ di
+2 ). The multi-level
+spatial pooling is then used to pool the second-order features. The multi-level spatial pyramid pooling calculation is
+identical to that implemented by Chan et al. [3]. Again, different post-processing procedures can be utilised; however, in
+our experiments (Section 3), we found these steps to be the most effective for achieving high accuracy in the databases
+we used.
+2.3.3
+Residual Mechanism
+This mechanism is non-iterative; we add layers sequentially. Each layer is trained in a single pass with new labels learnt
+from the residual information of all its previous layers. The first layer’s features are classified using an LDA classifier
+trained with the original classes. To produce the first layer’s posteriors ( ˜Y (1) in Figure 1), we use the following sigmoid
+function on the output of the LDA classifier:
+S(x) =
+1
+1 + e−x/σ ,
+(4)
+where σ is the sigmoid scale parameter. To identify the new labels required to train the second layer (T (2) in Figure 1),
+we first find the residual errors between the current layer’s predicted outputs ( ˜Y (1)) and the original classes (Y ∈ RN×C)
+in one-hot encoding, as follows:
+R(1) = λY − ˜Y (1),
+(5)
+where 0 ≤ λ ≤ 1 controls the maximum likelihood a class may attain. T (2) can then be defined as the classes with the
+maximum absolute residual errors, as follows:
+T (2)
+i
+= class(|R(1)
+ij |, ∀j)
+i = [1, . . . , N],
+(6)
+where class(x) denotes the name of the class whose value has the largest residual error magnitude. Since our network
+is developed primarily for classification tasks, it concentrates on labelling rather than regressing the correction of
+posteriors.
+The second layer, as shown in Figure 1, receives an input X(2) ∈ Rm×n×(d+d1)×N that is a concatenation of the first
+layer’s feature maps and the original images. After finding the second layer’s features, our objective is to learn a
+correction term based on the second layer classification results, which is then added to the posteriors of the first layer
+( ˜Y (1)) to provide more accurate predictions ( ˜Y (2)). The correction term can either be positive or negative to maintain
+the network probabilities at the second layer ( ˜Y (2)) to have values in the range of 0 to λ (Equation 5). In other words,
+after training the second layer’s LDA classifier using the new classes (T (2)), the posteriors acquired can be added to or
+subtracted from the first layer’s posteriors to correct them. The indicator variable (s(2)) indicates whether to add to or
+subtract from the first layer’s posteriors and is defined as the signum function of the maximum absolute residual errors
+of the previous layers, as follows:
+s(2)
+i
+= sign(R(1)
+i∗ ),
+i∗ = argmaxi |R(1)
+i |.
+(7)
+When the indicator values are positive, it indicates that the probabilities of the first and second layers are added,
+and when they are negative, the probabilities of the second layer are subtracted from those of the first. Consider a
+classification task of three classes A, B and C. Assume that, given a single image whose actual class is A, the predicted
+probabilities of the first layer using this image are 0.4, 0.6 and 0 for the three classes A, B and C, respectively. To add a
+second layer, the residual error using λ = 0.8 (Equation 5) is computed as 0.4, −0.6 and 0 for each of the three classes.
+Hence, the new label to train the second layer for this image is B, as it has the largest magnitude, and the indicator
+is −1 because the maximum absolute residual error has a negative sign. Assume that the second layer’s features are
+trained using class B and provided perfect prediction with probabilities of 0, 1 and 0 for the three classes. Because
+4
+
+arXiv Template
+A PREPRINT
+the indicator value is negative, we subtract them from the predicted outputs of the first layer, resulting in 0.4, −0.4
+and 0 for the three classes. Consequently, we reduce the error, and the predicted class of the second layer is A, which
+corresponds to the actual class of the image. To implement that and generalise it for the test set, as we do not know the
+classes, we divide the database into positive and negative samples based on their indicator variable values. The positive
+samples have positive indicator values, whereas the negative samples have negative indicator values. We then train two
+LDA classifiers for each layer; one is trained on the negative samples using {T (2)
+n
+⊂ T (2) : s(2) = −1}, and the other
+is trained with the positive samples and their classes {T (2)
+p
+⊂ T (2) : s(2) = 1}. The negative classifier is trained by
+assuming that each class in the positive samples is a negative class (its class is zero). Similarly, during the training of
+the positive classifier, each class in the negative samples is treated as a negative class. Thus, both classifiers have access
+to all training data. The network’s outputs at the second layer ( ˜Y (2)) can then be expressed as follows:
+˜Y (2) = ˜Y (1) + α[np
+N
+˜Yp
+(2) − nn
+N
+˜
+Yn
+(2)],
+(8)
+where ˜
+Yn
+(2) and ˜Yp
+(2) are the N predictions made by the classifiers trained on negative and positive samples, nn and
+np denote the number of negative and positive examples, respectively, N is the total number of training samples and α
+is a learning rate. The learning rate, similar to that used in neural networks, is introduced to reduce oscillations and
+provide faster convergence. However, the learning rate in our network also acts as a weight to integrate the probabilities
+of multiple layers, similar to weighted sum techniques.
+To add more layers, we repeat the steps used to add the second one. Algorithm 1 summarises the procedures of the
+network’s training with L layers, assuming that the second-order features are known. To add layer i, the new labels T (i)
+and indicator variable (s(i)) are computed based on the previous layer’s residual error R(i−1), as shown in Equation 10.
+The network’s output at that layer can then be defined using Equation 11. In general, for deeper residual compensation
+layers, the new classes (T (L)) learnt from the residual errors of the previous layers can be defined as follows:
+T (L) = class(|R(L−1)|)
+= class(|Y − ˜Y (L−1)|)
+= class(|Y − [ ˜Y L−2 + α
+N (n(L−2)
+p
+˜Y (L−2)
+p
+− n(L−2)
+n
+˜Y (L−2)
+n
+)])
+= ...
+= class(|Y − [ ˜Y (1) + α
+N
+L−1
+�
+i=2
+(ni
+p ˜Y i
+p − ni
+n ˜Y i
+n)]),
+(9)
+where class(x) is the name of the class whose value has the largest residual error magnitude, Y is the original classes in
+one-hot encoding, ni
+p and ni
+n are the number of positive and negative samples in layer i, respectively, and N denotes
+the total number of samples in the database.
+3
+EXPERIMENTS
+3.1
+Databases
+We used four standard benchmarks in our experiments: CIFAR-10 [13], CIFAR-100 [13], MNIST [4] and TinyImageNet
+[15]. The MNIST database comprises 60,000 training examples and 10,000 test images of size 28 × 28, drawn from the
+same distribution, normalised and centred in a fixed-size image. The CIFAR-10 database consists of 10 classes with
+50,000 images for training and 10,000 test images. The images of size 32 × 32 × 3 have a low resolution with different
+poses and angles. CIFAR-100 is similar to CIFAR-10 but with 100 classes. The TinyImageNet database consists of
+100,000 training images of size 64 × 64 × 3. The images are divided into 200 categories, with 500 images each. The
+validation and the test sets contain 10,000 images each, with 50 images per class. The test set is not labelled, and our
+experiments’ performance is reported on the validation set.
+3.2
+Image Classification without Data Augmentation
+In this section, we evaluate the proposed network on the MNIST, CIFAR-10, CIFAR-100 and TinyImageNet databases
+(Section 3.1) without data augmentation. To determine the network parameters, we examine a variety of configurations,
+each of which has different parameters, and report the results of the configuration that works the best.
+5
+
+arXiv Template
+A PREPRINT
+Algorithm 1 Deep Residual Compensation Convolutional Network Training
+Input: Second-order features: {F (i), i = [1, 2, . . . , L]}, L number of layers, C classes T (1) ∈ RN×1, learning rate:
+α and λ to determine the highest probability a class can reach.
+Output: Model’s accuracy: accuracy
+1: Generate Y ∈ RN×C, the one-hot encoding of T (1).
+2: i ← 1 and ˜Y (0) ← 0.
+3: s(i) = 1N×1 {fill the first layer’s indicator variable with 1. }
+4: while i < L do
+5:
+if i > 1 then
+6:
+Find the residual errors, new classes and indicator variable, as follows:
+R(i−1) = λY − ˜Y (i−1).
+T (i)
+j
+= class(|R(i−1)
+jk
+|, ∀k),
+s(i)
+j
+= sign(R(i−1)
+j∗
+), j∗ = argmaxj |R(i−1)
+j
+|,
+j = [1, 2, . . . , N].
+(10)
+7:
+end if
+8:
+Find F (i)
+n
+⊂ F (i) and T (i)
+n
+⊂ T (i), for which their indicator values are negative.
+9:
+Find F (i)
+p
+⊂ F (i) and T (i)
+p
+⊂ T (i), for which their indicator values are positive.
+10:
+if T (i)
+p
+̸= ∅ then
+11:
+L1 = LDA(F (i)
+p , T (i)
+p ).
+12:
+˜Y (i)
+p
+= prediction(L1, F (i)).
+13:
+else
+14:
+˜Y (i)
+p
+← 0.
+15:
+end if
+16:
+if T (i)
+n
+̸= ∅ then
+17:
+L2 = LDA(F (i)
+n , T (i)
+n ).
+18:
+˜Y (i)
+n
+= prediction(L2, F (i)).
+19:
+else
+20:
+˜Y (i)
+n
+← 0.
+21:
+end if
+22:
+Compute the current’s layer output ˜Y (i), as
+˜Y (i) = ˜Y (i−1) + α[np
+N
+˜Yp
+(i) − nn
+N
+˜
+Yn
+(i)],
+(11)
+where np and nn represent the number of positive and negative samples, respectively.
+23:
+i ← i + 1
+24: end while
+25: Compute the model’s accuracy at layer L using ˜Y (L).
+3.2.1
+Parameter Settings
+The optimal settings identified for the MNIST, CIFAR-10, CIFAR-100 and TinyImageNet databases are listed in Table 1.
+All architectures, except the MNIST database, used 3 × 3-pixel filters created by combining PCA [17] and stacked-LDA
+filters (Appendix A) at a 50% ratio. The MNIST database used 13 × 13 PCA filters in its first layer and 3 × 3 PCA
+filters in its residual layers. In this section and throughout the rest of this article, we used the same number of filters for
+all layers and stopped adding layers when the training error rate approached 0%. Therefore, the number of filters per
+layer is 60 in the MNIST’s architecture, 50 in the CIFAR’s architectures and 40 in TinyImageNet’s architecture. The
+number of layers beyond which an accuracy gain was no longer observed was 937 for the CIFAR-10 database, 436 for
+the CIFAR-100 database, 231 for the MNIST database and 512 for the TinyImageNet database. We utilised 7 × 7-block
+second-order pooling for MNIST with a stride of four pixels, 16 × 16 for CIFAR-100 with a four-pixel stride, 16 × 16
+for CIFAR-10 with a one-pixel stride, and 32 × 32 for TinyImageNet with an eight-pixel stride. In all architectures, we
+pooled the second-order features using three-level spatial pyramid pooling of 4 × 4, 2 × 2 and 1 × 1 subregions. The
+only data preprocessing was min-max normalisation applied to the input of each convolutional layer, and probabilities
+were retrieved from all datasets except MNIST using the sigmoid function with a scale value of 16 (Equation 4). In the
+6
+
+arXiv Template
+A PREPRINT
+MNIST database, we used the following softmax function to generate the probabilities in each layer:
+softmax(yi) =
+exp(βyi)
+�C
+j=1 exp(βyj)
+,
+(12)
+where β is assigned to 0.001, C denotes the number of classes, and y represents the outputs of the LDA classifier.
+During the training phase, λ (Equation 5) was set to 0.8, and the learning rate was fixed at α = 1 for the MNIST
+database and α = 0.4 for the CIFAR-10 database. The remaining databases used an initial learning rate of 1, which was
+dropped by 10% every 10 layers as follows:
+α = α − 10
+100α.
+(13)
+We stopped reducing the learning rate when it reached 0.387 and 0.478 for CIFAR-100 and TinyImageNet databases,
+respectively.
+Table 1: Network architectures using the MNIST, CIFAR-10, CIFAR-100 and TinyImageNet databases
+The MNIST database: 28 × 28 × 1
+Filter size
+SOP
+Output size
+13 × 13 × 1 × 60
+7 × 7, Stride = 4
+28 × 28 × 60
+[3 × 3 × 60 × 60] × 230
+7 × 7, Stride = 4
+28 × 28 × 60
+The CIFAR-10 database: 32 × 32 × 3
+Filter size
+SOP
+Output size
+3 × 3 × 3 × 50
+16 × 16, Stride = 1
+32 × 32 × 50
+[3 × 3 × 50 × 50] × 936
+16 × 16, Stride = 1
+32 × 32 × 50
+The CIFAR-100 database: 32 × 32 × 3
+Filter size
+SOP
+Output size
+3 × 3 × 3 × 50
+16 × 16, Stride = 4
+32 × 32 × 50
+[3 × 3 × 50 × 50] × 435
+16 × 16, Stride = 4
+32 × 32 × 50
+The TinyImageNet database: 64 × 64 × 3
+Filter size
+SOP
+Output size
+3 × 3 × 3 × 40
+32 × 32, Stride = 8
+64 × 64 × 40
+[3 × 3 × 40 × 40] × 511
+32 × 32, Stride = 8
+64 × 64 × 40
+3.2.2
+Performance Analysis
+Table 2 reports the accuracy of ResCNet compared with some gradient-based and non-gradient-based networks on the
+MNIST, CIFAR-10, CIFAR-100 and TinyImageNet databases without data augmentation. The non-gradient-based
+models reported in Table 2 are the best-performing PCANet-like models from the perspective of the datasets we
+considered. Such networks include PCANet [3], LDANet [3], DFSNet [8] and Multi-layer PCANet [2]. To show where
+our network fits in the literature of commonly used networks, we compared the performance achieved by our network
+to those obtained by standard gradient-based convolutional networks. Maxout [9], Network in network [16], stochastic
+pooling [21] and ResNet [10] are the gradient-based networks listed in the table. The accuracy of 164-ResNet with
+pre-activation was reported by Huang et al. [11], while the results of ResNet-18 and ResNet-34 were reported by Jeevan
+[12].
+According to Table 2, for the MNIST database, our model with 231 layers achieved an accuracy of 99.52%, making it
+superior to all non-gradient-based models, such as PCANet, Multi-Layer PCANet and LDANet, in terms of accuracy. It
+improved on the best results of the non-gradient-based networks by roughly 0.12%. In addition, the findings presented
+in the table demonstrated that our network produced results comparable with standard gradient-based networks, such as
+the Maxout network, Network in network and stochastic pooling.
+As shown in Table 2, for the CIFAR-100 database, the accuracy attained by our proposed network was the highest
+among all the networks. The accuracy of 64.91% was around 8% higher than that of Multi-Layer PCANet and stochastic
+pooling, more than 9% higher than that of ResNet-110 and ResNet-32, 14% higher than the original PCANet and
+more than 2% higher than that of Maxout and ResNet with stochastic depth. Moreover, the results achieved by our
+network were roughly equivalent to those obtained by ResNet with 164 layers and Network in network using the
+dropout technique.
+For TinyImageNet, as shown in Table 2, our network achieved the highest accuracy among all the networks without any
+data augmentation. This performance was around 2% higher than that of the ResNet-34 model and 1% better than that
+of the ResNet-18 model.
+7
+
+arXiv Template
+A PREPRINT
+According to the results shown in Table 2 for the CIFAR-10 database, our model outperformed PCANet and all of its
+variants, in terms of accuracy. The accuracy of the proposed network was around 10% better than the performance
+achieved by the original PCANet and 6% higher than that of Multi-Layer PCANet. Although ResCNet’s accuracy
+was around 1% lower than that of the Maxout network and 2% worse than that of Network in network, our model
+achieved accuracy comparable with that of ResNet-32, 1% higher than ResNet-18 and ResNet-110, and 3% greater than
+stochastic pooling. In general, ResCNet with more than 900 layers showed an excellent performance of 87.54% on the
+CIFAR-10 database with no data augmentation, making it the first PCANet-like network to reach such a number of
+layers and such a performance.
+In general, Table 2 demonstrates that ResCNet outperformed all PCANet-like networks in terms of accuracy and the
+number of layers required for training. It also shows that our model, which is trained without complicated non-linear
+functions or regularisation techniques, achieves accuracy similar to that of standard convolutional networks such as
+Network-in-Network, Maxout, stochastic pooling, and several residual networks. Even though the number of layers in
+our network is relatively large, these layers are added sequentially, one after the other, without iterations or intensive
+parameter updates. In addition, we used a small number of filters in each layer for all of our architectures, with no
+configuration exceeding 60 filters per layer. To the best of our knowledge, ResCNet is the first non-gradient-based
+propagation-free network to be trained with hundreds of layers.
+Table 2: Accuracy (%) of ResCNet compared with different methods on the CIFAR-10 (C10), CIFAR-100 (C100),
+MNIST and TinyImageNet (T200) databases without data augmentation
+Non-gradient-based networks
+Method
+C10
+C100
+MNIST
+T200
+PCANet-2
+77.14
+51.62
+99.34
+30
+LDANet
+78.33
+–
+99.38
+–
+DFSNet-3
+81.06
+–
+–
+–
+Multi-Layer PCANet
+81.72
+57.86
+99.40
+40.87
+ResCNet (ours)
+87.54
+64.9
+99.52
+44.37
+Gradient-based networks
+Method
+C10
+C100
+MNIST
+T200
+Stochastic pooling
+84.87
+57.49
+99.53
+–
+Maxout network, with dropout
+88.32
+61.43
+99.55
+–
+Network in network, with dropout
+89.59
+64.32
+99.53
+–
+Network in network, without dropout
+85.49
+–
+–
+–
+110 ResNet
+86.82
+55.26
+–
+–
+ResNet stochastic depth
+-
+62.20
+–
+–
+164-ResNet (pre-activation)
+-
+64.42
+–
+–
+ResNet-18
+86.29
+59.15
+–
+43.02
+ResNet-32
+87.97
+56.05
+–
+42.65
+3.3
+Network Parameters
+This section explains how to configure ResCNet’s parameters by analysing their impact on the model’s accuracy using
+the CIFAR-10 database. The experiments in this section are divided into two subsections, as follows:
+• Experiment 1: Testing the effect of the number of filters on the accuracy of ResCNet
+• Experiment 2: Studying the impact of learning rate on the model’s performance
+3.3.1
+Number of Filters
+This experiment aimed to determine how changing the number of filters affects the accuracy of ResCNet. For this, we
+designed two networks with the same settings but different numbers of filters. The first, referred to as ResCNet-30,
+used 30 filters in each of its layers, while the second (ResCNet-50) used 50 filters in all of its layers. The two networks
+followed the same parameters settings as those described in Section 3.2 for the CIFAR-10 database. However, we used
+an 8x8 second-order pooling block size.
+Figure 2 shows the accuracy of training and testing ResCNet-30 and ResCNet-50 on the CIFAR-10 database without
+data augmentation. According to the figure, both networks achieved the same level of accuracy, although ResCNet-30
+required more layers. ResCNet-50 obtained a training accuracy of 100% at layer 127, while ResCNet-30 achieved
+8
+
+arXiv Template
+A PREPRINT
+100% at layer 781. The testing accuracy for both 127- and 781-layer networks was around 86.28%. The best testing
+accuracy of 86.33% was obtained with 100-layer ResCNet-50 and 500-layer ResCNet-30. The findings in this section
+suggested that, unlike previous PCANet-like networks in which the number of filters in each layer should be determined
+in advance, we could obtain the same performance by employing any number of filters based on the available resources.
+However, we needed more filters in each layer to achieve the desired performance faster. The results in this section also
+showed that the second-order pooling block size affected the accuracy of ResCNet. For instance, the highest accuracy
+reached for the CIFAR-10 database (87.54% in Section 3.2) was around 1% better than that achieved in this section
+(86.33%), with the only difference between the models being the modification of the second-order pooling block size.
+0
+200
+400
+600
+800
+Layers
+75
+80
+85
+90
+95
+100
+Accuracy (%)
+ResCNet-30
+Training
+Testing
+0
+50
+100
+Layers
+75
+80
+85
+90
+95
+100
+Accuracy (%)
+ResCNet-50
+Training
+Testing
+Figure 2: The accuracy (%) of ResCNet-30 and ResCNet-50 on the training and testing sets of the CIFAR-10 database
+with no data augmentation.
+3.3.2
+Learning Rate
+This experiment aimed to show the impact of the learning rate (α in Equation 9) on ResCNet’s accuracy. Similar to
+neural networks, the learning rate is introduced to prevent oscillations and promote faster convergence. We designed
+two networks with the same parameters but different learning rates and evaluated them on the CIFAR-10 database. Both
+networks used 50 filters in all of their layers. The first network (ResCNet-50–1) was trained with a learning rate of
+α = 1, whereas the second (ResCNet-50-0.4) was trained using α = 0.4. The other network’s parameters were the
+same as those described in Section 3.2 for the CIFAR-10 database.
+Table 3 compares the accuracy of ResCNet-50–1 with that of ResCNet-50–0.4 using a different number of layers on the
+CIFAR-10 database. The training accuracy of ResCNet-50–1 reached 100% at layer 384, with a testing accuracy of
+86.91%. On the other hand, ResCNet-50–0.4 obtained a 100% training accuracy at layer 970, with a testing accuracy
+of 87.41%. The optimal testing accuracy of ResCNet-50–1 was achieved at layer 208, with 87.2% testing accuracy,
+while ResCNet-50–0.4 obtained its best performance of 87.54% at layer 937. The results achieved in this section
+demonstrated that the learning rate α = 0.4 was small, as we needed more than 600 layers for the network to reach its
+convergence. If the training error rate oscillates, which is not the case in CIFAR-10, the learning rate must be decreased
+to avoid oscillations.
+3.4
+Image Classification with Data Augmentation
+This section aims to enhance the classification accuracy of ResCNet using data augmentation on three databases, namely,
+CIFAR-10, CIFAR-100 and TinyImageNet. The ResCNet architectures in this section shared the same design and
+parameters as those in Section 3.2 but had 490, 507 and 480 layers for the CIFAR-10, CIFAR-100 and TinyImageNet
+9
+
+arXiv Template
+A PREPRINT
+Table 3: Accuracy (%) on the CIFAR-10 database test set using ResCNet-50–1 and ResCNet-50–0.4
+Network
+# Layers
+Accuracy (%)
+ResCNet-50–1
+348
+86.91
+ResCNet-50–1
+208
+87.02
+ResCNet-50–0.4
+970
+87.41
+ResCNet-50–0.4
+937
+87.54
+ResCNet-50–0.4
+600
+87.09
+databases, respectively. For CIFAR-10, we modified the pooling stride to four pixels and re-implemented a 193-layer
+ResCNet without data augmentation for adequate comparison.
+Table 4 compares the accuracy of ResCNet with and without data augmentation on the CIFAR-10, CIFAR-100 and
+TinyImageNet databases. With horizontal flipping being the only data augmentation used, the accuracy of ResCNet was
+enhanced across all three databases. This improvement was around 1% for the TinyImageNet database, 2% for the
+CIFAR-10 database and 3% for the CIFAR-100 database. For the CIFAR-10 database, the accuracy achieved with data
+augmentation was also 1% higher than the best result (87.54%) reported in Section 3.2.
+This section’s results showed the importance of data augmentation for improving the model’s generalisation and
+accuracy. Incorporating other data augmentation types are needed to increase the model’s accuracy further. Since
+ResCNet is not currently being trained in a batch-based manner, adding more data augmentation types is challenging as
+they need to be computed in advance. In future works, we will investigate the possibility of converting the current work
+into a batch-based system as in gradient-based models.
+Table 4: Accuracy (%) of ResCNet on the CIFAR-10, CIFAR-100 and TinyImageNet databases
+Data augmentation
+CIFAR-10
+CIFAR-100
+TinyImageNet
+
+86.82
+64.9
+44.37
+
+88.35
+67.8
+45.91
+4
+CONCLUSIONS
+In this article, we proposed ResCNet, a PCANet-like network that trains each layer with new labels derived from the
+residual data of all preceding layers. Our proposed network increased the network depth to more than 950 layers, making
+it the first non-gradient-based propagation-free network to achieve this number. Moreover, ResCNet’s performance was
+comparable to that of standard gradient-based models and superior to PCANet and all of its variants. Increasing the size
+of the databases by one type of data augmentation resulted in a considerable improvement in accuracy, particularly in
+CIFAR-100, where it reached 3%. However, increasing the number of samples leads to higher computational costs. To
+overcome this issue and in future work, we will investigate the possibility of transforming the current network into a
+batch-based system, similar to the neural network. In addition, the network may be developed further by modifying the
+filter types or by using other metrics to define the residual errors and, thus, the new classes.
+References
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+(SSPR), pages 292–301. Springer, 2021.
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+Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 4700–4708, 2017.
+[12] P. Jeevan. Convolutional xformers for vision. arXiv Preprint arXiv:2201.10271, 2022.
+[13] A. Krizhevsky, G. Hinton, et al. Learning multiple layers of features from tiny images. 2009, 2009.
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+recognition. EURASIP Journal on Image and Video Processing, 2018(1):1–7, 2018.
+11
+
+arXiv Template
+A PREPRINT
+A
+STACKED-LDA
+In this appendix, we introduce the stacked-LDA method, a model that stacks two Linear discriminant analysis layers.
+The first layer of the model is trained using new labels that are produced by clustering similar instances of a certain class.
+The second layer, on the other hand, is trained using the original classes. This general concept of the stacked-LDA
+model is described in Section A.1. We propose an iterative method to create clusters of similar groups. The description
+of our method is explained in Section A.2. In addition, Section A.3 discusses the procedures for using the stacked-LDA
+algorithm as convolution filters. Finally, the parameter settings we used in our experiments in the main paper were
+described in Section A.4.
+A.1
+General Concept
+The stacked-LDA algorithm relies on stacking two linear discriminant analysis (LDA) layers. The first LDA layer is
+trained using labels created from the original classes, while the second LDA layer is trained using the actual labels.
+To generate the classes of the first layer, similar samples from a given class are grouped to create a new class. For
+example, a class A with s instances can be subdivided into c new classes, each with a different number of instances. The
+minimum number of samples needed to represent a class is one. Suppose the first LDA can differentiate between the
+new classes ideally. In that case, the second LDA will receive the posteriors of the first LDA and be able to differentiate
+between the actual classes. For instance, if class A is subdivided into A1 and A2, the subsequent LDA will identify that
+both A1 and A2 are members of class A.
+A.2
+Stacked-LDA Algorithm
+Let X ∈ RN×M represent N training samples, each with M dimensions, and Target ∈ RN×1 represent their original
+classes. Algorithm 2 describes the procedures for applying the stacked-LDA to the training set. The algorithm starts by
+picking a random class c from the actual classes. We then choose Npositive random instances that belong to class c
+and Nnegative random examples that are not in class c, where Npositive and Nnegative are the number of positive and
+negative samples and are user-predefined parameters. Next, an LDA classifier discriminates between the positive and
+the negative samples. After that, we check if our chosen random samples are linearly separable, which can be done by
+comparing the error rate of the LDA classifier with a small value of nearly zero called tolerance (tol) and is chosen by
+the user. If the LDA’s error rate is lower than the tolerance, we consider the positive class to be a new class and collect
+the weights of the LDA. On the other hand, if the LDA’s error rate is greater than the tolerance, the chosen samples
+are not similar and cannot be grouped. Therefore, the algorithm proceeds to find other classes that separate the data
+accurately in the same way until reaching the required number of classes (N_classes). When algorithm 2 terminates,
+we can use the generated LDA’s weights to find the output of the first LDA. Another LDA can then be applied to the
+output of the first LDA to classify them back using the original classes.
+A.3
+Stacked-LDA Filters
+For N training samples X: {Xi ∈ Rm×n×c} with actual classes Target ∈ RN×1, where m and n are the spatial
+dimensions of the image and c is the number of channels, we compute the stacked-LDA filters as follows:
+1. Given a single image Xi ∈ Rm×n×c and a filter size kL × kL, we extract and vectorise all overlapping
+patches of size kL × kL × c each. The resulting matrix is Pi ∈ R(k2
+L×c)× ˜m˜n, where ˜m = (m − kL) + 1,
+˜n = (n − kL) + 1 and m and n are the spatial dimensions of the image, respectively;
+2. We repeat the previous step for all images in the dataset to obtain P ∈ R(k2
+L×c)× ˜m˜nN;
+3. We create a vector T ∈ R1× ˜m˜nN that contains the class labels of the patches. A single patch is assigned a
+label equivalent to the class of its full image;
+4. Using random samples and specific tolerance, we apply Algorithm 2 on P and T to obtain the stacked-LDA
+filters’ weights W L
+s and bias BL
+s ;
+5. We can express the Stacked-LDA filters as follows:
+W L
+s =
+mat
+kL×kL×cqs, s = 1, 2, . . . , dL,
+BL
+s = mat
+1×1×cqs, s = 1, 2, . . . , dL,
+(14)
+where dL is the number of filters chosen by the user, which is equivalent to the number of classes in Algorithm
+2;
+12
+
+arXiv Template
+A PREPRINT
+Algorithm 2 Stacked-LDA Algorithm
+Input: Training set: X ∈ RN×M, where {xN
+i , xi ∈ RM}, original classes: Target ∈ RN×1, number of classes user
+wants to generate: N_classes, number of positive samples: Npositives, number of negative samples: Nnegatives
+and tolerance of performance user can afford: tol
+Output: LDA’s weights: weights ∈ RN×N_classes and LDA’s bias or constant: bias ∈ RN_classes
+1: weights ← [ ].
+2: bias ← [ ].
+3: i ← 1.
+4: while i < N_classes do
+5:
+Pick a random class c from the Target.
+6:
+Pick random Npositives samples from class c (Spositives).
+7:
+Choose Nnegatives samples that are not in class c (Snegatives) randomly.
+8:
+Combine the negative and positive samples: S ← [Spositives, Snegatives].
+9:
+T ← [ones(Npositives), zeros(Nnegatives)].
+10:
+Find the linear discriminant analysis (LDA) between S and T: L = LDA(S, T).
+11:
+Find the perfromance (ErrorRate) of S using L.
+12:
+if ErrorRate < tol then
+13:
+weights ← [weights, LDA′sweights].
+14:
+bias ← [bias, LDA′sbias].
+15:
+i ← i + 1.
+16:
+end if
+17: end while
+6. We convolve the original images with the filters as follows:
+XL
+i = XL−1
+i
+∗ W L
+s + BL
+s ∈ Rm×n×dL,
+(15)
+where s = 1, 2, . . . , dL and XL−1
+i
+is zero-padded to obtain the same image size;
+A.4
+Parameter Settings
+To compute the stacked-LDA filters in our experiments in the main paper, we set the number of positive samples
+(Npositives) to 2, while the number of negative samples (Nnegatives) was 32. The tolerance (tol) in Algorithm 2 was
+set to zero. We used the LDA classifier with the one-versus-all decomposition method.
+13
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf,len=651
+page_content='DEEP RESIDUAL COMPENSATION CONVOLUTIONAL NETWORK  WITHOUT BACKPROPAGATION  A PREPRINT  Mubarakah M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Alotaibi  Department of Computer Science  University of York, Taif University  York, UK, Taif, Saudi Arabia  mmma512@york.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='uk  Richard C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Wilson  Department of Computer Science  University of York  York, UK  richard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='wilson@york.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='uk  January 30, 2023  ABSTRACT  PCANet and its variants provided good accuracy results for classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' However, despite the  importance of network depth in achieving good classification accuracy, these networks were trained  with a maximum of nine layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In this paper, we introduce a residual compensation convolutional  network, which is the first PCANet-like network trained with hundreds of layers while improving  classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The design of the proposed network consists of several convolutional  layers, each followed by post-processing steps and a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' To correct the classification errors  and significantly increase the network’s depth, we train each layer with new labels derived from  the residual information of all its preceding layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' This learning mechanism is accomplished  by traversing the network’s layers in a single forward pass without backpropagation or gradient  computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Our experiments on four distinct classification benchmarks (MNIST, CIFAR-10,  CIFAR-100, and TinyImageNet) show that our deep network outperforms all existing PCANet-like  networks and is competitive with several traditional gradient-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  Keywords PCANet · DCTNet · DCCNet · CCANet · OSNet · LDANet · Multi-layer PCANet · classification  1  INTRODUCTION  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='1  Background and Related Works  Deep learning is a non-task-specific learning technique that uses hierarchical structures to automatically learn represen-  tations from raw data [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Since Alexnet [14] won the 2012 ImageNet challenge, convolutional neural networks (CNNs)  have been widely used for image classification with great success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' VGG [19], ResNet [10] and Network in network  [16] are a few examples of standard CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Despite their success, CNNs are usually trained with a large number of  parameters, requiring intensive parameter updates and a lot of data for training, which might increase the computational  cost even with GPU-equipped computing resources [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' [3] presented PCANet as an alternative to deep CNNs for classification tasks on small datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The network  structure comprises two cascaded principal component analysis (PCA) layers followed by binary hashing, block-wise  histogram and a classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Unlike CNNs, PCANet learns its filter bank in a non-iterative layer-by-layer fashion through  PCA applied to image-based patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' This training mechanism provides a faster training time advantage to PCANet  over conventional CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' With its simple architecture, PCANet has achieved state-of-the-art performance across several  datasets, including MNIST, extended Yale B, AR and FERET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  The success of PCANet has inspired a family of related strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Generally speaking, PCANet-related research has  fallen into one of two categories: those focusing on improving the features representation process or those attempting to  increase network depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The work in this paper fits within the second category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='11663v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='CV]  27 Jan 2023  arXiv Template  A PREPRINT  In an effort to enhance PCANet’s feature representation, several articles studied networks that maintained PCANet’s  fundamental structure while producing different features using different convolutional filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' DCTNet [18], CCANet  [20] and ICANet [22] are examples of PCANet-like networks that create their filter banks using unsupervised approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  The LDANet [3], DCCNet [6] and OSNet [7] are a few examples of those that use supervised approaches to produce  their filter banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' DFSNet [8] is a good example of a network that uses semi-supervised filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  In order to increase the network depth, several studies have attempted to address PCANet’s primary issue, the features  explosion problem, which limits its depth to only two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The block-wise histogram is one factor that contributes to  this problem, as the number of bins required to calculate the histogram features grows exponentially with the number of  filters in the second stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' [5] significantly reduced PCANet’s features by replacing the histogram pooling  with second-order pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The per-channel convolution mechanism is another factor that contributes to the increasing  dimensionality problem in PCANet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' PCANet+ [17] provided an alternative solution to this issue by proposing a filter  ensemble mechanism that aggregates the feature maps over all channels, similar to CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' By adopting CNN-like  filters, as in [17], replacing PCANet’s binarisation step with the z-score method and using second-order pooling and late  fusion, Alotaibi and Wilson [2] were able to expand PCANet depth to nine layers and outperform the original design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  In contrast to prior studies that aimed to increase network depth by tackling the dimensionality issue, our objective is to  explore the design of a hundreds-layer PCANet-like network by optimising the classification process at each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2  Proposed Network  In this article, we present a residual compensation convolutional learning framework to achieve accuracy from a  considerably increased depth while simultaneously correcting network errors as we traverse it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The proposed network  inherits the simplicity of PCANet-like networks and is trained in a single forward-pass without gradient computations  or backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' A comprehensive description of the network architecture, along with its training procedures, is  described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The experimental section (3) consists of three subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2, we evaluate the  performance of the proposed network on MNIST, CIFAR-10, CIFAR-100 and TinyImageNet against gradient-based  and non-gradient architectures without data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3 examines the influence of several network  parameters, such as the number of filters (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='1) and the learning rate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2), on the accuracy of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  The final subsection (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4) addresses the use of data augmentation to improve the model’s accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Finally, the  conclusions and future works are discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  2  NETWORK ARCHITECTURE  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='1  Problem Formulation  Consider a classification problem with N training images X(1) ∈ Rm×n×d×N, where m and n are the images’ spatial  dimensions, and d ∈ {1, 3} is the number of channels for greyscale or colour images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Let T represent the C classes to  which the images originally belonged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The objective of our deep residual compensation convolution network (ResCNet)  is to construct a PCANet-like network that achieves high accuracy from a considerable network depth and is trained  without using gradient descent or backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Precisely, the network structure should be structured hierarchically  such that, as network depth increases, succeeding layers compensate for the classification errors of previous layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' By  combining the predicted probabilities of the network layers, the model should obtain a high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2  Design Overview  The network architecture, as shown in Figure 1, consists of multiple convolutional layers, each followed by post-  processing steps and a linear discriminant analysis (LDA) classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Each convolutional layer, except the first layer,  receives as input the concatenation of its previous layer’s outputs and the original features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' After post-processing the  feature maps of the first layer, an LDA classifier is trained to categorise the extracted features using the original classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  By contrast, the LDA classifiers of the subsequent layers, referred to as compensation layers, are trained using new  classes learnt from the residual information of the preceding layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In fact, each layer produces two outputs ( ˜Y (i) and  T (i+1) in Figure 1, where i represents the ith layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The first provides the predicted probabilities of the network at  that layer, whereas the second represents the new classes produced for training the subsequent layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' To produce the  network’s outputs in each layer ( ˜Y (i) in Figure 1), the predicted probabilities of that layer are added to or subtracted  from those of its preceding layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' This combination mechanism maintains the network’s predicted probabilities of  being in the range of between 0 and 1 and reduces the error of preceding layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3 describes in detail the  network’s components, including the convolutional layers (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='1), post-processing procedures (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2)  and residual mechanism (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  2  arXiv Template  A PREPRINT  filters  filters  +  filters  +  Post-processing  Residual  mechanism  Post-processing  Post-processing  Residual  mechanism  Residual  mechanism  Figure 1: The deep residual compensation convolutional network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' T (1) represents the original classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3  Network Components  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='1  Convolutional Layers  Let X(1) ∈ Rm×n×d×N represent N training images, T is their C classes, and O(i) ∈ Rm×n×di×N represents the  feature maps produced by the ith layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The i + 1 layer receives as input the output of the ith layer concatenated with  the original features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' this input is represented as X(i+1) ∈ Rm×n×(d+di)×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' We then divide the input images (X(i+1))  into patches of k × k size and a stride of 1 pixel, where k is the filter size and is a user-predefined parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The  resulting matrix is P ∈ Rk2(di+d)× ˜m˜nN, where ˜m = (m − k) + 1, ˜n = (n − k) + 1 and m and n are the width and  the height of the images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The filter learning process relies on applying any non-gradient-based method to  the extracted local patches P and collecting their weights to represent the filters used in the i + 1 layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The PCA filter  bank by Low et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' [17] is an example of such filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The authors first centralised the extracted patches to their mean to  obtain ¯P ∈ Rk2(di+d)× ˜m˜nN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' They then applied PCA to the centralised patches, where the principal components of  ¯P ¯P T can be computed by solving the following optimisation problem:  min  V ∈R(k2)×(di+d) || ¯P − V V T ¯P||2  F ,  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' V T V = I,  (1)  where I represents the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The convolutional output of the i + 1 layer can then be expressed as follows:  O(i+1) = Xi+1 ∗ W (i) ∈ Rm×n×di+1,  (2)  where di+1 represents the number of filters in layer i + 1, X(i+1) is the input images that are zero padded to obtain the  same image size outputs, ∗ represents the convolution operation, and W (i) denotes the PCA filters expressed as follows:  W (i) =  mat  k×k×(di+d)qs, s = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' , di+1,  (3)  where qs is the sth principal eigenvector of ¯P ¯P T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  The stacked-LDA filter bank is another example of filters generated without relying on gradient descent or backpropaga-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Appendix A discusses these filters in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' These filters are computed using an iterative process that involves  selecting a subset of the localised patches P and then applying an LDA classifier to train the selected patches with  their classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The algorithm searches for patches with separable classes and accumulates their weights, which are  subsequently used as stacked-LDA filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  This article focuses mainly on the network architecture rather than investigating the filter type used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In our experiments  (Section 3), we use semi-supervised stacked-LDA filters created by combining 50% of the supervised stacked-LDA  filters with 50% of the unsupervised PCA filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' However, different filter types can be employed, such as those used by  Ng and Teoh [18], Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' [20] and Gatto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  3  arXiv Template  A PREPRINT  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2  Post-processing Steps  The feature maps of each convolutional layer are post-processed using a ReLU non-linear activation function, followed  by second-order pooling and a multi-level spatial pyramid pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The ReLU function is applied to the feature maps  of each layer but not between the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The feature maps are then pooled locally using the second-order pooling  mechanism described by Alotaibi and Wilson [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Let X(i)  j  ∈ Rm×n×di denote the jth activations map in the ith  layer, where m and n are the spatial dimensions of the images, and di represents the number of filters in layer i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The  calculation of the second-order pooling starts by dividing the tensors of X(i)  j  into patches of the same size, which could  be overlapped, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' (r × c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Each of these patches is then normalised using the z-score method, defined as z = x−µ  σ ,  where µ and σ represent the mean and the standard deviation of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Next, the channel-wise covariance matrix of  each patch, after reshaping each of them to rc × di, is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Because of the symmetry property of the covariance  matrix, the number of second-order features is calculated as the number of patches×( di×di  2  + di  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The multi-level  spatial pooling is then used to pool the second-order features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The multi-level spatial pyramid pooling calculation is  identical to that implemented by Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Again, different post-processing procedures can be utilised;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' however, in  our experiments (Section 3), we found these steps to be the most effective for achieving high accuracy in the databases  we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3  Residual Mechanism  This mechanism is non-iterative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' we add layers sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Each layer is trained in a single pass with new labels learnt  from the residual information of all its previous layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The first layer’s features are classified using an LDA classifier  trained with the original classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' To produce the first layer’s posteriors ( ˜Y (1) in Figure 1), we use the following sigmoid  function on the output of the LDA classifier:  S(x) =  1  1 + e−x/σ ,  (4)  where σ is the sigmoid scale parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' To identify the new labels required to train the second layer (T (2) in Figure 1),  we first find the residual errors between the current layer’s predicted outputs ( ˜Y (1)) and the original classes (Y ∈ RN×C)  in one-hot encoding, as follows:  R(1) = λY − ˜Y (1),  (5)  where 0 ≤ λ ≤ 1 controls the maximum likelihood a class may attain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' T (2) can then be defined as the classes with the  maximum absolute residual errors, as follows:  T (2)  i  = class(|R(1)  ij |, ∀j)  i = [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' , N],  (6)  where class(x) denotes the name of the class whose value has the largest residual error magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Since our network  is developed primarily for classification tasks, it concentrates on labelling rather than regressing the correction of  posteriors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  The second layer, as shown in Figure 1, receives an input X(2) ∈ Rm×n×(d+d1)×N that is a concatenation of the first  layer’s feature maps and the original images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' After finding the second layer’s features, our objective is to learn a  correction term based on the second layer classification results, which is then added to the posteriors of the first layer  ( ˜Y (1)) to provide more accurate predictions ( ˜Y (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The correction term can either be positive or negative to maintain  the network probabilities at the second layer ( ˜Y (2)) to have values in the range of 0 to λ (Equation 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In other words,  after training the second layer’s LDA classifier using the new classes (T (2)), the posteriors acquired can be added to or  subtracted from the first layer’s posteriors to correct them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The indicator variable (s(2)) indicates whether to add to or  subtract from the first layer’s posteriors and is defined as the signum function of the maximum absolute residual errors  of the previous layers, as follows:  s(2)  i  = sign(R(1)  i∗ ),  i∗ = argmaxi |R(1)  i |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  (7)  When the indicator values are positive, it indicates that the probabilities of the first and second layers are added,  and when they are negative, the probabilities of the second layer are subtracted from those of the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Consider a  classification task of three classes A, B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Assume that, given a single image whose actual class is A, the predicted  probabilities of the first layer using this image are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='6 and 0 for the three classes A, B and C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' To add a  second layer, the residual error using λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='8 (Equation 5) is computed as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='6 and 0 for each of the three classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  Hence, the new label to train the second layer for this image is B, as it has the largest magnitude, and the indicator  is −1 because the maximum absolute residual error has a negative sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Assume that the second layer’s features are  trained using class B and provided perfect prediction with probabilities of 0, 1 and 0 for the three classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Because  4  arXiv Template  A PREPRINT  the indicator value is negative, we subtract them from the predicted outputs of the first layer, resulting in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4  and 0 for the three classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Consequently, we reduce the error, and the predicted class of the second layer is A, which  corresponds to the actual class of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' To implement that and generalise it for the test set, as we do not know the  classes, we divide the database into positive and negative samples based on their indicator variable values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The positive  samples have positive indicator values, whereas the negative samples have negative indicator values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' We then train two  LDA classifiers for each layer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' one is trained on the negative samples using {T (2)  n  ⊂ T (2) : s(2) = −1}, and the other  is trained with the positive samples and their classes {T (2)  p  ⊂ T (2) : s(2) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The negative classifier is trained by  assuming that each class in the positive samples is a negative class (its class is zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Similarly, during the training of  the positive classifier, each class in the negative samples is treated as a negative class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Thus, both classifiers have access  to all training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The network’s outputs at the second layer ( ˜Y (2)) can then be expressed as follows:  ˜Y (2) = ˜Y (1) + α[np  N  ˜Yp  (2) − nn  N  ˜  Yn  (2)],  (8)  where ˜  Yn  (2) and ˜Yp  (2) are the N predictions made by the classifiers trained on negative and positive samples, nn and  np denote the number of negative and positive examples, respectively, N is the total number of training samples and α  is a learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The learning rate, similar to that used in neural networks, is introduced to reduce oscillations and  provide faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' However, the learning rate in our network also acts as a weight to integrate the probabilities  of multiple layers, similar to weighted sum techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  To add more layers, we repeat the steps used to add the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Algorithm 1 summarises the procedures of the  network’s training with L layers, assuming that the second-order features are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' To add layer i, the new labels T (i)  and indicator variable (s(i)) are computed based on the previous layer’s residual error R(i−1), as shown in Equation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  The network’s output at that layer can then be defined using Equation 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In general, for deeper residual compensation  layers, the new classes (T (L)) learnt from the residual errors of the previous layers can be defined as follows:  T (L) = class(|R(L−1)|)  = class(|Y − ˜Y (L−1)|)  = class(|Y − [ ˜Y L−2 + α  N (n(L−2)  p  ˜Y (L−2)  p  − n(L−2)  n  ˜Y (L−2)  n  )])  = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  = class(|Y − [ ˜Y (1) + α  N  L−1  �  i=2  (ni  p ˜Y i  p − ni  n ˜Y i  n)]),  (9)  where class(x) is the name of the class whose value has the largest residual error magnitude, Y is the original classes in  one-hot encoding, ni  p and ni  n are the number of positive and negative samples in layer i, respectively, and N denotes  the total number of samples in the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  3  EXPERIMENTS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='1  Databases  We used four standard benchmarks in our experiments: CIFAR-10 [13], CIFAR-100 [13], MNIST [4] and TinyImageNet  [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The MNIST database comprises 60,000 training examples and 10,000 test images of size 28 × 28, drawn from the  same distribution, normalised and centred in a fixed-size image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The CIFAR-10 database consists of 10 classes with  50,000 images for training and 10,000 test images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The images of size 32 × 32 × 3 have a low resolution with different  poses and angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' CIFAR-100 is similar to CIFAR-10 but with 100 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The TinyImageNet database consists of  100,000 training images of size 64 × 64 × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The images are divided into 200 categories, with 500 images each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The  validation and the test sets contain 10,000 images each, with 50 images per class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The test set is not labelled, and our  experiments’ performance is reported on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2  Image Classification without Data Augmentation  In this section, we evaluate the proposed network on the MNIST, CIFAR-10, CIFAR-100 and TinyImageNet databases  (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='1) without data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' To determine the network parameters, we examine a variety of configurations,  each of which has different parameters, and report the results of the configuration that works the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  5  arXiv Template  A PREPRINT  Algorithm 1 Deep Residual Compensation Convolutional Network Training  Input: Second-order features: {F (i), i = [1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' , L]}, L number of layers, C classes T (1) ∈ RN×1, learning rate:  α and λ to determine the highest probability a class can reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  Output: Model’s accuracy: accuracy  1: Generate Y ∈ RN×C, the one-hot encoding of T (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  2: i ← 1 and ˜Y (0) ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  3: s(i) = 1N×1 {fill the first layer’s indicator variable with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' }  4: while i < L do  5:  if i > 1 then  6:  Find the residual errors, new classes and indicator variable, as follows:  R(i−1) = λY − ˜Y (i−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  T (i)  j  = class(|R(i−1)  jk  |, ∀k),  s(i)  j  = sign(R(i−1)  j∗  ), j∗ = argmaxj |R(i−1)  j  |,  j = [1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' , N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  (10)  7:  end if  8:  Find F (i)  n  ⊂ F (i) and T (i)  n  ⊂ T (i), for which their indicator values are negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  9:  Find F (i)  p  ⊂ F (i) and T (i)  p  ⊂ T (i), for which their indicator values are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  10:  if T (i)  p  ̸= ∅ then  11:  L1 = LDA(F (i)  p , T (i)  p ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  12:  ˜Y (i)  p  = prediction(L1, F (i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  13:  else  14:  ˜Y (i)  p  ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  15:  end if  16:  if T (i)  n  ̸= ∅ then  17:  L2 = LDA(F (i)  n , T (i)  n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  18:  ˜Y (i)  n  = prediction(L2, F (i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  19:  else  20:  ˜Y (i)  n  ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  21:  end if  22:  Compute the current’s layer output ˜Y (i), as  ˜Y (i) = ˜Y (i−1) + α[np  N  ˜Yp  (i) − nn  N  ˜  Yn  (i)],  (11)  where np and nn represent the number of positive and negative samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  23:  i ← i + 1  24: end while  25: Compute the model’s accuracy at layer L using ˜Y (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='1  Parameter Settings  The optimal settings identified for the MNIST, CIFAR-10, CIFAR-100 and TinyImageNet databases are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  All architectures, except the MNIST database, used 3 × 3-pixel filters created by combining PCA [17] and stacked-LDA  filters (Appendix A) at a 50% ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The MNIST database used 13 × 13 PCA filters in its first layer and 3 × 3 PCA  filters in its residual layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In this section and throughout the rest of this article, we used the same number of filters for  all layers and stopped adding layers when the training error rate approached 0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Therefore, the number of filters per  layer is 60 in the MNIST’s architecture, 50 in the CIFAR’s architectures and 40 in TinyImageNet’s architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The  number of layers beyond which an accuracy gain was no longer observed was 937 for the CIFAR-10 database, 436 for  the CIFAR-100 database, 231 for the MNIST database and 512 for the TinyImageNet database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' We utilised 7 × 7-block  second-order pooling for MNIST with a stride of four pixels, 16 × 16 for CIFAR-100 with a four-pixel stride, 16 × 16  for CIFAR-10 with a one-pixel stride, and 32 × 32 for TinyImageNet with an eight-pixel stride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In all architectures, we  pooled the second-order features using three-level spatial pyramid pooling of 4 × 4, 2 × 2 and 1 × 1 subregions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The  only data preprocessing was min-max normalisation applied to the input of each convolutional layer, and probabilities  were retrieved from all datasets except MNIST using the sigmoid function with a scale value of 16 (Equation 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In the  6  arXiv Template  A PREPRINT  MNIST database, we used the following softmax function to generate the probabilities in each layer:  softmax(yi) =  exp(βyi)  �C  j=1 exp(βyj)  ,  (12)  where β is assigned to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='001, C denotes the number of classes, and y represents the outputs of the LDA classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  During the training phase, λ (Equation 5) was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='8, and the learning rate was fixed at α = 1 for the MNIST  database and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4 for the CIFAR-10 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The remaining databases used an initial learning rate of 1, which was  dropped by 10% every 10 layers as follows:  α = α − 10  100α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  (13)  We stopped reducing the learning rate when it reached 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='387 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='478 for CIFAR-100 and TinyImageNet databases,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  Table 1: Network architectures using the MNIST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' CIFAR-10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' CIFAR-100 and TinyImageNet databases  The MNIST database: 28 × 28 × 1  Filter size  SOP  Output size  13 × 13 × 1 × 60  7 × 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Stride = 4  28 × 28 × 60  [3 × 3 × 60 × 60] × 230  7 × 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Stride = 4  28 × 28 × 60  The CIFAR-10 database: 32 × 32 × 3  Filter size  SOP  Output size  3 × 3 × 3 × 50  16 × 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Stride = 1  32 × 32 × 50  [3 × 3 × 50 × 50] × 936  16 × 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Stride = 1  32 × 32 × 50  The CIFAR-100 database: 32 × 32 × 3  Filter size  SOP  Output size  3 × 3 × 3 × 50  16 × 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Stride = 4  32 × 32 × 50  [3 × 3 × 50 × 50] × 435  16 × 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Stride = 4  32 × 32 × 50  The TinyImageNet database: 64 × 64 × 3  Filter size  SOP  Output size  3 × 3 × 3 × 40  32 × 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Stride = 8  64 × 64 × 40  [3 × 3 × 40 × 40] × 511  32 × 32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Stride = 8  64 × 64 × 40  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2  Performance Analysis  Table 2 reports the accuracy of ResCNet compared with some gradient-based and non-gradient-based networks on the  MNIST, CIFAR-10, CIFAR-100 and TinyImageNet databases without data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The non-gradient-based  models reported in Table 2 are the best-performing PCANet-like models from the perspective of the datasets we  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Such networks include PCANet [3], LDANet [3], DFSNet [8] and Multi-layer PCANet [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' To show where  our network fits in the literature of commonly used networks, we compared the performance achieved by our network  to those obtained by standard gradient-based convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Maxout [9], Network in network [16], stochastic  pooling [21] and ResNet [10] are the gradient-based networks listed in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The accuracy of 164-ResNet with  pre-activation was reported by Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' [11], while the results of ResNet-18 and ResNet-34 were reported by Jeevan  [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  According to Table 2, for the MNIST database, our model with 231 layers achieved an accuracy of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='52%, making it  superior to all non-gradient-based models, such as PCANet, Multi-Layer PCANet and LDANet, in terms of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' It  improved on the best results of the non-gradient-based networks by roughly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='12%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In addition, the findings presented  in the table demonstrated that our network produced results comparable with standard gradient-based networks, such as  the Maxout network, Network in network and stochastic pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  As shown in Table 2, for the CIFAR-100 database, the accuracy attained by our proposed network was the highest  among all the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The accuracy of 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='91% was around 8% higher than that of Multi-Layer PCANet and stochastic  pooling, more than 9% higher than that of ResNet-110 and ResNet-32, 14% higher than the original PCANet and  more than 2% higher than that of Maxout and ResNet with stochastic depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Moreover, the results achieved by our  network were roughly equivalent to those obtained by ResNet with 164 layers and Network in network using the  dropout technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  For TinyImageNet, as shown in Table 2, our network achieved the highest accuracy among all the networks without any  data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' This performance was around 2% higher than that of the ResNet-34 model and 1% better than that  of the ResNet-18 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  7  arXiv Template  A PREPRINT  According to the results shown in Table 2 for the CIFAR-10 database, our model outperformed PCANet and all of its  variants, in terms of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The accuracy of the proposed network was around 10% better than the performance  achieved by the original PCANet and 6% higher than that of Multi-Layer PCANet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Although ResCNet’s accuracy  was around 1% lower than that of the Maxout network and 2% worse than that of Network in network, our model  achieved accuracy comparable with that of ResNet-32, 1% higher than ResNet-18 and ResNet-110, and 3% greater than  stochastic pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In general, ResCNet with more than 900 layers showed an excellent performance of 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='54% on the  CIFAR-10 database with no data augmentation, making it the first PCANet-like network to reach such a number of  layers and such a performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  In general, Table 2 demonstrates that ResCNet outperformed all PCANet-like networks in terms of accuracy and the  number of layers required for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' It also shows that our model, which is trained without complicated non-linear  functions or regularisation techniques, achieves accuracy similar to that of standard convolutional networks such as  Network-in-Network, Maxout, stochastic pooling, and several residual networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Even though the number of layers in  our network is relatively large, these layers are added sequentially, one after the other, without iterations or intensive  parameter updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In addition, we used a small number of filters in each layer for all of our architectures, with no  configuration exceeding 60 filters per layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' To the best of our knowledge, ResCNet is the first non-gradient-based  propagation-free network to be trained with hundreds of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  Table 2: Accuracy (%) of ResCNet compared with different methods on the CIFAR-10 (C10), CIFAR-100 (C100),  MNIST and TinyImageNet (T200) databases without data augmentation  Non-gradient-based networks  Method  C10  C100  MNIST  T200  PCANet-2  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='14  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='62  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='34  30  LDANet  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='33  –  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='38  –  DFSNet-3  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='06  –  –  –  Multi-Layer PCANet  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='72  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='86  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='40  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='87  ResCNet (ours)  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='54  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='9  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='52  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='37  Gradient-based networks  Method  C10  C100  MNIST  T200  Stochastic pooling  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='87  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='49  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='53  –  Maxout network, with dropout  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='32  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='43  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='55  –  Network in network, with dropout  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='59  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='32  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='53  –  Network in network, without dropout  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='49  –  –  –  110 ResNet  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='82  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='26  –  –  ResNet stochastic depth 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='20  –  –  164-ResNet (pre-activation) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='42  –  –  ResNet-18  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='29  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='15  –  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='02  ResNet-32  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='97  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='05  –  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='65  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3  Network Parameters  This section explains how to configure ResCNet’s parameters by analysing their impact on the model’s accuracy using  the CIFAR-10 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The experiments in this section are divided into two subsections, as follows:  Experiment 1: Testing the effect of the number of filters on the accuracy of ResCNet  Experiment 2: Studying the impact of learning rate on the model’s performance  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='1  Number of Filters  This experiment aimed to determine how changing the number of filters affects the accuracy of ResCNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' For this, we  designed two networks with the same settings but different numbers of filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The first, referred to as ResCNet-30,  used 30 filters in each of its layers, while the second (ResCNet-50) used 50 filters in all of its layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The two networks  followed the same parameters settings as those described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2 for the CIFAR-10 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' However, we used  an 8x8 second-order pooling block size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  Figure 2 shows the accuracy of training and testing ResCNet-30 and ResCNet-50 on the CIFAR-10 database without  data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' According to the figure, both networks achieved the same level of accuracy, although ResCNet-30  required more layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' ResCNet-50 obtained a training accuracy of 100% at layer 127, while ResCNet-30 achieved  8  arXiv Template  A PREPRINT  100% at layer 781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The testing accuracy for both 127- and 781-layer networks was around 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='28%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The best testing  accuracy of 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='33% was obtained with 100-layer ResCNet-50 and 500-layer ResCNet-30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The findings in this section  suggested that, unlike previous PCANet-like networks in which the number of filters in each layer should be determined  in advance, we could obtain the same performance by employing any number of filters based on the available resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  However, we needed more filters in each layer to achieve the desired performance faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The results in this section also  showed that the second-order pooling block size affected the accuracy of ResCNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' For instance, the highest accuracy  reached for the CIFAR-10 database (87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='54% in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2) was around 1% better than that achieved in this section  (86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='33%), with the only difference between the models being the modification of the second-order pooling block size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  0  200  400  600  800  Layers  75  80  85  90  95  100  Accuracy (%)  ResCNet-30  Training  Testing  0  50  100  Layers  75  80  85  90  95  100  Accuracy (%)  ResCNet-50  Training  Testing  Figure 2: The accuracy (%) of ResCNet-30 and ResCNet-50 on the training and testing sets of the CIFAR-10 database  with no data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2  Learning Rate  This experiment aimed to show the impact of the learning rate (α in Equation 9) on ResCNet’s accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Similar to  neural networks, the learning rate is introduced to prevent oscillations and promote faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' We designed  two networks with the same parameters but different learning rates and evaluated them on the CIFAR-10 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Both  networks used 50 filters in all of their layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The first network (ResCNet-50–1) was trained with a learning rate of  α = 1, whereas the second (ResCNet-50-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4) was trained using α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The other network’s parameters were the  same as those described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2 for the CIFAR-10 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  Table 3 compares the accuracy of ResCNet-50–1 with that of ResCNet-50–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4 using a different number of layers on the  CIFAR-10 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The training accuracy of ResCNet-50–1 reached 100% at layer 384, with a testing accuracy of  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='91%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' On the other hand, ResCNet-50–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4 obtained a 100% training accuracy at layer 970, with a testing accuracy  of 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='41%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The optimal testing accuracy of ResCNet-50–1 was achieved at layer 208, with 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2% testing accuracy,  while ResCNet-50–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4 obtained its best performance of 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='54% at layer 937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The results achieved in this section  demonstrated that the learning rate α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4 was small, as we needed more than 600 layers for the network to reach its  convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' If the training error rate oscillates, which is not the case in CIFAR-10, the learning rate must be decreased  to avoid oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4  Image Classification with Data Augmentation  This section aims to enhance the classification accuracy of ResCNet using data augmentation on three databases, namely,  CIFAR-10, CIFAR-100 and TinyImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The ResCNet architectures in this section shared the same design and  parameters as those in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2 but had 490, 507 and 480 layers for the CIFAR-10, CIFAR-100 and TinyImageNet  9  arXiv Template  A PREPRINT  Table 3: Accuracy (%) on the CIFAR-10 database test set using ResCNet-50–1 and ResCNet-50–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4  Network  # Layers  Accuracy (%)  ResCNet-50–1  348  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='91  ResCNet-50–1  208  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='02  ResCNet-50–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4  970  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='41  ResCNet-50–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4  937  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='54  ResCNet-50–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4  600  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='09  databases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' For CIFAR-10, we modified the pooling stride to four pixels and re-implemented a 193-layer  ResCNet without data augmentation for adequate comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  Table 4 compares the accuracy of ResCNet with and without data augmentation on the CIFAR-10, CIFAR-100 and  TinyImageNet databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' With horizontal flipping being the only data augmentation used, the accuracy of ResCNet was  enhanced across all three databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' This improvement was around 1% for the TinyImageNet database, 2% for the  CIFAR-10 database and 3% for the CIFAR-100 database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' For the CIFAR-10 database, the accuracy achieved with data  augmentation was also 1% higher than the best result (87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='54%) reported in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  This section’s results showed the importance of data augmentation for improving the model’s generalisation and  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Incorporating other data augmentation types are needed to increase the model’s accuracy further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Since  ResCNet is not currently being trained in a batch-based manner, adding more data augmentation types is challenging as  they need to be computed in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In future works, we will investigate the possibility of converting the current work  into a batch-based system as in gradient-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  Table 4: Accuracy (%) of ResCNet on the CIFAR-10, CIFAR-100 and TinyImageNet databases  Data augmentation  CIFAR-10  CIFAR-100  TinyImageNet  \x17  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='82  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='9  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='37  \x13  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='35  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='8  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='91  4  CONCLUSIONS  In this article, we proposed ResCNet, a PCANet-like network that trains each layer with new labels derived from the  residual data of all preceding layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Our proposed network increased the network depth to more than 950 layers, making  it the first non-gradient-based propagation-free network to achieve this number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Moreover, ResCNet’s performance was  comparable to that of standard gradient-based models and superior to PCANet and all of its variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Increasing the size  of the databases by one type of data augmentation resulted in a considerable improvement in accuracy, particularly in  CIFAR-100, where it reached 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' However, increasing the number of samples leads to higher computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' To  overcome this issue and in future work, we will investigate the possibility of transforming the current network into a  batch-based system, similar to the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In addition, the network may be developed further by modifying the  filter types or by using other metrics to define the residual errors and, thus, the new classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
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+page_content=' The description  of our method is explained in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In addition, Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3 discusses the procedures for using the stacked-LDA  algorithm as convolution filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Finally, the parameter settings we used in our experiments in the main paper were  described in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='1  General Concept  The stacked-LDA algorithm relies on stacking two linear discriminant analysis (LDA) layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The first LDA layer is  trained using labels created from the original classes, while the second LDA layer is trained using the actual labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  To generate the classes of the first layer, similar samples from a given class are grouped to create a new class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' For  example, a class A with s instances can be subdivided into c new classes, each with a different number of instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The  minimum number of samples needed to represent a class is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Suppose the first LDA can differentiate between the  new classes ideally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' In that case, the second LDA will receive the posteriors of the first LDA and be able to differentiate  between the actual classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' For instance, if class A is subdivided into A1 and A2, the subsequent LDA will identify that  both A1 and A2 are members of class A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='2  Stacked-LDA Algorithm  Let X ∈ RN×M represent N training samples, each with M dimensions, and Target ∈ RN×1 represent their original  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Algorithm 2 describes the procedures for applying the stacked-LDA to the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The algorithm starts by  picking a random class c from the actual classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' We then choose Npositive random instances that belong to class c  and Nnegative random examples that are not in class c, where Npositive and Nnegative are the number of positive and  negative samples and are user-predefined parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Next, an LDA classifier discriminates between the positive and  the negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' After that, we check if our chosen random samples are linearly separable, which can be done by  comparing the error rate of the LDA classifier with a small value of nearly zero called tolerance (tol) and is chosen by  the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' If the LDA’s error rate is lower than the tolerance, we consider the positive class to be a new class and collect  the weights of the LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' On the other hand, if the LDA’s error rate is greater than the tolerance, the chosen samples  are not similar and cannot be grouped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Therefore, the algorithm proceeds to find other classes that separate the data  accurately in the same way until reaching the required number of classes (N_classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' When algorithm 2 terminates,  we can use the generated LDA’s weights to find the output of the first LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Another LDA can then be applied to the  output of the first LDA to classify them back using the original classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='3  Stacked-LDA Filters  For N training samples X: {Xi ∈ Rm×n×c} with actual classes Target ∈ RN×1, where m and n are the spatial  dimensions of the image and c is the number of channels, we compute the stacked-LDA filters as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Given a single image Xi ∈ Rm×n×c and a filter size kL × kL, we extract and vectorise all overlapping  patches of size kL × kL × c each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The resulting matrix is Pi ∈ R(k2  L×c)× ˜m˜n, where ˜m = (m − kL) + 1,  ˜n = (n − kL) + 1 and m and n are the spatial dimensions of the image, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' We repeat the previous step for all images in the dataset to obtain P ∈ R(k2  L×c)× ˜m˜nN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' We create a vector T ∈ R1× ˜m˜nN that contains the class labels of the patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' A single patch is assigned a  label equivalent to the class of its full image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' Using random samples and specific tolerance, we apply Algorithm 2 on P and T to obtain the stacked-LDA  filters’ weights W L  s and bias BL  s ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' We can express the Stacked-LDA filters as follows:  W L  s =  mat  kL×kL×cqs, s = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' , dL,  BL  s = mat  1×1×cqs, s = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' , dL,  (14)  where dL is the number of filters chosen by the user, which is equivalent to the number of classes in Algorithm  2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  12  arXiv Template  A PREPRINT  Algorithm 2 Stacked-LDA Algorithm  Input: Training set: X ∈ RN×M, where {xN  i , xi ∈ RM}, original classes: Target ∈ RN×1, number of classes user  wants to generate: N_classes, number of positive samples: Npositives, number of negative samples: Nnegatives  and tolerance of performance user can afford: tol  Output: LDA’s weights: weights ∈ RN×N_classes and LDA’s bias or constant: bias ∈ RN_classes  1: weights ← [ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  2: bias ← [ ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  3: i ← 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  4: while i < N_classes do  5:  Pick a random class c from the Target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  6:  Pick random Npositives samples from class c (Spositives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  7:  Choose Nnegatives samples that are not in class c (Snegatives) randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  8:  Combine the negative and positive samples: S ← [Spositives, Snegatives].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  9:  T ← [ones(Npositives), zeros(Nnegatives)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  10:  Find the linear discriminant analysis (LDA) between S and T: L = LDA(S, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  11:  Find the perfromance (ErrorRate) of S using L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  12:  if ErrorRate < tol then  13:  weights ← [weights, LDA′sweights].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  14:  bias ← [bias, LDA′sbias].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  15:  i ← i + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  16:  end if  17: end while  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' We convolve the original images with the filters as follows:  XL  i = XL−1  i  ∗ W L  s + BL  s ∈ Rm×n×dL,  (15)  where s = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' , dL and XL−1  i  is zero-padded to obtain the same image size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='4  Parameter Settings  To compute the stacked-LDA filters in our experiments in the main paper, we set the number of positive samples  (Npositives) to 2, while the number of negative samples (Nnegatives) was 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' The tolerance (tol) in Algorithm 2 was  set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content=' We used the LDA classifier with the one-versus-all decomposition method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
+page_content='  13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtFJT4oBgHgl3EQf3y2A/content/2301.11663v1.pdf'}
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+N3AS-23-001
+Celestial Objects as Strongly-Interacting Asymmetric Dark Matter Detectors
+Anupam Ray
+ID
+1, 2, a
+1Department of Physics, University of California Berkeley, Berkeley, California 94720, USA
+2School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA
+(Dated: January 11, 2023)
+Non-annihilating dark matter particles, owing to their interactions with ordinary baryonic matter,
+can efficiently accumulate inside celestial objects. For heavy mass, they gravitate toward the core of
+the celestial objects, thermalize in a small core region, and eventually form tiny black holes via core
+collapse, resulting destruction of the host objects. We demonstrate that the existence of a variety of
+celestial objects provides stringent constraints on strongly-interacting heavy dark matter, a blind-
+spot for the terrestrial dark matter detectors as well as for the cosmological probes. Celestial objects
+with larger sizes and lower core temperatures, such as Jupiter, are the most optimal detectors to
+probe the strongly-interacting heavy asymmetric dark matter.
+I.
+INTRODUCTION
+The existence of dark matter (DM) has been firmly es-
+tablished through its gravitational interactions with the
+ordinary baryonic matter [1]. However, its identity still
+remains a mystery. The ongoing terrestrial and cosmo-
+logical searches are trying to pinpoint its mass and hy-
+pothesized non-gravitational interactions with the bary-
+onic matter [2, 3]. While some DM parameters have been
+excluded by these searches, many well-motivated candi-
+dates are yet to be explored. Strongly-interacting heavy
+asymmetric DM is one such regime that remains largely
+untested.
+In this work, we demonstrate that celestial objects are
+excellent laboratories to search for strongly-interacting
+heavy asymmetric DM. More specifically, we point out
+that the continued existence of a variety of stellar ob-
+jects provides stringent exclusions on asymmetric DM
+interactions over a wide mass range. We mainly answer
+two basic questions: why celestial objects are superior to
+search for strongly-interacting heavy asymmetric DM as
+compared to the terrestrial detectors, and which celestial
+objects are the most optimal targets?
+Traditional direct detection experiments are not well-
+suited for heavy DM searches. The flux of Galactic DM
+at terrestrial detectors falls off linearly with higher DM
+mass, and the constraints weaken accordingly. Whereas,
+because of the enormous sizes of astrophysical objects
+and their long lifetimes, the effective exposure time (∼
+M⊙ Gyr) is orders of magnitude larger than human-made
+direct detection experiments (∼ kT yr), naturally provid-
+ing sensitivity to the tiny flux of heavy DM.
+Strongly-interacting DM is yet another blind-spot for
+typical direct detection experiments. If DM particles in-
+teract too strongly with baryonic matter, they lose a sig-
+nificant fraction of their energies via interactions with the
+material in the atmosphere as well as in the Earth-cover
+above the underground detectors.
+As a consequence,
+a anupam.ray@berkeley.edu
+they slow down significantly and can not deposit suffi-
+cient amounts of energy for detection. Whereas, stellar
+objects are ideal probes for strongly-interacting DM as
+almost all the DM particles that transit through the stel-
+lar objects get trapped, leading to a maximal sensitivity.
+Accumulation of particle DM in celestial objects is a
+novel astrophysical probe of non-gravitational interac-
+tions of DM with the ordinary baryonic matter.
+DM
+particles from the galactic halo, owing to their inter-
+actions with the stellar constituents, can down-scatter
+to energies below the local escape energy, and become
+gravitationally bound to the stellar objects [4–6]. These
+bound DM particles lose more energy via repeated scat-
+terings with the stellar constituents and eventually ther-
+malize inside the stellar volume. Such bound thermalized
+DM particles can become abundant inside the stellar vol-
+ume if they have sufficiently strong interactions with the
+baryonic matter, and possess intriguing phenomenologi-
+cal signatures.
+For non-annihilating DM, such bound DM particles
+gradually accumulate, and for heavy DM masses, they
+settle in a small region around the stellar core.
+Be-
+cause of their prodigious abundance in a tiny core vol-
+ume, their number density within the stellar core be-
+comes quite large, allowing dark core collapse and sub-
+sequent black hole (BH) formation. This nascent BH, if
+not too light, can rapidly swallow the host, and the exis-
+tence of stellar objects provides stringent constraints on
+non-annihilating DM interactions.
+This scenario has been extensively studied in the con-
+text of old neutron stars [7–27] primarily because of their
+enormously large baryonic density as well as high com-
+pactness. More specifically, neutron stars can capture a
+significant number of DM particles from the Galactic halo
+even for the low DM-nucleon scattering cross-sections as
+the single-collision capture rate scales linearly with the
+compactness (∼ M/R). Quantitatively, for a solar mass
+neutron star with a typical radius of 10 km, the capture
+rate is O(105) times larger than the Sun for low DM in-
+teractions. Apart from that, because of their large bary-
+onic density (∼ M/R3), the accumulated DM particles
+thermalize in a tiny region around the core, implying a
+arXiv:2301.03625v1  [hep-ph]  9 Jan 2023
+
+2
+huge core-density favorable for transmutation. This indi-
+cates neutron stars are the most optimal targets to probe
+weakly interacting heavy asymmetric DM, and the exis-
+tence of old neutron stars in the solar neighborhood ex-
+cludes mχ = 106 GeV and σχn = 10−48 cm2, the leading
+constraints on non-annihilating DM interactions [11, 20].
+However, in the optically thick (large DM-nucleon scat-
+tering cross-section) regime, non-compact objects such
+as the Sun, Jupiter, and Earth are more suitable detec-
+tors to probe non-annihilating DM interactions. This is
+simply because in the optically thick regime, almost all
+of the DM particles that transit through a stellar ob-
+ject get trapped, and therefore, the accumulation rate
+increases with the larger size. Since the non-compact ob-
+jects possess much larger radii, the accumulation rate
+for non-compact objects becomes comparable or even
+larger than the neutron stars in the strongly-interacting
+regime.
+Quantitatively, for mχ = 106 GeV, and suffi-
+ciently high σχn, the transit (accumulation) rate for a
+typical neutron star is 1019 s−1, comparable to the Earth,
+and 10−2 (10−5) smaller than the Jupiter (Sun).
+This
+naturally motivates us to explore the potential of non-
+compact stellar objects as strongly-interacting asymmet-
+ric DM detectors.
+We found that stellar objects with
+relatively large radii and low core-temperature (such as
+Jupiter), are the most optimal detectors. This is simply
+because the total number of accumulated DM particles
+increases with a larger radius, and the BH formation be-
+comes easier with a lower core-temperature, implying the
+most favorable transmutation criterion. Prior works [28–
+30], more particularly Ref. [30] has recently explored the
+transmutation scenario for the Earth and the Sun. We
+systematically revisit the issue to gain more insight on the
+constraints, and we show that stellar objects with larger
+sizes and small core temperatures (such as Jupiter) pro-
+vide the leading constraints on strongly-interacting heavy
+asymmetric DM.
+The rest of the paper is organized as follows. In Sec-
+tion II, we discuss different stages of DM-induced collapse
+of the celestial objects. In Section III, we present our
+exclusion limits from the existence of several stellar ob-
+jects, demonstrating that the constraints obtained in this
+analysis cover new parts of the DM parameter space and
+bridge the gap between the cosmological probes [31–34],
+and the terrestrial detectors [35–39]. Finally, we summa-
+rize and conclude in Section IV.
+II.
+DARK MATTER INDUCED COLLAPSE OF
+STELLAR OBJECTS
+Non-annihilating DM particles can accumulate effi-
+ciently inside the stellar volume if they possess suffi-
+ciently strong interactions with the stellar nuclei.
+For
+heavy DM mass, they gravitate towards the stellar core
+and settle in a tiny core-region. Because of their prodi-
+gious abundance, and tiny core volume, their number
+density within the stellar core becomes tantalizingly
+larger, eventually resulting in a BH formation inside the
+stellar core.
+This nascent BH, if not sufficiently light,
+can rapidly swallow the host, transmuting them to com-
+parable mass BHs. In the following, we discuss different
+stages of DM-induced collapse of stellar objects, and a
+schematic diagram for this process is depicted in Fig. 1.
+A.
+Dark Matter Accumulation
+We first estimate the total number of captured DM
+particles inside the stellar volume.
+For clarity, we de-
+fine the maximal capture rate as saturation capture rate
+(Csat), and it occurs when all of the DM particles that
+transit through the stellar objects get trapped.
+For a
+particular velocity distribution of the incoming DM par-
+ticles, the saturation capture rate is [6]
+Csat = ρχ
+mχ
+πR2
+� f(u)du
+u
+(u2 + v2
+esc) ,
+(1)
+where ρχ = 0.4 GeV/cm3 is the Galactic DM density,
+mχ is the DM mass, and R is the radius of the stellar
+object. f(u) denotes the velocity distribution of the in-
+coming DM particles, with vesc being the escape velocity
+of the stellar objects. For a Maxwell-Boltzmann velocity
+distribution, Csat simplifies to
+Csat = ρχ
+mχ
+πR2
+�
+8
+3π ¯v
+�
+1 + 3v2
+esc
+2¯v2
+�
+,
+(2)
+where ¯v = 270 km/s denotes the average velocity of the
+DM particles in the Galactic halo.
+A certain fraction of the DM particles will be cap-
+tured by interacting with the stellar constituents, and
+we aim to estimate this capture fraction (fc). Since we
+are mostly interested in the optically thick regime (large
+DM-nucleon scattering cross-section), fc behaves differ-
+ently for heavier and lighter DM. For heavier DM, i.e.,
+when the DM mass (mχ) is larger than the target mass
+(mA), scatterings do not alter the direction of the in-
+coming DM particles. As a consequence, the trajectories
+of the incoming DM particles are not randomized, and
+they follow almost a linear trajectory that ends inside
+the stellar interior when the final velocity falls below the
+escape velocity. In this case, in the limit of multiple col-
+lisions, the DM particles are essentially guaranteed to be
+captured, resulting fc = 1 [40]. Whereas, in the opposite
+regime (mχ < mA), the direction of the DM particles gets
+randomized after each collision, and as a result, a certain
+fraction of the DM particles can always escape from the
+stellar volume via reflection. This implies that for lighter
+DM, even for arbitrarily large cross-sections, the capture
+rate never reaches its saturation value (fc < 1) [40].
+For this analysis, we are interested in heavy DM cap-
+ture inside stellar objects, and hence, we take fc = 1.
+It implies that in the optically thick regime, we take the
+capture rate to its saturation value, and it does not de-
+pend on the DM-nucleon scattering cross-section.
+For
+
+3
+light DM capture in celestial objects, fc can be deter-
+mined by the recent MCMC results [40], or from analyt-
+ical estimates [41], both of which agree reasonably well.
+In the optically thin regime (small DM-nucleon scat-
+tering cross-section), capture occurs via single scatter-
+ing. This is simply because for smaller values of DM-
+nucleon scattering cross-sections, the mean free path of
+the incoming DM particles becomes larger and becomes
+comparable to the size of the stellar objects, and as a re-
+sult, they scatter once while transiting through the host.
+For this regime, we use the single-collision capture treat-
+ment [6], and fc becomes substantially smaller.
+B.
+Spatial Distribution of Dark Matter inside
+Stellar Objects
+Captured DM particles rapidly thermalize inside the
+celestial objects for sufficiently high DM-nuclei scatter-
+ing cross-sections [11–13, 20, 30, 42], and the spatial dis-
+tribution of the thermalized DM particles inside the stel-
+lar volume depends on the effects of diffusion and grav-
+ity [43–45]. By considering the effects of diffusion and
+gravity in a self-consistent manner, the spatial distribu-
+tion of the thermalized DM particles is [45]
+∇nχ(r)
+nχ(r) + (κ + 1) ∇T(r)
+T(r) + mχg(r)
+T(r)
+=
+Φ
+nχ(r)Dχn(r)
+R2
+⊕
+r2 ,
+(3)
+where nχ(r) denotes the number density of the thermal-
+ized DM particles within the stellar volume. T(r) denotes
+the temperature profile of the celestial objects, and Φ =
+C/πR2
+⊕ is the incoming flux of the DM particles, with C
+denotes the capture rate. κ ∼ −1/
+�
+2(1 + mχ/mA)3/2�
+and Dχn ∼ λvth are the diffusion coefficients, where λ
+denotes the mean free path of the DM particles and vth
+denotes their thermal velocity.
+It is evident that, for
+very heavy DM mass, the diffusion co-efficient (κ) be-
+comes smaller as it scales as m−3/2
+χ
+, and gravity (scales
+proportional to mχ) dominates over the diffusion pro-
+cesses.
+Therefore, heavy DM tends to settle down to-
+wards the core of the stellar objects. Quantitatively, by
+solving Eq. (3) for nχ(r) with the boundary condition
+that the volume integral of nχ(r) provides the total num-
+ber of captured DM particles, one can demonstrate that
+the captured DM particles mostly concentrate around the
+stellar core if they are heavy [45].
+Concentration of heavy DM around the stellar core
+can also be explained from the radius of the thermaliza-
+tion sphere
+�
+rth =
+�
+9kBT/ (4πGρmχ)
+�
+, which results
+from the balance between the thermal pressure and the
+gravitational potential [11, 12]. Since, the radius of the
+thermalization sphere scales as m−1/2
+χ
+, for heavy DM,
+rth becomes smaller, indicating the concentration of the
+captured DM particles primarily occurs around the core.
+C.
+Dark Core Collapse & Black Hole Formation
+For non-annihilating DM, accumulation grows linearly
+in time. As a consequence, the number density of the
+captured DM particles inside the thermalization volume
+becomes tantalizingly large. Quantitatively, for DM mass
+of 106 GeV, and sufficiently high DM-nuclei scattering
+cross-section (say 10−28 cm2), O(1036) number of DM
+particles thermalize inside the Earth within a radius of
+∼ 6 km, indicating a number density of ∼ 2×1018 cm−3.
+This corresponds to a core density of ∼ 2 × 1024 GeV
+cm−3, around 25 orders of magnitude higher than the lo-
+cal Galactic DM density, and it further increases as m3/2
+χ
+for heavier DM masses. Once the core-density exceeds
+its critical threshold value, it undergoes a gravitational
+collapse, and eventually results in a BH formation inside
+the stellar core [7–24, 30]. In the following, we quantify
+the critical core-density for the stellar objects.
+The BH formation criterion via dark core collapse has
+been extensively discussed in the literature for compact
+stars [7–27, 30], and is essentially determined by two con-
+ditions. The first one is within the stable thermal ra-
+dius, the DM density has to exceed the corresponding
+baryonic density (ρb). It leads to a self-gravitational col-
+lapse of the thermalized DM particles and is determined
+by [11, 13, 20]
+mχN self
+χ
+4
+3πr3
+th
+≥ ρb ,
+(4)
+where N self
+χ
+denotes the critical number of DM particles
+for ensuing self-gravitating collapse1. It is independent
+of the spin of the DM particles and only depends on the
+DM mass as well as properties of the stellar objects such
+as core-density and core-temperature. For Earth, it cor-
+responds to
+N self
+χ
+∼ 7×1036
+�
+ρcore
+13 g/cm3
+� � Tcore
+5800 K
+�3/2 �107 GeV
+mχ
+�5/2
+,
+(5)
+where, ρcore denotes the core density, and Tcore denotes
+the core temperature.
+The second condition is deter-
+mined by the maximum number of DM particles that can
+be stabilized by the degeneracy pressure and is commonly
+known as Chandrasekhar limit (N cha
+χ
+). Chandrasekhar
+limit depends on the spin-statistics of the DM particles as
+degeneracy pressure for bosonic DM stems from the Un-
+certainty principle, whereas, for fermionic DM, it arises
+from the Pauli exclusion principle. N cha
+χ
+solely depends
+on the DM mass, and quantitatively, it corresponds to
+M 2
+pl/m2
+χ (M 3
+pl/m3
+χ), where Mpl = 1.22 × 1019 GeV de-
+notes the Planck mass [11, 13, 20]. To summarize, for
+1 This self-gravitating criterion is essentially equivalent to the
+Jeans instability criterion in Ref. [30] up to O(1) factors.
+
+4
+Dark Matter Accumulation
+Dark Core Collapse
+Transmutation
+FIG. 1. Transmutation of stellar objects via gradual accumulation of non-annihilating DM. Heavy DM gravitate towards the
+core of the stellar objects and form tiny black holes via dark core collapse. These nascent BHs rapidly transmute the hosts by
+swallowing them, resulting in comparable low mass BHs.
+dark core collapse, the total number of captured DM par-
+ticles inside a stellar object within its lifetime (tage) has
+to satisfy the following [11, 13, 20, 24]
+Nχ|tage = C × tage ≥ max
+�
+N self
+χ
+, N cha
+χ
+�
+.
+(6)
+Note that, for stellar objects with high core temperature
+(such as Sun), the dark core collapse is essentially de-
+termined by N self
+χ
+for bosonic as well as fermionic DM,
+leading to identical exclusion limits for bosonic/fermionic
+DM. Whereas, for stellar objects with low core tempera-
+ture, the dark core collapse is determined by N self
+χ
+(N cha
+χ
+)
+for bosonic (fermionic) DM, leading to distinct exclusion
+limits for bosonic/fermionic DM.
+D.
+Growth and Evaporation of Nascent Black
+Holes
+It is important to stress that dark core collapse does
+not ensure the successful transmutation of the hosts.
+If the nascent BH is sufficiently light, transmutation
+can cease for two different reasons. Firstly, lighter BH
+takes a much longer time to swallow the hosts, and
+the swallow time (τswallow) can even be larger than the
+lifetime of the hosts. Secondly, and more importantly,
+Hawking radiation becomes significant for lighter BH
+masses (∼ 1/M 2
+BH), causing a rapid evaporation of the
+nascent BH. Since the mass of the nascent BH becomes
+smaller for heavier DM mass, this provides an upper
+limit on DM mass that can be probed via transmuta-
+tion [11, 13, 20, 24]. We quantify the upper limits of mχ
+for several stellar objects in the following, and it ranges
+around O(1010) GeV for the stellar objects under consid-
+eration.
+For the time-evolution of the nascent BH, we conser-
+vatively consider the baryonic matter accretion from the
+host (ignoring the possible DM accretion by the nascent
+BH) [11, 13, 46]
+dMBH
+dt
+= 4πρcoreG2M 2
+BH
+c3s
+− P (MBH)
+G2M 2
+BH
+,
+(7)
+where cs =
+�
+Tcore/mn denotes the sound speed at
+the core of the stellar object, and P (MBH) denotes the
+Page factor [47, 48]. Page factor properly accounts into
+gray-body corrections of the Hawking evaporation spec-
+trum, as well as the number of Standard Model (SM)
+species emission from an evaporating BH. In the classi-
+cal black-body radiation limit, the Page factor evaluates
+to 1/ (15360π), and is commonly used in the literature.
+Considering the gray-body corrections, the Page factor
+can be written as 2.8 × 10−4f(MBH) where f(MBH) en-
+codes the number of SM species emission from an evapo-
+rating BH [48]. For MBH ≥ 1017 g (which only emit mass-
+less particles, such as photons and neutrinos), f(MBH) is
+normalized to unity [48], and therefore, Page factor eval-
+uates to 1/(1135π), an order of magnitude larger than
+the classical black-body limit. For MBH ≤ 1017 g, the
+number of SM species emission from an evaporating BH
+crucially depends on its temperature (mass), and hence,
+f(MBH) varies with BH mass. We use the semi-analytic
+form of f(MBH) from Ref. [48] to estimate the Page fac-
+tor in this regime.
+Quantitatively, for light BHs, i.e.,
+MBH ≤ 1010 g, which can emit all SM species, f(MBH)
+evaluates to 15.35, and for 1015 g ≤ MBH ≤ 1017 g (which
+can emit electrons, positrons, photons, and neutrinos),
+f(MBH) evaluates to 1.569. To summarize, depending on
+BH mass, f(MBH) ranges from (1 − 15.35), implying the
+range of Page factor from 1/(1135π) to 1/(74π). We ver-
+ify that the Page factor obtained from the semi-analytic
+form of f(MBH) from Ref. [48] is in excellent agreement
+with the publicly available BlackHawk package [49].
+Since, the accretion term scales as M 2
+BH, and the evap-
+
+5
+oration term scales as 1/M 2
+BH, for low BH masses, evap-
+oration dominates over the accretion process. Quantita-
+tively, for Sun, Jupiter, Earth, and Moon, Hawking evap-
+oration dominates over the Bondi-Hoyle accretion for
+mχ ≥
+�
+�
+�
+�
+�
+�
+�
+�
+�
+7.1 × 1011 GeV
+6.2 × 109 GeV
+2.4 (5.3) × 109 GeV
+1.0 (6.0) × 109 GeV
+(8)
+for non-annihilating bosonic (fermionic) DM. On the
+other hand, by requiring that the τswallow has to be less
+than 1 Gyr, we obtain
+mχ ≥
+�
+�
+�
+�
+�
+�
+�
+�
+�
+2.1 × 1010 GeV
+1.1 × 1010 GeV
+0.9 (1.5) × 1010 GeV
+0.8 (3.1) × 1010 GeV
+(9)
+for non-annihilating bosonic (fermionic) DM. We note
+that, for stellar objects with high core-temperature,
+τswallow essentially determines the wash-out of the trans-
+mutation, whereas, for stellar objects with low core-
+temperature, it is determined by the efficient Hawking
+evaporation. This can simply be explained by the fol-
+lowing.
+For high core-temperature stellar objects, the
+nascent BH becomes relatively larger (MBH,init ∼ T 3/2),
+and therefore, the effects of Hawking evaporation become
+relatively sub-dominant, implying accretion determines
+the termination of the transmutation.
+E.
+Drift time and Maximal Possible Scattering
+Cross-section
+Transmutation of stellar objects also ceases at very
+high DM-nucleon scattering cross-sections. This is sim-
+ply because, at very high DM-nucleon cross-sections, DM
+particles lose a significant amount of energy in the outer
+shells of these stellar objects, and might not reach the
+stellar core to form a micro BH. In other words, the
+viscous drag force that drives the DM particles toward
+the core results in a long drift time, and therefore, pro-
+hibits transmutation. We estimate the drift time by us-
+ing the stellar density, temperature, and compositional
+profiles [8, 30, 50]
+tdrift =
+1
+Gmχ
+�
+j
+σχj
+� R
+0
+nj(r)
+�
+3AjT(r)
+� r
+0 d3r′ρj(r′)
+dr ,
+(10)
+where σχj
+denotes the DM-nuclei scattering cross-
+section, and it is related to the DM-nucleon scattering
+cross-section via σχj = σχn A2
+j
+�
+µχAj/µχn
+�2 with Aj is
+the mass number of the j-th nuclei, and µχn is the re-
+duced mass of the DM-nucleon system. We determine the
+ceilings of our results by demanding that tdrift ≤ 1 Gyr.
+Quantitatively, for (Sun) Earth, it corresponds to
+σχn ≤ (10−17) 10−21 cm2 �
+mχ
+107 GeV
+�
+,
+(11)
+and is same for bosonic/fermionic DM.
+F.
+Properties of Stellar Objects
+We accurately estimate the capture rate and the trans-
+mutation criterion for these stellar objects by utilizing
+stellar object properties, such as density profiles, tem-
+perature profiles, and detailed chemical compositions. In
+the following, we provide the inputs that have been con-
+sidered for this analysis. The density and temperature
+profiles for Sun, Earth, and Jupiter which have been used
+in this analysis have also complied in Ref. [45].
+• Sun: We use the solar density and temperature
+profiles from [51]. For the chemical composition, we
+assume that the Sun is entirely made up of 1H [45].
+• Jupiter: We use the Jupiter density and tempera-
+ture profiles from the Jovian model J11-4a [52]. For
+the chemical composition, we assume that Jupiter
+is entirely made up of 1H [45].
+• Earth: We use the Preliminary Reference Earth
+Model from [53] for the density profile, and we
+take the temperature profile from [54] under the
+assumption of a hydro-static equilibrium. For the
+chemical composition, we use the tabulated val-
+ues from Ref. [50] with the core-mantle bound-
+ary at 3480 km, and the mantle-crust boundary at
+6346 km. The core is dominantly made up of 56Fe,
+whereas, the mantle and the crust are dominantly
+made up of 16O.
+• Moon: We use the MAX model for density and the
+chemical compositions of the Moon [55]. We take
+the lunar core-mantle boundary at 450 km, and the
+mantle-crust boundary at 1650 km. The lunar core
+is dominantly made up of 56Fe, whereas, the mantle
+and the crust is dominantly made up of 16O. For
+the temperature profile, we consider the Moon as
+an isothermal sphere with T = 1700 K [55].
+III.
+RESULTS
+We consider a variety of stellar objects, such as the
+Sun, Jupiter, Earth, and Moon, and demonstrate their
+potential as asymmetric DM detectors.
+These choices
+are well-motivated by the fact that each of these stel-
+lar objects is an optimal detector in a different regime.
+More specifically, they cover a wide range of size, den-
+sity, and temperature, making them sensitive to different
+parts of the DM parameter space. Sun has the largest
+size, and the highest core temperature, and as a con-
+sequence, the total number of captured DM particles
+as well as the threshold for transmutation both become
+higher. Jupiter has a somewhat larger size, but possesses
+
+6
+CMB
+Bosonic / Fermionic DM
+Skylab
+Chicago
+MW Satellites
+Terrestrial
+detectors
+Sun
+(This analysis)
+105
+106
+107
+108
+109
+1010
+10-14
+10-16
+10-18
+10-20
+10-22
+10-24
+10-26
+10-28
+10-30
+10-32
+10-34
+mχ [GeV]
+σχn [cm2]
+CMB
+Bosonic DM
+Fermionic DM
+Skylab
+Chicago
+MW Satellites
+Terrestrial
+detectors
+Jupiter
+(This analysis)
+105
+106
+107
+108
+109
+1010
+10-14
+10-16
+10-18
+10-20
+10-22
+10-24
+10-26
+10-28
+10-30
+10-32
+10-34
+mχ [GeV]
+σχn [cm2]
+CMB
+Bosonic DM
+Fermionic DM
+Skylab
+Chicago
+MW Satellites
+Terrestrial
+detectors
+Earth
+(This analysis)
+105
+106
+107
+108
+109
+1010
+10-14
+10-16
+10-18
+10-20
+10-22
+10-24
+10-26
+10-28
+10-30
+10-32
+10-34
+mχ [GeV]
+σχn [cm2]
+CMB
+Bosonic DM
+Fermionic DM
+Skylab
+Chicago
+MW Satellites
+Terrestrial
+detectors
+Moon
+(This analysis)
+105
+106
+107
+108
+109
+1010
+10-14
+10-16
+10-18
+10-20
+10-22
+10-24
+10-26
+10-28
+10-30
+10-32
+10-34
+mχ [GeV]
+σχn [cm2]
+FIG. 2. Exclusion limits on spin-independent DM-nucleon scattering cross-sections from the existence of several stellar objects.
+The regions shaded by the solid red lines correspond to non-annihilating bosonic DM, whereas, the regions shaded by dashed
+red lines correspond to non-annihilating fermionic DM. The top left (right) panel corresponds to the Sun (Jupiter), whereas,
+the bottom left (right) panel corresponds to the Earth (Moon). The existence of the stellar objects provides unprecedented
+sensitivity to strongly-interacting heavy asymmetric DM. We show the existing exclusion limits (gray shaded regions) from
+terrestrial searches [35–38] (collected in [39, 56]), Skylab space station [57, 58], a recent shallow-depth experiment carried out
+at University of Chicago [59], and cosmological measurements [31, 33] for comparison. The existing constraints (gray-shaded
+regions) apply to both non-annihilating and annihilating DM, whereas, the constraints obtained from this analysis (red-shaded
+regions) apply solely to non-annihilating/very feebly annihilating DM.
+a much lower core temperature, implying a higher cap-
+ture rate but a lower threshold for transmutation. Earth
+and Moon have relatively smaller sizes and much lower
+core-temperatures, and hence, the capture rate as well as
+the threshold for transmutation both becomes smaller.
+We show our main results in Fig. 2 for non-annihilating
+bosonic and fermionic DM (spin-independent interac-
+tions).
+The top left (right) panel corresponds to Sun
+(Jupiter) as a DM detector, whereas, the bottom left
+(right) panel corresponds to Earth (Moon) as a DM de-
+tector. For stellar objects with low core temperatures,
+such as Moon, Earth, and Jupiter, the exclusion limit
+for bosonic DM is significantly stringent as compared
+to the fermionic DM. This is simply because for non-
+annihilating bosonic DM, the dark core collapse crite-
+rion is essentially determined by N self
+χ
+, whereas, for non-
+annihilating fermionic DM, it is determined by N cha
+χ
+,
+which is much higher than the self-gravitating criterion,
+i.e, N cha
+χ, fermion ≫ N self
+χ
+. This implies that transmutation
+for low core-temperature stellar objects is much harder
+to attain for non-annihilating fermionic DM, explaining
+the weaker exclusions. However, for stellar objects with
+higher core temperature, such as the Sun, the dark core
+collapse criterion for bosonic as well as fermionic DM is
+
+7
+set by N self
+χ
+(as it scales as T 3/2
+core), explaining identical
+exclusion limits for bosonic and fermionic DM.
+The exclusion limits in Fig. 2 can be explained quali-
+tatively from the following. For non-annihilating bosonic
+(fermionic) DM, the total number of captured DM par-
+ticles linearly increases with lighter mχ, whereas, the
+threshold for transmutation increases as m5/2
+χ
+(m3
+χ), and
+hence, for light DM masses, transmutation can not be
+achieved. This explains the sharp vertical cut-offs in the
+low mχ regime. For heavier DM masses, the mass of the
+nascent BH becomes smaller, resulting in two distinct
+effects.
+Firstly, the nascent BH takes a substantially
+longer time to consume the host, and secondly, Hawk-
+ing evaporation becomes crucial. Because of these two
+effects, transmutation ceases, providing the vertical cut-
+offs around mχ = 1010 GeV in Fig. 2.
+For the exact
+numerical values, see Section II D.
+Next, we discuss the σχn dependence of the exclu-
+sion limits in Fig. 2.
+For low DM-nucleon scattering
+cross-sections, the total number of captured DM par-
+ticles within the stellar objects decreases, and eventu-
+ally falls below the threshold for transmutation, indi-
+cating that low σχn can not be probed via transmuta-
+tion. Quantitatively, for non-annihilating bosonic DM,
+σχn ≤ 10−33 cm2 (σχn ≤ 10−28 cm2) can not be probed
+by the existence of the Sun (Moon).
+Very high DM-
+nucleon scattering cross-sections are also inaccessible via
+transmutations.
+This is simply because for very high
+σχn, the drift time of the DM particles becomes much
+longer, and they can not reach the stellar core for large
+interactions. In Fig. 2, we show the ceilings of our re-
+sults by demanding that tdrift ≤ 1 Gyr. For the Sun, it
+corresponds to σχn ≤ 10−17 cm2 for mχ = 107 GeV, and
+linearly increases with heavier DM mass.
+A.
+Comparison with the Existing Constraints
+In Fig. 2, we show the existing constraints on DM-
+nucleon scattering cross-section (gray-shaded regions) for
+comparison.
+The existing constraints can be classified
+into three broad categories: astrophysical, cosmological,
+and terrestrial. Constraints obtained from the terrestrial
+direct detection experiments (labeled as Terrestrial de-
+tectors) [35–38] are primarily based on non-observation
+of any anomalous scattering signature in the under-
+ground as well as in the surface detectors.
+We take
+these constraints from the summary plots in [39, 56].
+Cosmological constraints, such as Planck measurements
+of temperature and polarization anisotropy of the cos-
+mic microwave background (labeled as CMB) [31, 32],
+and Milky Way satellite observations (labeled as MW
+satellites) [33, 34] are also shown for comparison.
+As-
+trophysical constraints, such as disk stability [28], inter-
+stellar gas cooling [60], and Galactic Center gas-cloud
+heating [61, 62] are typically weaker than the constraints
+obtained from the Milky Way satellite observations, and
+therefore, are not shown for clarity.
+Very recently, a
+shallow-depth experiment carried out at the University
+of Chicago (labeled as Chicago) [59] provides a novel
+constraint on strongly-interacting heavy DM, and we
+show it in Fig. 2. Large panels of etched plastic, situ-
+ated aboard the Skylab Space Station also provide strin-
+gent exclusion limits on DM-nucleon interactions (labeled
+as Skylab) [28, 57, 58].
+Rocket-based X-ray Quantum
+Calorimetry (XQC) experiment [63], and searches for
+DM tracks in ancient underground mica [64, 65] probe
+strongly-interacting heavy DM. However, the XQC limit
+and the mica limit mostly apply for mχ ≤ 105 GeV and
+mχ ≥ 1010 GeV, respectively, and hence, are not shown.
+Finally, constraints obtained from cosmic ray silicon de-
+tector satellite (IMP7/8), and balloon-borne experiment
+(IMAX) [66] are not based on detailed analyses in peer-
+reviewed papers, and therefore, are not shown in Fig. 2.
+Comparing with the existing exclusion limits, it is ev-
+ident that the existence of a variety of stellar objects
+provides novel constraints on strongly-interacting heavy
+asymmetric DM. It provides unprecedented sensitivity to
+some regions in the parameter space as compared to the
+existing searches. Quantitatively, existence of the stel-
+lar objects exclude σχn = 10−24 cm2 for mχ = 107 GeV,
+not ruled by any other probes.
+This leading sensitiv-
+ity simply stems from the fact that stellar objects, ow-
+ing to their gigantic size and long lifetime, can capture
+a copious amount of Galactic DM particles for suffi-
+ciently high σχn, eventually causing an implosion. We
+note that, stellar objects with relatively larger sizes with
+much lower core temperatures, such as Jupiter, are the
+optimal targets to probe transmutation.
+This is sim-
+ply because the total number of accumulated DM par-
+ticles quadratically increases with a larger radius, and
+the threshold for transmutation falls off with lower core-
+temperature, implying the most favorable transmutation
+criterion. It is also important to stress that the existing
+constraints from cosmological and terrestrial searches ap-
+ply for both non-annihilating and annihilating DM inter-
+actions, as they are solely based on scatterings. Whereas,
+the constraints obtained in this analysis apply only to
+non-annihilating/very-feebly annihilating DM particles.
+We also note that, the exclusions obtained in Ref. [30]
+for Earth are weaker than our analysis, however for Sun,
+the constraints are similar.
+This can be explained in
+the following. Ref. [30] derived their results for bosonic
+DM with non-negligible quartic self-interactions (repul-
+sive). Because of the repulsive self-interactions among
+the DM particles, the effective Chandrasekhar limit sub-
+stantially increases [14], and it essentially determines the
+dark core collapse criterion for Earth. Whereas, in our
+analysis, we consider non-annihilating bosonic DM, and
+the dark core collapse criterion for Earth is determined by
+N self
+χ
+, explaining the difference between the two analyses.
+For Sun, because of its much larger core-temperature,
+N self
+χ
+(∼ T 3/2
+core) always exceeds over the Chandrasekhar
+limit, and sets the dark core collapse criterion, resulting
+in a similar exclusion in both the analyses.
+
+8
+B.
+Anomalous Heating Signatures
+Non-annihilating DM can also heat up the stellar ob-
+jects via successive scatterings with the stellar nuclei
+while getting captured. Such anomalous heating, com-
+monly known as dark kinetic heating, can be observed
+in neutron stars with very low surface temperatures [67].
+Cold neutron stars (TNS = O(1000) K) are ideal targets
+to search for dark kinetic heating as they have much
+higher escape velocities. Because of the large escape ve-
+locities, the velocity of the incoming DM particles gets
+significantly enhanced while falling into the steep gravita-
+tional potential of the neutron stars, and hence, they can
+transfer their kinetic energy to the stellar nuclei via col-
+lisions, heating up the cold neutron stars. We estimate
+the same for non-compact objects, concluding that be-
+cause of their sufficiently low escape velocity, and larger
+size, such anomalous heating signatures are too low to
+observe.
+IV.
+SUMMARY & CONCLUSION
+Celestial objects, owing to their enormous size and
+long lifetime, naturally act as novel DM detectors. We
+demonstrate that the continued existence of the Sun,
+Jupiter, Earth, and Moon provides stringent constraints
+on strongly-interacting asymmetric DM interactions over
+a wide mass range. These choices are well-motivated by
+the fact that each of these stellar objects is an optimal de-
+tector in a different regime as they cover a different range
+of size, density, and temperatures. We probe regions in
+the DM parameter space, which are entirely inaccessible
+to the existing DM searches, demonstrating the novelty
+of our analysis.
+Our proposal is simply based on the
+fact that non-annihilating DM particles from the Galactic
+halo get captured inside the stellar objects if they interact
+sufficiently with the nucleons. For heavy DM, these cap-
+tured DM particles rapidly thermalize within a very small
+region around the stellar core, resulting in a tantalizingly
+large core-density. Once the core-density exceeds its crit-
+ical threshold value, nascent BH forms inside the stellar
+core, eventually destroying the hosts. Mere existence of
+these stellar objects excludes such DM mass and cross-
+sections which predict the successful destruction of the
+hosts. We consider several stellar objects to demonstrate
+their potential as DM detectors and conclude that stel-
+lar objects with relatively larger sizes and low core tem-
+peratures, such as Jupiter, can probe the maximal DM
+mass window. Heavier DM masses are the most optimal
+for transmutation. However, as the nascent BH becomes
+smaller with an increase in DM mass, accretion becomes
+inefficient, and the Hawking evaporation becomes crucial,
+ceasing the transmutation for heavy DM masses. Overall,
+given these stringent exclusion limits, our work naturally
+inspires similar analysis for other celestial objects, such
+as Brown Dwarfs, Red Giants, and Exoplanets, once we
+have a better understanding of their density, tempera-
+ture, and compositional properties. Finally, we point out
+an intriguing direction that the transmutation of celestial
+objects can lead to planetary mass BHs, possibly explain-
+ing the six ultrashort microlensing events observed in the
+OGLE data [68], NANOGrav detection of stochastic GW
+background [69], as well as the BH hypothesis of Planet-
+9 [70]. Since, planet mass primordial BHs provide the
+viable solutions to this anomalies [68–70], it would be
+interesting to explore the alternative solutions via trans-
+mutation.
+V.
+ACKNOWLEDGMENTS
+It is my pleasure to thank Basudeb Dasgupta for help-
+ful discussions and useful comments on the manuscript.
+I also thank Sulagna Bhattacharya and Maxim Pospelov
+for helpful exchanges.
+AR acknowledges support from
+the National Science Foundation (Grant No.
+PHY-
+2020275), and to the Heising-Simons Foundation (Grant
+2017-228).
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diff --git a/WdE2T4oBgHgl3EQfDgZf/content/tmp_files/load_file.txt b/WdE2T4oBgHgl3EQfDgZf/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,934 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf,len=933
+page_content='N3AS-23-001  Celestial Objects as Strongly-Interacting Asymmetric Dark Matter Detectors  Anupam Ray  ID  1, 2, a  1Department of Physics, University of California Berkeley, Berkeley, California 94720, USA  2School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA  (Dated: January 11, 2023)  Non-annihilating dark matter particles, owing to their interactions with ordinary baryonic matter,  can efficiently accumulate inside celestial objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' For heavy mass, they gravitate toward the core of  the celestial objects, thermalize in a small core region, and eventually form tiny black holes via core  collapse, resulting destruction of the host objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We demonstrate that the existence of a variety of  celestial objects provides stringent constraints on strongly-interacting heavy dark matter, a blind-  spot for the terrestrial dark matter detectors as well as for the cosmological probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Celestial objects  with larger sizes and lower core temperatures, such as Jupiter, are the most optimal detectors to  probe the strongly-interacting heavy asymmetric dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  INTRODUCTION  The existence of dark matter (DM) has been firmly es-  tablished through its gravitational interactions with the  ordinary baryonic matter [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' However, its identity still  remains a mystery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' The ongoing terrestrial and cosmo-  logical searches are trying to pinpoint its mass and hy-  pothesized non-gravitational interactions with the bary-  onic matter [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' While some DM parameters have been  excluded by these searches, many well-motivated candi-  dates are yet to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Strongly-interacting heavy  asymmetric DM is one such regime that remains largely  untested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  In this work, we demonstrate that celestial objects are  excellent laboratories to search for strongly-interacting  heavy asymmetric DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' More specifically, we point out  that the continued existence of a variety of stellar ob-  jects provides stringent exclusions on asymmetric DM  interactions over a wide mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We mainly answer  two basic questions: why celestial objects are superior to  search for strongly-interacting heavy asymmetric DM as  compared to the terrestrial detectors, and which celestial  objects are the most optimal targets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Traditional direct detection experiments are not well-  suited for heavy DM searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' The flux of Galactic DM  at terrestrial detectors falls off linearly with higher DM  mass, and the constraints weaken accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Whereas,  because of the enormous sizes of astrophysical objects  and their long lifetimes, the effective exposure time (∼  M⊙ Gyr) is orders of magnitude larger than human-made  direct detection experiments (∼ kT yr), naturally provid-  ing sensitivity to the tiny flux of heavy DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Strongly-interacting DM is yet another blind-spot for  typical direct detection experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' If DM particles in-  teract too strongly with baryonic matter, they lose a sig-  nificant fraction of their energies via interactions with the  material in the atmosphere as well as in the Earth-cover  above the underground detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  As a consequence,  a anupam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='ray@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='edu  they slow down significantly and can not deposit suffi-  cient amounts of energy for detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Whereas, stellar  objects are ideal probes for strongly-interacting DM as  almost all the DM particles that transit through the stel-  lar objects get trapped, leading to a maximal sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Accumulation of particle DM in celestial objects is a  novel astrophysical probe of non-gravitational interac-  tions of DM with the ordinary baryonic matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  DM  particles from the galactic halo, owing to their inter-  actions with the stellar constituents, can down-scatter  to energies below the local escape energy, and become  gravitationally bound to the stellar objects [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' These  bound DM particles lose more energy via repeated scat-  terings with the stellar constituents and eventually ther-  malize inside the stellar volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Such bound thermalized  DM particles can become abundant inside the stellar vol-  ume if they have sufficiently strong interactions with the  baryonic matter, and possess intriguing phenomenologi-  cal signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  For non-annihilating DM, such bound DM particles  gradually accumulate, and for heavy DM masses, they  settle in a small region around the stellar core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Be-  cause of their prodigious abundance in a tiny core vol-  ume, their number density within the stellar core be-  comes quite large, allowing dark core collapse and sub-  sequent black hole (BH) formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' This nascent BH, if  not too light, can rapidly swallow the host, and the exis-  tence of stellar objects provides stringent constraints on  non-annihilating DM interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  This scenario has been extensively studied in the con-  text of old neutron stars [7–27] primarily because of their  enormously large baryonic density as well as high com-  pactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' More specifically, neutron stars can capture a  significant number of DM particles from the Galactic halo  even for the low DM-nucleon scattering cross-sections as  the single-collision capture rate scales linearly with the  compactness (∼ M/R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Quantitatively, for a solar mass  neutron star with a typical radius of 10 km, the capture  rate is O(105) times larger than the Sun for low DM in-  teractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Apart from that, because of their large bary-  onic density (∼ M/R3), the accumulated DM particles  thermalize in a tiny region around the core, implying a  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='03625v1  [hep-ph]  9 Jan 2023  2  huge core-density favorable for transmutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' This indi-  cates neutron stars are the most optimal targets to probe  weakly interacting heavy asymmetric DM, and the exis-  tence of old neutron stars in the solar neighborhood ex-  cludes mχ = 106 GeV and σχn = 10−48 cm2, the leading  constraints on non-annihilating DM interactions [11, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  However, in the optically thick (large DM-nucleon scat-  tering cross-section) regime, non-compact objects such  as the Sun, Jupiter, and Earth are more suitable detec-  tors to probe non-annihilating DM interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' This is  simply because in the optically thick regime, almost all  of the DM particles that transit through a stellar ob-  ject get trapped, and therefore, the accumulation rate  increases with the larger size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Since the non-compact ob-  jects possess much larger radii, the accumulation rate  for non-compact objects becomes comparable or even  larger than the neutron stars in the strongly-interacting  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Quantitatively, for mχ = 106 GeV, and suffi-  ciently high σχn, the transit (accumulation) rate for a  typical neutron star is 1019 s−1, comparable to the Earth,  and 10−2 (10−5) smaller than the Jupiter (Sun).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  This  naturally motivates us to explore the potential of non-  compact stellar objects as strongly-interacting asymmet-  ric DM detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  We found that stellar objects with  relatively large radii and low core-temperature (such as  Jupiter), are the most optimal detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' This is simply  because the total number of accumulated DM particles  increases with a larger radius, and the BH formation be-  comes easier with a lower core-temperature, implying the  most favorable transmutation criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Prior works [28–  30], more particularly Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' [30] has recently explored the  transmutation scenario for the Earth and the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We  systematically revisit the issue to gain more insight on the  constraints, and we show that stellar objects with larger  sizes and small core temperatures (such as Jupiter) pro-  vide the leading constraints on strongly-interacting heavy  asymmetric DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' In Sec-  tion II, we discuss different stages of DM-induced collapse  of the celestial objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' In Section III, we present our  exclusion limits from the existence of several stellar ob-  jects, demonstrating that the constraints obtained in this  analysis cover new parts of the DM parameter space and  bridge the gap between the cosmological probes [31–34],  and the terrestrial detectors [35–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Finally, we summa-  rize and conclude in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  DARK MATTER INDUCED COLLAPSE OF  STELLAR OBJECTS  Non-annihilating DM particles can accumulate effi-  ciently inside the stellar volume if they possess suffi-  ciently strong interactions with the stellar nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  For  heavy DM mass, they gravitate towards the stellar core  and settle in a tiny core-region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Because of their prodi-  gious abundance, and tiny core volume, their number  density within the stellar core becomes tantalizingly  larger, eventually resulting in a BH formation inside the  stellar core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  This nascent BH, if not sufficiently light,  can rapidly swallow the host, transmuting them to com-  parable mass BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' In the following, we discuss different  stages of DM-induced collapse of stellar objects, and a  schematic diagram for this process is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Dark Matter Accumulation  We first estimate the total number of captured DM  particles inside the stellar volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  For clarity, we de-  fine the maximal capture rate as saturation capture rate  (Csat), and it occurs when all of the DM particles that  transit through the stellar objects get trapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  For a  particular velocity distribution of the incoming DM par-  ticles, the saturation capture rate is [6]  Csat = ρχ  mχ  πR2  � f(u)du  u  (u2 + v2  esc) ,  (1)  where ρχ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='4 GeV/cm3 is the Galactic DM density,  mχ is the DM mass, and R is the radius of the stellar  object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' f(u) denotes the velocity distribution of the in-  coming DM particles, with vesc being the escape velocity  of the stellar objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' For a Maxwell-Boltzmann velocity  distribution, Csat simplifies to  Csat = ρχ  mχ  πR2  �  8  3π ¯v  �  1 + 3v2  esc  2¯v2  �  ,  (2)  where ¯v = 270 km/s denotes the average velocity of the  DM particles in the Galactic halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  A certain fraction of the DM particles will be cap-  tured by interacting with the stellar constituents, and  we aim to estimate this capture fraction (fc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Since we  are mostly interested in the optically thick regime (large  DM-nucleon scattering cross-section), fc behaves differ-  ently for heavier and lighter DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' For heavier DM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=',  when the DM mass (mχ) is larger than the target mass  (mA), scatterings do not alter the direction of the in-  coming DM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' As a consequence, the trajectories  of the incoming DM particles are not randomized, and  they follow almost a linear trajectory that ends inside  the stellar interior when the final velocity falls below the  escape velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' In this case, in the limit of multiple col-  lisions, the DM particles are essentially guaranteed to be  captured, resulting fc = 1 [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Whereas, in the opposite  regime (mχ < mA), the direction of the DM particles gets  randomized after each collision, and as a result, a certain  fraction of the DM particles can always escape from the  stellar volume via reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' This implies that for lighter  DM, even for arbitrarily large cross-sections, the capture  rate never reaches its saturation value (fc < 1) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  For this analysis, we are interested in heavy DM cap-  ture inside stellar objects, and hence, we take fc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  It implies that in the optically thick regime, we take the  capture rate to its saturation value, and it does not de-  pend on the DM-nucleon scattering cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  For  3  light DM capture in celestial objects, fc can be deter-  mined by the recent MCMC results [40], or from analyt-  ical estimates [41], both of which agree reasonably well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  In the optically thin regime (small DM-nucleon scat-  tering cross-section), capture occurs via single scatter-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' This is simply because for smaller values of DM-  nucleon scattering cross-sections, the mean free path of  the incoming DM particles becomes larger and becomes  comparable to the size of the stellar objects, and as a re-  sult, they scatter once while transiting through the host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  For this regime, we use the single-collision capture treat-  ment [6], and fc becomes substantially smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Spatial Distribution of Dark Matter inside  Stellar Objects  Captured DM particles rapidly thermalize inside the  celestial objects for sufficiently high DM-nuclei scatter-  ing cross-sections [11–13, 20, 30, 42], and the spatial dis-  tribution of the thermalized DM particles inside the stel-  lar volume depends on the effects of diffusion and grav-  ity [43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' By considering the effects of diffusion and  gravity in a self-consistent manner, the spatial distribu-  tion of the thermalized DM particles is [45]  ∇nχ(r)  nχ(r) + (κ + 1) ∇T(r)  T(r) + mχg(r)  T(r)  =  Φ  nχ(r)Dχn(r)  R2  ⊕  r2 ,  (3)  where nχ(r) denotes the number density of the thermal-  ized DM particles within the stellar volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' T(r) denotes  the temperature profile of the celestial objects, and Φ =  C/πR2  ⊕ is the incoming flux of the DM particles, with C  denotes the capture rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' κ ∼ −1/  �  2(1 + mχ/mA)3/2�  and Dχn ∼ λvth are the diffusion coefficients, where λ  denotes the mean free path of the DM particles and vth  denotes their thermal velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  It is evident that, for  very heavy DM mass, the diffusion co-efficient (κ) be-  comes smaller as it scales as m−3/2  χ  , and gravity (scales  proportional to mχ) dominates over the diffusion pro-  cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Therefore, heavy DM tends to settle down to-  wards the core of the stellar objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Quantitatively, by  solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' (3) for nχ(r) with the boundary condition  that the volume integral of nχ(r) provides the total num-  ber of captured DM particles, one can demonstrate that  the captured DM particles mostly concentrate around the  stellar core if they are heavy [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Concentration of heavy DM around the stellar core  can also be explained from the radius of the thermaliza-  tion sphere  �  rth =  �  9kBT/ (4πGρmχ)  �  , which results  from the balance between the thermal pressure and the  gravitational potential [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Since, the radius of the  thermalization sphere scales as m−1/2  χ  , for heavy DM,  rth becomes smaller, indicating the concentration of the  captured DM particles primarily occurs around the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Dark Core Collapse & Black Hole Formation  For non-annihilating DM, accumulation grows linearly  in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' As a consequence, the number density of the  captured DM particles inside the thermalization volume  becomes tantalizingly large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Quantitatively, for DM mass  of 106 GeV, and sufficiently high DM-nuclei scattering  cross-section (say 10−28 cm2), O(1036) number of DM  particles thermalize inside the Earth within a radius of  ∼ 6 km, indicating a number density of ∼ 2×1018 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  This corresponds to a core density of ∼ 2 × 1024 GeV  cm−3, around 25 orders of magnitude higher than the lo-  cal Galactic DM density, and it further increases as m3/2  χ  for heavier DM masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Once the core-density exceeds  its critical threshold value, it undergoes a gravitational  collapse, and eventually results in a BH formation inside  the stellar core [7–24, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' In the following, we quantify  the critical core-density for the stellar objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  The BH formation criterion via dark core collapse has  been extensively discussed in the literature for compact  stars [7–27, 30], and is essentially determined by two con-  ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' The first one is within the stable thermal ra-  dius, the DM density has to exceed the corresponding  baryonic density (ρb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' It leads to a self-gravitational col-  lapse of the thermalized DM particles and is determined  by [11, 13, 20]  mχN self  χ  4  3πr3  th  ≥ ρb ,  (4)  where N self  χ  denotes the critical number of DM particles  for ensuing self-gravitating collapse1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' It is independent  of the spin of the DM particles and only depends on the  DM mass as well as properties of the stellar objects such  as core-density and core-temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' For Earth, it cor-  responds to  N self  χ  ∼ 7×1036  �  ρcore  13 g/cm3  � � Tcore  5800 K  �3/2 �107 GeV  mχ  �5/2  ,  (5)  where, ρcore denotes the core density, and Tcore denotes  the core temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  The second condition is deter-  mined by the maximum number of DM particles that can  be stabilized by the degeneracy pressure and is commonly  known as Chandrasekhar limit (N cha  χ  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Chandrasekhar  limit depends on the spin-statistics of the DM particles as  degeneracy pressure for bosonic DM stems from the Un-  certainty principle, whereas, for fermionic DM, it arises  from the Pauli exclusion principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' N cha  χ  solely depends  on the DM mass, and quantitatively, it corresponds to  M 2  pl/m2  χ (M 3  pl/m3  χ), where Mpl = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='22 × 1019 GeV de-  notes the Planck mass [11, 13, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' To summarize, for  1 This self-gravitating criterion is essentially equivalent to the  Jeans instability criterion in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' [30] up to O(1) factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  4  Dark Matter Accumulation  Dark Core Collapse  Transmutation  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Transmutation of stellar objects via gradual accumulation of non-annihilating DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Heavy DM gravitate towards the  core of the stellar objects and form tiny black holes via dark core collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' These nascent BHs rapidly transmute the hosts by  swallowing them, resulting in comparable low mass BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  dark core collapse, the total number of captured DM par-  ticles inside a stellar object within its lifetime (tage) has  to satisfy the following [11, 13, 20, 24]  Nχ|tage = C × tage ≥ max  �  N self  χ  , N cha  χ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  (6)  Note that, for stellar objects with high core temperature  (such as Sun), the dark core collapse is essentially de-  termined by N self  χ  for bosonic as well as fermionic DM,  leading to identical exclusion limits for bosonic/fermionic  DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Whereas, for stellar objects with low core tempera-  ture, the dark core collapse is determined by N self  χ  (N cha  χ  )  for bosonic (fermionic) DM, leading to distinct exclusion  limits for bosonic/fermionic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Growth and Evaporation of Nascent Black  Holes  It is important to stress that dark core collapse does  not ensure the successful transmutation of the hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  If the nascent BH is sufficiently light, transmutation  can cease for two different reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Firstly, lighter BH  takes a much longer time to swallow the hosts, and  the swallow time (τswallow) can even be larger than the  lifetime of the hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Secondly, and more importantly,  Hawking radiation becomes significant for lighter BH  masses (∼ 1/M 2  BH), causing a rapid evaporation of the  nascent BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Since the mass of the nascent BH becomes  smaller for heavier DM mass, this provides an upper  limit on DM mass that can be probed via transmuta-  tion [11, 13, 20, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We quantify the upper limits of mχ  for several stellar objects in the following, and it ranges  around O(1010) GeV for the stellar objects under consid-  eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  For the time-evolution of the nascent BH, we conser-  vatively consider the baryonic matter accretion from the  host (ignoring the possible DM accretion by the nascent  BH) [11, 13, 46]  dMBH  dt  = 4πρcoreG2M 2  BH  c3s  − P (MBH)  G2M 2  BH  ,  (7)  where cs =  �  Tcore/mn denotes the sound speed at  the core of the stellar object, and P (MBH) denotes the  Page factor [47, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Page factor properly accounts into  gray-body corrections of the Hawking evaporation spec-  trum, as well as the number of Standard Model (SM)  species emission from an evaporating BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' In the classi-  cal black-body radiation limit, the Page factor evaluates  to 1/ (15360π), and is commonly used in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Considering the gray-body corrections, the Page factor  can be written as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='8 × 10−4f(MBH) where f(MBH) en-  codes the number of SM species emission from an evapo-  rating BH [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' For MBH ≥ 1017 g (which only emit mass-  less particles, such as photons and neutrinos), f(MBH) is  normalized to unity [48], and therefore, Page factor eval-  uates to 1/(1135π), an order of magnitude larger than  the classical black-body limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' For MBH ≤ 1017 g, the  number of SM species emission from an evaporating BH  crucially depends on its temperature (mass), and hence,  f(MBH) varies with BH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We use the semi-analytic  form of f(MBH) from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' [48] to estimate the Page fac-  tor in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Quantitatively, for light BHs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=',  MBH ≤ 1010 g, which can emit all SM species, f(MBH)  evaluates to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='35, and for 1015 g ≤ MBH ≤ 1017 g (which  can emit electrons, positrons, photons, and neutrinos),  f(MBH) evaluates to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' To summarize, depending on  BH mass, f(MBH) ranges from (1 − 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='35), implying the  range of Page factor from 1/(1135π) to 1/(74π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We ver-  ify that the Page factor obtained from the semi-analytic  form of f(MBH) from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' [48] is in excellent agreement  with the publicly available BlackHawk package [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Since, the accretion term scales as M 2  BH, and the evap-  5  oration term scales as 1/M 2  BH, for low BH masses, evap-  oration dominates over the accretion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Quantita-  tively, for Sun, Jupiter, Earth, and Moon, Hawking evap-  oration dominates over the Bondi-Hoyle accretion for  mχ ≥  �  �  �  �  �  �  �  �  �  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='1 × 1011 GeV  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='2 × 109 GeV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='4 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='3) × 109 GeV  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='0 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='0) × 109 GeV  (8)  for non-annihilating bosonic (fermionic) DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' On the  other hand, by requiring that the τswallow has to be less  than 1 Gyr, we obtain  mχ ≥  �  �  �  �  �  �  �  �  �  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='1 × 1010 GeV  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='1 × 1010 GeV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='9 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='5) × 1010 GeV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='8 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='1) × 1010 GeV  (9)  for non-annihilating bosonic (fermionic) DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We note  that, for stellar objects with high core-temperature,  τswallow essentially determines the wash-out of the trans-  mutation, whereas, for stellar objects with low core-  temperature, it is determined by the efficient Hawking  evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' This can simply be explained by the fol-  lowing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  For high core-temperature stellar objects, the  nascent BH becomes relatively larger (MBH,init ∼ T 3/2),  and therefore, the effects of Hawking evaporation become  relatively sub-dominant, implying accretion determines  the termination of the transmutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Drift time and Maximal Possible Scattering  Cross-section  Transmutation of stellar objects also ceases at very  high DM-nucleon scattering cross-sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' This is sim-  ply because, at very high DM-nucleon cross-sections, DM  particles lose a significant amount of energy in the outer  shells of these stellar objects, and might not reach the  stellar core to form a micro BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' In other words, the  viscous drag force that drives the DM particles toward  the core results in a long drift time, and therefore, pro-  hibits transmutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We estimate the drift time by us-  ing the stellar density, temperature, and compositional  profiles [8, 30, 50]  tdrift =  1  Gmχ  �  j  σχj  � R  0  nj(r)  �  3AjT(r)  � r  0 d3r′ρj(r′)  dr ,  (10)  where σχj  denotes the DM-nuclei scattering cross-  section, and it is related to the DM-nucleon scattering  cross-section via σχj = σχn A2  j  �  µχAj/µχn  �2 with Aj is  the mass number of the j-th nuclei, and µχn is the re-  duced mass of the DM-nucleon system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We determine the  ceilings of our results by demanding that tdrift ≤ 1 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Quantitatively, for (Sun) Earth, it corresponds to  σχn ≤ (10−17) 10−21 cm2 �  mχ  107 GeV  �  ,  (11)  and is same for bosonic/fermionic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Properties of Stellar Objects  We accurately estimate the capture rate and the trans-  mutation criterion for these stellar objects by utilizing  stellar object properties, such as density profiles, tem-  perature profiles, and detailed chemical compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' In  the following, we provide the inputs that have been con-  sidered for this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' The density and temperature  profiles for Sun, Earth, and Jupiter which have been used  in this analysis have also complied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Sun: We use the solar density and temperature  profiles from [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' For the chemical composition, we  assume that the Sun is entirely made up of 1H [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Jupiter: We use the Jupiter density and tempera-  ture profiles from the Jovian model J11-4a [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' For  the chemical composition, we assume that Jupiter  is entirely made up of 1H [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Earth: We use the Preliminary Reference Earth  Model from [53] for the density profile, and we  take the temperature profile from [54] under the  assumption of a hydro-static equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' For the  chemical composition, we use the tabulated val-  ues from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' [50] with the core-mantle bound-  ary at 3480 km, and the mantle-crust boundary at  6346 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' The core is dominantly made up of 56Fe,  whereas, the mantle and the crust are dominantly  made up of 16O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Moon: We use the MAX model for density and the  chemical compositions of the Moon [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We take  the lunar core-mantle boundary at 450 km, and the  mantle-crust boundary at 1650 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' The lunar core  is dominantly made up of 56Fe, whereas, the mantle  and the crust is dominantly made up of 16O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' For  the temperature profile, we consider the Moon as  an isothermal sphere with T = 1700 K [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  RESULTS  We consider a variety of stellar objects, such as the  Sun, Jupiter, Earth, and Moon, and demonstrate their  potential as asymmetric DM detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  These choices  are well-motivated by the fact that each of these stel-  lar objects is an optimal detector in a different regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  More specifically, they cover a wide range of size, den-  sity, and temperature, making them sensitive to different  parts of the DM parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Sun has the largest  size, and the highest core temperature, and as a con-  sequence, the total number of captured DM particles  as well as the threshold for transmutation both become  higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Jupiter has a somewhat larger size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' but possesses  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='CMB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Bosonic / Fermionic DM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Skylab  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Chicago  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='MW Satellites  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Terrestrial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='detectors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Sun  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='(This analysis)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='105  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='106  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='107  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='108  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='109  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='1010  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-24  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-26  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-28  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='mχ [GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='σχn [cm2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='CMB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Bosonic DM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Fermionic DM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Skylab  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Chicago  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='MW Satellites  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Terrestrial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='detectors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Jupiter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='(This analysis)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='105  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='106  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='107  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='108  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='109  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='1010  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-24  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-26  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-28  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='mχ [GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='σχn [cm2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='CMB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Bosonic DM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Fermionic DM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Skylab  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Chicago  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='MW Satellites  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Terrestrial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='detectors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Earth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='(This analysis)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='105  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='106  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='107  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='108  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='109  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='1010  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-24  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-26  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-28  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='mχ [GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='σχn [cm2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='CMB  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Bosonic DM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Fermionic DM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Skylab  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Chicago  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='MW Satellites  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Terrestrial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='detectors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='Moon  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='(This analysis)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='105  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='106  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='107  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='108  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='109  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='1010  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-22  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-24  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-26  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-28  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='10-34  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='mχ [GeV]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='σχn [cm2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Exclusion limits on spin-independent DM-nucleon scattering cross-sections from the existence of several stellar objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  The regions shaded by the solid red lines correspond to non-annihilating bosonic DM, whereas, the regions shaded by dashed  red lines correspond to non-annihilating fermionic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' The top left (right) panel corresponds to the Sun (Jupiter), whereas,  the bottom left (right) panel corresponds to the Earth (Moon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' The existence of the stellar objects provides unprecedented  sensitivity to strongly-interacting heavy asymmetric DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We show the existing exclusion limits (gray shaded regions) from  terrestrial searches [35–38] (collected in [39, 56]), Skylab space station [57, 58], a recent shallow-depth experiment carried out  at University of Chicago [59], and cosmological measurements [31, 33] for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' The existing constraints (gray-shaded  regions) apply to both non-annihilating and annihilating DM, whereas, the constraints obtained from this analysis (red-shaded  regions) apply solely to non-annihilating/very feebly annihilating DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  a much lower core temperature, implying a higher cap-  ture rate but a lower threshold for transmutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Earth  and Moon have relatively smaller sizes and much lower  core-temperatures, and hence, the capture rate as well as  the threshold for transmutation both becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  We show our main results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' 2 for non-annihilating  bosonic and fermionic DM (spin-independent interac-  tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  The top left (right) panel corresponds to Sun  (Jupiter) as a DM detector, whereas, the bottom left  (right) panel corresponds to Earth (Moon) as a DM de-  tector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' For stellar objects with low core temperatures,  such as Moon, Earth, and Jupiter, the exclusion limit  for bosonic DM is significantly stringent as compared  to the fermionic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' This is simply because for non-  annihilating bosonic DM, the dark core collapse crite-  rion is essentially determined by N self  χ  , whereas, for non-  annihilating fermionic DM, it is determined by N cha  χ  ,  which is much higher than the self-gravitating criterion,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='e, N cha  χ, fermion ≫ N self  χ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' This implies that transmutation  for low core-temperature stellar objects is much harder  to attain for non-annihilating fermionic DM, explaining  the weaker exclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' However, for stellar objects with  higher core temperature, such as the Sun, the dark core  collapse criterion for bosonic as well as fermionic DM is  7  set by N self  χ  (as it scales as T 3/2  core), explaining identical  exclusion limits for bosonic and fermionic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  The exclusion limits in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' 2 can be explained quali-  tatively from the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' For non-annihilating bosonic  (fermionic) DM, the total number of captured DM par-  ticles linearly increases with lighter mχ, whereas, the  threshold for transmutation increases as m5/2  χ  (m3  χ), and  hence, for light DM masses, transmutation can not be  achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' This explains the sharp vertical cut-offs in the  low mχ regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' For heavier DM masses, the mass of the  nascent BH becomes smaller, resulting in two distinct  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Firstly, the nascent BH takes a substantially  longer time to consume the host, and secondly, Hawk-  ing evaporation becomes crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Because of these two  effects, transmutation ceases, providing the vertical cut-  offs around mχ = 1010 GeV in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  For the exact  numerical values, see Section II D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Next, we discuss the σχn dependence of the exclu-  sion limits in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  For low DM-nucleon scattering  cross-sections, the total number of captured DM par-  ticles within the stellar objects decreases, and eventu-  ally falls below the threshold for transmutation, indi-  cating that low σχn can not be probed via transmuta-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Quantitatively, for non-annihilating bosonic DM,  σχn ≤ 10−33 cm2 (σχn ≤ 10−28 cm2) can not be probed  by the existence of the Sun (Moon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Very high DM-  nucleon scattering cross-sections are also inaccessible via  transmutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  This is simply because for very high  σχn, the drift time of the DM particles becomes much  longer, and they can not reach the stellar core for large  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' 2, we show the ceilings of our re-  sults by demanding that tdrift ≤ 1 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' For the Sun, it  corresponds to σχn ≤ 10−17 cm2 for mχ = 107 GeV, and  linearly increases with heavier DM mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Comparison with the Existing Constraints  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' 2, we show the existing constraints on DM-  nucleon scattering cross-section (gray-shaded regions) for  comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  The existing constraints can be classified  into three broad categories: astrophysical, cosmological,  and terrestrial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Constraints obtained from the terrestrial  direct detection experiments (labeled as Terrestrial de-  tectors) [35–38] are primarily based on non-observation  of any anomalous scattering signature in the under-  ground as well as in the surface detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  We take  these constraints from the summary plots in [39, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Cosmological constraints, such as Planck measurements  of temperature and polarization anisotropy of the cos-  mic microwave background (labeled as CMB) [31, 32],  and Milky Way satellite observations (labeled as MW  satellites) [33, 34] are also shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  As-  trophysical constraints, such as disk stability [28], inter-  stellar gas cooling [60], and Galactic Center gas-cloud  heating [61, 62] are typically weaker than the constraints  obtained from the Milky Way satellite observations, and  therefore, are not shown for clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Very recently, a  shallow-depth experiment carried out at the University  of Chicago (labeled as Chicago) [59] provides a novel  constraint on strongly-interacting heavy DM, and we  show it in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Large panels of etched plastic, situ-  ated aboard the Skylab Space Station also provide strin-  gent exclusion limits on DM-nucleon interactions (labeled  as Skylab) [28, 57, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Rocket-based X-ray Quantum  Calorimetry (XQC) experiment [63], and searches for  DM tracks in ancient underground mica [64, 65] probe  strongly-interacting heavy DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' However, the XQC limit  and the mica limit mostly apply for mχ ≤ 105 GeV and  mχ ≥ 1010 GeV, respectively, and hence, are not shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Finally, constraints obtained from cosmic ray silicon de-  tector satellite (IMP7/8), and balloon-borne experiment  (IMAX) [66] are not based on detailed analyses in peer-  reviewed papers, and therefore, are not shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Comparing with the existing exclusion limits, it is ev-  ident that the existence of a variety of stellar objects  provides novel constraints on strongly-interacting heavy  asymmetric DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' It provides unprecedented sensitivity to  some regions in the parameter space as compared to the  existing searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Quantitatively, existence of the stel-  lar objects exclude σχn = 10−24 cm2 for mχ = 107 GeV,  not ruled by any other probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  This leading sensitiv-  ity simply stems from the fact that stellar objects, ow-  ing to their gigantic size and long lifetime, can capture  a copious amount of Galactic DM particles for suffi-  ciently high σχn, eventually causing an implosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We  note that, stellar objects with relatively larger sizes with  much lower core temperatures, such as Jupiter, are the  optimal targets to probe transmutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  This is sim-  ply because the total number of accumulated DM par-  ticles quadratically increases with a larger radius, and  the threshold for transmutation falls off with lower core-  temperature, implying the most favorable transmutation  criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' It is also important to stress that the existing  constraints from cosmological and terrestrial searches ap-  ply for both non-annihilating and annihilating DM inter-  actions, as they are solely based on scatterings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Whereas,  the constraints obtained in this analysis apply only to  non-annihilating/very-feebly annihilating DM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  We also note that, the exclusions obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' [30]  for Earth are weaker than our analysis, however for Sun,  the constraints are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  This can be explained in  the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' [30] derived their results for bosonic  DM with non-negligible quartic self-interactions (repul-  sive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Because of the repulsive self-interactions among  the DM particles, the effective Chandrasekhar limit sub-  stantially increases [14], and it essentially determines the  dark core collapse criterion for Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Whereas, in our  analysis, we consider non-annihilating bosonic DM, and  the dark core collapse criterion for Earth is determined by  N self  χ  , explaining the difference between the two analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  For Sun, because of its much larger core-temperature,  N self  χ  (∼ T 3/2  core) always exceeds over the Chandrasekhar  limit, and sets the dark core collapse criterion, resulting  in a similar exclusion in both the analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  8  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Anomalous Heating Signatures  Non-annihilating DM can also heat up the stellar ob-  jects via successive scatterings with the stellar nuclei  while getting captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Such anomalous heating, com-  monly known as dark kinetic heating, can be observed  in neutron stars with very low surface temperatures [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Cold neutron stars (TNS = O(1000) K) are ideal targets  to search for dark kinetic heating as they have much  higher escape velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Because of the large escape ve-  locities, the velocity of the incoming DM particles gets  significantly enhanced while falling into the steep gravita-  tional potential of the neutron stars, and hence, they can  transfer their kinetic energy to the stellar nuclei via col-  lisions, heating up the cold neutron stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We estimate  the same for non-compact objects, concluding that be-  cause of their sufficiently low escape velocity, and larger  size, such anomalous heating signatures are too low to  observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  SUMMARY & CONCLUSION  Celestial objects, owing to their enormous size and  long lifetime, naturally act as novel DM detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We  demonstrate that the continued existence of the Sun,  Jupiter, Earth, and Moon provides stringent constraints  on strongly-interacting asymmetric DM interactions over  a wide mass range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' These choices are well-motivated by  the fact that each of these stellar objects is an optimal de-  tector in a different regime as they cover a different range  of size, density, and temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We probe regions in  the DM parameter space, which are entirely inaccessible  to the existing DM searches, demonstrating the novelty  of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  Our proposal is simply based on the  fact that non-annihilating DM particles from the Galactic  halo get captured inside the stellar objects if they interact  sufficiently with the nucleons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' For heavy DM, these cap-  tured DM particles rapidly thermalize within a very small  region around the stellar core, resulting in a tantalizingly  large core-density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Once the core-density exceeds its crit-  ical threshold value, nascent BH forms inside the stellar  core, eventually destroying the hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Mere existence of  these stellar objects excludes such DM mass and cross-  sections which predict the successful destruction of the  hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' We consider several stellar objects to demonstrate  their potential as DM detectors and conclude that stel-  lar objects with relatively larger sizes and low core tem-  peratures, such as Jupiter, can probe the maximal DM  mass window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Heavier DM masses are the most optimal  for transmutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' However, as the nascent BH becomes  smaller with an increase in DM mass, accretion becomes  inefficient, and the Hawking evaporation becomes crucial,  ceasing the transmutation for heavy DM masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Overall,  given these stringent exclusion limits, our work naturally  inspires similar analysis for other celestial objects, such  as Brown Dwarfs, Red Giants, and Exoplanets, once we  have a better understanding of their density, tempera-  ture, and compositional properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Finally, we point out  an intriguing direction that the transmutation of celestial  objects can lead to planetary mass BHs, possibly explain-  ing the six ultrashort microlensing events observed in the  OGLE data [68], NANOGrav detection of stochastic GW  background [69], as well as the BH hypothesis of Planet-  9 [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content=' Since, planet mass primordial BHs provide the  viable solutions to this anomalies [68–70], it would be  interesting to explore the alternative solutions via trans-  mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  It is my pleasure to thank Basudeb Dasgupta for help-  ful discussions and useful comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  I also thank Sulagna Bhattacharya and Maxim Pospelov  for helpful exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  AR acknowledges support from  the National Science Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
+page_content='  PHY-  2020275), and to the Heising-Simons Foundation (Grant  2017-228).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/WdE2T4oBgHgl3EQfDgZf/content/2301.03625v1.pdf'}
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+1
+Aerial Base Station Placement via
+Propagation Radio Maps
+Daniel Romero, Pham Q. Viet, and Raju Shrestha
+Abstract—The technology of base stations on board unmanned
+aerial vehicles, also known as aerial base stations (ABSs),
+promises to deliver cellular connectivity in areas where the
+terrestrial infrastructure is overloaded, damaged, or inexistent.
+A central problem in this context is to determine the locations
+where these ABSs must be deployed to serve a set of users on
+the ground given the positions of the latter. However, existing
+schemes assume that the channel gain depends only on the
+length and (possibly) the elevation of the link. To alleviate this
+limitation, this paper proposes a scheme that accommodates
+arbitrary channel gains by means of a propagation radio map of
+the air-to-ground channel. The algorithm finds the locations of
+an approximately minimal number of ABSs to serve all ground
+terminals with a target rate while meeting the given constraints
+on the capacity of the backhaul links and respecting no-fly
+regions. A convex-relaxation formulation ensures convergence
+and the alternating-direction method of multipliers is utilized
+to derive an implementation whose complexity is linear in the
+number of ground terminals. Numerical results with tomographic
+as well as ray-tracing channel models corroborate the strengths
+of the proposed scheme.
+Index Terms—Aerial base stations, radio maps, spectrum
+cartography, radio tomography, aerial base station placement,
+ray tracing.
+I. INTRODUCTION
+Aerial base stations (ABSs), namely unmanned aerial ve-
+hicles (UAVs) equipped with on-board base stations, were
+conceived as a means to deliver cellular connectivity in areas
+where the terrestrial infrastructure is absent, overloaded, or
+damaged [2]. This may occur for example in remote areas,
+in the vicinity of a crowded event, or after a natural disaster,
+such as a wildfire or a flood. Users on the ground, here referred
+to as ground terminals (GTs), are served by ABSs, which in
+turn connect to the terrestrial infrastructure through backhaul
+links, possibly in multiple hops through other UAVs that act
+as relays. Deploying ABSs involves addressing the problem
+of ABS placement, where one is given the locations of the
+GTs and must decide on a suitable set of spatial positions
+for the ABSs to effectively serve the GTs [3]. This task is
+typically hindered by several challenges, remarkably (C1) the
+uncertainty in the gain of the propagation channel between
+the GTs and the potential ABS locations, (C2) the limited
+The authors are with the Dept. of Information and Communication Technol-
+ogy, University of Agder, Jon Lilletunsvei 9, 4879 Grimstad, Norway. Email
+{daniel.romero,viet.q.pham, raju.shrestha}@uia.no.
+This work was supported by the Research Council of Norway through the
+IKTPLUSS Grant 311994.
+The present paper extends its conference precursor [1] to accommodate
+constraints in the backhaul and more general propagation radio maps. It also
+includes a much more comprehensive simulation study where, among others,
+simulations using ray-tracing software are carried out.
+capacity of the backhaul links, and (C3) constraints on the
+positions that the ABSs may adopt, often due to no-fly zones
+such as airports, embassies, or prisons.
+The problem of placing a single ABS has been extensively
+investigated in the literature; see e.g. [4]–[8]. However, in
+general, the number of ABSs required in a practical scenario
+need not equal one. For this reason, a large number of works,
+including the present one, focus on placing multiple ABSs.
+A usual approach is to regard the height of the ABSs as
+given and address the problem of 2D placement, where the
+ABSs must be placed on a horizontal plane of the given
+height; see e.g. [9]–[16]. Nevertheless, since the heights of
+the ABSs are useful degrees of freedom to optimize the target
+communication metric, the focus here is on 3D placement.
+Existing algorithms for 3D placement of multiple ABSs can
+be classified according to how they handle uncertainty in the
+air-to-ground channel. The first category comprises schemes
+that do not explicitly model the channel. For example, in
+[17], each ground user associates with the ABS from which
+it receives the strongest beacons, but nothing is known or
+assumed about the channel gain from the GT locations to a
+given location until an ABS is physically there. This means
+that every iteration of the placement algorithm involves placing
+the ABSs in a particular set of locations, which drastically
+limits convergence speed. The second class includes works that
+assume free-space propagation and, therefore, the coverage
+area of each ABS is a circle; see e.g. [18]. Unfortunately, this
+assumption is too inaccurate in practice. The third category is
+made up of works that rely on the empirical model from [19],
+[20]; see e.g. [21]–[25]. These works use the mean provided
+by such a model as the predictor of the channel gain, which
+is equivalent to assuming that the gain of a link depends
+only on its length and elevation. Again, this assumption is
+not very realistic since two links with the same length and
+elevation may exhibit totally different gains depending on
+whether there are obstructions such as buildings between the
+transmitter and the receiver. The fourth category, which can
+be termed channel-aware, is composed of works that rely
+on gain predictions that do depend on the locations of the
+endpoints of the link. To the best of our knowledge, only
+[26], [27] fall in this category. Unfortunately, these schemes
+entail prohibitive complexity, assume an unlimited backhaul
+connection between the ABSs and the terrestrial infrastructure,
+and cannot guarantee a minimum service to the GTs.
+The main contribution of this paper is a scheme that relies
+on radio propagation maps [28] to solve the problem of
+channel-aware 3D placement of multiple ABSs. Recall that
+a propagation map is a special kind of radio map [28]–[32]
+arXiv:2301.04966v1  [math.OC]  12 Jan 2023
+
+2
+that provides a channel metric of interest for every pair of
+transmitter and receiver locations. In this paper, this metric is
+the channel gain, which can be used to predict the capacity of
+the communication link between each candidate ABS location
+and every GT without deploying an ABS at that location to
+measure the channel. Two classes of propagation maps with
+complementary strengths will be considered, namely those
+obtained via ray-tracing and those that rely on the radio
+tomographic model [33], [34]. The former are more suitable to
+frequency bands where the dominating propagation phenom-
+ena beyond free-space loss are reflection and diffraction. The
+latter are suitable to bands where the dominating propagation
+phenomenon is the absorption introduced by obstacles such
+as buildings. In this context, a secondary contribution of this
+paper is to adapt existing radio tomographic techniques to air-
+to-ground channels.
+To the best of our knowledge, the proposed scheme is the
+first for channel-aware ABS placement that can guarantee a
+minimum rate for all GTs. Formally, the algorithm can find
+a feasible placement if it exists, where a feasible placement
+is an assignment of ABSs to spatial locations that ensures a
+target rate for all GTs according to the given propagation map.
+Besides, unlike the vast majority of works in the literature,
+constraints in the backhaul link between the ABSs and the ter-
+restrial infrastructure as well as no-fly zones can be enforced.
+With these constraints, the proposed algorithm approximately
+minimizes the number of ABSs required to serve all GTs.
+Note that this is of special interest in emergency scenarios,
+which is one of the main use cases of ABSs. The algorithm
+relies on a sparse optimization formulation that naturally
+arises from a discretization of the space of candidate ABS
+positions, as required to be able to utilize propagation maps
+in a tractable fashion. To counteract the high-dimensionality
+of the 3D placement problem, a linear-complexity and highly
+parallelizable algorithm is developed based on the alternating-
+direction method of multipliers (ADMM) [35].
+Experiments with tomographic and ray-tracing models
+showcase a great reduction in the number of required ABSs
+as compared to existing algorithms. To complement this
+manuscript, an open-source simulator was released to allow
+developing and testing algorithms for ABS placement. This
+simulator and the code needed to reproduce all experiments
+is available at https://github.com/uiano/ABS placement via
+propagation maps.
+Paper structure. The rest of the paper is organized as
+follows. Sec. II presents the model and formulates the problem.
+Two approaches for predicting the capacity of a link between
+arbitrary pairs of endpoints of the air-to-ground channel are
+then described in Sec. III. The problem of ABS placement
+and rate allocation is then addressed in Sec. IV and a solver
+with linear complexity is developed in Sec. V. Finally, Secs. VI
+and VII respectively present numerical results and conclusions.
+Notation. R+ is set of non-negative real numbers and R++
+is the set of positive real numbers. Boldface uppercase (lower-
+case) letters denote matrices (column vectors). a[i] represents
+the i-th entry of vector a. Notation 0 (respectively 1) refers to
+the matrix of the appropriate dimensions with all zeros (ones).
+∥A∥F denotes the Frobenius norm of matrix A, whereas ∥a∥p
+Fig. 1: Example of ABS placement in an urban environment.
+GTs are represented by markers on the ground, flight grid
+points by blue dots, and ABS positions by green circles.
+denotes the ℓp-norm of vector a. With no subscript, ∥a∥ stands
+for the ℓ2-norm. Inequalities between vectors or matrices must
+be understood entrywise. The Kronecker product is denoted by
+⊗. If a and b are vectors of the same dimension, then a ⊙ b
+is the entrywise product of a and b, whereas a ÷ b is the
+entrywise quotient of a and b.
+II. MODEL AND PROBLEM FORMULATION
+Consider M users located at positions {xGT
+1 , . . . , xGT
+M } ⊂
+X ⊂ R3, where the region X represents an arbitrary set of
+spatial locations, including for example points on the street,
+inside buildings, inside vehicles, and so on. The proposed
+scheme carries over unaltered to the scenario where X includes
+points in the airspace and some or all users are airborne,
+which can be of interest e.g. to deploy auxiliary ABSs as
+picocells. However, to simplify the exposition, this possibility
+is neglected and the users will be referred to throughout as
+ground terminals (GTs).
+To provide connectivity to these GTs, N ABSs are deployed
+at locations {xABS
+1
+, . . . , xABS
+N } ⊂ F ⊂ R3, where F comprises
+all spatial positions where a UAV is allowed to fly. This
+excludes no-fly zones, airspace occupied by buildings, and
+altitudes out of legal limits.
+For the sake of specificity, it will be assumed that data
+packets originated in a remote location are sent from the
+terrestrial infrastructure to the ABSs through a backhaul link
+and the ABSs forward these packets to their intended users
+through the downlink of the radio access network. However,
+the entire discussion applies also to the uplink, i.e., when the
+data packets are originated at the GTs and sent through the
+ABSs to the terrestrial infrastructure.
+Ignoring frequency-selective effects for simplicity, the ca-
+pacity of the communication link between an ABS at position
+xABS ∈ X and the m-th GT is given by
+Cm(xABS) = W log2
+�
+1 + PTX10γm(xABS)/10
+σ2
+�
+,
+(1)
+where W denotes bandwidth, PTX the transmit power spectral
+density (PSD), σ2 the noise PSD, and γm(xABS) the channel
+gain, which is described in Sec. III.
+Unlike most schemes in the literature of ABS placement, the
+present work can accommodate constraints in the backhaul.
+
+3
+To formalize such constraints, let CBH(xABS) denote the max-
+imum rate of the link between the terrestrial ground station(s)
+that serve(s) an ABS at xABS and the ABS at xABS. An
+equation like (1) can also be established to express CBH(xABS)
+in terms of the gain of the relevant channel(s). Note that, as
+the notation suggests, in general CBH(xABS) depends on the
+ABS position xABS – typically, the greater the distance from
+xABS to the terrestrial ground stations, the lower CBH(xABS).
+With Rm(xABS) denoting the downlink rate that an ABS at
+xABS allocates to the m-th GT, the backhaul rate constraint
+imposes that �
+m Rm(xABS) ≤ CBH(xABS).
+The problem is to find a minimal number of ABS locations
+that guarantee that every user receives a rate of at least
+Rmin. This criterion arises naturally in some of the main use
+cases of UAV-assisted networks such as emergency response
+or disaster management. Motivated by this scenario and to
+enhance flexibility in the deployment, each GT may be served
+by multiple ABSs. This means that the rate that the m-th user
+receives is �
+n Rm(xABS
+n
+), where xABS
+n
+denotes the location
+of the n-th ABS.
+To summarize, the problem can be formulated as follows:
+minimize
+N,{xABS
+n
+}N
+n=1,{rm[n]}M,N
+m=1,n=1
+N
+(2a)
+s.t.
+�
+m
+rm[n] ≤ CBH(xABS
+n
+),
+(2b)
+�
+n
+rm[n] ≥ Rmin,
+(2c)
+0 ≤ rm[n] ≤ Cm(xABS
+n
+),
+(2d)
+xABS
+n
+∈ F,
+(2e)
+where the constraints need to hold for all m and n and the
+earlier notation Rm(xABS
+n
+) has been replaced with rm[n] :=
+Rm(xABS
+n
+) to emphasize that it refers to optimization vari-
+ables, not to functions. Observe that Problem (2) constitutes a
+joint placement and rate-allocation problem. Note also that the
+same minimum rate Rmin is imposed for all GTs, but different
+rates can be set up to straightforward modifications.
+From a practical perspective, solving (2) involves two chal-
+lenges. First, Cm(xABS
+n
+) and CBH(xABS
+n
+) depend on the chan-
+nel gain of the corresponding downlink and backhaul links,
+which is generally unknown. This issue will be addressed in
+Sec. III. Second, given Cm(xABS
+n
+) and CBH(xABS
+n
+), one needs
+to find the positions of the ABSs and the rate allocations that
+solve (2). This will be the subject of Sec. IV.
+III. CAPACITY PREDICTION VIA PROPAGATION RADIO
+MAPS
+As described in Sec. I, nearly all existing schemes for ABS
+placement assume that Cm(xABS
+n
+) depends on xGT
+m and xABS
+n
+only through the length and (possibly) the elevation angle of
+the line segment between these two points. However, one can
+expect that this simplification entails a significant performance
+degradation since channel gain in reality is heavily affected by
+the environment: two links with the same length and elevation
+may experience very different channel gain depending on the
+position, shape, and material of the surrounding obstacles and
+scatterers. The same observation applies to CBH(xABS
+n
+).
+For this reason, this section proposes the utilization of radio
+propagation maps to obtain Cm(xABS
+n
+) and CBH(xABS
+n
+). A
+radio propagation map is a function of two spatial locations
+that provides a certain metric of interest, in this case the
+gain, for the channel between those spatial locations [28].
+Without loss of generality, the rest of this section focuses on
+Cm(xABS
+n
+), but the same approaches and considerations apply
+to CBH(xABS
+n
+).
+A. Ray-tracing Models
+Ray-tracing
+techniques
+[36]
+can
+be
+used
+to
+predict
+γm(xABS
+n
+) for arbitrary pairs (xGT
+m , xABS
+n
+) given a 3D model of
+the propagation environment, which includes buildings, terrain
+elements, and other objects. The idea is to geometrically obtain
+the relevant propagation paths between the transmitter and the
+receiver and subsequently calculate the attenuation and delay
+or phase change of each path.
+Ray-tracing models are especially attuned to high-frequency
+bands, where the dominating propagation phenomena are re-
+flection and diffraction. The limitation is that, to track changes
+in the channel, one needs to track changes in the 3D model,
+which may not be viable. Thus, in the case of ABS placement,
+it makes sense to use ray-tracing to precompute the values of
+γm(xABS
+n
+) (cf. Sec. VI-B) for the given scenario and ignore the
+effects of changes in the channel. If the impact of the changes
+(e.g. moving vehicles) is not too large, using a ray-tracing
+propagation map would still yield better placement solutions
+than adopting the approaches in the literature, which either
+assume free space or assume that the gain is a function of just
+the distance and elevation of the link.
+B. Radio Tomographic Models
+As opposed to ray-tracing models, which account for reflec-
+tion and diffraction effects, the radio tomographic model [33]
+(see also [37] and the references therein) focuses on the
+absorption undergone by radio waves on the straight path
+between the transmitter and the receiver. This model is best
+suited to traditional cellular communication frequencies, where
+radio waves readily penetrate structures such as buildings.
+However, the existing works in this context focus on ground-
+to-ground channels. To the best of our knowledge, this is the
+first work to apply radio tomography to air-to-ground channels.
+This entails special challenges due to the high dimensionality
+of the underlying space, which render existing techniques
+unsuitable. After describing the radio tomographic model, this
+section proposes an algorithm to bypass these difficulties.
+The radio tomographic model dictates that the channel gain
+between xGT
+m and xABS can be decomposed into a free-space
+loss component and a shadowing component as1
+γm(xABS) = 20 log10
+�
+λ
+4π∥xGT
+m − xABS∥
+�
+− ξ(xGT
+m , xABS),
+(3)
+1Expression (3) assumes that small-scale fading has been averaged out.
+Otherwise, a random term can be added to the right-hand side.
+
+4
+where λ is the wavelength associated with the carrier fre-
+quency and the shadowing function ξ is given by [33]
+ξ(x1, x2) =
+1
+∥x1 − x2∥1/2
+2
+� x2
+x1
+l(x)dx.
+(4)
+The non-negative function l inside the line integral is termed
+spatial loss field (SLF) and quantifies the local attenuation
+(absorption) that a signal suffers at each position.
+Two tasks are of interest: (T1) evaluate γm(xABS) for
+a pair of locations (xGT
+m , xABS), which involves evaluat-
+ing the integral in (4) and substituting the result into (3);
+(T2) estimate l given a set of measurements of the form
+(xABS, xGT
+m , γm(xABS)) collected beforehand.
+In both tasks, l needs to be discretized, which can be
+accomplished by storing its values l(x ¯
+X
+1 ), . . . , l(x ¯
+X
+Q) on a
+regular grid of Q points ¯
+X := {x ¯
+X
+1 , . . . , x ¯
+X
+Q}. The rest of
+this section explains how ξ(x1, x2) can be expressed in terms
+of these values. This addresses (T1) and enables (T2) in
+combination with standard estimators; see e.g. [37]–[39]
+The conventional approach approximates the right-hand side
+of (4) as the weighted sum [40]
+ξ(x1, x2) ≈
+�
+q
+w(x1, x2, x
+¯
+X
+q )l(x
+¯
+X
+q ),
+(5)
+where the weight function w(x1, x2, x ¯
+X ) aims at assigning
+a non-zero weight only to those grid points x ¯
+X lying close
+to the line segment between x1 and x2. Although there are
+some variations, the functions w adopted in the literature are
+non-zero only when x ¯
+X lies inside an ellipsoid with foci at
+x1 and x2 [37], [40], [41]; see the ellipses in Fig. 2 for a
+depiction in 2D.
+Such an approximation suffers from several limitations that
+render it impractical for the application at hand. First, since
+existing choices of w(x1, x2, x ¯
+X ) are discontinuous functions,
+the resulting approximations of ξ(x1, x2) are also discon-
+tinuous, which may lead to erratic behavior. For example,
+it may well happen that �
+q w(x1, x2, x ¯
+X
+q )l(x ¯
+X
+q ) = 0 even
+when x1 ̸= x2 and l(x ¯
+X
+q ) ̸= 0 ∀q. This happens when x1
+and x2 are such that no grid point falls inside the elliptical
+support of w(x1, x2, x ¯
+X ); see the upper ellipse in Fig. 2.
+Since w is typically chosen so that its support coincides with
+the first Fresnel ellipsoid, the length of its minor axis is
+roughly proportional to
+√
+λ. If the carrier frequency is low,
+the ellipsoid is large and, therefore, likely to contain a large
+number of grid points, which somehow mitigates the effects of
+the discontinuity. Conversely, if the carrier frequency is high,
+one could think of creating a sufficiently dense grid so that the
+distance between grid points is small relative to λ. However,
+it is easy to see that this may entail a prohibitively high Q.
+For example, if one wishes to set the grid points, say, 5 cm
+apart and the region of interest is 1 km × 1 km × 100 m,
+then the total number of grid points is 1011, which is obvi-
+ously impractical. Another critical limitation is computational
+complexity. Observe that (5) generally requires evaluating
+w(x1, x2, x ¯
+X
+q ) and the product w(x1, x2, x ¯
+X
+q )l(x ¯
+X
+q ) for each
+grid point. Thus, if the grid is Q0 × Q0 × Q0, the complexity
+of approximating ξ(x1, x2) is O(Q3
+0). This is computationally
+Algorithm 1: Tomographic Integral Approximation
+1
+1: Input: x1, x2, grid spacing vector δ ¯
+X ∈ R3,
+SLF tensor L ∈ RQx×Qy×Qz.
+2: Initialize ∆x = x2 − x1, binc = sign(∆x),
+I = 0, t = 0.
+3: Set zero entries of ∆x to 1 # To avoid dividing by 0
+4: Set icurrent = round(x1 ÷ δ ¯
+X ) # Current voxel indices
+5: while t < 1 do
+6:
+Set tcand = (δ ¯
+X ⊙ (icurrent + binc/2) − x1) ÷ ∆x
+7:
+Set inext = arg mini tcand[i] s.t. binc[i] ̸= 0
+8:
+Set tnext = min(1, tcand[inext])
+9:
+Set I = I + (tnext − t)L[icurrent]
+10:
+Set t = tnext
+11:
+Set icurrent[inext] = icurrent[inext] + binc[inext]
+12: end while
+13: return ∥x2 − x1∥1/2I
+problematic for the application at hand since ξ(x1, x2) must
+be approximated at a large number of locations to solve (2).
+To remedy these issues, this paper advocates approximating
+the integral in (4) as a line integral of a piecewise constant
+approximation of l. The resulting approximation is continuous,
+can be used with large grid point spacing, and can be computed
+with complexity only O(Q0) for a Q0 × Q0 × Q0 grid. This
+technique, commonly used in other disciplines (see references
+in [42]), involves splitting the 3D space into voxels centered
+at the grid points ¯
+X := {x ¯
+X
+1 , . . . , x ¯
+X
+Q} and approximating l
+by a function that takes the value l(x ¯
+X
+q ) at all points of the q-
+th voxel. The resulting piecewise constant approximation of l
+can be integrated by determining the positions of the crossings
+between the voxel boundaries and the line segment between
+x1 and x2; see Fig. 2.
+Algorithm 1, derived in Appendix C, is our implementation
+of this approach. This algorithm solves the limitations of the
+conventional approximation outlined earlier. First, Algorithm 1
+yields an approximation of ξ(x1, x2) that is a continuous
+function of x1 and x2 since the line integral of a piecewise
+constant function is a continuous function of the endpoints.
+Besides, the algorithm does not suffer from the issue of
+the approximation becoming zero when the elliptical support
+of the weight function in (5) misses all grid points. For
+this reason, the voxels can now be kept large regardless of
+the wavelength and, therefore, the total number of voxels
+can be kept low enough to be handled given the available
+computational resources. Finally, as indicated in Sec. III, the
+computational complexity of Algorithm 1 is much smaller
+than the one of the conventional approximation. Specifically,
+one can observe in Algorithm 1 that a constant number of
+products and additions is required for each crossing. The total
+number of crossings is at most Qz + Qy + Qz, which means
+that, if Qx = Qy = Qz = Q0, then the total complexity of
+Algorithm 1 is O(Q0), whereas the complexity of the standard
+approximation is O(Q3
+0).
+IV. PLACEMENT WITH MIN-RATE GUARANTEES
+The approaches in Sec. III make it possible to predict the
+
+5
+Weight-function 
+approximations
+Piecewise linear 
+approximation
+Voxel
+Centroid
+Fig. 2: 2D illustration of the conventional weight-function
+approximation of the tomographic integral (4) (orange ellipses)
+and the approximation adopted here (colored line segment).
+Observe that the upper ellipse contains no centroid and, there-
+fore, the approximation will yield zero attenuation regardless
+of the values of the SLF.
+channel gain γm(xABS
+n
+) for all required pairs of user location
+xm and ABS location xABS
+n
+. These predictions can then be
+substituted into (1) to obtain the capacity values Cm(xABS
+n
+)
+that appear in the constraint (2d). A similar observation
+applies to CBH(xABS
+n
+) in (2b). Unfortunately, solving (2) is
+challenging: even if N were known and one just needed to find
+feasible {xABS
+n
+}N
+n=1, the problem would still be non-convex
+since the right-hand sides of (2b) and (2d) are, in general,
+non-concave functions of xABS
+n
+. To bypass this difficulty,
+the flight region F will be discretized into a flight grid
+¯F := {x ¯
+F
+1 , . . . , x ¯
+F
+G} ⊂ F ⊂ R3; see Fig. 1.
+Replacing xABS
+n
+∈ F in (2e) with xABS
+n
+∈
+¯F means
+that optimizing with respect to N and the ABS positions
+{xABS
+n
+}N
+n=1 is equivalent to choosing the smallest subset of
+points in ¯F for which there exists a feasible rate allocation,
+i.e., for which there exist {rm[n]}M,N
+m=1,n=1 satisfying the
+constraints in (2). Equivalently, one can consider all grid points
+by setting each xABS
+n
+to be one of the grid points and then
+“disable” those xABS
+n
+where no ABS is going to be present.
+To this end, let αg be 1 if there is an ABS at x ¯
+F
+g and 0
+otherwise, in which case the rates from that location need to
+be zero. By following this approach, Problem (2) becomes
+minimize
+{αg}G
+g=1,{rg[m]}M,G
+m=1,g=1
+G
+�
+g=1
+αg
+(6a)
+s.t.
+�
+m
+rg[m] ≤ CBH(x
+¯
+F
+g ),
+(6b)
+�
+g
+rg[m] ≥ Rmin,
+(6c)
+0 ≤ rg[m] ≤ αgCm(x
+¯
+F
+g ),
+(6d)
+αg ∈ {0, 1},
+(6e)
+where now the constraints need to hold for all m and g. Note
+the presence of αg in the right-hand side of (6d), which forces
+the rate to be 0 at grid points without an actual ABS.
+To simplify notation, it is convenient to express Prob-
+lem (6) in terms of rg
+:=
+[rg[1], . . . , rg[M]]⊤, R
+:=
+[r1, . . . , rG], cBH := [CBH(x ¯
+F
+1 ), . . . , CBH(x ¯
+F
+G)]⊤, and cg :=
+[C1(x ¯
+F
+g ), . . . , CM(x ¯
+F
+g )]⊤ as
+minimize
+{αg}G
+g=1,R
+G
+�
+g=1
+αg
+(7a)
+s.t. R⊤1 ≤ cBH
+(7b)
+R1 ≥ Rmin1
+(7c)
+0 ≤ rg ≤ αgcg
+(7d)
+αg ∈ {0, 1}.
+(7e)
+Observe that, if {αg}G
+g=1 were given, then Problem (7)
+would be a convex feasibility problem in R. However, since
+one needs to optimize over {αg}G
+g=1 as well, Problem (7)
+becomes combinatorial. In fact, the following result establishes
+the NP-hardness of (7).
+Theorem 1: Problem (7) is NP-hard unless P=NP.
+Proof: See Appendix E.
+It is useful to reduce Problem (7) to the optimization with
+respect to R alone. To this end, one can note that, for an
+arbitrary R, the corresponding optimal {αg}G
+g=1 satisfy that
+αg = 0 if rg = 0 and αg = 1 otherwise. This means that (7)
+can be written as
+minimize
+R
+G
+�
+g=1
+I[rg ̸= 0]
+(8a)
+s.t. R⊤1 ≤ cBH
+(8b)
+R1 ≥ Rmin1
+(8c)
+0 ≤ rg ≤ cg,
+(8d)
+where I[·] equals 1 if the condition inside brackets is true and
+0 otherwise.
+The objective �G
+g=1 I[rg ̸= 0] can be equivalently ex-
+pressed as �G
+g=1 I[∥rg∥∞ ̸= 0], where the ℓ∞-norm ∥v∥∞
+equals the largest absolute value of the entries of vector v.
+Clearly, �G
+g=1 I[∥rg∥∞ ̸= 0] = ∥[∥r1∥∞, . . . , ∥rG∥∞]⊤∥0,
+which suggests the relaxation ∥[∥r1∥∞, . . . , ∥rG∥∞]⊤∥1 =
+�
+g ∥rg∥∞, or its reweighted version �
+g wg∥rg∥∞, where
+{wg}g are non-negative constants set as in [43]. The problem
+thereby becomes
+minimize
+R
+�
+g
+wg∥rg∥∞
+(9a)
+s.t. R⊤1 ≤ cBH
+(9b)
+R1 ≥ Rmin1
+(9c)
+0 ≤ R ≤ C,
+(9d)
+where the (m, g)-th entry of C := [c1, . . . , cG] ∈ RM×G
++
+is given by cm,g := Cm(x ¯
+F
+g ), i.e., the capacity of the link
+between the m-th user and the g-th grid point. The (m, g)-th
+entry of R therefore satisfies 0 ≤ rm,g ≤ cm,g, which means
+that it can be interpreted as the rate at which a virtual ABS [14]
+placed at grid point x ¯
+F
+g communicates with the m-th user. In
+case that rm,g = 0 for all m, then no actual ABS needs to
+be deployed at x ¯
+F
+g . In other words, the virtual ABS at x ¯
+F
+g
+corresponds to an actual ABS if and only if rm,g ̸= 0 for
+some m.
+
+6
+V. NUMERICAL SOLVER
+Observe that (9) is a convex optimization problem and
+therefore it can be numerically solved in polynomial time.
+In view of the inequality constraints, the first possibility one
+may consider is to apply an interior-point solver, as described
+in Appendix A.
+Unfortunately, such an approach is only
+suitable for relatively small values of M and G given the
+poor scalability of interior-point methods with the number of
+variables and constraints [44]. Indeed, in this application, G
+can be in the order of tens of thousands, which would render
+the (at least cubic; cf. Appendix A) complexity of interior-
+point methods prohibitive. In contrast, the rest of this section
+presents a solver whose complexity is linear in G and M by
+building upon the so-called alternating-direction method of
+multipliers (ADMM) [35] and exploiting the special structure
+of the problem.
+A. ADMM Decomposition
+As outlined below, ADMM alternately solves two opti-
+mization subproblems in the primal variables and performs
+a gradient step along the dual variables [35]. But before
+decomposing (9) into such subproblems, a few manipulations
+are in order.
+The first is to replace the inequality constraint (9c) with
+an equality constraint. To this end, let ¯rm denote the m-th
+column of R⊤ and suppose that R is feasible. Then, it follows
+from (9c) that ¯r⊤
+m1 ≥ Rmin. If one replaces ¯rm with ¯r′
+m :=
+(Rmin/(¯r⊤
+m1))¯rm, the entries of ¯r′
+m are non-negative and not
+greater than the entries of ¯rm since 0 ≤ (Rmin/(¯r⊤
+m1)) ≤ 1.
+Hence, the resulting R still satisfies all other constraints and
+the m-th constraint in (9c) now holds with equality. Besides,
+the resulting objective will not be greater. Therefore, if R is
+optimal, scaling any of its rows in this way yields another
+optimal R. Applying this reasoning to all rows (i.e., for all
+m) shows that (9c) can be replaced with an equality constraint
+without loss of optimality.
+Second, due to (9d), the entries of rg are non-negative and,
+thus, ∥rg∥∞ equals the largest entry of rg. This means that
+the objective can be replaced with �
+g wgsg upon introducing
+the auxiliary variables sg and constraints rg ≤ sg1 for each
+g. This gives rise to
+minimize
+R,s
+w⊤s
+(10a)
+s.t. R⊤1 ≤ cBH
+(10b)
+R1 = Rmin1
+(10c)
+0 ≤ R ≤ C
+(10d)
+R ≤ 1s⊤,
+(10e)
+where w := [w1, . . . , wG]⊤ and s := [s1, . . . , sG]⊤.
+The next step is to express (10) in a form amenable to
+application of ADMM. Specifically, (10) will be expressed in
+the homogeneous form
+minimize
+X,Z
+f(X) + h(Z)
+(11a)
+s.t.
+A1XA2 + B1ZB2 = 0,
+(11b)
+for which the ADMM iteration becomes [35, Sec. 3.1.1]
+Xk+1 = arg min
+X
+f(X) + ρ
+2∥A1XA2 + B1ZkB2 + U k∥2
+F
+(12a)
+Zk+1 = arg min
+Z
+h(Z) + ρ
+2∥A1Xk+1A2 + B1ZB2 + U k∥2
+F
+(12b)
+U k+1 = U k + A1Xk+1A2 + B1Zk+1B2
+(12c)
+for k = 1, 2 . . .. Here, Xk and Zk collect the primal variables,
+U k is a matrix of scaled dual variables, and ρ > 0 is the step-
+size parameter.
+Each possible correspondence that one may establish be-
+tween the variables, constants, and functions of (11) and those
+of (10) results in a different ADMM algorithm. Finding a
+good correspondence is typically the most critical step and
+takes multiple attempts since, unless properly accomplished,
+the complexity of the subproblems (12a) and (12b) will be
+comparable to the complexity of the original problem. For the
+problem at hand, the following assignment was found to yield
+subproblems that separate along the rows and columns of R:
+X → [R⊤, s]⊤
+(13a)
+Z → R
+(13b)
+f(X) → w⊤s +
+�
+g
+I[rg ≤ sg1]
++
+�
+g
+I[1⊤rg ≤ cBH
+g ]
+(13c)
+h(Z) → I[R1 = Rmin1] + I[0 ≤ R ≤ C]
+(13d)
+A1 → [IM, 0], A2 → IG,
+(13e)
+B1 → −IM, B2 → IG.
+(13f)
+Here, cBH := [cBH
+1 , . . . , cBH
+G ]⊤ and I[·] is a function that takes
+the value 0 when the condition inside brackets holds and ∞
+otherwise. Note that, given (13), it follows that A1XA2 +
+B1ZB2 = R−Z and, therefore, (11b) imposes that R = Z.
+Thus, intuitively speaking, each subproblem (12a) and (12b)
+tries to find values for their respective variables that satisfy the
+structure promoted by the first terms in (12a) and (12b) while,
+at the same, the second terms in these expressions as well
+as (12c) push towards an agreement between the solutions of
+both subproblems.
+The next two subsections will be respectively concerned
+with finding the solutions of (12a) and (12b). Afterwards, both
+solutions are put together in Sec. V-D to obtain the desired
+algorithm.
+B. The X-subproblem
+This section decomposes the X-update (12a) into G smaller
+problems. The latter can be efficiently solved by finding a root
+of a scalar equation through the bisection algorithm.
+
+7
+In view of (13), (12a) can be expressed as
+(Rk+1, sk+1) = arg min
+R,s w⊤s +
+�
+g
+I[rg ≤ sg1]
+(14a)
++
+�
+g
+I[1⊤rg ≤ cBH
+g ] + ρ
+2∥R − Zk + U k∥2
+F
+= arg min
+R,s
+�
+g
+�
+wgsg + I[rg ≤ sg1]
+(14b)
++I[1⊤rg ≤ cBH
+g ] + ρ
+2∥rg − zk
+g + uk
+g∥2
+2
+�
+,
+where zk
+g and uk
+g respectively denote the g-th column of Zk
+and U k. This problem clearly separates into G problems of
+the form
+(rk+1
+g
+, sk+1
+g
+) = arg min
+rg,sg wgsg + ρ
+2∥rg − zk
+g + uk
+g∥2
+2 (15a)
+s.t.
+rg ≤ sg1
+(15b)
+1⊤rg ≤ cBH
+g .
+(15c)
+There are two cases: C1) constraint (15c) is active (i.e.
+it holds with equality) at the optimal solution; C2) (15c) is
+inactive (i.e. it holds with strict inequality) at the optimal
+solution. Thus, to solve (15), one can apply the following
+strategy. First, solve the problem that results from removing
+(15c):
+(rk+1
+g
+, sk+1
+g
+) = arg min
+rg,sg wgsg + ρ
+2∥rg − zk
+g + uk
+g∥2
+2 (16a)
+s.t.
+rg ≤ sg1.
+(16b)
+If the solution to (16) satisfies (15c), then it is also the optimal
+solution of (15). Else, due to the convexity of the problem,
+(15c) must necessarily be active at the optimum. In this case,
+the optimal solution of (15) can be found by replacing (15c)
+with an equality constraint. Thus, let us start by solving (16).
+Proposition 1: Let rk+1
+g
+and sk+1
+g
+be given by (16). It holds
+that
+rk+1
+g
+= min(zk
+g − uk
+g, sk+1
+g
+1)
+(17a)
+1⊤ max(zk
+g − uk
+g − sk+1
+g
+1, 0) = wg
+ρ ,
+(17b)
+where min and max operate entrywise.
+Proof: See Appendix F
+Observe that (17a) can be used to obtain rk+1
+g
+if sk+1
+g
+is
+given, whereas (17b) does not depend on rk+1
+g
+. Therefore, a
+solution to (17) can be found by first solving (17b) for sk+1
+g
+and then substituting the result into (17a) to recover rk+1
+g
+. To
+this end, we have the following:
+Proposition 2: Equation (17b) has a unique root. This root
+lies in the interval [ˇsk
+g, ˆsk
+g], where
+ˇsk
+g := min
+m
+�
+zk
+g[m] − uk
+g[m]
+�
+− wg
+Mρ
+(18a)
+ˆsk
+g := max
+m
+�
+zk
+g[m] − uk
+g[m]
+�
+− wg
+Mρ.
+(18b)
+Proof: See Appendix G.
+Observe that Proposition 2 essentially provides the bounds
+that are required to find the unique root of (17b) via the well-
+known bisection algorithm. Recall that the latter is a very
+efficient algorithm as it geometrically reduces the uncertainty
+at every iteration.
+In accordance with the strategy mentioned earlier, one also
+necessitates a means to solve (15) when (15c) is replaced with
+an equality constraint:
+(rk+1
+g
+, sk+1
+g
+) = arg min
+rg,sg wgsg + ρ
+2∥rg − zk
+g + uk
+g∥2
+2 (19a)
+s.t.
+rg ≤ sg1
+(19b)
+1⊤rg = cBH
+g .
+(19c)
+Proposition 3: Let rk+1
+g
+and sk+1
+g
+be given by (19). Then,
+it holds that
+rk+1
+g
+= min(zk
+g − uk
+g − (µ/ρ)1, sk+1
+g
+1)
+(20a)
+1⊤ max(µ1, ρ(zk
+g − uk
+g − sk+1
+g
+1)) = wg + µM
+(20b)
+where
+µ := −ρcBH
+g
++ ρ1⊤(zk
+g − uk
+g) − wg
+M
+.
+(21)
+Proof: See Appendix H.
+Observe that (20a) and (21) can be used to obtain rk+1
+g
+if sk+1
+g
+is given, whereas (20b) does not depend on rk+1
+g
+.
+Therefore, a solution to (19) can be found by first solving
+(20b) for sk+1
+g
+and then substituting the result into (20a) to
+recover rk+1
+g
+. To this end, we have the following:
+Proposition 4: Equation (20b) has a unique root. This root
+lies in the interval [ˇsk
+g, ˆsk
+g], where
+ˇsk
+g := min
+m
+�
+zk
+g[m] − uk
+g[m]
+�
+− wg
+Mρ − µ
+ρ
+(22a)
+ˆsk
+g := max
+m
+�
+zk
+g[m] − uk
+g[m]
+�
+− wg
+Mρ − µ
+ρ .
+(22b)
+Proof: See Appendix I.
+To sum up, (14) can be solved separately for each column
+of R and entry of s. Each of these G problems can be solved
+by first solving (16) and checking whether (15c) holds for
+the obtained solution. If it does not hold, one must solve
+(19). Proposition 1 and Proposition 3 respectively establish
+that a solution can be found for each of these problems just
+by solving a scalar equation. Proposition 2 and Proposition 4
+prove uniqueness of the solutions of these two equations
+and provide an interval where they can be sought using the
+bisection algorithm.
+C. The Z-subproblem
+This section describes how a solution to (12b) can be found
+by solving a bisection problem per row of Z. To this end, start
+by noting that it follows from (12b) and (13) that
+Zk+1 = arg min
+Z
+�
+I[Z1 = Rmin1] + I[0 ≤ Z ≤ C]
++ ρ
+2∥Rk+1 − Z + U k∥2
+F
+�
+(23a)
+= arg min
+Z
+�
+m
+�
+I[¯z⊤
+m1 = Rmin] + I[0 ≤ ¯zm ≤ ¯cm]
++ ρ
+2∥¯rk+1
+m
+− ¯zm + ¯uk
+m∥2
+F
+�
+,
+(23b)
+
+8
+Algorithm
+2: Group-sparse Placement Algorithm
+(GSPA).
+1 Input: C ∈ RM×G
++
+, Rmin ∈ R+,
+{wg}g ⊂ R+,{cBH
+g }g ⊂ R+, ρ > 0
+2 Initialize U 1 ∈ RM×G and Z1 ∈ RM×G
++
+3 for k = 1, 2, . . . do
+4
+for g = 1, 2, . . . , G do
+5
+Bisection: find sk+1
+g
+s.t.
+1⊤ max(0, ρ(zk
+g − uk
+g − sk+1
+g
+1)) = wg
+6
+Set rk+1
+g
+= min(zk
+g − uk
+g, sk+1
+g
+1)
+7
+if 1⊤rk+1
+g
+> cBH
+g
+then
+8
+Set
+µ = (−ρcBH
+g
++ ρ1⊤(zk
+g − uk
+g) − wg)/M
+9
+Bisection: find sk+1
+g
+s.t.
+1⊤ max(µ1, ρ(zk
+g − uk
+g − sk+1
+g
+1)) = wg + µM
+10
+Set rk+1
+g
+= min(zk
+g − uk
+g − (µ/ρ)1, sk+1
+g
+1)
+11
+for m = 1, 2, . . . , M do
+12
+Bisection: find λ s.t.
+1⊤ max(0, min(¯cm, ¯rk+1
+m
++ ¯uk
+m − λ1)) = Rmin
+13
+Set ¯zk+1
+m
+= max(0, min(¯cm, ¯rk+1
+m
++ ¯uk
+m − λ1))
+14
+Set U k+1 = U k + Rk+1 − Zk+1
+15
+If convergence( ) then return Rk+1
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Number of GTs (M)
+5
+10
+15
+20
+25
+30
+35
+40
+45
+Mean number of ABSs
+Lower bound
+K-means Alg. (Galkin et al.)
+Space rate Alg. (Hammouti et al.)
+Genetic Alg. (Shehzad et al.)
+GSPA (proposed)
+Fig. 3: Mean number of ABSs vs. number of GTs (Rmin = 20
+Mbps, cBH = 99 Mbps).
+where ¯zm, ¯cm, and ¯uk
+m respectively denote the m-th column
+of Z⊤, C⊤, and (U k)⊤. Clearly, this separates into M
+problems of the form
+¯zk+1
+m
+= arg min
+¯zm
+1
+2∥¯rk+1
+m
+− ¯zm + ¯uk
+m∥2
+F
+(24a)
+s.t.
+1⊤¯zm = Rmin,
+0 ≤ ¯zm ≤ ¯cm.
+(24b)
+Proposition 5: If 1⊤¯cm < Rmin, then (24) is infeasible. If
+1⊤¯cm ≥ Rmin, the solution to (24) is given by
+¯zk+1
+m
+= max(0, min(¯cm, ¯rk+1
+m
++ ¯uk
+m − λ1)),
+(25)
+where λ satisfies
+1⊤ max(0, min(¯cm, ¯rk+1
+m
++ ¯uk
+m − λ1)) = Rmin.
+(26)
+Proof: See Appendix J.
+Thus, as in Sec. V-B, one needs to solve the scalar equation
+(26). The following result is the counterpart of Proposition 2
+for the Z-subprblem.
+Proposition 6: If 1⊤¯cm < Rmin, then equation (26) has no
+roots. If 1⊤¯cm ≥ Rmin, then (26) has a unique root. This root
+lies in the interval [ˇλ
+k
+m, ˆλ
+k
+m], where
+ˇλ
+k
+m = min
+g [¯rk+1
+m [g] + ¯uk
+m[g] − ¯cm[g]]
+(27a)
+ˆλ
+k
+m = max{¯rk+1
+m [g] + ¯uk
+m[g] : g ∈ {g : ¯cm[g] > Rmin
+G }}
+− Rmin
+G
+(27b)
+Proof: See Appendix K.
+Proposition 6 establishes uniqueness and provides the
+bounds needed to solve (26), and therefore (24), via the
+bisection algorithm.
+D. The Proposed Solver
+Having addressed both X- and Z- subproblems, it remains
+only to obtain the U-update in (12c), which for the assign-
+ments in (13) becomes
+U k+1 = U k + Rk+1 − Zk+1.
+(28)
+This completes the derivation of the proposed scheme,
+summarized as Algorithm 2 and referred to as the group-
+sparse placement algorithm (GSPA) since it promotes group
+sparsity [45] in the columns of R, that is, only a few columns
+of the matrix Rk+1 returned by the algorithm are expected to
+be non-zero. Recall that the non-zero columns indicate which
+grid points will be occupied by an ABSs.
+In the notation used in Algorithm 2, if A is a matrix, then
+am is its m-th column and ¯a⊤
+n its n-th row. Furthermore,
+superscripts indicate the iteration index, ρ > 0 is the step
+size, and the min and max operators act entrywise. The
+criterion on line 15, which determines whether the algorithm
+has converged, is detailed in Appendix D.
+Observe that the main strategy in the previous two subsec-
+tions was to exploit the structure of Problem (9) to decompose
+it into one subproblem per row and column of R. Each of
+these subproblems involves solving a bisection task of a 1D
+monotonically decreasing function and therefore can be solved
+with O(1) evaluations. The total complexity is O(GM), much
+smaller than the O(G3M 3) complexity per inner iteration of
+interior-point methods; cf. Appendix A.
+VI. NUMERICAL EXPERIMENTS
+This section empirically validates the performance of the
+proposed algorithm by means of numerical experiments
+with channel data generated using the tomographic model
+(Sec. VI-A) and ray-tracing software (Sec. VI-B). The code
+and data necessary to reproduce the experiments is available
+at https://github.com/uiano/ABS placement via propagation
+maps.
+
+9
+39
+49
+59
+69
+79
+89
+99
+Total rate of each ABS [Mbps]
+10
+20
+30
+40
+50
+60
+70
+Mean number of ABSs
+Lower bound
+K-means Alg. (Galkin et al.)
+Space rate Alg. (Hammouti et al.)
+Genetic Alg. (Shehzad et al.)
+GSPA (proposed)
+Fig. 5: Mean number of ABSs vs. the capacity cBH of the
+backhaul link (Rmin = 20 Mbps).
+10
+15
+20
+25
+30
+35
+40
+45
+Minimum GT rate [Mbps]
+10
+20
+30
+40
+50
+60
+70
+Mean number of ABSs
+Lower bound
+K-means Alg. (Galkin et al.)
+Space rate Alg. (Hammouti et al.)
+Genetic Alg. (Shehzad et al.)
+GSPA (proposed)
+Fig. 4: Mean number of ABSs vs. Rmin (cBH = 100 Mbps).
+The channel gain, obtained from the aforementioned models
+when the carrier frequency is 2.4 GHz, is substituted into (1)
+with W = 20 MHz, PTX = 20 dBm/Hz, and σ2 = −96
+dBm/Hz to form the capacity matrix C. For simplicity, the
+backhaul capacity is set to a common value cBH
+1
+= . . . =
+cBH
+G = cBH.
+The proposed placement algorithm is compared with three
+benchmarks: i) the K-means placement algorithm by Galkin
+et al. [11], ii) the iterative space rate K-means placement
+algorithm by Hammouti et al. [22], and iii) the genetic
+placement algorithm by Shehzad et al. [25] with 50 solutions
+per generation. Since we did not manage to obtain the code
+used by the authors of these works, we implemented their algo-
+rithms ourselves. The resulting implementations are available
+in the repository mentioned earlier. The first two algorithms are
+unable to enforce no-fly zones, which results in ABSs placed
+inside buildings. To avoid this behavior, the ABS locations
+provided by these algorithms are projected onto the same flight
+grid as the rest of algorithms. The proposed GSPA algorithm
+utilizes, unless otherwise stated, a step size of ρ = 10−7
+and stopping criterion parameters ϵabs = ϵrel = 10−4; cf.
+Appendix D.
+To quantify performance, the number of ABSs required
+by each algorithm to guarantee a rate Rmin for every GT is
+considered as a performance metric. This metric is averaged
+using Monte Carlo simulation across realizations of the GT
+locations. As a reference, figures will also include a lower
+bound on this metric; cf. Appendix B.
+A. Experiments with the Tomographic Model
+In the experiments of this section, the channel gain is
+generated using Algorithm 1 in an environment like the one
+in Fig. 1. The SLF takes a constant value, termed building ab-
+sorption, inside the buildings and 0 outside. Unless otherwise
+stated, the simulation parameters in this section are as follows.
+Area size: 500 m × 400 m × 150 m; dimensions of the SLF
+grid: 50 × 40 × 15; dimensions of the flight grid: 9 × 9 ×
+5; minimum flight height: 50 m; height of the buildings: 63
+m; building absorption: 1 dB/m; number of GTs: M = 70.
+Fig. 3 investigates the influence of the number of GTs
+(M) on the performance of the compared algorithms. Several
+observations are in order.
+First, the mean number of ABSs
+is seen to increase roughly proportionally to M for all the
+algorithms. Second, the proposed GSPA algorithm not only
+yields a lower mean number of ABSs than the competing
+algorithms, but its slope is smaller, which means that the
+margin by which GSPA outperforms the benchmarks increases
+with M. Third, GSPA asymptotically approaches the lower
+bound (cf. Appendix B), which means that its efficiency,
+quantified as the number of GTs per ABS, increases with M.
+In contrast, the opposite is true for the other algorithms.
+To investigate the influence of the GT requirements on
+performance, Fig. 4 depicts the mean number of ABSs vs.
+Rmin when cBH = 100 Mbps. As in Fig. 3, the mean number
+of ABSs seems to increase roughly linearly. However, in Fig. 4
+a saturation phenomenon arises: the mean number of ABSs
+cannot be greater than the number of GTs M = 70 since
+one ABS placed approximately above each GT suffices to
+serve all GTs. It is also observed that the performance of
+the genetic placement algorithm degrades faster than the rest
+of algorithms for high Rmin. The reason may be that this
+algorithm essentially tests multiple placements and the number
+of placements increases drastically with the number of ABSs,
+which is larger when Rmin is larger.
+Fig. 5 plots the mean number of ABSs vs. the backhaul
+capacity cBH when Rmin = 20 Mbps. The values of cBH on the
+horizontal axis are selected so that each cBH is not an integer
+multiple of Rmin. This gives rise to a “staircase” behavior for
+the benchmarks, which do not exploit the fact that a GT can be
+served by multiple ABSs. In contrast, GSPA exploits this fact,
+as corroborated by the smoothness of its curve. It is also seen
+that the mean number of ABSs required by GSPA is roughly
+half the one of the best competing alternative.
+To study the influence of the channel, Fig. 6 shows the
+mean number of ABSs vs. the absorption undergone by the
+
+10
+30
+40
+50
+60
+70
+80
+90
+100
+Number of GTs (M)
+20
+40
+60
+80
+100
+Mean number of ABSs
+Lower bound
+K-means Alg. (Galkin et al.)
+Space rate Alg. (Hammouti et al.)
+Genetic Alg. (Shehzad et al.)
+GSPA (proposed)
+Fig. 7: Mean number of ABSs vs. number of GTs (Rmin = 20
+Mbps, cBH = 74 Mbps).
+11
+15
+19
+23
+27
+31
+35
+40
+45
+Minimum GT rate [Mbps]
+10
+20
+30
+40
+50
+Mean number of ABSs
+Lower bound
+K-means Alg. (Galkin et al.)
+Space rate Alg. (Hammouti et al.)
+Genetic Alg. (Shehzad et al.)
+GSPA (proposed)
+Fig. 8: Mean number of ABSs vs. minimum GT rate Rmin
+(M = 50 GTs, cBH = 100 Mbps).
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+2.25
+Building absorption [dB/m]
+10
+15
+20
+25
+30
+35
+40
+45
+50
+55
+Mean number of ABSs
+Lower bound
+K-means Alg. (Galkin et al.)
+Space rate Alg. (Hammouti et al.)
+Genetic Alg. (Shehzad et al.)
+GSPA (proposed)
+Fig. 6: Mean number of ABSs vs. building absorption (Rmin =
+17 Mbps, cBH = 84 Mbps).
+communication signals when propagating through the build-
+ings. When the absorption is zero, the propagation conditions
+are those of free space. In this case, the proposed algorithm
+outperforms the rest only because of a better ability to perform
+the rate allocation. As the absorption increases, the benchmark
+algorithms are dramatically affected, which suggests that these
+algorithms are not well suited to scenarios without line-of-
+sight. In contrast, the proposed algorithm remains unaffected
+since there are always sufficiently good flight grid points
+regardless of the building absorption considered in the figure.
+Informally speaking, matrix R in (9) is upper bounded by cBH
+and C. When the former constraint is tighter than the latter,
+the resulting number of ABSs will not depend on C. This
+phenomenon is investigated further in Fig. 9.
+Additional experiments with the tomographic channel model
+are presented in Appendix L.
+B. Experiments with the Ray-Tracing Model
+This section corroborates the main findings of Sec. VI-A
+when the channel is obtained via the ray-tracing model, which
+is more accurate than the tomographic channel model for
+higher carrier frequencies.
+To this end, a data set was generated using the X3D ray-
+tracing software Wireless Insite with six reflections and one
+diffraction in a 400 × 600 m2 area of the city of Ottawa. The
+data set is also published in our repository. The channel was
+computed between all points of the flight grid and all points
+of a GT grid. The former comprises 35 points at each of the
+heights of 40, 60, and 80 m. The latter is a 2D regular grid of
+2501 GT locations at a height of 2 m spaced uniformly with a
+distance of 10 m on each axis. Points inside the buildings are
+removed. At each Monte Carlo realization, the GT locations
+are generated by drawing M points uniformly at random
+without replacement from the GT grid.
+Figs. 7 and 8 are the counterparts of Figs. 3 and 4 for ray-
+tracing channel data. It is observed that the mean number of
+ABSs is also an approximately linear function of M for all
+algorithms and that GSPA roughly attains the lower bound.
+However, here the differences among benchmarks are greater.
+The fact that the K-means algorithm outperforms the space
+rate algorithm suggests that the channel changes rapidly with
+respect to the GT location, since the latter algorithm relies on
+clustering vectors of channel gains. Despite this fact, GSPA
+performs almost optimally.
+Fig. 9 further investigates the phenomenon already dis-
+cussed regarding Fig. 6. Observe that the tightness of the
+bounds in Fig. 9 increases (i) for larger Rmin and (ii) for
+smaller cBH. Those are precisely the situations where the
+backhaul limitations become stricter. Indeed, it can be easily
+seen that the bound in Appendix B can be attained with
+equality when the entries of C approach infinity.
+Finally, the experiment in Fig. 5 is performed with ray-
+tracing channel data in Fig. 10. The proposed algorithm still
+outperforms the other three benchmarks by a wide margin,
+requiring roughly 50% of ABSs. However, relative to Fig. 5,
+
+11
+3
+5
+7
+9
+11
+Minimum GT rate [Mbps]
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+20.0
+22.5
+Mean number of ABSs
+cBH=34 Mbps
+Lower bound for cBH=34 Mbps
+cBH=44 Mbps
+Lower bound for cBH=44 Mbps
+cBH=54 Mbps
+Lower bound for cBH=54 Mbps
+cBH=64 Mbps
+Lower bound for cBH=64 Mbps
+Fig. 9: Mean number of ABSs vs. minimum GT rate Rmin for
+GSPA (M = 50 GTs).
+34
+44
+54
+64
+74
+80
+90
+100
+Total rate of each ABS [Mbps]
+5
+10
+15
+20
+25
+Mean number of ABSs
+Lower bound
+K-means Alg. (Galkin et al.)
+Space rate Alg. (Hammouti et al.)
+Genetic Alg. (Shehzad et al.)
+GSPA (proposed)
+Fig. 10: Mean number of ABSs vs. backhaul link capacity cBH
+(Rmin = 7 Mbps, M = 50 GTs).
+one can observe that the benchmark algorithms saturate for
+large cBH, which indicates that C imposes a more stringent
+constraint than cBH in the scenario of Fig. 10. When it comes
+to GSPA, the greater tightness of the bound for small cBH is
+a manifestation of the same effect, as discussed earlier.
+VII. CONCLUSIONS
+Whereas existing algorithms for ABS placement assume
+that the channel gain depends only on the length and (possibly)
+elevation of each link, this paper presents a scheme that can
+accommodate an arbitrary dependence of the gain on the posi-
+tion of the ABSs and GTs. This enables the utilization of radio
+maps for ABS placement. The proposed algorithm determines
+a set of ABS locations that approximately minimizes the num-
+ber of ABSs required to guarantee a minimum rate to all GTs.
+Relative to most existing schemes, the proposed algorithm has
+a low complexity, accounts for a limited backhaul capacity,
+and can accommodate flight restrictions such as no-fly zones
+or airspace occupied by buildings. A solver whose complexity
+is linear in the number of users was derived based on the
+alternating-direction method of multipliers and the problem of
+evaluating tomographic integrals was revisited and extended
+to air-to-ground channels. An extensive set of simulations
+demonstrate that the proposed GSPA algorithm outperforms
+competing algorithms by a wide margin both in tomographic
+and ray-tracing channels. Remarkably, it was observed in the
+numerical experiments that the proposed algorithm is the only
+one among the compared schemes whose efficiency, measured
+in terms of number of GTs served per ABS, increases with the
+number of GTs. This fundamental distinction renders GSPA
+especially suitable for scenarios with a large number of users.
+Future directions include approaches for tracking air-to-
+ground propagation maps, possibly based on online kernel
+methods [46], [47], and algorithms that can adapt to changes
+in the GT locations.
+VIII. ACKNOWLEDGEMENTS
+The authors would like to thank Prof. Geert Leus for
+insightful discussions.
+APPENDIX A
+INTERIOR-POINT SOLVER
+The present section illustrates how (10) can be solved
+using an interior-point algorithm. Although such a solver is
+not utilized in this paper, the ensuing derivation provides
+its computational complexity, which motivates the ADMM
+algorithm from Sec. V.
+It is convenient to start by expressing (10) in a canonical
+form with only non-negativity constraints and linear equality
+constraints. To this end, introduce the slack variables δ1, ∆2,
+and ∆3 to write (10) as
+minimize
+R,s,δ1,∆2,∆3 w⊤s
+(29a)
+s.t. R⊤1 + δ1 = cBH
+(29b)
+R1 = Rmin1
+(29c)
+R + ∆2 = C
+(29d)
+R + ∆3 = 1s⊤
+(29e)
+R ≥ 0, δ1 ≥ 0, ∆2 ≥ 0, ∆3 ≥ 0.
+(29f)
+With this formulation, it is easy to see that (29) is equivalent
+to
+minimize
+˜x
+˜w⊤˜x
+(30a)
+s.t. ˜A˜x = ˜b
+(30b)
+˜x[G + 1 : end] ≥ 0,
+(30c)
+where
+˜x
+:=
+[s⊤, r⊤, δ⊤
+1 , vec⊤(∆2), vec⊤(∆3)]⊤,
+r
+=
+vec(R),
+˜w
+:=
+[w1, . . . , wG, 0, . . . , 0]⊤,
+˜x[G + 1
+:
+end]
+:=
+[r⊤, δ⊤
+1 , vec⊤(∆2), vec⊤(∆3)]⊤,
+˜b := [(cBH)⊤, Rmin1⊤, vec⊤(C), 0⊤]⊤, and
+˜A :=
+�
+���
+0
+IG ⊗ 1⊤
+IG
+0
+0
+0
+1⊤ ⊗ IM
+0
+0
+0
+0
+IGM
+0
+IGM
+0
+−IG ⊗ 1
+IGM
+0
+0
+IGM
+�
+��� . (31)
+
+12
+Problem (30) can be solved by means of a standard interior-
+point algorithm. To derive a lower bound for its computational
+complexity, note that each inner iteration of the algorithm will
+involve solving a system of equations where the number of
+unknowns equals the number of variables of the optimization
+problem plus the number of linear constraints [48, Ch. 10
+and 11]. For (30), the former equals 2G + 3GM whereas
+the latter is given by G + M + 2GM. Solving this system
+of equations without any tailor-made approach that exploits
+the specific structure of ˜A in this problem therefore involves
+O(G3M 3) arithmetic operations.
+It is worth remarking that this complexity is prohibitive in
+practice: if, for example, G = M = 100, then 1012 operations
+would be required per inner iteration.
+APPENDIX B
+LOWER BOUND FOR THE NUMBER OF ABSS
+This appendix presents a lower bound for the number of
+ABSs and, therefore, also for the mean number of ABSs,
+which is the performance metric adopted in Sec. VI. This
+bound constitutes a fundamental limit for the problem of ABS
+placement and, hence, it applies regardless of the adopted
+algorithm.
+Let N denote the smallest number of ABSs required to
+serve all M users with rate at least Rmin. This means that the
+total backhaul rate available to all ABSs together, which is
+not greater than N maxg cBH
+g , cannot be less than the total rate
+MRmin demanded by the users. It follows that N maxg cBH
+g
+≥
+MRmin and, therefore, N ≥ ⌈MRmin/ maxg cBH
+g ⌉, where ⌈z⌉
+denotes the smallest integer greater than or equal to z.
+APPENDIX C
+DERIVATION OF ALGORITHM 1
+Algorithm 1, which can be classified as a parametric,
+floating point, and zeroth-order algorithm according to the
+terminology of [42, Sec. I-B-1], is our approach (yet others are
+possible) to approximate the tomographic integral by comput-
+ing exactly the integral of a piecewise constant approximation
+of the SLF. The idea is to parameterize the line segment
+between x1 and x2 as x(t) = x1 + t(x2 − x1), where
+t ∈ [0, 1], and identify the values t1 < t2 < . . . < tT for
+which the boundary between two adjacent voxels is crossed.
+Since ∥x(ti) − x(ti−1)∥ = (ti − ti−1)∥x2 − x1∥ whenever
+ti > ti−1, the approximation is then
+ξ(x1, x2) ≈
+�T
+i=2(ti − ti−1)∥x2 − x1∥l(x ¯
+X
+qi)
+∥x2 − x1∥1/2
+(32a)
+= ∥x2 − x1∥1/2
+T
+�
+i=2
+(ti − ti−1)l(x
+¯
+X
+qi),
+(32b)
+where qi is the index of the voxel that contains the i-th segment
+{x(t)
+:
+t ∈ (ti−1, ti)}. Since ¯
+X is a 3D grid, each point
+in {x ¯
+X
+1 , . . . , x ¯
+X
+Q} can also be indexed by a vector i of 3
+indices that lies in the set I := {1, . . . , Qx} × {1, . . . , Qy} ×
+{1, . . . , Qz}. The values of the SLF can also be collected in a
+tensor L ∈ RQx×Qy×Qz, whose entry L[i] is the value of l at
+the i-th grid point. If δ ¯
+X ∈ R3
+++ denotes a vector whose j-th
+entry δ ¯
+X [j] represents the spacing between grid points along
+the j-th axis, the coordinates of the i-th grid point are clearly
+i ⊙ δ ¯
+X , where ⊙ denotes entrywise product. Similarly, the
+boundaries between adjacent voxels along the j-th axis occur
+at values of the j-th coordinate given by δ ¯
+X [j](i±1/2), where
+i is an integer. It is then clear that steps 6-8 in Algorithm 1
+simply find the next value of t for which the segment crosses a
+voxel boundary along one of the axes by solving the equation
+x1[j] + t(x2[j] − x1[j]) = δ ¯
+X [j](icurrent[j] ± 1/2)
+(33)
+for t along each axis j and taking the minimum across axes.
+The ± becomes a plus sign for the j-th axis if the segment
+is increasing along this axis (i.e. x1[j] ≤ x2[j]) and a minus
+sign otherwise.
+An alternative implementation of the same integral approx-
+imation with smaller computational complexity but greater
+memory complexity could be obtained by creating 3 lists
+corresponding to the values of t for which the line segment
+between x1 and x2 intersects each axis and then merging those
+lists into a list with non-decreasing values of t.
+APPENDIX D
+STOPPING CRITERION
+The stopping criterion of Algorithm 2 follows the frame-
+work in [35]. Particularly, given the absolute and relative
+tolerance parameters ϵabs and ϵrel, let ϵk+1
+pri
+and ϵk+1
+dual be
+ϵk+1
+pri
+:=
+√
+MGϵabs
+(34a)
++ ϵrel max{∥A1Xk+1A2∥F, ∥B1Zk+1B2∥F},
+ϵk+1
+dual :=
+√
+MGϵabs + ϵrel∥ρA⊤
+1 U k+1A⊤
+2 ∥F.
+(34b)
+Algorithm 2 stops when both conditions
+∥Qk+1∥2
+F ⩽ ϵk+1
+pri
+and ∥P k+1∥2
+F ⩽ ϵk+1
+dual
+(35a)
+are satisfied, where
+Qk+1 := A1Xk+1A2 + B1Zk+1B2
+(36a)
+P k+1 := ρA⊤
+1 B1(Zk+1 − Zk)B2A⊤
+2
+(36b)
+are the so-called primal and dual residuals.
+APPENDIX E
+PROOF OF THEOREM 1
+The idea is to establish that a special case of (7) is a
+multidimensional knapsack problem. To this end, let the g-
+th entry of cBH be at least as large as 1⊤cg and note that,
+due to (7d), constraint (7b) holds regardless of the choice of
+{αg}G
+g=1 and R, meaning that (7b) can be removed.
+Next, note that if {αg}G
+g=1 and R are feasible, then replac-
+ing any rg with αgcg yields another feasible point that attains
+the same cost. This is because none of the entries of the left-
+hand side of (7c) decreases after modifying rg in this way.
+The left-hand side of (7c) can then be written as R1 =
+�
+g rg = �
+g αgcg, which yields the following problem
+minimize
+{αg}G
+g=1
+�
+g
+αg
+(37a)
+s.t.
+�
+g
+αgcg ≥ Rmin1
+(37b)
+αg ∈ {0, 1}.
+(37c)
+
+13
+Finally, applying the change of variables βg ← 1 − αg, the
+objective becomes G − �
+g βg and the left-hand side of (37b)
+becomes �
+g(1 − βg)cg = �
+g cg − �
+g βgcg, which implies
+that (37) reads as
+maximize
+{βg}G
+g=1
+�
+g
+βg
+(38a)
+s.t.
+�
+g
+βgcg ≤
+�
+g
+cg − Rmin1
+(38b)
+βg ∈ {0, 1}.
+(38c)
+This problem is an instance of the so-called multidimensional
+knapsack problem, which has been shown to be NP-hard
+unless P=NP [49].
+APPENDIX F
+PROOF OF PROPOSITION 1
+Since Problem (16) is convex differentiable and Slater’s con-
+ditions are satisfied, it follows that the Karush-Kuhn-Tucker
+(KKT) conditions are sufficient and necessary [48, Sec. 5.5.3].
+To obtain these conditions, observe that the Lagrangian of (16)
+is given by
+L(rg, sg; ν) = wgsg + ρ
+2∥rg − zk
+g + uk
+g∥2
+2 + ν⊤(rg − sg1)
+(39)
+and note that the KKT conditions can be stated as
+∇rgL(rg, sg; ν) = ρ(rg − zk
+g + uk
+g) + ν = 0
+(40a)
+∇sgL(rg, sg; ν) = wg − 1⊤ν = 0
+(40b)
+rg ≤ sg1
+(40c)
+ν ≥ 0,
+ν[m](rg[m] − sg) = 0 ∀m.
+(40d)
+From (40a) and the inequality in (40d), it follows that
+ν = −ρ(rg − zk
+g + uk
+g) ≥ 0.
+(41)
+This implies that rg ≤ zk
+g − uk
+g. Combining this inequality
+with (40c) yields
+rg ≤ min(zk
+g − uk
+g, sg1).
+(42)
+On the other hand, from the equality in (41) and the equality
+in (40d), one finds that
+−ρ(rg[m] − zk
+g[m] + uk
+g[m])(rg[m] − sg) = 0 ∀m.
+(43)
+This holds if and only if either rg[m] = zk
+g[m] − uk
+g[m] or
+rg[m] = sg. Therefore, it follows from (42) that
+rg = min(zk
+g − uk
+g, sg1),
+(44)
+which establishes (17a). Finally, combine this expression with
+(40b) and (41) to arrive at
+wg = −ρ1⊤(rg − zk
+g + uk
+g)
+(45a)
+= −ρ1⊤(min(zk
+g − uk
+g, sg1) − zk
+g + uk
+g)
+(45b)
+= −ρ1⊤ min(0, sg1 − zk
+g + uk
+g)
+(45c)
+= ρ1⊤ max(0, zk
+g − uk
+g − sg1),
+(45d)
+thereby recovering (17b). The proof is complete by noting that
+(40) holds if and only if (44) and (45d) hold.
+APPENDIX G
+PROOF OF PROPOSITION 2
+Consider the function F(s) := 1⊤ max(zk
+g −uk
+g −s1, 0) =
+�
+m max(zk
+g[m] − uk
+g[m] − s, 0). Since F is the sum of non-
+increasing piecewise linear functions, so is F. Since F(s) →
+∞ as s → −∞ and F(s) = 0 for a sufficiently large s, it
+follows that (17b) has at least one root. Uniqueness of the
+root follows readily by noting that F is strictly decreasing
+whenever F(s) > 0.
+It remains to be shown that F(ˇsk
+g) ≥ wg/(Mρ) whereas
+F(ˆsk
+g) ≤ wg/(Mρ). For the first of these inequalities, observe
+that ˇsk
+g ≤ zk
+g[m] − uk
+g[m] − wg/(Mρ) for all m, which in
+turn implies that zk
+g[m] − uk
+g[m] − ˇsk
+g ≥ wg/(Mρ). Thus,
+max(zk
+g[m] − uk
+g[m] − ˇsk
+g, 0) = zk
+g[m] − uk
+g[m] − ˇsk
+g ≥
+wg/(Mρ), which yields F(ˇsk
+g) ≥ �
+m wg/(Mρ) = wg/ρ.
+For the second inequality, note similarly that zk
+g[m]−uk
+g[m]−
+ˆsk
+g
+≤
+wg/(Mρ) for all m. This means that F(ˆsk
+g)
+≤
+�
+m max(wg/(Mρ), 0) = wg/ρ.
+APPENDIX H
+PROOF OF PROPOSITION 3
+Again, the KKT conditions are sufficient and necessary.
+Since the Lagrangian is
+L(rg, sg; ν) = wgsg + ρ
+2∥rg − zk
+g + uk
+g∥2
+2
++ ν⊤(rg − sg1) + µ(1⊤rg − cBH
+g ),
+(46)
+the KKT conditions read as
+∇rgL(rg, sg; ν) = ρ(rg − zk
+g + uk
+g) + ν + µ1 = 0
+(47a)
+∇sgL(rg, sg; ν) = wg − 1⊤ν = 0
+(47b)
+rg ≤ sg1
+(47c)
+ν ≥ 0,
+ν[m](rg[m] − sg) = 0 ∀m
+(47d)
+1⊤rg = cBH
+g .
+(47e)
+From (47a) and the inequality in (47d), it follows that
+ν = −ρ(rg − zk
+g + uk
+g) − µ1 ≥ 0.
+(48)
+This implies that rg ≤ zk
+g − uk
+g − (µ/ρ)1. Combining this
+inequality with (47c) yields
+rg ≤ min(zk
+g − uk
+g − (µ/ρ)1, sg1).
+(49)
+On the other hand, from the equality in (48) and the equality
+in (47d), one finds that
+[−ρ(rg[m] − zk
+g[m] + uk
+g[m]) − µ](rg[m] − sg) = 0 ∀m.
+(50)
+This holds if and only if either rg[m] = zk
+g[m]−uk
+g[m]−µ/ρ
+or rg[m] = sg. Therefore, it follows that
+rg = min(zk
+g − uk
+g − (µ/ρ)1, sg1),
+(51)
+which establishes (20a).
+To find µ, substitute the equality in (48) into (47b) to obtain
+1⊤[−ρ(rg − zk
+g + uk
+g) − µ1] = wg.
+(52)
+
+14
+Solving for µ yields
+µ = −ρ1⊤rg + ρ1⊤(zk
+g − uk
+g) − wg
+M
+(53)
+and using (47e) results in (21).
+Finally, substitute (51) into (52) to arrive at
+wg = 1⊤ max(µ1, ρ(zk
+g − uk
+g − sg1)) − µM,
+(54)
+thereby recovering (20b). The proof is complete by noting that
+(47) holds if and only if (20a) and (20b) hold.
+APPENDIX I
+PROOF OF PROPOSITION 4
+Consider the function F(s) := 1⊤ max(µ1, ρ(zk
+g − uk
+g −
+s1)) = �
+m max(µ, ρ(zk
+g[m] − uk
+g[m] − s)). Due to the same
+argument as in the proof of Proposition 2, this function has a
+unique root.
+To show that F(ˇsk
+g) ≥ wg + µM, observe that
+F(ˇsk
+g) ≥
+�
+m
+ρ(zk
+g[m] − uk
+g[m] − ˇsk
+g)
+(55a)
+≥ M min
+m [ρ(zk
+g[m] − uk
+g[m] − ˇsk
+g)]
+(55b)
+= Mρ
+� wg
+Mρ + µ
+ρ
+�
+(55c)
+= wg + µM.
+(55d)
+To show that F(ˆsk
+g) ≤ wg + µM, observe that
+F(ˆsk
+g) ≤ M max
+m [max(µ, ρ(zk
+g[m] − uk
+g[m] − ˆsk
+g))]
+(56a)
+= M max(µ, ρ(max
+m [zk
+g[m] − uk
+g[m]] − ˆsk
+g))
+(56b)
+= M max
+�
+µ, ρ
+� wg
+Mρ + µ
+ρ
+��
+(56c)
+≤ wg + µM.
+(56d)
+APPENDIX J
+PROOF OF PROPOSITION 5
+The fact that 1⊤¯cm < Rmin implies that (24) is infeasible
+is trivial and, therefore, the rest of the proof focuses on the
+case where 1⊤¯cm ≥ Rmin.
+As before, the KKT conditions are sufficient and necessary
+in this case. Noting that the Lagrangian is given by
+L(¯zm; λ, ν, µ) = 1
+2∥¯rk+1
+m
+− ¯zm + ¯uk
+m∥2
+F
++ λ(1⊤¯zm − Rmin) − ν⊤¯zm + µ⊤(¯zm − ¯cm)
+(57)
+yields the KKT conditions
+∇¯zmL(¯zm; λ, ν, µ) =
+− (¯rk+1
+m
+− ¯zm + ¯uk
+m) + λ1 − ν + µ = 0,
+(58a)
+1⊤¯zm = Rmin,
+(58b)
+¯zm ≥ 0, ν ≥ 0, ν[g]¯zm[g] = 0 ∀g,
+(58c)
+¯zm ≤ ¯cm, µ ≥ 0, µ[g](¯zm[g] − ¯cm[g]) = 0 ∀g.
+(58d)
+From (58a) and the second inequality in (58d), it follows that
+µ = ¯rk+1
+m
+− ¯zm + ¯uk
+m − λ1 + ν ≥ 0,
+(59)
+which in turn implies that
+¯zm ≤ ¯rk+1
+m
++ ¯uk
+m − λ1 + ν.
+(60)
+Combining this expression with the first inequality in (58d)
+yields
+¯zm ≤ min(¯cm, ¯rk+1
+m
++ ¯uk
+m − λ1 + ν).
+(61)
+To show that this expression holds with equality, substitute
+(59) into the equality of (58d) to obtain
+(¯rk+1
+m [g] − ¯zm[g] + ¯uk
+m[g] − λ + ν[g])(¯zm[g] − ¯cm[g]) = 0,
+(62)
+which implies that either ¯zm[g] = ¯rk+1
+m [g] + ¯uk
+m[g] − λ + ν[g]
+or ¯zm[g] = ¯cm[g]. Therefore,
+¯zm = min(¯cm, ¯rk+1
+m
++ ¯uk
+m − λ1 + ν).
+(63)
+To obtain an expression for ¯zm that does not depend on ν,
+one may consider three cases for each g:
+• C1: ¯rk+1
+m [g] + ¯uk
+m[g] − λ < 0. In this case, if ν[g] = 0,
+expression (63) would imply that ¯zm[g] < 0, which
+would violate the first inequality in (58c). Therefore,
+ν[g] > 0 and, due to the equality in (58c), ¯zm[g] = 0.
+If ¯cm[g] > 0, it is then clear from (63) that ν[g] =
+−(¯rk+1
+m [g]+ ¯uk
+m[g]−λ). If ¯cm[g] = 0, then greater values
+of ν[g] will also satisfy the KKT conditions but this is
+not relevant since the only feasible ¯zm[g] in case C1 is
+¯zm[g] = 0.
+• C2: ¯rk+1
+m [g] + ¯uk
+m[g] − λ = 0. In this case, (63) becomes
+¯zm[g] = min(¯cm[g], ν[g]). Due to the equality in (58c),
+it then follows that either ¯cm[g] = 0 and ν[g] ≥ 0, or
+¯zm[g] = ν[g] = 0.
+• C3: ¯rk+1
+m [g] + ¯uk
+m[g] − λ > 0. If ¯cm[g] = 0, then
+necessarily ¯zm[g] = 0 and any ν[g] ≥ 0 satisfies the
+KKT conditions. On the other hand, if ¯cm[g] > 0, then
+it is clear that ¯zm[g] > 0 and, due to the equality in
+(58c), one has that ν[g] = 0, which in turn implies that
+¯zm[g] = min(¯cm[g], ¯rk+1
+m [g] + ¯uk
+m[g] − λ).
+Combining C1-C3 yields
+¯zm[g] = max(0, min(¯cm[g], ¯rk+1
+m [g] + ¯uk
+m[g] − λ)),
+(64)
+which is just the scalar version of (25). Finally, to obtain λ,
+one may substitute (64) into (58b), which produces (26).
+
+15
+10
+20
+30
+40
+50
+60
+70
+80
+90
+Minimum flight height [m]
+10
+15
+20
+25
+30
+35
+40
+45
+50
+55
+Mean number of ABSs
+Lower bound
+K-means Alg. (Galkin et al.)
+Space rate Alg. (Hammouti et al.)
+Genetic Alg. (Shehzad et al.)
+GSPA (proposed)
+Fig. 12: Mean number of ABSs vs. minimum flight height
+(Rmin = 17 Mbps, cBH = 84 Mbps).
+0
+10
+20
+30
+40
+50
+Height of the buildings [m]
+12
+14
+16
+18
+20
+22
+24
+26
+28
+30
+32
+Mean number of ABSs
+Lower bound
+K-means Alg. (Galkin et al.)
+Space rate Alg. (Hammouti et al.)
+Genetic Alg. (Shehzad et al.)
+GSPA (proposed)
+Fig. 13: Mean number of ABSs vs. height of the buildings
+(Rmin = 17 Mbps, cBH = 84 Mbps).
+5
+7
+9
+11
+13
+15
+17
+Minimum GT rate [Mbps]
+0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+Mean number of ABSs
+cBH = 40 Mbps
+Lower bound for cBH = 40 Mbps
+cBH = 60 Mbps
+Lower bound for cBH = 60 Mbps
+cBH = 80 Mbps
+Lower bound for cBH = 80 Mbps
+cBH = 100 Mbps
+Lower bound for cBH = 100 Mbps
+Fig. 11: Mean number of ABSs vs. Rmin of the proposed GSPA
+algorithm.
+APPENDIX K
+PROOF OF PROPOSITION 6
+Denote by G(λ) the left-hand side of (26), i.e.,
+G(λ) :=
+�
+g
+max(0, min(¯cm[g], ¯rk+1
+m [g] + ¯uk
+m[g] − λ)).
+(65)
+This is a sum of non-increasing piecewise continuous func-
+tions and therefore G is also non-increasing piecewise contin-
+uous. The maximum value is attained for sufficiently small
+λ and equals �
+g ¯cm[g] = 1⊤¯cm. If 1⊤¯cm < Rmin, then
+G(λ) < Rmin ∀λ and (26) admits no solution. Conversely,
+if 1⊤¯cm > Rmin, then a solution can be found since G(λ) >
+Rmin for sufficiently small λ and G(λ) = 0 for sufficiently
+large λ. Uniqueness follows from the fact that G is strictly
+decreasing except when G(λ) = 0 or G(λ) = 1⊤¯cm.
+To show that G(ˇλ
+k
+m) ≥ Rmin just note from (27a) that
+ˇλ
+k
+m ≤ ¯rk+1
+m [g] + ¯uk
+m[g] − ¯cm[g] or, equivalently, ¯cm[g] ≤
+¯rk+1
+m [g] + ¯uk
+m[g] − ˇλ
+k
+m, for all g. This clearly yields G(ˇλ
+k
+m) =
+�
+g max(0, ¯cm[g]) = �
+g ¯cm[g], which is greater than or equal
+to Rmin by assumption.
+To show that G(ˆλ
+k
+m) ≤ Rmin, note from (27b) that ˆλ
+k
+m ≥
+¯rk+1
+m [g]+¯uk
+m[g]−Rmin/G for all g such that ¯cm[g] > Rmin/G.
+This clearly implies that ¯rk+1
+m [g] + ¯uk
+m[g] − ˆλ
+k
+m ≤ Rmin/G
+for all g such that ¯cm[g] > Rmin/G and, as a consequence,
+min(¯cm[g], ¯rk+1
+m [g] + ¯uk
+m[g] − ˆλ
+k
+m)) ≤ Rmin/G and the in-
+equality G(ˆλ
+k
+m) ≤ Rmin follows.
+APPENDIX L
+ADDITIONAL EXPERIMENTS WITH THE TOMOGRAPHIC
+MODEL
+Fig. 11 is the counterpart of Fig. 9 for tomographic chan-
+nels. The purpose of this simulation is to confirm that the
+approximate linearity and proximity to the bound of GSPA
+observed in Fig. 4 take place for a wide range of parameters.
+Fig. 12 investigates the influence of the minimum flight
+height on the performance of the considered algorithms. The
+staircase behavior of the benchmarks can be explained by
+noting that the flight grid initially comprises points with
+heights 0, 30 m, 60 m, 90 m, etc. Then, the points inside
+buildings and the points below the minimum flight height are
+removed. Thus, the allowed flight points are the same e.g.
+when the minimum flight height is 10 m as when it is 20 m.
+As already observed in Sec. VI, the performance of GSPA
+is not degraded for increasing flight height because the back-
+haul capacity poses a more stringent constraint than the one
+imposed by C even for the maximum flight height considered
+in the figure.
+Fig. 13 studies the impact of the height of the buildings.
+To reduce the spatial quantization effect of the tomographic
+integral approximation, the numbers of SLF grid points in the
+x, y, and z axes were set to 50, 40, and 150, respectively. It is
+observed that the K-means and space rate K-means algorithms
+
+16
+are negatively affected by the increased attenuation introduced
+by higher buildings. In contrast, the genetic placement algo-
+rithm and GSPA exhibit a milder dependence on the building
+height. The reason is the greater capacity of these algorithms
+to select ABS locations with favorable propagation conditions.
+The backhaul capacity is again the limiting constraint, which
+explains why the performance of GSPA is unaffected by a
+greater building height.
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diff --git a/Z9E4T4oBgHgl3EQfOgzR/content/tmp_files/load_file.txt b/Z9E4T4oBgHgl3EQfOgzR/content/tmp_files/load_file.txt
new file mode 100644
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--- /dev/null
+++ b/Z9E4T4oBgHgl3EQfOgzR/content/tmp_files/load_file.txt
@@ -0,0 +1,1280 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf,len=1279
+page_content='1  Aerial Base Station Placement via  Propagation Radio Maps  Daniel Romero, Pham Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Viet, and Raju Shrestha  Abstract—The technology of base stations on board unmanned  aerial vehicles, also known as aerial base stations (ABSs),  promises to deliver cellular connectivity in areas where the  terrestrial infrastructure is overloaded, damaged, or inexistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  A central problem in this context is to determine the locations  where these ABSs must be deployed to serve a set of users on  the ground given the positions of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' However, existing  schemes assume that the channel gain depends only on the  length and (possibly) the elevation of the link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' To alleviate this  limitation, this paper proposes a scheme that accommodates  arbitrary channel gains by means of a propagation radio map of  the air-to-ground channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The algorithm finds the locations of  an approximately minimal number of ABSs to serve all ground  terminals with a target rate while meeting the given constraints  on the capacity of the backhaul links and respecting no-fly  regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' A convex-relaxation formulation ensures convergence  and the alternating-direction method of multipliers is utilized  to derive an implementation whose complexity is linear in the  number of ground terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Numerical results with tomographic  as well as ray-tracing channel models corroborate the strengths  of the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Index Terms—Aerial base stations, radio maps, spectrum  cartography, radio tomography, aerial base station placement,  ray tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' INTRODUCTION  Aerial base stations (ABSs), namely unmanned aerial ve-  hicles (UAVs) equipped with on-board base stations, were  conceived as a means to deliver cellular connectivity in areas  where the terrestrial infrastructure is absent, overloaded, or  damaged [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This may occur for example in remote areas,  in the vicinity of a crowded event, or after a natural disaster,  such as a wildfire or a flood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Users on the ground, here referred  to as ground terminals (GTs), are served by ABSs, which in  turn connect to the terrestrial infrastructure through backhaul  links, possibly in multiple hops through other UAVs that act  as relays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Deploying ABSs involves addressing the problem  of ABS placement, where one is given the locations of the  GTs and must decide on a suitable set of spatial positions  for the ABSs to effectively serve the GTs [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This task is  typically hindered by several challenges, remarkably (C1) the  uncertainty in the gain of the propagation channel between  the GTs and the potential ABS locations, (C2) the limited  The authors are with the Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' of Information and Communication Technol-  ogy, University of Agder, Jon Lilletunsvei 9, 4879 Grimstad, Norway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Email  {daniel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='romero,viet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='pham, raju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='shrestha}@uia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  This work was supported by the Research Council of Norway through the  IKTPLUSS Grant 311994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  The present paper extends its conference precursor [1] to accommodate  constraints in the backhaul and more general propagation radio maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' It also  includes a much more comprehensive simulation study where, among others,  simulations using ray-tracing software are carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  capacity of the backhaul links, and (C3) constraints on the  positions that the ABSs may adopt, often due to no-fly zones  such as airports, embassies, or prisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  The problem of placing a single ABS has been extensively  investigated in the literature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' [4]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' However, in  general, the number of ABSs required in a practical scenario  need not equal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' For this reason, a large number of works,  including the present one, focus on placing multiple ABSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  A usual approach is to regard the height of the ABSs as  given and address the problem of 2D placement, where the  ABSs must be placed on a horizontal plane of the given  height;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' [9]–[16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Nevertheless, since the heights of  the ABSs are useful degrees of freedom to optimize the target  communication metric, the focus here is on 3D placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Existing algorithms for 3D placement of multiple ABSs can  be classified according to how they handle uncertainty in the  air-to-ground channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The first category comprises schemes  that do not explicitly model the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' For example, in  [17], each ground user associates with the ABS from which  it receives the strongest beacons, but nothing is known or  assumed about the channel gain from the GT locations to a  given location until an ABS is physically there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This means  that every iteration of the placement algorithm involves placing  the ABSs in a particular set of locations, which drastically  limits convergence speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The second class includes works that  assume free-space propagation and, therefore, the coverage  area of each ABS is a circle;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Unfortunately, this  assumption is too inaccurate in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The third category is  made up of works that rely on the empirical model from [19],  [20];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' [21]–[25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' These works use the mean provided  by such a model as the predictor of the channel gain, which  is equivalent to assuming that the gain of a link depends  only on its length and elevation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Again, this assumption is  not very realistic since two links with the same length and  elevation may exhibit totally different gains depending on  whether there are obstructions such as buildings between the  transmitter and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The fourth category, which can  be termed channel-aware, is composed of works that rely  on gain predictions that do depend on the locations of the  endpoints of the link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' To the best of our knowledge, only  [26], [27] fall in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Unfortunately, these schemes  entail prohibitive complexity, assume an unlimited backhaul  connection between the ABSs and the terrestrial infrastructure,  and cannot guarantee a minimum service to the GTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  The main contribution of this paper is a scheme that relies  on radio propagation maps [28] to solve the problem of  channel-aware 3D placement of multiple ABSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Recall that  a propagation map is a special kind of radio map [28]–[32]  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='04966v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='OC]  12 Jan 2023  2  that provides a channel metric of interest for every pair of  transmitter and receiver locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' In this paper, this metric is  the channel gain, which can be used to predict the capacity of  the communication link between each candidate ABS location  and every GT without deploying an ABS at that location to  measure the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Two classes of propagation maps with  complementary strengths will be considered, namely those  obtained via ray-tracing and those that rely on the radio  tomographic model [33], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The former are more suitable to  frequency bands where the dominating propagation phenom-  ena beyond free-space loss are reflection and diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The  latter are suitable to bands where the dominating propagation  phenomenon is the absorption introduced by obstacles such  as buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' In this context, a secondary contribution of this  paper is to adapt existing radio tomographic techniques to air-  to-ground channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To the best of our knowledge, the proposed scheme is the  first for channel-aware ABS placement that can guarantee a  minimum rate for all GTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Formally, the algorithm can find  a feasible placement if it exists, where a feasible placement  is an assignment of ABSs to spatial locations that ensures a  target rate for all GTs according to the given propagation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Besides, unlike the vast majority of works in the literature,  constraints in the backhaul link between the ABSs and the ter-  restrial infrastructure as well as no-fly zones can be enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  With these constraints, the proposed algorithm approximately  minimizes the number of ABSs required to serve all GTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Note that this is of special interest in emergency scenarios,  which is one of the main use cases of ABSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The algorithm  relies on a sparse optimization formulation that naturally  arises from a discretization of the space of candidate ABS  positions, as required to be able to utilize propagation maps  in a tractable fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' To counteract the high-dimensionality  of the 3D placement problem, a linear-complexity and highly  parallelizable algorithm is developed based on the alternating-  direction method of multipliers (ADMM) [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Experiments with tomographic and ray-tracing models  showcase a great reduction in the number of required ABSs  as compared to existing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' To complement this  manuscript, an open-source simulator was released to allow  developing and testing algorithms for ABS placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This  simulator and the code needed to reproduce all experiments  is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='com/uiano/ABS placement via  propagation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Paper structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The rest of the paper is organized as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' II presents the model and formulates the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Two approaches for predicting the capacity of a link between  arbitrary pairs of endpoints of the air-to-ground channel are  then described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The problem of ABS placement  and rate allocation is then addressed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' IV and a solver  with linear complexity is developed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Finally, Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' VI  and VII respectively present numerical results and conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' R+ is set of non-negative real numbers and R++  is the set of positive real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Boldface uppercase (lower-  case) letters denote matrices (column vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' a[i] represents  the i-th entry of vector a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Notation 0 (respectively 1) refers to  the matrix of the appropriate dimensions with all zeros (ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  ∥A∥F denotes the Frobenius norm of matrix A, whereas ∥a∥p  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 1: Example of ABS placement in an urban environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  GTs are represented by markers on the ground, flight grid  points by blue dots, and ABS positions by green circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  denotes the ℓp-norm of vector a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' With no subscript, ∥a∥ stands  for the ℓ2-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Inequalities between vectors or matrices must  be understood entrywise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The Kronecker product is denoted by  ⊗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' If a and b are vectors of the same dimension, then a ⊙ b  is the entrywise product of a and b, whereas a ÷ b is the  entrywise quotient of a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' MODEL AND PROBLEM FORMULATION  Consider M users located at positions {xGT  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , xGT  M } ⊂  X ⊂ R3, where the region X represents an arbitrary set of  spatial locations, including for example points on the street,  inside buildings, inside vehicles, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The proposed  scheme carries over unaltered to the scenario where X includes  points in the airspace and some or all users are airborne,  which can be of interest e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' to deploy auxiliary ABSs as  picocells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' However, to simplify the exposition, this possibility  is neglected and the users will be referred to throughout as  ground terminals (GTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To provide connectivity to these GTs, N ABSs are deployed  at locations {xABS  1  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , xABS  N } ⊂ F ⊂ R3, where F comprises  all spatial positions where a UAV is allowed to fly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This  excludes no-fly zones, airspace occupied by buildings, and  altitudes out of legal limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  For the sake of specificity, it will be assumed that data  packets originated in a remote location are sent from the  terrestrial infrastructure to the ABSs through a backhaul link  and the ABSs forward these packets to their intended users  through the downlink of the radio access network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' However,  the entire discussion applies also to the uplink, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=', when the  data packets are originated at the GTs and sent through the  ABSs to the terrestrial infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Ignoring frequency-selective effects for simplicity, the ca-  pacity of the communication link between an ABS at position  xABS ∈ X and the m-th GT is given by  Cm(xABS) = W log2  �  1 + PTX10γm(xABS)/10  σ2  �  ,  (1)  where W denotes bandwidth, PTX the transmit power spectral  density (PSD), σ2 the noise PSD, and γm(xABS) the channel  gain, which is described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Unlike most schemes in the literature of ABS placement, the  present work can accommodate constraints in the backhaul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  3  To formalize such constraints, let CBH(xABS) denote the max-  imum rate of the link between the terrestrial ground station(s)  that serve(s) an ABS at xABS and the ABS at xABS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' An  equation like (1) can also be established to express CBH(xABS)  in terms of the gain of the relevant channel(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Note that, as  the notation suggests, in general CBH(xABS) depends on the  ABS position xABS – typically, the greater the distance from  xABS to the terrestrial ground stations, the lower CBH(xABS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  With Rm(xABS) denoting the downlink rate that an ABS at  xABS allocates to the m-th GT, the backhaul rate constraint  imposes that �  m Rm(xABS) ≤ CBH(xABS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  The problem is to find a minimal number of ABS locations  that guarantee that every user receives a rate of at least  Rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This criterion arises naturally in some of the main use  cases of UAV-assisted networks such as emergency response  or disaster management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Motivated by this scenario and to  enhance flexibility in the deployment, each GT may be served  by multiple ABSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This means that the rate that the m-th user  receives is �  n Rm(xABS  n  ), where xABS  n  denotes the location  of the n-th ABS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To summarize, the problem can be formulated as follows:  minimize  N,{xABS  n  }N  n=1,{rm[n]}M,N  m=1,n=1  N  (2a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  �  m  rm[n] ≤ CBH(xABS  n  ),  (2b)  �  n  rm[n] ≥ Rmin,  (2c)  0 ≤ rm[n] ≤ Cm(xABS  n  ),  (2d)  xABS  n  ∈ F,  (2e)  where the constraints need to hold for all m and n and the  earlier notation Rm(xABS  n  ) has been replaced with rm[n] :=  Rm(xABS  n  ) to emphasize that it refers to optimization vari-  ables, not to functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Observe that Problem (2) constitutes a  joint placement and rate-allocation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Note also that the  same minimum rate Rmin is imposed for all GTs, but different  rates can be set up to straightforward modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  From a practical perspective, solving (2) involves two chal-  lenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' First, Cm(xABS  n  ) and CBH(xABS  n  ) depend on the chan-  nel gain of the corresponding downlink and backhaul links,  which is generally unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This issue will be addressed in  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Second, given Cm(xABS  n  ) and CBH(xABS  n  ), one needs  to find the positions of the ABSs and the rate allocations that  solve (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This will be the subject of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' CAPACITY PREDICTION VIA PROPAGATION RADIO  MAPS  As described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' I, nearly all existing schemes for ABS  placement assume that Cm(xABS  n  ) depends on xGT  m and xABS  n  only through the length and (possibly) the elevation angle of  the line segment between these two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' However, one can  expect that this simplification entails a significant performance  degradation since channel gain in reality is heavily affected by  the environment: two links with the same length and elevation  may experience very different channel gain depending on the  position, shape, and material of the surrounding obstacles and  scatterers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The same observation applies to CBH(xABS  n  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  For this reason, this section proposes the utilization of radio  propagation maps to obtain Cm(xABS  n  ) and CBH(xABS  n  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' A  radio propagation map is a function of two spatial locations  that provides a certain metric of interest, in this case the  gain, for the channel between those spatial locations [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Without loss of generality, the rest of this section focuses on  Cm(xABS  n  ), but the same approaches and considerations apply  to CBH(xABS  n  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Ray-tracing Models  Ray-tracing  techniques  [36]  can  be  used  to  predict  γm(xABS  n  ) for arbitrary pairs (xGT  m , xABS  n  ) given a 3D model of  the propagation environment, which includes buildings, terrain  elements, and other objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The idea is to geometrically obtain  the relevant propagation paths between the transmitter and the  receiver and subsequently calculate the attenuation and delay  or phase change of each path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Ray-tracing models are especially attuned to high-frequency  bands, where the dominating propagation phenomena are re-  flection and diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The limitation is that, to track changes  in the channel, one needs to track changes in the 3D model,  which may not be viable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Thus, in the case of ABS placement,  it makes sense to use ray-tracing to precompute the values of  γm(xABS  n  ) (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' VI-B) for the given scenario and ignore the  effects of changes in the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' If the impact of the changes  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' moving vehicles) is not too large, using a ray-tracing  propagation map would still yield better placement solutions  than adopting the approaches in the literature, which either  assume free space or assume that the gain is a function of just  the distance and elevation of the link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Radio Tomographic Models  As opposed to ray-tracing models, which account for reflec-  tion and diffraction effects, the radio tomographic model [33]  (see also [37] and the references therein) focuses on the  absorption undergone by radio waves on the straight path  between the transmitter and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This model is best  suited to traditional cellular communication frequencies, where  radio waves readily penetrate structures such as buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  However, the existing works in this context focus on ground-  to-ground channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' To the best of our knowledge, this is the  first work to apply radio tomography to air-to-ground channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  This entails special challenges due to the high dimensionality  of the underlying space, which render existing techniques  unsuitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' After describing the radio tomographic model, this  section proposes an algorithm to bypass these difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  The radio tomographic model dictates that the channel gain  between xGT  m and xABS can be decomposed into a free-space  loss component and a shadowing component as1  γm(xABS) = 20 log10  �  λ  4π∥xGT  m − xABS∥  �  − ξ(xGT  m , xABS),  (3)  1Expression (3) assumes that small-scale fading has been averaged out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Otherwise, a random term can be added to the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  4  where λ is the wavelength associated with the carrier fre-  quency and the shadowing function ξ is given by [33]  ξ(x1, x2) =  1  ∥x1 − x2∥1/2  2  � x2  x1  l(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (4)  The non-negative function l inside the line integral is termed  spatial loss field (SLF) and quantifies the local attenuation  (absorption) that a signal suffers at each position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Two tasks are of interest: (T1) evaluate γm(xABS) for  a pair of locations (xGT  m , xABS), which involves evaluat-  ing the integral in (4) and substituting the result into (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (T2) estimate l given a set of measurements of the form  (xABS, xGT  m , γm(xABS)) collected beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  In both tasks, l needs to be discretized, which can be  accomplished by storing its values l(x ¯  X  1 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , l(x ¯  X  Q) on a  regular grid of Q points ¯  X := {x ¯  X  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , x ¯  X  Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The rest of  this section explains how ξ(x1, x2) can be expressed in terms  of these values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This addresses (T1) and enables (T2) in  combination with standard estimators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' [37]–[39]  The conventional approach approximates the right-hand side  of (4) as the weighted sum [40]  ξ(x1, x2) ≈  �  q  w(x1, x2, x  ¯  X  q )l(x  ¯  X  q ),  (5)  where the weight function w(x1, x2, x ¯  X ) aims at assigning  a non-zero weight only to those grid points x ¯  X lying close  to the line segment between x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Although there are  some variations, the functions w adopted in the literature are  non-zero only when x ¯  X lies inside an ellipsoid with foci at  x1 and x2 [37], [40], [41];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' see the ellipses in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 2 for a  depiction in 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Such an approximation suffers from several limitations that  render it impractical for the application at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' First, since  existing choices of w(x1, x2, x ¯  X ) are discontinuous functions,  the resulting approximations of ξ(x1, x2) are also discon-  tinuous, which may lead to erratic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' For example,  it may well happen that �  q w(x1, x2, x ¯  X  q )l(x ¯  X  q ) = 0 even  when x1 ̸= x2 and l(x ¯  X  q ) ̸= 0 ∀q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This happens when x1  and x2 are such that no grid point falls inside the elliptical  support of w(x1, x2, x ¯  X );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' see the upper ellipse in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Since w is typically chosen so that its support coincides with  the first Fresnel ellipsoid, the length of its minor axis is  roughly proportional to  √  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' If the carrier frequency is low,  the ellipsoid is large and, therefore, likely to contain a large  number of grid points, which somehow mitigates the effects of  the discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Conversely, if the carrier frequency is high,  one could think of creating a sufficiently dense grid so that the  distance between grid points is small relative to λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' However,  it is easy to see that this may entail a prohibitively high Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  For example, if one wishes to set the grid points, say, 5 cm  apart and the region of interest is 1 km × 1 km × 100 m,  then the total number of grid points is 1011, which is obvi-  ously impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Another critical limitation is computational  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Observe that (5) generally requires evaluating  w(x1, x2, x ¯  X  q ) and the product w(x1, x2, x ¯  X  q )l(x ¯  X  q ) for each  grid point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Thus, if the grid is Q0 × Q0 × Q0, the complexity  of approximating ξ(x1, x2) is O(Q3  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This is computationally  Algorithm 1: Tomographic Integral Approximation  1  1: Input: x1, x2, grid spacing vector δ ¯  X ∈ R3,  SLF tensor L ∈ RQx×Qy×Qz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  2: Initialize ∆x = x2 − x1, binc = sign(∆x),  I = 0, t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  3: Set zero entries of ∆x to 1 # To avoid dividing by 0  4: Set icurrent = round(x1 ÷ δ ¯  X ) # Current voxel indices  5: while t < 1 do  6:  Set tcand = (δ ¯  X ⊙ (icurrent + binc/2) − x1) ÷ ∆x  7:  Set inext = arg mini tcand[i] s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' binc[i] ̸= 0  8:  Set tnext = min(1, tcand[inext])  9:  Set I = I + (tnext − t)L[icurrent]  10:  Set t = tnext  11:  Set icurrent[inext] = icurrent[inext] + binc[inext]  12: end while  13: return ∥x2 − x1∥1/2I  problematic for the application at hand since ξ(x1, x2) must  be approximated at a large number of locations to solve (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To remedy these issues, this paper advocates approximating  the integral in (4) as a line integral of a piecewise constant  approximation of l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The resulting approximation is continuous,  can be used with large grid point spacing, and can be computed  with complexity only O(Q0) for a Q0 × Q0 × Q0 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This  technique, commonly used in other disciplines (see references  in [42]), involves splitting the 3D space into voxels centered  at the grid points ¯  X := {x ¯  X  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , x ¯  X  Q} and approximating l  by a function that takes the value l(x ¯  X  q ) at all points of the q-  th voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The resulting piecewise constant approximation of l  can be integrated by determining the positions of the crossings  between the voxel boundaries and the line segment between  x1 and x2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Algorithm 1, derived in Appendix C, is our implementation  of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This algorithm solves the limitations of the  conventional approximation outlined earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' First, Algorithm 1  yields an approximation of ξ(x1, x2) that is a continuous  function of x1 and x2 since the line integral of a piecewise  constant function is a continuous function of the endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Besides, the algorithm does not suffer from the issue of  the approximation becoming zero when the elliptical support  of the weight function in (5) misses all grid points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' For  this reason, the voxels can now be kept large regardless of  the wavelength and, therefore, the total number of voxels  can be kept low enough to be handled given the available  computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Finally, as indicated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' III, the  computational complexity of Algorithm 1 is much smaller  than the one of the conventional approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Specifically,  one can observe in Algorithm 1 that a constant number of  products and additions is required for each crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The total  number of crossings is at most Qz + Qy + Qz, which means  that, if Qx = Qy = Qz = Q0, then the total complexity of  Algorithm 1 is O(Q0), whereas the complexity of the standard  approximation is O(Q3  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' PLACEMENT WITH MIN-RATE GUARANTEES  The approaches in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' III make it possible to predict the  5  Weight-function  approximations  Piecewise linear  approximation  Voxel  Centroid  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 2: 2D illustration of the conventional weight-function  approximation of the tomographic integral (4) (orange ellipses)  and the approximation adopted here (colored line segment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Observe that the upper ellipse contains no centroid and, there-  fore, the approximation will yield zero attenuation regardless  of the values of the SLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  channel gain γm(xABS  n  ) for all required pairs of user location  xm and ABS location xABS  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' These predictions can then be  substituted into (1) to obtain the capacity values Cm(xABS  n  )  that appear in the constraint (2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' A similar observation  applies to CBH(xABS  n  ) in (2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Unfortunately, solving (2) is  challenging: even if N were known and one just needed to find  feasible {xABS  n  }N  n=1, the problem would still be non-convex  since the right-hand sides of (2b) and (2d) are, in general,  non-concave functions of xABS  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' To bypass this difficulty,  the flight region F will be discretized into a flight grid  ¯F := {x ¯  F  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , x ¯  F  G} ⊂ F ⊂ R3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Replacing xABS  n  ∈ F in (2e) with xABS  n  ∈  ¯F means  that optimizing with respect to N and the ABS positions  {xABS  n  }N  n=1 is equivalent to choosing the smallest subset of  points in ¯F for which there exists a feasible rate allocation,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=', for which there exist {rm[n]}M,N  m=1,n=1 satisfying the  constraints in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Equivalently, one can consider all grid points  by setting each xABS  n  to be one of the grid points and then  “disable” those xABS  n  where no ABS is going to be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To this end, let αg be 1 if there is an ABS at x ¯  F  g and 0  otherwise, in which case the rates from that location need to  be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' By following this approach, Problem (2) becomes  minimize  {αg}G  g=1,{rg[m]}M,G  m=1,g=1  G  �  g=1  αg  (6a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  �  m  rg[m] ≤ CBH(x  ¯  F  g ),  (6b)  �  g  rg[m] ≥ Rmin,  (6c)  0 ≤ rg[m] ≤ αgCm(x  ¯  F  g ),  (6d)  αg ∈ {0, 1},  (6e)  where now the constraints need to hold for all m and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Note  the presence of αg in the right-hand side of (6d), which forces  the rate to be 0 at grid points without an actual ABS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To simplify notation, it is convenient to express Prob-  lem (6) in terms of rg  :=  [rg[1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , rg[M]]⊤, R  :=  [r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , rG], cBH := [CBH(x ¯  F  1 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , CBH(x ¯  F  G)]⊤, and cg :=  [C1(x ¯  F  g ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , CM(x ¯  F  g )]⊤ as  minimize  {αg}G  g=1,R  G  �  g=1  αg  (7a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' R⊤1 ≤ cBH  (7b)  R1 ≥ Rmin1  (7c)  0 ≤ rg ≤ αgcg  (7d)  αg ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (7e)  Observe that, if {αg}G  g=1 were given, then Problem (7)  would be a convex feasibility problem in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' However, since  one needs to optimize over {αg}G  g=1 as well, Problem (7)  becomes combinatorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' In fact, the following result establishes  the NP-hardness of (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Theorem 1: Problem (7) is NP-hard unless P=NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Proof: See Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  It is useful to reduce Problem (7) to the optimization with  respect to R alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' To this end, one can note that, for an  arbitrary R, the corresponding optimal {αg}G  g=1 satisfy that  αg = 0 if rg = 0 and αg = 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This means that (7)  can be written as  minimize  R  G  �  g=1  I[rg ̸= 0]  (8a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' R⊤1 ≤ cBH  (8b)  R1 ≥ Rmin1  (8c)  0 ≤ rg ≤ cg,  (8d)  where I[·] equals 1 if the condition inside brackets is true and  0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  The objective �G  g=1 I[rg ̸= 0] can be equivalently ex-  pressed as �G  g=1 I[∥rg∥∞ ̸= 0], where the ℓ∞-norm ∥v∥∞  equals the largest absolute value of the entries of vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Clearly, �G  g=1 I[∥rg∥∞ ̸= 0] = ∥[∥r1∥∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , ∥rG∥∞]⊤∥0,  which suggests the relaxation ∥[∥r1∥∞, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , ∥rG∥∞]⊤∥1 =  �  g ∥rg∥∞, or its reweighted version �  g wg∥rg∥∞, where  {wg}g are non-negative constants set as in [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The problem  thereby becomes  minimize  R  �  g  wg∥rg∥∞  (9a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' R⊤1 ≤ cBH  (9b)  R1 ≥ Rmin1  (9c)  0 ≤ R ≤ C,  (9d)  where the (m, g)-th entry of C := [c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , cG] ∈ RM×G  +  is given by cm,g := Cm(x ¯  F  g ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=', the capacity of the link  between the m-th user and the g-th grid point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The (m, g)-th  entry of R therefore satisfies 0 ≤ rm,g ≤ cm,g, which means  that it can be interpreted as the rate at which a virtual ABS [14]  placed at grid point x ¯  F  g communicates with the m-th user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' In  case that rm,g = 0 for all m, then no actual ABS needs to  be deployed at x ¯  F  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' In other words, the virtual ABS at x ¯  F  g  corresponds to an actual ABS if and only if rm,g ̸= 0 for  some m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  6  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' NUMERICAL SOLVER  Observe that (9) is a convex optimization problem and  therefore it can be numerically solved in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  In view of the inequality constraints, the first possibility one  may consider is to apply an interior-point solver, as described  in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Unfortunately, such an approach is only  suitable for relatively small values of M and G given the  poor scalability of interior-point methods with the number of  variables and constraints [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Indeed, in this application, G  can be in the order of tens of thousands, which would render  the (at least cubic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Appendix A) complexity of interior-  point methods prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' In contrast, the rest of this section  presents a solver whose complexity is linear in G and M by  building upon the so-called alternating-direction method of  multipliers (ADMM) [35] and exploiting the special structure  of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' ADMM Decomposition  As outlined below, ADMM alternately solves two opti-  mization subproblems in the primal variables and performs  a gradient step along the dual variables [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' But before  decomposing (9) into such subproblems, a few manipulations  are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  The first is to replace the inequality constraint (9c) with  an equality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' To this end, let ¯rm denote the m-th  column of R⊤ and suppose that R is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Then, it follows  from (9c) that ¯r⊤  m1 ≥ Rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' If one replaces ¯rm with ¯r′  m :=  (Rmin/(¯r⊤  m1))¯rm, the entries of ¯r′  m are non-negative and not  greater than the entries of ¯rm since 0 ≤ (Rmin/(¯r⊤  m1)) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Hence, the resulting R still satisfies all other constraints and  the m-th constraint in (9c) now holds with equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Besides,  the resulting objective will not be greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Therefore, if R is  optimal, scaling any of its rows in this way yields another  optimal R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Applying this reasoning to all rows (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=', for all  m) shows that (9c) can be replaced with an equality constraint  without loss of optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Second, due to (9d), the entries of rg are non-negative and,  thus, ∥rg∥∞ equals the largest entry of rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This means that  the objective can be replaced with �  g wgsg upon introducing  the auxiliary variables sg and constraints rg ≤ sg1 for each  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This gives rise to  minimize  R,s  w⊤s  (10a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' R⊤1 ≤ cBH  (10b)  R1 = Rmin1  (10c)  0 ≤ R ≤ C  (10d)  R ≤ 1s⊤,  (10e)  where w := [w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , wG]⊤ and s := [s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , sG]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  The next step is to express (10) in a form amenable to  application of ADMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Specifically, (10) will be expressed in  the homogeneous form  minimize  X,Z  f(X) + h(Z)  (11a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  A1XA2 + B1ZB2 = 0,  (11b)  for which the ADMM iteration becomes [35, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='1]  Xk+1 = arg min  X  f(X) + ρ  2∥A1XA2 + B1ZkB2 + U k∥2  F  (12a)  Zk+1 = arg min  Z  h(Z) + ρ  2∥A1Xk+1A2 + B1ZB2 + U k∥2  F  (12b)  U k+1 = U k + A1Xk+1A2 + B1Zk+1B2  (12c)  for k = 1, 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='. Here, Xk and Zk collect the primal variables,  U k is a matrix of scaled dual variables, and ρ > 0 is the step-  size parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Each possible correspondence that one may establish be-  tween the variables, constants, and functions of (11) and those  of (10) results in a different ADMM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Finding a  good correspondence is typically the most critical step and  takes multiple attempts since, unless properly accomplished,  the complexity of the subproblems (12a) and (12b) will be  comparable to the complexity of the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' For the  problem at hand, the following assignment was found to yield  subproblems that separate along the rows and columns of R:  X → [R⊤, s]⊤  (13a)  Z → R  (13b)  f(X) → w⊤s +  �  g  I[rg ≤ sg1]  +  �  g  I[1⊤rg ≤ cBH  g ]  (13c)  h(Z) → I[R1 = Rmin1] + I[0 ≤ R ≤ C]  (13d)  A1 → [IM, 0], A2 → IG,  (13e)  B1 → −IM, B2 → IG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (13f)  Here, cBH := [cBH  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , cBH  G ]⊤ and I[·] is a function that takes  the value 0 when the condition inside brackets holds and ∞  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Note that, given (13), it follows that A1XA2 +  B1ZB2 = R−Z and, therefore, (11b) imposes that R = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Thus, intuitively speaking, each subproblem (12a) and (12b)  tries to find values for their respective variables that satisfy the  structure promoted by the first terms in (12a) and (12b) while,  at the same, the second terms in these expressions as well  as (12c) push towards an agreement between the solutions of  both subproblems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  The next two subsections will be respectively concerned  with finding the solutions of (12a) and (12b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Afterwards, both  solutions are put together in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' V-D to obtain the desired  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The X-subproblem  This section decomposes the X-update (12a) into G smaller  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The latter can be efficiently solved by finding a root  of a scalar equation through the bisection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  7  In view of (13), (12a) can be expressed as  (Rk+1, sk+1) = arg min  R,s w⊤s +  �  g  I[rg ≤ sg1]  (14a)  +  �  g  I[1⊤rg ≤ cBH  g ] + ρ  2∥R − Zk + U k∥2  F  = arg min  R,s  �  g  �  wgsg + I[rg ≤ sg1]  (14b)  +I[1⊤rg ≤ cBH  g ] + ρ  2∥rg − zk  g + uk  g∥2  2  �  ,  where zk  g and uk  g respectively denote the g-th column of Zk  and U k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This problem clearly separates into G problems of  the form  (rk+1  g  , sk+1  g  ) = arg min  rg,sg wgsg + ρ  2∥rg − zk  g + uk  g∥2  2 (15a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  rg ≤ sg1  (15b)  1⊤rg ≤ cBH  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (15c)  There are two cases: C1) constraint (15c) is active (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  it holds with equality) at the optimal solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' C2) (15c) is  inactive (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' it holds with strict inequality) at the optimal  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Thus, to solve (15), one can apply the following  strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' First, solve the problem that results from removing  (15c):  (rk+1  g  , sk+1  g  ) = arg min  rg,sg wgsg + ρ  2∥rg − zk  g + uk  g∥2  2 (16a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  rg ≤ sg1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (16b)  If the solution to (16) satisfies (15c), then it is also the optimal  solution of (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Else, due to the convexity of the problem,  (15c) must necessarily be active at the optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' In this case,  the optimal solution of (15) can be found by replacing (15c)  with an equality constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Thus, let us start by solving (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Proposition 1: Let rk+1  g  and sk+1  g  be given by (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' It holds  that  rk+1  g  = min(zk  g − uk  g, sk+1  g  1)  (17a)  1⊤ max(zk  g − uk  g − sk+1  g  1, 0) = wg  ρ ,  (17b)  where min and max operate entrywise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Proof: See Appendix F  Observe that (17a) can be used to obtain rk+1  g  if sk+1  g  is  given, whereas (17b) does not depend on rk+1  g  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Therefore, a  solution to (17) can be found by first solving (17b) for sk+1  g  and then substituting the result into (17a) to recover rk+1  g  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' To  this end, we have the following:  Proposition 2: Equation (17b) has a unique root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This root  lies in the interval [ˇsk  g, ˆsk  g], where  ˇsk  g := min  m  �  zk  g[m] − uk  g[m]  �  − wg  Mρ  (18a)  ˆsk  g := max  m  �  zk  g[m] − uk  g[m]  �  − wg  Mρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (18b)  Proof: See Appendix G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Observe that Proposition 2 essentially provides the bounds  that are required to find the unique root of (17b) via the well-  known bisection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Recall that the latter is a very  efficient algorithm as it geometrically reduces the uncertainty  at every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  In accordance with the strategy mentioned earlier, one also  necessitates a means to solve (15) when (15c) is replaced with  an equality constraint:  (rk+1  g  , sk+1  g  ) = arg min  rg,sg wgsg + ρ  2∥rg − zk  g + uk  g∥2  2 (19a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  rg ≤ sg1  (19b)  1⊤rg = cBH  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (19c)  Proposition 3: Let rk+1  g  and sk+1  g  be given by (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Then,  it holds that  rk+1  g  = min(zk  g − uk  g − (µ/ρ)1, sk+1  g  1)  (20a)  1⊤ max(µ1, ρ(zk  g − uk  g − sk+1  g  1)) = wg + µM  (20b)  where  µ := −ρcBH  g  + ρ1⊤(zk  g − uk  g) − wg  M  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (21)  Proof: See Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Observe that (20a) and (21) can be used to obtain rk+1  g  if sk+1  g  is given, whereas (20b) does not depend on rk+1  g  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Therefore, a solution to (19) can be found by first solving  (20b) for sk+1  g  and then substituting the result into (20a) to  recover rk+1  g  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' To this end, we have the following:  Proposition 4: Equation (20b) has a unique root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This root  lies in the interval [ˇsk  g, ˆsk  g], where  ˇsk  g := min  m  �  zk  g[m] − uk  g[m]  �  − wg  Mρ − µ  ρ  (22a)  ˆsk  g := max  m  �  zk  g[m] − uk  g[m]  �  − wg  Mρ − µ  ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (22b)  Proof: See Appendix I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To sum up, (14) can be solved separately for each column  of R and entry of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Each of these G problems can be solved  by first solving (16) and checking whether (15c) holds for  the obtained solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' If it does not hold, one must solve  (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Proposition 1 and Proposition 3 respectively establish  that a solution can be found for each of these problems just  by solving a scalar equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Proposition 2 and Proposition 4  prove uniqueness of the solutions of these two equations  and provide an interval where they can be sought using the  bisection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The Z-subproblem  This section describes how a solution to (12b) can be found  by solving a bisection problem per row of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' To this end, start  by noting that it follows from (12b) and (13) that  Zk+1 = arg min  Z  �  I[Z1 = Rmin1] + I[0 ≤ Z ≤ C]  + ρ  2∥Rk+1 − Z + U k∥2  F  �  (23a)  = arg min  Z  �  m  �  I[¯z⊤  m1 = Rmin] + I[0 ≤ ¯zm ≤ ¯cm]  + ρ  2∥¯rk+1  m  − ¯zm + ¯uk  m∥2  F  �  ,  (23b)  8  Algorithm  2: Group-sparse Placement Algorithm  (GSPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  1 Input: C ∈ RM×G  +  , Rmin ∈ R+,  {wg}g ⊂ R+,{cBH  g }g ⊂ R+, ρ > 0  2 Initialize U 1 ∈ RM×G and Z1 ∈ RM×G  +  3 for k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' do  4  for g = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , G do  5  Bisection: find sk+1  g  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  1⊤ max(0, ρ(zk  g − uk  g − sk+1  g  1)) = wg  6  Set rk+1  g  = min(zk  g − uk  g, sk+1  g  1)  7  if 1⊤rk+1  g  > cBH  g  then  8  Set  µ = (−ρcBH  g  + ρ1⊤(zk  g − uk  g) − wg)/M  9  Bisection: find sk+1  g  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  1⊤ max(µ1, ρ(zk  g − uk  g − sk+1  g  1)) = wg + µM  10  Set rk+1  g  = min(zk  g − uk  g − (µ/ρ)1, sk+1  g  1)  11  for m = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , M do  12  Bisection: find λ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  1⊤ max(0, min(¯cm, ¯rk+1  m  + ¯uk  m − λ1)) = Rmin  13  Set ¯zk+1  m  = max(0, min(¯cm, ¯rk+1  m  + ¯uk  m − λ1))  14  Set U k+1 = U k + Rk+1 − Zk+1  15  If convergence( ) then return Rk+1  10  20  30  40  50  60  70  80  90  Number of GTs (M)  5  10  15  20  25  30  35  40  45  Mean number of ABSs  Lower bound  K-means Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Galkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Space rate Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Hammouti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Genetic Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Shehzad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  GSPA (proposed)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 3: Mean number of ABSs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' number of GTs (Rmin = 20  Mbps, cBH = 99 Mbps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  where ¯zm, ¯cm, and ¯uk  m respectively denote the m-th column  of Z⊤, C⊤, and (U k)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Clearly, this separates into M  problems of the form  ¯zk+1  m  = arg min  ¯zm  1  2∥¯rk+1  m  − ¯zm + ¯uk  m∥2  F  (24a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  1⊤¯zm = Rmin,  0 ≤ ¯zm ≤ ¯cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (24b)  Proposition 5: If 1⊤¯cm < Rmin, then (24) is infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' If  1⊤¯cm ≥ Rmin, the solution to (24) is given by  ¯zk+1  m  = max(0, min(¯cm, ¯rk+1  m  + ¯uk  m − λ1)),  (25)  where λ satisfies  1⊤ max(0, min(¯cm, ¯rk+1  m  + ¯uk  m − λ1)) = Rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (26)  Proof: See Appendix J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Thus, as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' V-B, one needs to solve the scalar equation  (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The following result is the counterpart of Proposition 2  for the Z-subprblem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Proposition 6: If 1⊤¯cm < Rmin, then equation (26) has no  roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' If 1⊤¯cm ≥ Rmin, then (26) has a unique root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This root  lies in the interval [ˇλ  k  m, ˆλ  k  m], where  ˇλ  k  m = min  g [¯rk+1  m [g] + ¯uk  m[g] − ¯cm[g]]  (27a)  ˆλ  k  m = max{¯rk+1  m [g] + ¯uk  m[g] : g ∈ {g : ¯cm[g] > Rmin  G }}  − Rmin  G  (27b)  Proof: See Appendix K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Proposition 6 establishes uniqueness and provides the  bounds needed to solve (26), and therefore (24), via the  bisection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The Proposed Solver  Having addressed both X- and Z- subproblems, it remains  only to obtain the U-update in (12c), which for the assign-  ments in (13) becomes  U k+1 = U k + Rk+1 − Zk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (28)  This completes the derivation of the proposed scheme,  summarized as Algorithm 2 and referred to as the group-  sparse placement algorithm (GSPA) since it promotes group  sparsity [45] in the columns of R, that is, only a few columns  of the matrix Rk+1 returned by the algorithm are expected to  be non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Recall that the non-zero columns indicate which  grid points will be occupied by an ABSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  In the notation used in Algorithm 2, if A is a matrix, then  am is its m-th column and ¯a⊤  n its n-th row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Furthermore,  superscripts indicate the iteration index, ρ > 0 is the step  size, and the min and max operators act entrywise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The  criterion on line 15, which determines whether the algorithm  has converged, is detailed in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Observe that the main strategy in the previous two subsec-  tions was to exploit the structure of Problem (9) to decompose  it into one subproblem per row and column of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Each of  these subproblems involves solving a bisection task of a 1D  monotonically decreasing function and therefore can be solved  with O(1) evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The total complexity is O(GM), much  smaller than the O(G3M 3) complexity per inner iteration of  interior-point methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' NUMERICAL EXPERIMENTS  This section empirically validates the performance of the  proposed algorithm by means of numerical experiments  with channel data generated using the tomographic model  (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' VI-A) and ray-tracing software (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' VI-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The code  and data necessary to reproduce the experiments is available  at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='com/uiano/ABS placement via propagation  maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  9  39  49  59  69  79  89  99  Total rate of each ABS [Mbps]  10  20  30  40  50  60  70  Mean number of ABSs  Lower bound  K-means Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Galkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Space rate Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Hammouti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Genetic Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Shehzad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  GSPA (proposed)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 5: Mean number of ABSs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' the capacity cBH of the  backhaul link (Rmin = 20 Mbps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  10  15  20  25  30  35  40  45  Minimum GT rate [Mbps]  10  20  30  40  50  60  70  Mean number of ABSs  Lower bound  K-means Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Galkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Space rate Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Hammouti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Genetic Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Shehzad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  GSPA (proposed)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 4: Mean number of ABSs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Rmin (cBH = 100 Mbps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  The channel gain, obtained from the aforementioned models  when the carrier frequency is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='4 GHz, is substituted into (1)  with W = 20 MHz, PTX = 20 dBm/Hz, and σ2 = −96  dBm/Hz to form the capacity matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' For simplicity, the  backhaul capacity is set to a common value cBH  1  = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' =  cBH  G = cBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  The proposed placement algorithm is compared with three  benchmarks: i) the K-means placement algorithm by Galkin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' [11], ii) the iterative space rate K-means placement  algorithm by Hammouti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' [22], and iii) the genetic  placement algorithm by Shehzad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' [25] with 50 solutions  per generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Since we did not manage to obtain the code  used by the authors of these works, we implemented their algo-  rithms ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The resulting implementations are available  in the repository mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The first two algorithms are  unable to enforce no-fly zones, which results in ABSs placed  inside buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' To avoid this behavior, the ABS locations  provided by these algorithms are projected onto the same flight  grid as the rest of algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The proposed GSPA algorithm  utilizes, unless otherwise stated, a step size of ρ = 10−7  and stopping criterion parameters ϵabs = ϵrel = 10−4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To quantify performance, the number of ABSs required  by each algorithm to guarantee a rate Rmin for every GT is  considered as a performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This metric is averaged  using Monte Carlo simulation across realizations of the GT  locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' As a reference, figures will also include a lower  bound on this metric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Experiments with the Tomographic Model  In the experiments of this section, the channel gain is  generated using Algorithm 1 in an environment like the one  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The SLF takes a constant value, termed building ab-  sorption, inside the buildings and 0 outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Unless otherwise  stated, the simulation parameters in this section are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Area size: 500 m × 400 m × 150 m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' dimensions of the SLF  grid: 50 × 40 × 15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' dimensions of the flight grid: 9 × 9 ×  5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' minimum flight height: 50 m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' height of the buildings: 63  m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' building absorption: 1 dB/m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' number of GTs: M = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 3 investigates the influence of the number of GTs  (M) on the performance of the compared algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Several  observations are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  First, the mean number of ABSs  is seen to increase roughly proportionally to M for all the  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Second, the proposed GSPA algorithm not only  yields a lower mean number of ABSs than the competing  algorithms, but its slope is smaller, which means that the  margin by which GSPA outperforms the benchmarks increases  with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Third, GSPA asymptotically approaches the lower  bound (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Appendix B), which means that its efficiency,  quantified as the number of GTs per ABS, increases with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  In contrast, the opposite is true for the other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To investigate the influence of the GT requirements on  performance, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 4 depicts the mean number of ABSs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Rmin when cBH = 100 Mbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 3, the mean number  of ABSs seems to increase roughly linearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' However, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 4  a saturation phenomenon arises: the mean number of ABSs  cannot be greater than the number of GTs M = 70 since  one ABS placed approximately above each GT suffices to  serve all GTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' It is also observed that the performance of  the genetic placement algorithm degrades faster than the rest  of algorithms for high Rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The reason may be that this  algorithm essentially tests multiple placements and the number  of placements increases drastically with the number of ABSs,  which is larger when Rmin is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 5 plots the mean number of ABSs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' the backhaul  capacity cBH when Rmin = 20 Mbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The values of cBH on the  horizontal axis are selected so that each cBH is not an integer  multiple of Rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This gives rise to a “staircase” behavior for  the benchmarks, which do not exploit the fact that a GT can be  served by multiple ABSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' In contrast, GSPA exploits this fact,  as corroborated by the smoothness of its curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' It is also seen  that the mean number of ABSs required by GSPA is roughly  half the one of the best competing alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To study the influence of the channel, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 6 shows the  mean number of ABSs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' the absorption undergone by the  10  30  40  50  60  70  80  90  100  Number of GTs (M)  20  40  60  80  100  Mean number of ABSs  Lower bound  K-means Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Galkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Space rate Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Hammouti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Genetic Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Shehzad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  GSPA (proposed)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 7: Mean number of ABSs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' number of GTs (Rmin = 20  Mbps, cBH = 74 Mbps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  11  15  19  23  27  31  35  40  45  Minimum GT rate [Mbps]  10  20  30  40  50  Mean number of ABSs  Lower bound  K-means Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Galkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Space rate Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Hammouti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Genetic Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Shehzad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  GSPA (proposed)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 8: Mean number of ABSs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' minimum GT rate Rmin  (M = 50 GTs, cBH = 100 Mbps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='25  Building absorption [dB/m]  10  15  20  25  30  35  40  45  50  55  Mean number of ABSs  Lower bound  K-means Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Galkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Space rate Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Hammouti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Genetic Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Shehzad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  GSPA (proposed)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 6: Mean number of ABSs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' building absorption (Rmin =  17 Mbps, cBH = 84 Mbps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  communication signals when propagating through the build-  ings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' When the absorption is zero, the propagation conditions  are those of free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' In this case, the proposed algorithm  outperforms the rest only because of a better ability to perform  the rate allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' As the absorption increases, the benchmark  algorithms are dramatically affected, which suggests that these  algorithms are not well suited to scenarios without line-of-  sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' In contrast, the proposed algorithm remains unaffected  since there are always sufficiently good flight grid points  regardless of the building absorption considered in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Informally speaking, matrix R in (9) is upper bounded by cBH  and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' When the former constraint is tighter than the latter,  the resulting number of ABSs will not depend on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This  phenomenon is investigated further in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Additional experiments with the tomographic channel model  are presented in Appendix L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Experiments with the Ray-Tracing Model  This section corroborates the main findings of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' VI-A  when the channel is obtained via the ray-tracing model, which  is more accurate than the tomographic channel model for  higher carrier frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To this end, a data set was generated using the X3D ray-  tracing software Wireless Insite with six reflections and one  diffraction in a 400 × 600 m2 area of the city of Ottawa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The  data set is also published in our repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The channel was  computed between all points of the flight grid and all points  of a GT grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The former comprises 35 points at each of the  heights of 40, 60, and 80 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The latter is a 2D regular grid of  2501 GT locations at a height of 2 m spaced uniformly with a  distance of 10 m on each axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Points inside the buildings are  removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' At each Monte Carlo realization, the GT locations  are generated by drawing M points uniformly at random  without replacement from the GT grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 7 and 8 are the counterparts of Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 3 and 4 for ray-  tracing channel data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' It is observed that the mean number of  ABSs is also an approximately linear function of M for all  algorithms and that GSPA roughly attains the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  However, here the differences among benchmarks are greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  The fact that the K-means algorithm outperforms the space  rate algorithm suggests that the channel changes rapidly with  respect to the GT location, since the latter algorithm relies on  clustering vectors of channel gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Despite this fact, GSPA  performs almost optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 9 further investigates the phenomenon already dis-  cussed regarding Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Observe that the tightness of the  bounds in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 9 increases (i) for larger Rmin and (ii) for  smaller cBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Those are precisely the situations where the  backhaul limitations become stricter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Indeed, it can be easily  seen that the bound in Appendix B can be attained with  equality when the entries of C approach infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Finally, the experiment in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 5 is performed with ray-  tracing channel data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The proposed algorithm still  outperforms the other three benchmarks by a wide margin,  requiring roughly 50% of ABSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' However, relative to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 5,  11  3  5  7  9  11  Minimum GT rate [Mbps]  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='0  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='5  Mean number of ABSs  cBH=34 Mbps  Lower bound for cBH=34 Mbps  cBH=44 Mbps  Lower bound for cBH=44 Mbps  cBH=54 Mbps  Lower bound for cBH=54 Mbps  cBH=64 Mbps  Lower bound for cBH=64 Mbps  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 9: Mean number of ABSs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' minimum GT rate Rmin for  GSPA (M = 50 GTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  34  44  54  64  74  80  90  100  Total rate of each ABS [Mbps]  5  10  15  20  25  Mean number of ABSs  Lower bound  K-means Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Galkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Space rate Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Hammouti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Genetic Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Shehzad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  GSPA (proposed)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 10: Mean number of ABSs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' backhaul link capacity cBH  (Rmin = 7 Mbps, M = 50 GTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  one can observe that the benchmark algorithms saturate for  large cBH, which indicates that C imposes a more stringent  constraint than cBH in the scenario of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' When it comes  to GSPA, the greater tightness of the bound for small cBH is  a manifestation of the same effect, as discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' CONCLUSIONS  Whereas existing algorithms for ABS placement assume  that the channel gain depends only on the length and (possibly)  elevation of each link, this paper presents a scheme that can  accommodate an arbitrary dependence of the gain on the posi-  tion of the ABSs and GTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This enables the utilization of radio  maps for ABS placement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The proposed algorithm determines  a set of ABS locations that approximately minimizes the num-  ber of ABSs required to guarantee a minimum rate to all GTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Relative to most existing schemes, the proposed algorithm has  a low complexity, accounts for a limited backhaul capacity,  and can accommodate flight restrictions such as no-fly zones  or airspace occupied by buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' A solver whose complexity  is linear in the number of users was derived based on the  alternating-direction method of multipliers and the problem of  evaluating tomographic integrals was revisited and extended  to air-to-ground channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' An extensive set of simulations  demonstrate that the proposed GSPA algorithm outperforms  competing algorithms by a wide margin both in tomographic  and ray-tracing channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Remarkably, it was observed in the  numerical experiments that the proposed algorithm is the only  one among the compared schemes whose efficiency, measured  in terms of number of GTs served per ABS, increases with the  number of GTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This fundamental distinction renders GSPA  especially suitable for scenarios with a large number of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Future directions include approaches for tracking air-to-  ground propagation maps, possibly based on online kernel  methods [46], [47], and algorithms that can adapt to changes  in the GT locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' ACKNOWLEDGEMENTS  The authors would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Geert Leus for  insightful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  APPENDIX A  INTERIOR-POINT SOLVER  The present section illustrates how (10) can be solved  using an interior-point algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Although such a solver is  not utilized in this paper, the ensuing derivation provides  its computational complexity, which motivates the ADMM  algorithm from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  It is convenient to start by expressing (10) in a canonical  form with only non-negativity constraints and linear equality  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' To this end, introduce the slack variables δ1, ∆2,  and ∆3 to write (10) as  minimize  R,s,δ1,∆2,∆3 w⊤s  (29a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' R⊤1 + δ1 = cBH  (29b)  R1 = Rmin1  (29c)  R + ∆2 = C  (29d)  R + ∆3 = 1s⊤  (29e)  R ≥ 0, δ1 ≥ 0, ∆2 ≥ 0, ∆3 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (29f)  With this formulation, it is easy to see that (29) is equivalent  to  minimize  ˜x  ˜w⊤˜x  (30a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' ˜A˜x = ˜b  (30b)  ˜x[G + 1 : end] ≥ 0,  (30c)  where  ˜x  :=  [s⊤, r⊤, δ⊤  1 , vec⊤(∆2), vec⊤(∆3)]⊤,  r  =  vec(R),  ˜w  :=  [w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , wG, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , 0]⊤,  ˜x[G + 1  :  end]  :=  [r⊤, δ⊤  1 , vec⊤(∆2), vec⊤(∆3)]⊤,  ˜b := [(cBH)⊤, Rmin1⊤, vec⊤(C), 0⊤]⊤, and  ˜A :=  �  ���  0  IG ⊗ 1⊤  IG  0  0  0  1⊤ ⊗ IM  0  0  0  0  IGM  0  IGM  0  −IG ⊗ 1  IGM  0  0  IGM  �  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (31)  12  Problem (30) can be solved by means of a standard interior-  point algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' To derive a lower bound for its computational  complexity, note that each inner iteration of the algorithm will  involve solving a system of equations where the number of  unknowns equals the number of variables of the optimization  problem plus the number of linear constraints [48, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 10  and 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' For (30), the former equals 2G + 3GM whereas  the latter is given by G + M + 2GM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Solving this system  of equations without any tailor-made approach that exploits  the specific structure of ˜A in this problem therefore involves  O(G3M 3) arithmetic operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  It is worth remarking that this complexity is prohibitive in  practice: if, for example, G = M = 100, then 1012 operations  would be required per inner iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  APPENDIX B  LOWER BOUND FOR THE NUMBER OF ABSS  This appendix presents a lower bound for the number of  ABSs and, therefore, also for the mean number of ABSs,  which is the performance metric adopted in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This  bound constitutes a fundamental limit for the problem of ABS  placement and, hence, it applies regardless of the adopted  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Let N denote the smallest number of ABSs required to  serve all M users with rate at least Rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This means that the  total backhaul rate available to all ABSs together, which is  not greater than N maxg cBH  g , cannot be less than the total rate  MRmin demanded by the users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' It follows that N maxg cBH  g  ≥  MRmin and, therefore, N ≥ ⌈MRmin/ maxg cBH  g ⌉, where ⌈z⌉  denotes the smallest integer greater than or equal to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  APPENDIX C  DERIVATION OF ALGORITHM 1  Algorithm 1, which can be classified as a parametric,  floating point, and zeroth-order algorithm according to the  terminology of [42, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' I-B-1], is our approach (yet others are  possible) to approximate the tomographic integral by comput-  ing exactly the integral of a piecewise constant approximation  of the SLF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The idea is to parameterize the line segment  between x1 and x2 as x(t) = x1 + t(x2 − x1), where  t ∈ [0, 1], and identify the values t1 < t2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' < tT for  which the boundary between two adjacent voxels is crossed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Since ∥x(ti) − x(ti−1)∥ = (ti − ti−1)∥x2 − x1∥ whenever  ti > ti−1, the approximation is then  ξ(x1, x2) ≈  �T  i=2(ti − ti−1)∥x2 − x1∥l(x ¯  X  qi)  ∥x2 − x1∥1/2  (32a)  = ∥x2 − x1∥1/2  T  �  i=2  (ti − ti−1)l(x  ¯  X  qi),  (32b)  where qi is the index of the voxel that contains the i-th segment  {x(t)  :  t ∈ (ti−1, ti)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Since ¯  X is a 3D grid, each point  in {x ¯  X  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , x ¯  X  Q} can also be indexed by a vector i of 3  indices that lies in the set I := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , Qx} × {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , Qy} ×  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' , Qz}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The values of the SLF can also be collected in a  tensor L ∈ RQx×Qy×Qz, whose entry L[i] is the value of l at  the i-th grid point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' If δ ¯  X ∈ R3  ++ denotes a vector whose j-th  entry δ ¯  X [j] represents the spacing between grid points along  the j-th axis, the coordinates of the i-th grid point are clearly  i ⊙ δ ¯  X , where ⊙ denotes entrywise product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Similarly, the  boundaries between adjacent voxels along the j-th axis occur  at values of the j-th coordinate given by δ ¯  X [j](i±1/2), where  i is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' It is then clear that steps 6-8 in Algorithm 1  simply find the next value of t for which the segment crosses a  voxel boundary along one of the axes by solving the equation  x1[j] + t(x2[j] − x1[j]) = δ ¯  X [j](icurrent[j] ± 1/2)  (33)  for t along each axis j and taking the minimum across axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  The ± becomes a plus sign for the j-th axis if the segment  is increasing along this axis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' x1[j] ≤ x2[j]) and a minus  sign otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  An alternative implementation of the same integral approx-  imation with smaller computational complexity but greater  memory complexity could be obtained by creating 3 lists  corresponding to the values of t for which the line segment  between x1 and x2 intersects each axis and then merging those  lists into a list with non-decreasing values of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  APPENDIX D  STOPPING CRITERION  The stopping criterion of Algorithm 2 follows the frame-  work in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Particularly, given the absolute and relative  tolerance parameters ϵabs and ϵrel, let ϵk+1  pri  and ϵk+1  dual be  ϵk+1  pri  :=  √  MGϵabs  (34a)  + ϵrel max{∥A1Xk+1A2∥F, ∥B1Zk+1B2∥F},  ϵk+1  dual :=  √  MGϵabs + ϵrel∥ρA⊤  1 U k+1A⊤  2 ∥F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (34b)  Algorithm 2 stops when both conditions  ∥Qk+1∥2  F ⩽ ϵk+1  pri  and ∥P k+1∥2  F ⩽ ϵk+1  dual  (35a)  are satisfied, where  Qk+1 := A1Xk+1A2 + B1Zk+1B2  (36a)  P k+1 := ρA⊤  1 B1(Zk+1 − Zk)B2A⊤  2  (36b)  are the so-called primal and dual residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  APPENDIX E  PROOF OF THEOREM 1  The idea is to establish that a special case of (7) is a  multidimensional knapsack problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' To this end, let the g-  th entry of cBH be at least as large as 1⊤cg and note that,  due to (7d), constraint (7b) holds regardless of the choice of  {αg}G  g=1 and R, meaning that (7b) can be removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Next, note that if {αg}G  g=1 and R are feasible, then replac-  ing any rg with αgcg yields another feasible point that attains  the same cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This is because none of the entries of the left-  hand side of (7c) decreases after modifying rg in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  The left-hand side of (7c) can then be written as R1 =  �  g rg = �  g αgcg, which yields the following problem  minimize  {αg}G  g=1  �  g  αg  (37a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  �  g  αgcg ≥ Rmin1  (37b)  αg ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (37c)  13  Finally, applying the change of variables βg ← 1 − αg, the  objective becomes G − �  g βg and the left-hand side of (37b)  becomes �  g(1 − βg)cg = �  g cg − �  g βgcg, which implies  that (37) reads as  maximize  {βg}G  g=1  �  g  βg  (38a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  �  g  βgcg ≤  �  g  cg − Rmin1  (38b)  βg ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (38c)  This problem is an instance of the so-called multidimensional  knapsack problem, which has been shown to be NP-hard  unless P=NP [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  APPENDIX F  PROOF OF PROPOSITION 1  Since Problem (16) is convex differentiable and Slater’s con-  ditions are satisfied, it follows that the Karush-Kuhn-Tucker  (KKT) conditions are sufficient and necessary [48, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To obtain these conditions, observe that the Lagrangian of (16)  is given by  L(rg, sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' ν) = wgsg + ρ  2∥rg − zk  g + uk  g∥2  2 + ν⊤(rg − sg1)  (39)  and note that the KKT conditions can be stated as  ∇rgL(rg, sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' ν) = ρ(rg − zk  g + uk  g) + ν = 0  (40a)  ∇sgL(rg, sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' ν) = wg − 1⊤ν = 0  (40b)  rg ≤ sg1  (40c)  ν ≥ 0,  ν[m](rg[m] − sg) = 0 ∀m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (40d)  From (40a) and the inequality in (40d), it follows that  ν = −ρ(rg − zk  g + uk  g) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (41)  This implies that rg ≤ zk  g − uk  g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Combining this inequality  with (40c) yields  rg ≤ min(zk  g − uk  g, sg1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (42)  On the other hand, from the equality in (41) and the equality  in (40d), one finds that  −ρ(rg[m] − zk  g[m] + uk  g[m])(rg[m] − sg) = 0 ∀m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (43)  This holds if and only if either rg[m] = zk  g[m] − uk  g[m] or  rg[m] = sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Therefore, it follows from (42) that  rg = min(zk  g − uk  g, sg1),  (44)  which establishes (17a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Finally, combine this expression with  (40b) and (41) to arrive at  wg = −ρ1⊤(rg − zk  g + uk  g)  (45a)  = −ρ1⊤(min(zk  g − uk  g, sg1) − zk  g + uk  g)  (45b)  = −ρ1⊤ min(0, sg1 − zk  g + uk  g)  (45c)  = ρ1⊤ max(0, zk  g − uk  g − sg1),  (45d)  thereby recovering (17b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The proof is complete by noting that  (40) holds if and only if (44) and (45d) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  APPENDIX G  PROOF OF PROPOSITION 2  Consider the function F(s) := 1⊤ max(zk  g −uk  g −s1, 0) =  �  m max(zk  g[m] − uk  g[m] − s, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Since F is the sum of non-  increasing piecewise linear functions, so is F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Since F(s) →  ∞ as s → −∞ and F(s) = 0 for a sufficiently large s, it  follows that (17b) has at least one root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Uniqueness of the  root follows readily by noting that F is strictly decreasing  whenever F(s) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  It remains to be shown that F(ˇsk  g) ≥ wg/(Mρ) whereas  F(ˆsk  g) ≤ wg/(Mρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' For the first of these inequalities, observe  that ˇsk  g ≤ zk  g[m] − uk  g[m] − wg/(Mρ) for all m, which in  turn implies that zk  g[m] − uk  g[m] − ˇsk  g ≥ wg/(Mρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Thus,  max(zk  g[m] − uk  g[m] − ˇsk  g, 0) = zk  g[m] − uk  g[m] − ˇsk  g ≥  wg/(Mρ), which yields F(ˇsk  g) ≥ �  m wg/(Mρ) = wg/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  For the second inequality, note similarly that zk  g[m]−uk  g[m]−  ˆsk  g  ≤  wg/(Mρ) for all m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This means that F(ˆsk  g)  ≤  �  m max(wg/(Mρ), 0) = wg/ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  APPENDIX H  PROOF OF PROPOSITION 3  Again, the KKT conditions are sufficient and necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Since the Lagrangian is  L(rg, sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' ν) = wgsg + ρ  2∥rg − zk  g + uk  g∥2  2  + ν⊤(rg − sg1) + µ(1⊤rg − cBH  g ),  (46)  the KKT conditions read as  ∇rgL(rg, sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' ν) = ρ(rg − zk  g + uk  g) + ν + µ1 = 0  (47a)  ∇sgL(rg, sg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' ν) = wg − 1⊤ν = 0  (47b)  rg ≤ sg1  (47c)  ν ≥ 0,  ν[m](rg[m] − sg) = 0 ∀m  (47d)  1⊤rg = cBH  g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (47e)  From (47a) and the inequality in (47d), it follows that  ν = −ρ(rg − zk  g + uk  g) − µ1 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (48)  This implies that rg ≤ zk  g − uk  g − (µ/ρ)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Combining this  inequality with (47c) yields  rg ≤ min(zk  g − uk  g − (µ/ρ)1, sg1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (49)  On the other hand, from the equality in (48) and the equality  in (47d), one finds that  [−ρ(rg[m] − zk  g[m] + uk  g[m]) − µ](rg[m] − sg) = 0 ∀m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (50)  This holds if and only if either rg[m] = zk  g[m]−uk  g[m]−µ/ρ  or rg[m] = sg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Therefore, it follows that  rg = min(zk  g − uk  g − (µ/ρ)1, sg1),  (51)  which establishes (20a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To find µ, substitute the equality in (48) into (47b) to obtain  1⊤[−ρ(rg − zk  g + uk  g) − µ1] = wg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (52)  14  Solving for µ yields  µ = −ρ1⊤rg + ρ1⊤(zk  g − uk  g) − wg  M  (53)  and using (47e) results in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Finally, substitute (51) into (52) to arrive at  wg = 1⊤ max(µ1, ρ(zk  g − uk  g − sg1)) − µM,  (54)  thereby recovering (20b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The proof is complete by noting that  (47) holds if and only if (20a) and (20b) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  APPENDIX I  PROOF OF PROPOSITION 4  Consider the function F(s) := 1⊤ max(µ1, ρ(zk  g − uk  g −  s1)) = �  m max(µ, ρ(zk  g[m] − uk  g[m] − s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Due to the same  argument as in the proof of Proposition 2, this function has a  unique root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To show that F(ˇsk  g) ≥ wg + µM, observe that  F(ˇsk  g) ≥  �  m  ρ(zk  g[m] − uk  g[m] − ˇsk  g)  (55a)  ≥ M min  m [ρ(zk  g[m] − uk  g[m] − ˇsk  g)]  (55b)  = Mρ  � wg  Mρ + µ  ρ  �  (55c)  = wg + µM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (55d)  To show that F(ˆsk  g) ≤ wg + µM, observe that  F(ˆsk  g) ≤ M max  m [max(µ, ρ(zk  g[m] − uk  g[m] − ˆsk  g))]  (56a)  = M max(µ, ρ(max  m [zk  g[m] − uk  g[m]] − ˆsk  g))  (56b)  = M max  �  µ, ρ  � wg  Mρ + µ  ρ  ��  (56c)  ≤ wg + µM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (56d)  APPENDIX J  PROOF OF PROPOSITION 5  The fact that 1⊤¯cm < Rmin implies that (24) is infeasible  is trivial and, therefore, the rest of the proof focuses on the  case where 1⊤¯cm ≥ Rmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  As before, the KKT conditions are sufficient and necessary  in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Noting that the Lagrangian is given by  L(¯zm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' λ, ν, µ) = 1  2∥¯rk+1  m  − ¯zm + ¯uk  m∥2  F  + λ(1⊤¯zm − Rmin) − ν⊤¯zm + µ⊤(¯zm − ¯cm)  (57)  yields the KKT conditions  ∇¯zmL(¯zm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' λ, ν, µ) =  − (¯rk+1  m  − ¯zm + ¯uk  m) + λ1 − ν + µ = 0,  (58a)  1⊤¯zm = Rmin,  (58b)  ¯zm ≥ 0, ν ≥ 0, ν[g]¯zm[g] = 0 ∀g,  (58c)  ¯zm ≤ ¯cm, µ ≥ 0, µ[g](¯zm[g] − ¯cm[g]) = 0 ∀g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (58d)  From (58a) and the second inequality in (58d), it follows that  µ = ¯rk+1  m  − ¯zm + ¯uk  m − λ1 + ν ≥ 0,  (59)  which in turn implies that  ¯zm ≤ ¯rk+1  m  + ¯uk  m − λ1 + ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (60)  Combining this expression with the first inequality in (58d)  yields  ¯zm ≤ min(¯cm, ¯rk+1  m  + ¯uk  m − λ1 + ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (61)  To show that this expression holds with equality, substitute  (59) into the equality of (58d) to obtain  (¯rk+1  m [g] − ¯zm[g] + ¯uk  m[g] − λ + ν[g])(¯zm[g] − ¯cm[g]) = 0,  (62)  which implies that either ¯zm[g] = ¯rk+1  m [g] + ¯uk  m[g] − λ + ν[g]  or ¯zm[g] = ¯cm[g].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Therefore,  ¯zm = min(¯cm, ¯rk+1  m  + ¯uk  m − λ1 + ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (63)  To obtain an expression for ¯zm that does not depend on ν,  one may consider three cases for each g:  C1: ¯rk+1  m [g] + ¯uk  m[g] − λ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' In this case, if ν[g] = 0,  expression (63) would imply that ¯zm[g] < 0, which  would violate the first inequality in (58c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Therefore,  ν[g] > 0 and, due to the equality in (58c), ¯zm[g] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  If ¯cm[g] > 0, it is then clear from (63) that ν[g] =  −(¯rk+1  m [g]+ ¯uk  m[g]−λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' If ¯cm[g] = 0, then greater values  of ν[g] will also satisfy the KKT conditions but this is  not relevant since the only feasible ¯zm[g] in case C1 is  ¯zm[g] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  C2: ¯rk+1  m [g] + ¯uk  m[g] − λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' In this case, (63) becomes  ¯zm[g] = min(¯cm[g], ν[g]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Due to the equality in (58c),  it then follows that either ¯cm[g] = 0 and ν[g] ≥ 0, or  ¯zm[g] = ν[g] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  C3: ¯rk+1  m [g] + ¯uk  m[g] − λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' If ¯cm[g] = 0, then  necessarily ¯zm[g] = 0 and any ν[g] ≥ 0 satisfies the  KKT conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' On the other hand, if ¯cm[g] > 0, then  it is clear that ¯zm[g] > 0 and, due to the equality in  (58c), one has that ν[g] = 0, which in turn implies that  ¯zm[g] = min(¯cm[g], ¯rk+1  m [g] + ¯uk  m[g] − λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Combining C1-C3 yields  ¯zm[g] = max(0, min(¯cm[g], ¯rk+1  m [g] + ¯uk  m[g] − λ)),  (64)  which is just the scalar version of (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Finally, to obtain λ,  one may substitute (64) into (58b), which produces (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  15  10  20  30  40  50  60  70  80  90  Minimum flight height [m]  10  15  20  25  30  35  40  45  50  55  Mean number of ABSs  Lower bound  K-means Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Galkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Space rate Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Hammouti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Genetic Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Shehzad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  GSPA (proposed)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 12: Mean number of ABSs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' minimum flight height  (Rmin = 17 Mbps, cBH = 84 Mbps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  0  10  20  30  40  50  Height of the buildings [m]  12  14  16  18  20  22  24  26  28  30  32  Mean number of ABSs  Lower bound  K-means Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Galkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Space rate Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Hammouti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  Genetic Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' (Shehzad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=')  GSPA (proposed)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 13: Mean number of ABSs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' height of the buildings  (Rmin = 17 Mbps, cBH = 84 Mbps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  5  7  9  11  13  15  17  Minimum GT rate [Mbps]  0  5  10  15  20  25  30  35  40  45  Mean number of ABSs  cBH = 40 Mbps  Lower bound for cBH = 40 Mbps  cBH = 60 Mbps  Lower bound for cBH = 60 Mbps  cBH = 80 Mbps  Lower bound for cBH = 80 Mbps  cBH = 100 Mbps  Lower bound for cBH = 100 Mbps  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 11: Mean number of ABSs vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Rmin of the proposed GSPA  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  APPENDIX K  PROOF OF PROPOSITION 6  Denote by G(λ) the left-hand side of (26), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=',  G(λ) :=  �  g  max(0, min(¯cm[g], ¯rk+1  m [g] + ¯uk  m[g] − λ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  (65)  This is a sum of non-increasing piecewise continuous func-  tions and therefore G is also non-increasing piecewise contin-  uous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The maximum value is attained for sufficiently small  λ and equals �  g ¯cm[g] = 1⊤¯cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' If 1⊤¯cm < Rmin, then  G(λ) < Rmin ∀λ and (26) admits no solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Conversely,  if 1⊤¯cm > Rmin, then a solution can be found since G(λ) >  Rmin for sufficiently small λ and G(λ) = 0 for sufficiently  large λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Uniqueness follows from the fact that G is strictly  decreasing except when G(λ) = 0 or G(λ) = 1⊤¯cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To show that G(ˇλ  k  m) ≥ Rmin just note from (27a) that  ˇλ  k  m ≤ ¯rk+1  m [g] + ¯uk  m[g] − ¯cm[g] or, equivalently, ¯cm[g] ≤  ¯rk+1  m [g] + ¯uk  m[g] − ˇλ  k  m, for all g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' This clearly yields G(ˇλ  k  m) =  �  g max(0, ¯cm[g]) = �  g ¯cm[g], which is greater than or equal  to Rmin by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To show that G(ˆλ  k  m) ≤ Rmin, note from (27b) that ˆλ  k  m ≥  ¯rk+1  m [g]+¯uk  m[g]−Rmin/G for all g such that ¯cm[g] > Rmin/G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  This clearly implies that ¯rk+1  m [g] + ¯uk  m[g] − ˆλ  k  m ≤ Rmin/G  for all g such that ¯cm[g] > Rmin/G and, as a consequence,  min(¯cm[g], ¯rk+1  m [g] + ¯uk  m[g] − ˆλ  k  m)) ≤ Rmin/G and the in-  equality G(ˆλ  k  m) ≤ Rmin follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  APPENDIX L  ADDITIONAL EXPERIMENTS WITH THE TOMOGRAPHIC  MODEL  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 11 is the counterpart of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 9 for tomographic chan-  nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The purpose of this simulation is to confirm that the  approximate linearity and proximity to the bound of GSPA  observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 4 take place for a wide range of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 12 investigates the influence of the minimum flight  height on the performance of the considered algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The  staircase behavior of the benchmarks can be explained by  noting that the flight grid initially comprises points with  heights 0, 30 m, 60 m, 90 m, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Then, the points inside  buildings and the points below the minimum flight height are  removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' Thus, the allowed flight points are the same e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  when the minimum flight height is 10 m as when it is 20 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  As already observed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' VI, the performance of GSPA  is not degraded for increasing flight height because the back-  haul capacity poses a more stringent constraint than the one  imposed by C even for the maximum flight height considered  in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' 13 studies the impact of the height of the buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content='  To reduce the spatial quantization effect of the tomographic  integral approximation, the numbers of SLF grid points in the  x, y, and z axes were set to 50, 40, and 150, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' It is  observed that the K-means and space rate K-means algorithms  16  are negatively affected by the increased attenuation introduced  by higher buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' In contrast, the genetic placement algo-  rithm and GSPA exhibit a milder dependence on the building  height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
+page_content=' The reason is the greater capacity of these algorithms  to select ABS locations with favorable propagation conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfOgzR/content/2301.04966v1.pdf'}
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+Strategy evolution on dynamic networks
+Qi Su1,2,3, Alex McAvoy1,2, and Joshua B. Plotkin1,2,3
+1Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA
+2Center for Mathematical Biology, University of Pennsylvania, Philadelphia, PA 19104, USA
+3Department of Biology, University of Pennsylvania, Philadelphia, PA 19104, USA
+Abstract
+Models of strategy evolution on static networks help us understanding how population struc-
+ture can promote the spread of traits like cooperation. One key mechanism is the formation
+of altruistic spatial clusters, where neighbors of a cooperative individual are likely to recip-
+rocate, which protects prosocial traits from exploitation. But most real-world interactions
+are ephemeral and subject to exogenous restructuring, resulting in dynamic social networks.
+Strategic behavior on dynamic networks is difficult to study, and much less is known about
+the resulting evolutionary dynamics. Here, we provide an analytical treatment of cooperation
+on dynamic networks, allowing for arbitrary spatial and temporal heterogeneity. We show
+that transitions among network structures can favor the spread of cooperation, even if each
+individual social network would inhibit cooperation when static. Furthermore, we show that
+spatial heterogeneity tends to inhibit cooperation while temporal heterogeneity tends to pro-
+mote it. Dynamic networks can therefore have profound effects on the evolution of prosocial
+traits, even in cases where individuals have no agency over network structure.
+1
+Introduction
+The geographic locations of individuals, together with their social or physical connections, con-
+strain interactions and shape behavioral evolution in a population. A network is a useful model
+of a population’s structure, where nodes represent individuals and edges capture interactions.
+How network structure affects evolutionary dynamics has been extensively investigated over the
+last several decades, using techniques including computer simulations, mathematical analysis,
+and experimental studies with human subjects. A well-known and illustrative finding1 is that
+population structure can favor cooperation provided the ratio of the benefit from cooperative be-
+havior, b, to its cost, c, exceeds the average number of neighbors, d. The mechanism underlying
+this cooperation-promoting effect is that spatial structure enables the formation of cooperative
+clusters of individuals, who have high payoffs and are capable of resisting invasion by defectors.
+Most existing studies are based on a static network, where the duration and intensity of inter-
+actions remain unchanged throughout the evolutionary process. In contrast, empirical networks
+1
+arXiv:2301.11982v1  [physics.soc-ph]  27 Jan 2023
+
+frequently vary over time2. Representative examples include communication networks involv-
+ing telephone calls or emails3,4; networks of physical proximity, where individuals encounter
+different people as they move through space5,6; and ecological networks that change with the
+seasons as organisms go through different phases of their life cycles7–9. Temporal features can
+even reverse the evolutionary outcomes. For example, whether an idea or information diffuses
+throughout a society depends not only on the structure of the network guiding interactions but
+also on the timing of those interactions, as the the coexistence of individuals with different active
+timing maximizes diffusion10. In the context of epidemics, high concurrency (the number of
+neighbors of a node) leads to a lower epidemic threshold under susceptible-infected-susceptible
+dynamics, while low concurrency can suppress epidemics11.
+Despite the attention that other dynamical processes have received on time-varying networks,
+the evolution of cooperation in this setting remains much less studied. One reason to discount
+any positive effect of dynamic structures comes from intuition on static networks: since coop-
+erators spread via clusters, network transitions will tend break up these clusters, likely leading
+to diminished reciprocity and exploitation by defectors. Another impediment to undertaking
+research in this area is the lack of mathematical tools for analyzing strategic interactions on dy-
+namic networks. In static networks, mathematical approaches provide general conditions for how
+structure affects evolutionary dynamics12,13. They also allow for extensive, efficient numerical
+explorations into example networks, both artificial and empirical14. Whether these approaches
+can be extended to dynamic networks remains unknown.
+Endogenous network transitions often produce predictable results for the evolution of coopera-
+tion. For example, if cooperators can selectively seek out new connections with other cooperators
+(“cooperation begets friends”) and sever ties with defectors, then it is not surprising to find that
+these endogenous networks changes favor the spread cooperation. But it is much less clear how
+exogenous transitions in network structure will affect the evolution of cooperation, and so this is
+the main focus of our study. There is also substantial evidence for the prevalence of exogenous
+network transitions in nature, ranging from weather fluctuations to human-induced changes to
+ecosystems15. The scope of models with dynamic networks is broad and can include environ-
+mental feedback and ecosystem engineering16. And even when an organism has some agency
+over the structure of their environment, the behavioral trait of interest might be unrelated to these
+changes (e.g. movement between cities need not be tied to altruistic tendencies). Finally, exoge-
+nous network transitions that are not dependent on individual behavior provide the most natural
+point of comparison to static structures.
+In this paper, we study the evolution of strategic behavior in a population whose structure of
+social interactions change over time. At any point in time, the population structure is described by
+a network whose nodes represent individuals and edges represent interactions. Individuals may
+change their strategies over time, imitating neighbors who have higher payoffs; and the network
+of interactions itself may also change over time. The interaction network changes at random
+times, unrelated to the current composition of strategies in the population. We derive general
+mathematical results for when cooperative behavior is favored, which apply to any stochastic
+transition pattern among any number of networks, each with arbitrary structure. Surprisingly, we
+2
+
+find that in a large class of networks, stochastic transitions among networks can strongly promote
+cooperation, even though they tend to disrupt cooperative clusters in each network. In fact, even
+if each individual static network would disfavor cooperation, transitions among them can rescue
+cooperation. We conclude by analyzing spatial and temporal burstiness, which we show have
+opposite effects on the evolution of cooperation.
+2
+Model
+Our model consists of a finite population of size N, with individuals engaged in pairwise social
+interactions. The structure of the population varies over time, and at each discrete time it is
+represented by one of L weighted networks, each with N nodes. For network β ∈ {1, . . . , L},
+we let w[β]
+ij denote the weight of the edge between nodes i and j. We assume that all networks
+are undirected, meaning w[β]
+ij = w[β]
+ji for all i, j ∈ {1, . . . , N} and β ∈ {1, . . . , L}.
+Each individual in the population can adopt one of two types, or strategies: “cooperator” (C) or
+“defector” (D). Individuals interact in pairwise donation games, with cooperators paying a cost c
+to generate benefit b for their co-player. Defectors pay no costs and generate no benefits. In each
+time step, everyone plays a donation game with each of their neighbors in the current network,
+β. We denote the state of the population by x, where xi ∈ {0, 1} indicates the type of individual
+i, with 0 and 1 representing types D and C, respectively. The accumulated payoff to individual i
+in network β is then
+ui (x, β) =
+N
+∑
+j=1
+w[β]
+ij
+�−cxi + bxj
+�
+.
+(1)
+In other words, individual i receives a benefit w[β]
+ij b from of each of its neighbors j who are
+cooperators (xj = 1), and i pays a cost w[β]
+ij c to each j if i is itself a cooperator (xi = 1). An
+individual’s accumulated payoff in network β is transformed into fecundity, which represents i’s
+propensity to reproduce or, equivalently, to be imitated by another individual. The fecundity is
+given by Fi (x, β) = 1 + δui (x, β), where δ is called the selection intensity, which we assume
+to be small (δ ≪ 1). This assumption, called “weak selection,” is common in the literature and
+it aims to capture scenarios in which the social trait (C or D) has a small effect on reproductive
+success.
+After all pairwise games are played in network β and individuals accumulate payoffs, a random
+individual i is selected uniformly from the population to update his or her strategy. This individ-
+ual then imitates the type of a neighbor, j, with probability proportional to j’s fecundity. In other
+words, in network β, the probability that i copies j’s type is
+eji (x, β) = 1
+N
+Fj (x, β) w[β]
+ji
+∑N
+k=1 Fk (x, β) w[β]
+ki
+.
+(2)
+3
+
+a
+Network 1
+Network 2
+b
+c
+Strategy updating
+d
+Network 1
+Network 2
+Network updating
+e
+C
+C
+C
+C
+D
+D
+D
+C
+C
+?
+C
+D
+D
+D
+C
+C
+D
+C
+D
+D
+D
+1
+2
+6
+4
+7
+5
+3
+1
+2
+6
+4
+7
+5
+3
+1
+2
+6
+4
+7
+5
+3
+-2c
+2b
+2b
+b-2c
+2b-4c
+2b
+b-2c
+b-2c
+b-2c
+2b
+0
+-2c
+b
+b
+1
+2
+6
+4
+7
+5
+3
+q11
+q12
+q21
+q22
+Interaction at time n+1
+Interaction at time n
+Figure 1: Evolutionary games on dynamic networks. a, The population structure at any time is described by a
+network, which may change from one time point to the next. (The figure illustrates an example with two possible
+networks.) b, Each individual (node) in the population adopts the strategy cooperate (C) or defect (D) in games
+played with each neighbor. Each individual i accumulates a total payoff ui across pairwise interactions with neigh-
+bors, which determine their reproductive rate Fi = 1 + δui. c, An individual (marked by “?”) is selected uniformly
+at random to update its strategy, and all neighboring individuals, indicated by black circles, compete to be imitated
+by the focal node, with probability proportional to reproductive rates. d, After an individual updates its strategy,
+the population structure itself either changes (from network 1 to network 2 with probability q12, or from network
+2 to network 1 with probability q21) or remains the same. e, Social interactions and strategy updates repeat on the
+population structure at the next time step, n + 1.
+Here, the factor of 1/N represents the probability that i is chosen to update in the first place.
+After each strategic update, the population structure itself then undergoes a transition step. The
+probability of moving from network β to network γ is independent of the strategic composition
+of the population, and it depends only on the current network state, β. The stochastic process
+governing these transitions is described by an L × L matrix Q =
+�
+qβγ
+�
+, where qβγ is the prob-
+ability of transitioning from network β to network γ. Note that there may be (and we often
+assume) a positive chance that the network will remain unchanged at the transition stage, e.g.
+qββ > 0. The pairwise social interactions, strategic update, and network transition, which com-
+prise a single time step, are depicted in Fig. 1.
+4
+
+3
+Results
+Without mutation, the population must eventually reach a monomorphic strategic state in which
+all individuals have the same type, either cooperate or defect. The duration that the population
+spends in each network is proportional to the corresponding value in stationary distribution υ,
+which is determined by the network transition matrix Q (see Methods). We assume that a mutant
+appears in network β with probability υ (β), and it is located at a node chosen uniformly at
+random. We let ρC denote the probability that a single cooperator mutant eventually takes over
+a resident population of defectors. Likewise, we let ρD be the probability that a single defector
+mutant takes over a resident population of cooperators. We use the condition ρC > ρD to
+measure whether selection favors cooperation relative to defection17.
+3.1
+Selection condition for the evolution of cooperation
+We first derive a general result applicable to almost any transition pattern, Q, among any finite
+number of networks, each with arbitrary spatial structure. This result combines several different
+quantities describing the dynamics under neutral drift (δ = 0), together with the payoffs for the
+game13,18.
+Let p[β]
+ij
+:= w[β]
+ij / ∑N
+k=1 w[β]
+ik be the one-step random-walk probability of moving from i to j on
+network β. This quantity can be interpreted as the probability that i imitates the strategy of j
+under neutral drift, conditioned on i being chosen for an update. In other words, p can be seen
+as defining an ancestral process, tracking replacement backwards in time under neutral drift.
+The most fundamental neutral quantity is the reproductive value of individual i in network β,
+which can be interpreted as the probability that a mutant introduced at node i in network β
+generates a lineage that eventually takes over the population. This quantity, denoted by π[β]
+i
+is independent of the payoffs and thus independent of the particular mutant that arises in the
+population. The version of reproductive value that we use is a generalization of Fisher’s clas-
+sical notion19,20 that also takes into account environmental changes. It can be calculated using
+Equation 5 in Methods.
+Another neutral quantity we use is related to coalescence times. Under neutral drift, we can look
+backward in time and ask how long it takes, on average, before two or more lineages meet at a
+common ancestor. Starting in network β, let T[β] be the expected number of steps to the most
+recent common ancestor of the entire population. If τ[β]
+ij
+is the expected time to the most recent
+common ancestor of i and j, then the mean amount of time that i and j are identical by descent is
+T[β] − τ[β]
+ij . The pairwise times to a common ancestor, τ, can be calculated using Equation 8 in
+Methods.
+In terms of the neutral quantities π, τ, and T, the general condition for cooperation to be favored
+5
+
+over defection under weak selection is given by
+N
+∑
+i,j=1
+L
+∑
+β=1
+υ (β)
+�
+L
+∑
+γ=1
+qβγπ[γ]
+i
+�
+p[β]
+ij
+N
+∑
+ℓ=1
+�
+−
+�
+T[β]−τ[β]
+jj
+�
+w[β]
+jℓ c
++
+�
+T[β]−τ[β]
+jℓ
+�
+w[β]
+ℓj b
+�
+>
+N
+∑
+i,j,k=1
+L
+∑
+β=1
+υ (β)
+�
+L
+∑
+γ=1
+qβγπ[γ]
+i
+�
+p[β]
+ij p[β]
+ik
+N
+∑
+ℓ=1
+�
+−
+�
+T[β]−τ[β]
+jk
+�
+w[β]
+kℓ c
++
+�
+T[β]−τ[β]
+jℓ
+�
+w[β]
+ℓk b
+�
+.
+(3)
+Broadly speaking, what Equation 3 says is that an individual i is chosen, a cooperator is placed
+at a neighbor j of i, and another neighbor k of i is chosen to compare its (weighted) payoff with
+that of the cooperator. If j’s weighted payoff exceeds that of k, then selection favors the evolution
+of cooperation.
+The condition above reflects a similar intuition behind the corresponding condition for static
+networks (L = 1; see Allen et al.14 or Fig. 1 of McAvoy & Wakeley21), but there are a few
+notable effects of network transitions in Equation 3. The first effect is that the network β is chosen
+with probability υ (β), where υ is the stationary distribution of the structure-transition chain
+defined by Q. Moreover, whereas individual i is chosen with probability based on reproductive
+value πi on a static network, here i is chosen based on reproductive value in the next network
+following imitation, ∑L
+γ=1 qβγπ[γ]
+i
+. The reason for this is natural, because once an individual
+replaces i in network β, the network immediately transitions to network γ, and so the resulting
+reproductive value of i must be understood within the context of γ. Once β and i are chosen,
+the probabilities of choosing neighbors j and k are p[β]
+ij and p[β]
+ik , respectively. Moreover, if j is a
+cooperator, then individual k is also a cooperator for T[β] − τ[β]
+jk time steps, and during each such
+step k pays cw[β]
+kℓ to provide ℓ with a benefit of bw[β]
+kℓ . This property accounts for the weighting
+of benefits and costs in Equation 3. Note that the term T[β] cancels out in Equation 3, and
+so although this quantity is helpful for gathering intuition, it is not strictly needed to evaluate
+whether cooperators are favored by selection.
+Given the vast number of networks with N nodes, as well as the vast space of possible transitions
+among them, we focus most of our analysis on transitions between a pair of networks (i.e. L =
+2). For a given network transition matrix Q, the value 1/q12 (resp. 1/q21) gives the expected
+time during which the population remains in network 1 (resp. network 2) before transitioning
+to network 2 (resp. network 1). We denote 1/q12 and 1/q21 by t1N and t2N, respectively, so
+that t1 and t2 correspond to the expected number of times each individual updates prior to a
+transition to a different network. Small values of t1 and t2 correspond to frequent changes in the
+population structure. Sufficiently large values of t1 and t2 indicate that the population structure is
+nearly fixed, so that the population will reach an absorbing strategic state (all C or all D) before
+the network transitions to a different state. The regime t1 = 1 (resp. t2 = 1) means that, on
+average, each individual updates their strategy once in network 1 (resp. network 2) before the
+network structure changes.
+6
+
+a
+Network 1
+Network 2
+5
+6
+7
+8
+9
+10
+11
+Benefit, b
+-1.680
+-0.800
+-0.080
+-0.040
+-0.020
+0
+0.020
+0.040
+0.060
+Rescaled fixation probability, N(
+C-
+D)
+b
+//
+//
+αN
+(1−α)N
+Dynamic (α=0.5)
+Static network 1 or 2
+          (α=0.5)
+Dynamic (α=0.8)
+Static network 1 
+      (α=0.8)
+Static network 2 
+       (α=0.8)
+tN
+1
+tN
+1
+tN
+1
+1−
+tN
+1
+1−
+Figure 2: Transitions between networks that contain dense and sparse cliques. We consider dynamic transitions
+between two networks, each of which is comprised of two cliques containing aN and (1 − a) N nodes, respectively.
+a, Each network has a star graph comprising one clique and a complete graph comprising the other clique, with
+a single edge connecting the two cliques. When network 1 transitions to network 2, the star clique becomes the
+complete clique and vice versa. b, The fixation probability of cooperation versus defection, ρC − ρD, as a function
+of the benefit b in the donation game. Selection favors cooperation over defection if ρC − ρD exceeds the horizontal
+line, i.e., ρC > ρD. Dots indicate the results of Monte Carlo simulations on dynamic networks (solid dots) and
+on a static network (open dots). The vertical lines correspond to analytical predictions for the critical benefit-to-
+cost ratio (b/c)∗ on dynamic networks, above which we predict cooperation will be favored. The results show
+that cooperation is always disfavored in both static network 1 and static network 2, but dynamic transition between
+these networks can favor cooperation. Here, we show two examples with different clique sizes, a = 0.5 (blue) and
+a = 0.7 (green). The beneficial effect of structure transitions is strongest when cliques have equal size (a = 0.5;
+see Supplementary Figure 1). Parameter values: N = 40, t = 1, and c = 1.0. Fixation probabilities are computed
+across an ensemble of 107 runs with selection intensity δ = 0.002.
+3.2
+Dynamic networks with dense and sparse cliques
+We begin by studying dynamic transitions between a pair of networks where each network is
+comprised of two cliques. One clique is a star graph, which is sparse, and the other clique is
+a complete graph, which is dense. In each network, the two cliques are connected by a single
+edge. When the population transitions from one network to another, the star clique becomes the
+complete clique and vice versa (see Figure 2a). This kind of dynamic network models a situation
+in which a portion of the population is densely connected while the remainder of the population
+is connected to only a single node; and which portion is dense versus sparse changes over time,
+as the state transitions between the two networks.
+When the population evolves on either network 1 or network 2 alone, the fixation probability
+of cooperators is always lower than that of defectors, i.e. ρC < ρD,meaning that cooperation
+is disfavored by selection regardless of the benefit-to-cost ratio b/c (Figure 2b). Nonetheless,
+when the population transitions dynamically between networks 1 and 2, cooperation is favored
+7
+
+provided the benefit-to-cost ratio b/c exceeds the critical value (b/c)∗ ≈ 7. As a result, we
+see that dynamic population structures can favor cooperation, even when all networks involved
+would each individually suppress cooperation were they static.
+Dynamic population structure facilities cooperation across a wide range of population sizes for
+the pair of networks shown in Figure 2a. When t = 1, which means that individuals each
+update their strategy once, on average, before the network changes, cooperation can be favored
+by selection regardless of network size N (Figure 3a). By contrast, if the network is static,
+then cooperation is favored only when the population size is very small (N < 17)–and, even
+then, only if the benefit-to-cost ratio is large. For larger population sizes, N ⩾ 17, the critical
+benefit-to-cost ratio is negative on a static network, (b/c)∗ < 0, which means that selection
+actually favors the evolution of spite, a behavior in which individuals pay a cost c to decrease
+the fitness of their opponent by b. For this static network we can prove that (b/c)∗ ≈ −N/2 in
+large populations (see Methods), compared to (b/c)∗ ≈ 7 for any population size in a dynamic
+network. Consequently, we see that the effects of dynamic population structures are dramatic,
+capable of converting a spiteful outcome into a cooperative one, and they persist across a wide
+range of population sizes.
+Dynamic networks also facilitate cooperation across a wide range of structural transition rates.
+For a sufficiently large population size, N, on a single static network of the type shown in Fig-
+ure 2a, the critical benefit-to-cost ratio is negative ((b/c)∗ ≈ −N/2), which means that selec-
+tion favors the evolution of spite. By contrast, dynamic transitions between networks 1 and 2
+can favor cooperation, especially when they occur rapidly (Figure 3b). When the transition rate
+is very slow–in particular, when t exceeds
+�√
+2 + 1
+�
+N–the population stays in one network for
+so long that the evolutionary dynamics are similar to those of a static network, and the critical
+benefit-to-cost ratio becomes negative (Figure 3b). In the limit of the transition rate approaching
+zero (t → ∞), the “dynamic” network is actually static and our dynamic calculations agree with
+those of a static network.
+3.3
+How dynamic structures can facilitate the spread of cooperation
+To further understand how dynamic structures can favor cooperation more than their static coun-
+terparts, we inspect evolutionary trajectories on the dense-sparse graph of Figure 2a. When the
+network is static, the process is depicted in Figure 4a. Starting from a specific configuration
+of cooperators in both hubs and two leaf nodes, cooperation will initially tend to spread in the
+star clique while shrinking in the complete clique. After cooperation fixes within the star clique,
+selection strongly suppresses further spread to the complete clique because the node connected
+to the star clique is exploited by multiple defectors. If ever a defector manages to diffuse to the
+hub of the star clique, however, defection will then rapidly spread within the star and ultimately
+fix in the entire network.
+By contrast, if the population undergoes structural transitions between networks (e.g. n2 → n3
+in Figure 4b), the star clique of network 1 will transition into the complete clique of network
+2, which promotes the exploitation of cooperators and allows defectors to spread (n3 → n4).
+8
+
+101
+102
+103
+104
+Network size, N
+-106
+-104
+-102
+0
+102
+104
+106
+Critical benefit-to-cost ratio, (b/c)*
+a
+Dynamic, Eq. 3
+7.00
+Static network 1, Eq. 3
+-N/2
+5
+10
+15
+20
+25
+-103
+-102
+-101
+0
+101
+102
+103
+10-1
+100
+101
+102
+103
+104
+105
+106
+Rescaled duration, t
+-106
+-104
+-102
+0
+102
+104
+106
+Critical benefit-to-cost ratio, (b/c)*
+b
+Dynamic, Eq. 3
+Static network 1, Eq. 3
+-N/2
+104
+105
+106
+-107
+-106
+-105
+-104
+-103
+t3+10t2+26t+19
+t2+4t+3
+t2N+tN2
+−t2+2tN+N2
+Figure 3: Dynamic structures facilitate cooperation for a broad range of population sizes and network transi-
+tion rates. We consider transitions between the two networks shown in Figure 2a, each composed of a sparse clique
+and a dense clique. a, The critical benefit-to-cost ratio required to favor cooperation as a function of population size,
+N, for a = 0.5 and t = 1. Dynamic networks can favor cooperation for any population size, N, provided b/c > 7.
+In contrast, the corresponding static networks favor cooperation only in small populations (N < 17), and they favor
+the evolution of spite ((b/c)∗ < 0) in larger populations. Dots show exact analytical computations for finite N
+(Equation 3), and lines show analytical approximations for large N. b, The critical benefit-to-cost ratio as a func-
+tion of the mean duration between network transitions, t, for a = 0.5 and N = 10,000. Whereas a static network
+always disfavors cooperation, dynamic networks can favor cooperation provided they do not transition too slowly
+(t <
+�√
+2 + 1
+�
+N). Dots show exact analytical computations for arbitrary t; the blue line shows an analytical
+approximation in the regime t ≪ N; and the red line shows an analytical approximation in the regime t = O (N).
+9
+
+Meanwhile, the complete clique of network 1 transitions into the star clique of network 2, which
+stimulates the expansion of cooperators. The rate of cooperator expansion in one clique exceeds
+their exploitation in the other clique so that, overall, network transitions facilitate cooperation.
+3.4
+Other dynamic structures
+The examples of dynamic structure considered so far may seem highly specialized because the
+networks each contain two stylized cliques with a single edge between them. But we find similar
+results on networks with many cliques and with more complicated connections between them. In
+Figure 5a,b, we analyze networks comprised of multiple star and complete cliques, connected by
+either hub nodes or by leaf nodes. In both cases, we again find that dynamic transitions between
+networks reduce the critical benefit-to-cost ratio for the evolution of cooperation, compared to
+any single static network. This effect is increasingly strong as the network size grows (see
+Supplementary Figure 2). For the networks in Figure 5a with N = 1,200, for example, the
+critical benefit-to-cost ratio to favor cooperation is (b/c)∗ ≈ 188.1 when the network is static,
+which is reduced to (b/c)∗ ≈ 3.49 when the network is dynamic.
+In addition to networks comprised of star and complete cliques, we also investigated networks
+with cliques defined by various types of random graphs, such as Erd¨os-R´enyi and scale-free
+networks. In the former case, node degrees within a clique do not vary substantially, while the
+latter exhibits large variation in degree. For both classes of random networks, we still find that
+dynamic transitions between random networks tends to promote cooperation, compared to each
+static network (Figure 5c,d).
+In all examples of dynamic networks considered thus far, transitions between networks involve
+dense regions of a network swapping with sparse regions. Regardless of the exact structure of
+the cliques, this general feature of structural transitions conforms to the underlying intuition for
+why dynamic networks can facilitate cooperation (Figure 4). Dynamic structures can still facil-
+itate cooperation even when networks differ in only a small fraction of connections, although
+the strength of the effect is weakened. Furthermore, these effects also persist (and can be quite
+strong) when populations transition between three or more network structures. We give illustra-
+tions in Supplementary Figure 3.
+3.5
+The probability and time to fixation of cooperation
+We have studied dynamic structures by comparing the fixation probability of a cooperator to
+that of a defector, and by calculating the critical benefit-to-cost ratio (b/c)∗ that ensures ρC >
+ρD. We can also study the fixation probability ρC in absolute terms. We find that a dynamic
+population structure increases the fixation probability of cooperators, making them more likely
+to overtake the population, compared to a static network. Dynamic population structures also
+tends to decrease the duration before one type or another fixes (see Supplementary Figure 4),
+as well as shorten the mean conditional time until cooperators fix. The underlying intuition for
+these results is evident in Figure 4: on a static network, the population will tend to be stuck
+at stage n3 for a long time, before defectors eventually diffuse to the sparse clique; whereas
+10
+
+Evolution on static network
+Evolution on dynamic network
+Network 1
+Network 1
+Network 2
+Cooperator
+Defector
+n1
+n2
+n3
+n4
+n1
+n2
+n5
+n6
+n3
+n4
+n7
+n8
+Strategy 
+updating
+Network
+updating
+a
+b
+Figure 4: Intuition for how dynamic structures can facilitate cooperation. Starting from a configuration in
+which the hub and two leaf nodes are cooperators (time point n1 in a and b), we illustrate how cooperation can be
+favored in dynamic structures even when it is inhibited in each static structure. Initially, cooperators are expected
+to spread in the star clique and shrink in the complete clique, and the rate of spreading exceeds that of shrinking. a,
+The evolutionary process on a static network. Cooperators rapidly take over the star clique and nearly die out in the
+complete clique (n1 → n3). The system tends to stay in this state until defectors spread throughout the star clique
+(n4). b, The evolutionary process with network transitions. Initially, cooperators spread in the star clique and shrink
+in the complete clique (n1 → n2). However, when the network changes, the star clique transitions to the complete
+clique and vice versa (n2 → n3). This transition is followed by the rapid spread of cooperators in the star clique
+and (relatively slower) shrinking of cooperators in the complete clique (n3 → n4). From n1 to n5, the frequency of
+cooperators increases in both cliques so that, under dynamic structure transitions, the population tends to result in
+cooperators being fixed in both cliques (n8).
+11
+
+a
+Network 1, (b/c)*
+7.31
+Network 2, (b/c)*
+7.31
+Dynamic network, (b/c)*
+4.92
+Multiple cliques connected via hub nodes
+b
+Network 1, (b/c)*
+7.00
+Network 2, (b/c)*
+7.00
+Dynamic network, (b/c)*
+5.34
+Multiple cliques connected via leaf nodes
+c
+Network 1, (b/c)*
+39.85
+Network 2, (b/c)*
+40.80
+Dynamic network, (b/c)*
+32.86
+Multiple ER communities
+d
+Network 1, (b/c)*
+39.64
+Network 2, (b/c)*
+40.07
+Dynamic network, (b/c)*
+22.33
+Multiple GKK communities
+Figure 5: Evolution of cooperation on diverse dynamic structures. a, Each individual network comprises four
+star cliques and four complete cliques, where each star clique in one network corresponds to a complete clique
+in the other network, and clique hubs are fully connected to each other. b is similar to a, but cliques are now
+sparsely connected via leaf nodes. Network transitions facilitate cooperation compared to a static structure. c, Each
+individual network comprises two sparse and two dense cliques of Erd¨os-R´enyi (ER) random networks22, with
+cliques connected by random nodes. d, Each individual network comprises two sparse and two dense cliques of
+Goh-Kahng-Kim scale-free networks (GKK)23 with exponent 2.5, with cliques connected by nodes of the highest
+degree. In all these examples, network transitions reduce the benefit-to-cost ratio (b/c)∗ required for cooperation
+compared to each static network. Parameters: t = 1 and N = 64 for a and b. For panels c and d, in network 1,
+the two sparse cliques have 30 nodes and average degree 4, and the two dense cliques have 40 nodes and average
+degree 30; in network 2, the two sparse cliques have 40 nodes and average degree 4, and the two dense cliques have
+30 nodes and average degree 20.
+12
+
+on dynamic networks, cooperators spread rapidly by selection in both cliques. Thus, dynamic
+networks increase the likelihood that cooperators sweep the population as well as the rate at
+which they do so.
+3.6
+Spatial and temporal burstiness
+We can adapt our method of analysis to study the effects of spatial and temporal burstiness.
+For dynamically changing networks, spatial burstiness arises when there is temporal variation in
+the density of network edges (node degree), whereas temporal burstiness arises when there are
+periods of rapidly changing network structures along with periods in which structures change
+more slowly. Empirical networks of both human and non-human (e.g. honeybee) interactions
+are known to exhibit both spatial and temporal burstiness10,24, but the effects of these two forms
+of over-dispersion for behavior remains an active area of current research.
+To study spatial burstiness, we consider the following minimal model of dynamically varying
+networks that differ in their average node degree. We construct a pair of networks as follows
+(see Figure 6a): (i) we first generate a single network with N nodes and E edges drawn from one
+of several classical families of networks (e.g. Erdos-Reyni random networks22, Watts-Strogatz
+small-world networks25, Barab´asi-Albert scale free networks26, etc.); (ii) we decompose this
+network into two networks, by randomly selecting a fraction ε ∈ [0, 1/2] of the edges for net-
+work 1 and using the remaining (1 − ε) E edges for network 2. If ε = 1/2 then the resulting
+networks 1 and 2 have the same density of interactions, and there is no spatial burstiness. For all
+other values of ε ̸= 1/2, the network exhibits spatial burstiness, and we study a simple stochastic
+transition pattern between these networks, with t1 = t2 = 1 so that each individual updates his
+strategy once, on average, before the network switches.
+We find that spatial burstiness tends to inhibit the evolution of cooperation, whereas spatial reg-
+ularity (equal network densities) is more beneficial for cooperation (Figure 6c). In particular,
+regardless of the class of network from which networks 1 and 2 are derived, the critical ratio
+(b/c)∗ required to favor cooperation is substantially increased (roughly by a factor of two) in
+the regime ε → 0 compared to the spatially homogeneous regime ε = 1/2.
+We also study the effects of temporal burstiness, in which case networks 1 and 2 are chosen to
+have the same edge density (ε = 1/2), but there are periods of rapid transitions between the
+two networks, punctuated by periods of slow transitions. To construct this scenario, instead of
+having a single transition matrix, Q, we consider two such matrices, Q f and Qs, corresponding
+to fast and slow epochs. At any time, the population is either in hidden state f, so that network
+transitions occur according to Q f , or alternatively in hidden state s, so that network transitions
+occur according to Qs. Whenever the population transitions to a new network, the hidden state
+is drawn uniformly-at-random from { f, s} (see Figure 6b). (Note that the hidden state s or f is
+re-sampled only when the network changes, from 1 to 2 or from 2 to 1.)
+The speed of network transitions in each hidden state, s and f, is governed by a parameter
+¯t ∈ [0, 1], so that transitions are fast in state f and slow in state s. When the population enters
+13
+
+a
+b
+Hidden
+state s
+Hidden
+state f
+on transition to
+new structure
+Network 1
+(edge fraction, ε)
+Network 2
+(edge fraction, 1−ε)
+ε
+High spatial 
+ burstiness
+Low spatial 
+ burstiness
+High temporal
+   burstiness
+Low temporal
+  burstiness
+Figure 6: Effects of spatial and temporal burstiness on cooperation. We consider transitions between two
+networks, with either a, spatial burstiness (different edge densities) or b, temporal burstiness (periods of both rapid
+and slow transitions). c, The critical benefit-to-cost ratio (b/c)∗ as a function of spatial heterogeneity, ε. When
+the two networks have the same edge density, ε = 0.5, cooperation is most readily favored. When the networks
+that differ in their edge densities (ε ≪ 0.5), much larger values of b/c are required to support cooperation. d The
+critical benefit-to-cost ratio (b/c)∗ required to favor cooperation as a function of temporal heterogeneity, ¯t. The
+case ¯t = 1 means that networks transition at the same rate, regardless of the hidden state. When ¯t < 1, the networks
+transition more rapidly in state f than in state s, so that there is temporal burstiness. Results on spatial and temporal
+burtiness are shown for six classes of networks: random regular networks (RR), Erd¨os-R´enyi networks (ER)22,
+Watts-Strogatz small-world networks (SW)25 with rewiring probability 0.1, Barab´asi-Albert scale-free networks
+(BA)26, Goh-Kahng-Kim scale-free networks (GKK)23 with exponent 2.5, and Holme–Kim scale-free networks
+(HK)27 with triad formation probability 0.1. For each such class, we generate 2,000 networks, each with 100 nodes
+and average degree 20. We take ¯t = 1 in c and ε = 0.5 in d.
+14
+
+100
+30
+RR
+ER
+Critical benefit-to-cost ratio, (b/c)*
+sw
+80
+BA
+ratio,
+GKK
+HK
+60
+40 
+RR
+ER
+sw
+20 
+0008880888088809
+BA
+GKK ·
+HK
+10 
+0L
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Edge fraction, E
+Rescaled duration, tstate f, the expected duration before a network transition is small, namely ¯tN. Whereas when
+the population enters state s the expected duration of the current network is longer, (2 − ¯t) N
+(see Figure 6b). The case ¯t = 1 means that the current network has the same expected duration,
+regardless of the hidden state, and there is no temporal burstiness. When ¯t < 1, the networks
+transition more quickly in state f than they do in state s. Regardless of the value of ¯t, how-
+ever, the total accumulated time spent in network 1 is the same as in network 2, throughout the
+evolutionary process.
+Temporal burstiness tends to facilitate cooperation, regardless of the overall structure of underly-
+ing networks (Figure 6d). In particular, the critical benefit-to-cost required to favor cooperation
+is largest when temporal burstiness is absent (¯t = 1), and it is reduced (typically by 20%)
+when temporal burstiness is large (¯t = 0). Therefore, even when two networks have the same
+edge density (ε = 1/2) and the accumulated time is spent on each network is the same, tem-
+poral burstiness facilitates the spread of cooperation, in stark contrast to our findings for spatial
+burstiness.
+4
+Discussion
+Many real-world interactions are ephemeral, and the entire network of social interactions may
+be subject to exogenous changes. Seasonal changes in a species’ environment, for example, can
+lead to active and dormant periods, as can diurnal cycles. Such periodic transitions are widely
+used to model temporal networks28–31. Stochastic transitions in social structures can arise from
+the effects of weather, animal migration and movement, and role reversal32. Motivated by the
+ubiquity of structural variation in nature, we provide a treatment of dynamic social networks that
+allows for arbitrary stochastic transitions between structures, with arbitrary networks within each
+time step.
+Our main mathematical result (Equation 3) predicts when cooperation will evolve on dynamic
+networks, under weak selection. The population structure in every time step need not be con-
+nected; all that we require is that the population satisfy a coherence condition so that it does not
+become fragmented into multiple sub-populations (see §SI.1.1 in Supplementary Information).
+In addition to probabilistic transitions, our analysis also extends to deterministic and periodic net-
+work transitions (see Equation SI.33 in Supplementary Information). Our work can also cover
+other scenarios for changing structures, such as when the direction of public goods or informa-
+tion flow changes over time33; the number of active nodes or edges varies; or the population
+size fluctuates (in fact, the results in Supplementary Information allow for arbitrary patterns of
+replacement). Although prosocial behaviors in different strategic domains may manifest in dif-
+ferent ways, such as trust games or dictator games, the desire to pay costs to benefit others has a
+substantial degree of domain generality34. Our conclusions, based on donation games, are thus
+indicative of how dynamic networks may broadly impact prosocial behavior.
+In the donation game, we have seen that changing social structures can promote cooperation,
+and that these effects can be dramatic. Even if every network individually disfavors cooperators,
+15
+
+transitions between them can facilitate the evolution of cooperation – a result that is reminiscent
+of Parrondo’s paradox35. Figure 4 illustrates the mechanism for how this phenomenon arises,
+as transitions move individuals between regions of the network that are dense to those that are
+sparse. These types of changing social structures are common in real-world settings. Groups and
+communities are more likely to form among people with close geographical locations and similar
+religion, culture, and affiliations36,37; but connection density will be altered when individuals
+migrate or change social groups. Changes in connection densities in different communities may
+alternatively result from a phase difference, e.g. in online social networks across different time
+zones. Spatio-temporal heterogeneity of interaction density within a community also leads to
+time-varying connection densities, from sparse to dense and vice versa2. We find that each kind
+of burstiness has a clear effect on cooperation, either hindering it in the case of spatial burstiness
+or promoting it in the case of temporal burstiness. Broadly speaking, our work highlights the
+significance of integrating multiple communities into one system, since treating communities
+individually and independently may lead to erroneous conclusions about behavioral dynamics38.
+All of our results are based on exogenous network transitions, which means that individuals can-
+not selectively engineer their neighborhoods based on the traits of others. There are, of course,
+many interesting models involving endogenous transitions, in which cooperators can selectively
+form links with other cooperators and break links with defectors. In such models cooperation
+can flourish when structure transitions are rapid enough39–51, for the simple reason that this en-
+dogenous dynamic establishes cooperative clusters. Such “form follows function” models are
+frequently aimed at answering the question: what kinds of networks arise from certain traits, and
+how do these networks serve the greater good? By contrast, our focus is not the coevolutionary
+dynamics of trait and structure, but on a different question altogether: what is the impact of ex-
+ogenous structural changes for the evolution of behavior. This approach is more closely related
+to classical studies of network effects on cooperation: given a (dynamic) network, what behav-
+ioral traits evolve? Since exogenous structural changes do not provide any explicit advantage or
+disadvantage to cooperators relative to defectors, the resulting evolutionary dynamics of social
+traits are all the more intriguing.
+We have aimed for generality in framing our mathematical results, but a natural limitation of
+our study is the scope of networks we have analysed, compared to the vast space of possible
+population structures and transitions among them. For this reason, even static structures are still
+an active topic of current research in evolutionary game theory. We have therefore chosen to
+consider a limited number of representative examples of dynamic networks, which showcase the
+interesting effects they can have on the evolution of cooperation. Areas for future investigation
+include the effects of fluctuating resources on cooperation, alternative evolutionary update rules,
+stronger selection, and environments that involve both endogenous and exogenous transitions. In
+fact, although we use cooperation as an example, our analysis is framed quite generally to allow
+the study of other traits on dynamic structures. To the best of our knowledge, our analytical
+findings constitute the first general results for behavioral evolution on dynamic networks, and we
+hope that they will be valuable tools in future work.
+16
+
+5
+Methods
+5.1
+Analysis of weak selection
+Here, we outline a derivation of the critical benefit-to-cost ratio (b/c)∗ for selection to favor co-
+operation, based on an extension of the methods of McAvoy & Allen18. Complete mathematical
+details are provided in Supplementary Information.
+For i, j ∈ N , let w[β]
+ij be the weight of edge between nodes i and j in network β ∈ {1, . . . , L}.
+We assume that the network is undirected, meaning w[β]
+ij
+= w[β]
+ji
+for all i, j ∈ {1, . . . , N} and
+β ∈ {1, . . . , L}. If i and j share an edge, then they interact. The class of models we are interested
+in here involve social goods52 in which, on network β, an individual of type A at i pays a cost of
+C[β]
+ij to donate B[β]
+ij to the individual at j. In state (x, β), the total payoff to the individual at i is
+ui (x, β) =
+N
+∑
+j=1
+�
+−xiC[β]
+ij + xjB[β]
+ji
+�
+.
+(4)
+This net payoff is converted to reproductive rate via the formula Fk (x, β) = eδuk(x,β). If the pop-
+ulation structure is β, then a node in β is first selected uniformly-at-random to die. Subsequently,
+all neighboring nodes in β compete to produce an offspring to fill the vacancy at node i. The
+probability that j replaces i in state (x, β) is given by Equation 2.
+Let p[β]
+ij := w[β]
+ij / ∑N
+k=1 w[β]
+ik be the probability of moving from i to j in one step of a random walk
+on network β. Under neutral drift, the probability π[β]
+i
+that, starting in network β, i generates a
+lineage that takes over the population (i.e. the reproductive value of i in β) satisfies
+π[β]
+i
+= 1
+N
+N
+∑
+j=1
+p[β]
+ji
+L
+∑
+γ=1
+qβγπ[γ]
+j
++
+�
+1 − 1
+N
+N
+∑
+j=1
+p[β]
+ij
+�
+L
+∑
+γ=1
+qβγπ[γ]
+i
+,
+(5)
+subject to the constraint ∑N
+i=1 π[β]
+i
+= 1. π is thus determined by a linear system of size O (LN).
+For the initial state, we choose the network according to the stationary distribution of the network-
+transition chain, and a mutant appears uniformly-at-random within that network. There are two
+mutant-appearance distributions overall, one for C arising after the all-D state (denoted µC) and
+one for D arising after the all-C state (denoted µD). Associated to each µ ∈ {µC, µD} is a
+quantity η[β]
+I
+(µ) related to the co-occurrence of a trait in β among the nodes in I ⊆ {1, . . . , N},
+which is defined formally in §SI.1.5 of Supplementary Information. For our purposes, we need
+17
+
+η[β]
+I
+(µ) only for I containing one or two nodes, in which case η[β]
+I
+(µC) = η[β]
+I
+(µD) and
+η[β]
+ij =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0
+i = j,
+1
+N υ (β) +
+L
+∑
+γ=1
+qγβ
+�
+1
+N
+N
+∑
+k=1
+p[γ]
+ik η[γ]
+kj + 1
+N
+N
+∑
+k=1
+p[γ]
+jk η[γ]
+ik +
+�
+1 − 2
+N
+�
+η[γ]
+ij
+�
+i ̸= j.
+(6)
+We refer the reader to Equation SI.32 in Supplementary Information for details.
+It turns out that a scaled version of η, namely τ[β]
+ij
+:= η[β]
+ij /υ (β), allows for a more intuitive
+interpretation of the selection condition. Consider the time-reversed structure transition chain
+defined by
+�qβγ := υ (γ)
+υ (β)qγβ.
+(7)
+Using this time-reversed chain in conjunction with Equation 6, we see that
+τ[β]
+ij
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0
+i = j,
+1
+N +
+L
+∑
+γ=1
+�qβγ
+�
+1
+N
+N
+∑
+k=1
+p[γ]
+ik τ[γ]
+kj + 1
+N
+N
+∑
+k=1
+p[γ]
+jk τ[γ]
+ik +
+�
+1 − 2
+N
+�
+τ[γ]
+ij
+�
+i ̸= j.
+(8)
+In the ancestral process, looking backward in time under neutral drift, Nτ[β]
+ij
+has the interpre-
+tation as the expected number of update steps until i and j coalesce. Equivalently, since one of
+N individuals is updated in each time step, τ[β]
+ij
+can be seen as the mean number of generations
+needed for i and j to coalesce. If, conditioned on the population being in state β, T[β] is the mean
+time to reach the most recent common ancestor going backward in time, then the mean time that
+i and j spend identical by descent is T[β] − τ[β]
+ij . Finding τ for all structures and pairs of individ-
+uals involves solving a linear system of size O
+�
+LN2�
+. (Although T aids in the interpretation of
+τ as determining identity by descent, it does not need to be calculated in order to understand the
+first-order effects of selection on fixation probability.)
+We now have all of the neutral quantities we need to state the selection condition. The final piece
+is the connection between the payoffs and the replacement probabilities under weak selection. A
+straightforward calculation gives eji (x, β) = 1
+N p[β]
+ij + δ ∑N
+k=1 cji
+k (β) xk + O
+�
+δ2�
+, where
+cji
+k (β) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+N p[β]
+ij
+�
+−
+N
+∑
+ℓ=1
+C[β]
+jℓ + B[β]
+jj + p[β]
+ij
+N
+∑
+ℓ=1
+C[β]
+jℓ −
+N
+∑
+ℓ=1
+p[β]
+iℓ B[β]
+jℓ
+�
+k = j,
+1
+N p[β]
+ij
+�
+B[β]
+kj + p[β]
+ik
+N
+∑
+ℓ=1
+C[β]
+kℓ −
+N
+∑
+ℓ=1
+p[β]
+iℓ B[β]
+kℓ
+�
+k ̸= j.
+(9)
+18
+
+Putting everything together using Equation SI.31 in Supplementary Information, we see that
+d
+dδ
+�����
+δ=0
+ρC = 1
+N
+N
+∑
+i,j=1
+L
+∑
+β=1
+υ (β)
+�
+L
+∑
+γ=1
+qβγπ[γ]
+i
+�
+p[β]
+ij
+N
+∑
+ℓ=1
+�
+−
+�
+T[β]−τ[β]
+jj
+�
+C[β]
+jℓ
++
+�
+T[β]−τ[β]
+jℓ
+�
+B[β]
+ℓj
+�
+− 1
+N
+N
+∑
+i,j,k=1
+L
+∑
+β=1
+υ (β)
+�
+L
+∑
+γ=1
+qβγπ[γ]
+i
+�
+p[β]
+ij p[β]
+ik
+N
+∑
+ℓ=1
+�
+−
+�
+T[β]−τ[β]
+jk
+�
+C[β]
+kℓ
++
+�
+T[β]−τ[β]
+jℓ
+�
+B[β]
+ℓk
+�
+.
+(10)
+Moreover, an analogous calculation for D gives d
+dδ
+���
+δ=0ρD = − d
+dδ
+���
+δ=0ρC, which means that the
+condition d
+dδ
+���
+δ=0ρC > 0 is equivalent to d
+dδ
+���
+δ=0ρC > d
+dδ
+���
+δ=0ρD.
+In the donation game, we have B[β]
+ij = w[β]
+ij b and C[β]
+ij = w[β]
+ij c, and Equation 10 gives
+d
+dδ
+�����
+δ=0
+ρC > d
+dδ
+�����
+δ=0
+ρD ⇐⇒ bµ2 − cν2 > bµ0 − cν0,
+(11)
+where
+µ0 = 1
+N
+N
+∑
+i,j=1
+L
+∑
+β=1
+υ (β)
+�
+L
+∑
+γ=1
+qβγπ[γ]
+i
+�
+p[β]
+ij
+N
+∑
+ℓ=1
+w[β]
+ℓj τ[β]
+jℓ ;
+(12a)
+ν0 = 1
+N
+N
+∑
+i,j=1
+L
+∑
+β=1
+υ (β)
+�
+L
+∑
+γ=1
+qβγπ[γ]
+i
+�
+p[β]
+ij
+N
+∑
+ℓ=1
+w[β]
+jℓ τ[β]
+jj ;
+(12b)
+µ2 = 1
+N
+N
+∑
+i,j,k=1
+L
+∑
+β=1
+υ (β)
+�
+L
+∑
+γ=1
+qβγπ[γ]
+i
+�
+p[β]
+ij p[β]
+ik
+N
+∑
+ℓ=1
+w[β]
+ℓk τ[β]
+jℓ ;
+(12c)
+ν2 = 1
+N
+N
+∑
+i,j,k=1
+L
+∑
+β=1
+υ (β)
+�
+L
+∑
+γ=1
+qβγπ[γ]
+i
+�
+p[β]
+ij p[β]
+ik
+N
+∑
+ℓ=1
+w[β]
+kℓ τ[β]
+jk .
+(12d)
+The critical benefit-to-cost ratio is therefore (b/c)∗ = (ν2 − ν0) / (µ2 − µ0).
+Note that, for simplicity, we have assumed that any node can be selected for death in a given
+network. In reality, this assumption might not hold because each individual network need not
+be connected, which can lead to isolated nodes. If an isolated node is chosen for death, then
+the individual at this node cannot be immediately replaced by the offspring of a neighbor. All
+of our calculations can be modified to allow for only non-isolated nodes to be chosen for death,
+although in practice we do not need to do so in any of our examples.
+19
+
+5.2
+Specific examples
+We study the transition between two networks, with transition probabilities given by
+qβγ =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1 − p
+β = 1, γ = 1;
+p
+β = 1, γ = 2;
+q
+β = 2, γ = 1;
+1 − q
+β = 2, γ = 2.
+(13)
+The expected durations in networks 1 and 2 are q/ (p + q) and p/ (p + q), respectively.
+We study evolution on dynamic two-clique networks. The two-clique network is made up of a
+star clique and a complete clique, with the hubs connected (see Figure 2a). Let n and m denote
+the numbers of nodes in the star and complete cliques, respectively, so that n + m = N. We
+denote by 1, . . . , n the nodes in the star clique and by n + 1, . . . n + m the nodes in the complete
+clique, where n is the hub of the star and n + m is the node of the complete clique connected
+to the hub of the star. The other network is obtained by swapping the star and complete cliques.
+The adjacency matrix for the first network satisfies w[1]
+ij = 1 only if (i) i = n and j < n, or i < n
+and j = n; (ii) i = n and j = n + m, or i = n + m and j = n; or (iii) i, j ⩾ n + 1 and i ̸= j.
+The adjacency for the second network satisfies w[2]
+ij = 1 only if (i) i = n + 1 and j > n + 1, or
+i > n + 1 and j = n + 1; (ii) i = n and j = n + m, or i = n + m and j = n; or (iii) i, j ⩽ n
+and i ̸= j.
+Using the results of the previous section, we can directly calculate π and τ and calculate the
+critical benefit-to-cost ratio. Here, we provide explicit mathematical results for representative
+cases. Assuming p = q = 1/ (tN) and letting a := n/ (n + m), we find that
+�b
+c
+�∗
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(2a2−2a+1)t3+(8a2−8a+7)t2+(8a2−8a+15)t+2a2−2a+10
+2a(1−a)(t2+4t+3)
+N → ∞,
+t3+10t2+26t+19
+t2+4t+3
+N → ∞, a = 1
+2,
+20a2−20a+33
+16a(1−a)
+N → ∞ , t = 1,
+¯t(¯t+1)
+−2¯t2+2¯t+1N
+N → ∞, t/N = ¯t, a = 1
+2,
+�
+�
+�
+210N16−520N15−1034N14+1770N13
++14028N12−93440N11+300848N10−330944N9
+−663040N8+2230528N7−1096448N6−4570112N5
++10000384N4−9265152N3+4259840N2−786432N
+�
+�
+�
+�
+�
+�
+30N16−79N15+225N14+1756N13
+−15088N12−13128N11+247296N10−365152N9
+−849344N8+2987392N7−1801984N6−5024768N5
++11302912N4−9949184N3+3940352N2−327680N−131072
+�
+�
+�
+a = 1
+2, t = 1.
+(14)
+20
+
+In particular, when N → ∞ and a = 1/2, (b/c)∗ is a monotonically increasing function of t.
+We can compare this critical ratio to that of just a single network, which is the same for either
+network 1 or network 2 and satisfies
+�b
+c
+�∗
+=
+�
+�
+�
+�
+�
+− (1 − a) N
+N → ∞,
+−3N9−40N8−204N7−848N6−2464N5+1920N4+15872N3−40960N2+24576N
+6N8−64N7−520N6−1232N5+3872N4+6272N3−24320N2+22528N+4096
+a = 1
+2.
+(15)
+21
+
+0
+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
+1
+-106
+-104
+-102
+0
+102
+104
+106
+Supplementary Figure 1: The cooperation-promoting effects of structure transitions as the sizes of the two
+cliques vary. The dynamic network is illustrated in Figure 2a, with a fraction a (resp. 1 − a) of nodes in the top
+(resp. bottom) clique. The critical benefit-to-cost ratio, (b/c)∗, is shown as a function of a. The dots are the results
+of numerical calculations with N = 10,000 and the lines are analytical approximations for sufficiently large N. The
+rescaled duration is t = 1.
+22
+
+50
+100
+150
+200
+250
+300
+350
+400
+0
+10
+20
+30
+40
+50
+a
+Network 1
+Dynamic
+10-1
+100
+101
+102
+103
+104
+0
+5
+10
+15
+20
+25
+b
+50
+100
+150
+200
+250
+300
+350
+400
+0
+10
+20
+30
+40
+50
+60
+c
+10-1
+100
+101
+102
+103
+104
+0
+5
+10
+15
+20
+25
+30
+d
+Supplementary Figure 2: Cooperation-promoting effects of dynamic multi-clique networks. We consider
+networks made up of eight cliques connected via hub nodes (see Figure 5a; panels a and b here) and via leaf nodes
+(see Figure 5b; panels c and d here). a,c, The critical ratio (b/c)∗ as a function of population size N, for the rescaled
+duration t = 1. b,d, The critical ratio (b/c)∗ as a function of the rescaled duration t, for N = 200.
+23
+
+a
+b
+Supplementary Figure 3: Cooperation-promoting effects of structure transitions among more than two net-
+works, and when networks differ in a small fraction of connections. a, Structure transitions among three net-
+works. Every network transitions to another network with probability 1/ (2tN) and remains unchanged otherwise.
+b, Structure transitions between multi-clique networks in which the two networks differ in only two cliques. We
+take N = 150 in a and N = 64 in b, and the rescaled duration is t = 1.
+24
+
+10-1
+100
+101
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+a
+10-1
+100
+101
+0
+50
+100
+150
+200
+250
+300
+350
+400
+b
+Supplementary Figure 4: Dynamic networks promote and accelerate the fixation of cooperators. We consider
+the network with a star clique and a complete-graph clique with N = 16 and a = 0.5 (see Figure 2a). a, Fixation
+probability of cooperators as a function of the rescaled duration, t, in network 1 and in the dynamic network. The
+dynamic network leads to the larger fixation probability of cooperators than in network 1. b, Conditional and
+unconditional fixation times as functions of the rescaled duration, t. Both the conditional and unconditional times
+in the dynamic networks are smaller than in network 1.We take selection intensity δ = 0.1.
+25
+
+Supplementary Information
+SI.1
+Modeling evolution on dynamic networks
+SI.1.1
+Assumptions, definitions, and notation
+We consider a population of N individuals (labeled N = {1, 2, . . . , N}), residing at any point in
+time on one of L structures (labeled L = {1, 2, . . . , L}). Implicitly, this means that each of these
+L structures is a network on N nodes, although each network need not be connected and some
+nodes can be isolated. Each individual has type A or B, and the state of population is tracked by
+a pair (x, β) ∈ {0, 1}N × L, where xi = 1 means i has type A and xi = 0 means i has type B.
+At each time step, a set of individuals to be replaced, R ⊆ N , is chosen, together with an
+offspring-to-parent map, α : R → N . Let p(R,α) (x, β) denote the probability of replacement
+event (R, α) in state (x, β). Once (R, α) is chosen, the type configuration, x, is updated to y,
+where yi = xα(i) if i ∈ R and yi = xi if i ̸∈ R. This update can be specified more succinctly
+using an extended mapping �α : N → N defined by �α (j) = α (j) if j ∈ R and �α (j) = j if
+j ̸∈ R, which leads to the updated state x�α, where (x�α)i = x�α(i) for i ∈ N . The network, β, is
+updated via a transition matrix, Q =
+�
+qβγ
+�
+β,γ∈L, where qβγ is the probability of transitioning
+from network β to network γ. An important feature of the model is that network transitions are
+independent of x; thus, the population structure is exogenous and not influenced by traits. We
+assume that Q is irreducible, which guarantees that it has a unique stationary distribution, υ.
+We assume that for each replacement event, (R, α), type configuration, x, and network, β, the
+probability p(R,α) (x, β) is a smooth function of a selection intensity parameter, δ ⩾ 0, in a small
+neighborhood of δ = 0. Moreover, when δ = 0 (“neutral drift”), we assume that p(R,α) (x, β) is
+independent of x (but it can depend on β). We denote by p◦
+(R,α) (β) the probability of choosing
+(R, α) under neutral drift. The chain defined by Q does not depend on the selection intensity.
+We also make the following assumption, which ensures that for every starting configuration
+and network, there exists at least one individual whose lineage can take over the population:
+Fixation Axiom. For all network structures β0 ∈ L, there exists a location i ∈ N , an integer
+m ⩾ 1, and sequences of replacement events {(Rk, αk)}m
+k=1 and networks {βk}m−1
+k=1 for which
+(i) p(Rk,αk) (x, βk−1) > 0 for every k ∈ {1, . . . , m} and x ∈ {0, 1}N ;
+(ii) qβk−1βk > 0 for every k ∈ {1, . . . , m − 1};
+(iii) i ∈ Rk for some k ∈ {1, . . . , m};
+(iv) �α1 ◦ �α2 ◦ · · · ◦ �αm (j) = i for all locations j ∈ N .
+These conditions are similar to those used by Allen & McAvoy13 and McAvoy & Allen18,
+except here it is modified to account for dynamic networks. Informally, it guarantees that no
+individual lives forever and that the process eventually reaches a state in which all individuals
+are identical by descent. We note that here it does not require each network to be connected.
+26
+
+Since there is no mutation of traits, all individuals must have the same type when they
+are identical by descent. The configurations A := (1, 1, . . . , 1) and B := (0, 0, . . . , 0) are
+the only absorbing configurations. (Note that while the configuration of types cannot leave A
+or B, the state itself, which includes the network structure, can still change.) We denote by
+BN the set of all configurations, {0, 1}N , and by BN
+⊺ the set of all transient configurations,
+{0, 1}N − {A, B}. From the Fixation Axiom, we see that given any starting configuration-
+network pair, (x, β) ∈ BN × L, there is a well-defined probability, ρA (x, β) (resp. ρB (x, β)),
+that the population eventually reaches the monomorphic state A (resp. B). The behavior of these
+fixation probabilities (under weak selection, meaning δ ≪ 1) is the main focus of this study.
+We follow the workflow proposed by McAvoy & Allen18 for analyzing mutation-free evo-
+lutionary dynamics under weak selection. We first study the assortment of traits under neutral
+drift (δ = 0). Subsequently, we link these findings to the game using a martingale perturba-
+tion argument. We avoid reproducing the entire derivation in18; instead, we highlight the main
+modifications to those arguments necessary to accommodate stochastic network transitions.
+SI.1.2
+Network-mediated reproductive value
+With the main assumptions in place, we now introduce some derived, demographic quantities
+that we will refer to throughout the analysis of the model. If the population is in state (x, β), then
+the marginal probability that i produces an offspring that replaces j in the next update is
+eij (x, β) :=
+∑
+(R,α)
+j∈R, α(j)=i
+p(R,α) (x, β) .
+(SI.1)
+The expected change in the abundance of A in state (x, β) can be expressed as
+∆ (x, β) := ∑
+i∈N
+xi ∑
+j∈N
+eij (x, β) + ∑
+i∈N
+xi
+�
+1 − ∑
+j∈N
+eji (x, β)
+�
+− ∑
+i∈N
+xi
+= ∑
+i,j∈N
+eji (x, β)
+�
+xj − xi
+�
+.
+(SI.2)
+One inconvenient aspect of dealing with the true abundance of A is that it is generally not
+a martingale under neutral drift. This property is well-known even in models without dynamic
+structure13 and it necessitates working with a weighted frequency instead. The notion of repro-
+ductive value, which can be (informally) interpreted as the expected contribution of an individual
+to future generations, turns out to give the proper weighting. For our purposes, we interpret the
+reproductive value of i ∈ N as the probability that, under neutral drift, i generates a lineage
+that eventually takes over the population. Because our interest is in fixation probabilities in the
+first place, it is not surprising that such a quantity should appear. This quantity depends on the
+network structure, but it is independent of the type configuration due to the drift assumption.
+Formally, we define the reproductive value of i in network β, denoted π[β]
+i , to be the proba-
+bility that under neutral drift and starting in structure β, a mutant in node i eventually takes over
+the whole population. Let e◦
+ij (β) denote the probability, that under neutral drift and in structure
+27
+
+β, individual i spreads her strategy to j. A one-step analysis of the neutral Markov chain gives
+π[β]
+i
+= ∑
+j∈N
+e◦
+ij (β) ∑
+γ∈L
+qβγπ[γ]
+j
++
+�
+1 − ∑
+j∈N
+e◦
+ji (β)
+�
+∑
+γ∈L
+qβγπ[γ]
+i
+;
+(SI.3a)
+∑
+i∈N
+π[β]
+i
+= 1
+(SI.3b)
+for all i ∈ N and β ∈ L. There is one point of subtlety in relation to reproductive value on
+static networks, which relates to the normalization condition ∑i∈N π[β]
+i
+= 1 for all β ∈ L.
+The Fixation Axiom guarantees that there is a unique π satisfying Equation SI.3a up to a
+scalar multiple. In this case, for any fixed C ∈ R, requiring ∑i∈N ∑β∈L π[β]
+i
+= C yields
+a unique solution to Equation SI.3a.
+Summing both sides of Equation SI.3a over i ∈ N
+yields ∑i∈N π[β]
+i
+= ∑γ∈L qβγ ∑i∈N π[γ]
+i
+. Since the chain Q is irreducible, it follows that
+∑i∈N π[β]
+i
+is independent of β ∈ L, and thus it must be equal to C/L. Therefore, asserting
+that ∑i∈N ∑β∈L π[β]
+i
+= L is equivalent to the requirement that ∑i∈N π[β]
+i
+= 1 for all β ∈ L.
+As a result, π, which we refer to as network-mediated reproductive value due to its dependence
+on network transitions, is uniquely defined by Equation SI.3.
+Finally, the change in ∑i∈N π[β]
+i xi, the π-weighted abundance of A, is
+�∆ (x, β) = ∑
+i∈N
+xi ∑
+j∈N
+eij (x, β) ∑
+γ∈L
+qβγπ[γ]
+j
++ ∑
+i∈N
+xi
+�
+1 − ∑
+j∈N
+eji (x, β)
+�
+∑
+γ∈L
+qβγπ[γ]
+i
+− ∑
+i∈N
+π[β]
+i xi
+= ∑
+i,j∈N
+eji (x, β) ∑
+γ∈L
+qβγπ[γ]
+i
+�
+xj − xi
+� + ∑
+i∈N
+xi
+�
+∑
+γ∈L
+qβγπ[γ]
+i
+− π[β]
+i
+�
+.
+(SI.4)
+It follows from Equation SI.3 that, under neutral drift, �∆◦ (x, β) = 0, for all x ∈ BN and
+β ∈ L. This property will play a key role in our subsequent weak-selection analysis of the
+process (Equation SI.13).
+SI.1.3
+A mutation-modified evolutionary process
+The process under consideration is mutation-free. However, following Ref.18, in order to get
+an idea of the assortment of types prior to hitting an absorbing configuration, it is convenient
+to introduce an artificial mutation that makes the chain ergodic and gives it a unique stationary
+distribution. The idea is to choose a state (z, λ) with z ∈ BN
+⊺ , and let mutations bring absorbing
+configurations into (z, λ) with some small probability u > 0. If P(x,β)→(y,γ) denotes the proba-
+bility of transitioning from (x, β) to (y, γ) in the original (mutation-free) chain over the course
+28
+
+of one time step, then the transition probabilities for the mutation-modified chain are given by
+P⟳(z,λ)
+(x,β)→(y,γ) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+u
+x ∈ {A, B} , (y, γ) = (z, λ) ,
+(1 − u) P(x,β)→(y,γ)
+x ∈ {A, B} , (y, γ) ̸= (z, λ) ,
+P(x,β)→(y,γ)
+x ̸∈ {A, B} .
+(SI.5)
+As a result of the Fixation Axiom, there is a unique stationary distribution, π⟳(z,λ), such that
+π◦
+⟳(z,λ) (x, β) = ∑
+γ∈L
+�
+π◦
+⟳(z,λ) (A, γ) P⟳(z,λ)
+(A,γ)→(x,β) + π◦
+⟳(z,λ) (B, γ) P⟳(z,λ)
+(B,γ)→(x,β)
+�
++ ∑
+y∈BN
+⊺
+∑
+γ∈L
+π◦
+⟳(z,λ) (y, γ) P⟳(z,λ)
+(y,γ)→(x,β)
+= ∑
+γ∈L
+π◦
+⟳(z,λ) (A, γ)
+�
+uδz,xδλ,β + (1 − u) δA,xqγβ
+�
++ ∑
+γ∈L
+π◦
+⟳(z,λ) (B, γ)
+�
+uδz,xδλ,β + (1 − u) δB,xqγβ
+�
++ ∑
+y∈BN
+⊺
+∑
+γ∈L
+π◦
+⟳(z,λ) (y, γ) P(y,γ)→(x,β)
+(SI.6)
+for all x ∈ B and β ∈ L.
+In one step after state (x, β), the expected change in the π-weighted abundance of A is
+�∆⟳(z,λ) (x, β) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−u
+�
+1 − ∑i∈N π[λ]
+i
+zi
+�
+x = A,
+u ∑i∈N π[λ]
+i
+zi
+x = B,
+�∆ (x, β)
+x ̸∈ {A, B} .
+(SI.7)
+Averaging this expected change over the stationary distribution of the modified chain gives
+0 = E⟳(z,λ)
+�
+�∆⟳(z,λ)
+�
+= E⟳(z,λ)
+�
+�∆
+�
+− u ∑
+β∈L
+π⟳(z,λ) (A, β)
+�
+1 − ∑
+i∈N
+π[λ]
+i
+zi
+�
++ u ∑
+β∈L
+π⟳(z,λ) (B, β) ∑
+i∈N
+π[λ]
+i
+zi.
+(SI.8)
+Owing to a result of Fudenberg & Imhof53, we know that, in the low-mutation limit,
+lim
+u→0 ∑
+β∈L
+π⟳(z,λ) (A, β) = ρA (z, λ) ;
+(SI.9a)
+lim
+u→0 ∑
+β∈L
+π⟳(z,λ) (B, β) = ρB (z, λ) .
+(SI.9b)
+29
+
+Therefore, taking the derivative of both sides of Equation SI.8 with respect to u at u = 0 gives
+ρA (z, λ) = ∑
+i∈N
+π[λ]
+i
+zi + d
+du
+�����
+u=0
+E⟳(z,λ)
+�
+�∆
+�
+.
+(SI.10)
+Let ⟨·⟩(z,λ) :=
+d
+du
+���
+u=0E⟳(z,λ) [·]. By the argument given in Ref.18 Corollary 1, we see that
+for any function ϕ : BN × L → R satisfying ϕ (A, β) = ϕ (B, β) = 0 for all β ∈ L,
+⟨ϕ⟩(z,λ) =
+∞
+∑
+t=0
+E
+�
+ϕ
+�
+xt, βt� |
+�
+x0, β0�
+= (z, λ)
+�
+,
+(SI.11)
+where the summation on the right-hand side converges absolutely. In particular, this equation
+holds for the expected change in the π-weighted abundance of A, ϕ = �∆. Since we also have
+d
+dδ
+�����
+δ=0
+eij (x, β) = ∑
+I⊆N
+cij
+I (β) xI
+(SI.12)
+for unique coefficients cij
+I (β), where xI := ∏i∈I xi, it follows that
+d
+dδ
+�����
+δ=0
+ρA (z, λ) = d
+dδ
+�����
+δ=0
+�
+�∆
+�
+(z,λ)
+=
+�
+d
+dδ
+�����
+δ=0
+�∆
+�◦
+(z,λ)
+=
+�
+d
+dδ
+�����
+δ=0
+∑
+i,j∈N
+eji (x, β) ∑
+γ∈L
+qβγπ[γ]
+i
+�
+xj − xi
+�
+�◦
+(z,λ)
+= ∑
+i,j∈N ∑
+I⊆N
+�
+cji
+I (β) ∑
+γ∈L
+qβγπ[γ]
+i
+�
+xI∪{j} − xI∪{i}
+��◦
+(z,λ)
+,
+(SI.13)
+where the interchange of the two limits is possible due to Equation SI.11 and the absolute con-
+vergence of its summation. The second line of Equation SI.13 is where we use the fact that
+�∆0 (x, β) = 0 for all x ∈ BN and β ∈ L, highlighting the importance of network-mediated
+reproductive value.
+As a result of these calculations, what remains in order to understand the first-order effects
+of selection on a mutant type’s fixation probability is an analysis of the neutral operator ⟨·⟩◦
+(z,λ).
+SI.1.4
+Analysis of neutral drift
+Throughout this section, we denote the stationary distribution of the structure-transition chain,
+Q, by υ. We also suppress either the configuration or the network when we marginalize. For ex-
+ample, we write π⟳(z,λ) (x) for ∑β∈L π⟳(z,λ) (x, β) and π⟳(z,λ) (β) for ∑x∈BN π⟳(z,λ) (x, β).
+In the limit of low mutation, we know π◦
+⟳(z,λ) (A) converges to ρ◦
+A (z, λ) and π◦
+⟳(z,λ) (B)
+converges to ρ◦
+B (z, λ). The following lemma is a slightly stronger version of this result:
+30
+
+Lemma 1. For all networks β ∈ L,
+lim
+u→0 π◦
+⟳(z,λ) (A, β) = ρ◦
+A (z, λ) υ (β) ;
+(SI.14a)
+lim
+u→0 π◦
+⟳(z,λ) (B, β) = ρ◦
+B (z, λ) υ (β) .
+(SI.14b)
+Proof. Letting x = A in Equation SI.6 and taking u → 0 gives
+lim
+u→0 π◦
+⟳(z,λ) (A, β) = ∑
+γ∈L
+�
+lim
+u→0 π◦
+⟳(z,λ) (A, γ)
+�
+qγβ.
+(SI.15)
+It follows that limu→0 π◦
+⟳(z,λ) (A, β) is proportional to υ (β), for all β ∈ L. The constant of
+proportionality must be ρ◦
+A (z, λ) due to the fact that limu→0 π◦
+⟳(z,λ) (A) = ρ◦
+A (z, λ). The
+result for limu→0 π◦
+⟳(z,λ) (B, β) follows from analogous reasoning and is omitted here.
+Remark 1. Neutral fixation probabilities, ρ◦
+A (z, λ) and ρ◦
+B (z, λ), can be calculated using re-
+productive values and the identities ρ◦
+A (z, λ) = ∑i∈N π[λ]
+i
+zi and ρ◦
+B (z, λ) = 1 − ∑i∈N π[λ]
+i
+zi.
+The following is an immediate consequence of Lemma 1:
+Corollary 1. limu→0 π◦
+⟳(z,λ) (β) = υ (β).
+The next lemma establishes a recurrence for d
+du
+���
+u=0π◦
+⟳(z,λ) (β):
+Lemma 2. For every β, we have
+d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (β) = δβ,λ − υ (β) + ∑
+γ∈L
+�
+d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (γ)
+�
+qγβ.
+(SI.16)
+Proof. Summing both sides of Equation SI.6 over all x ∈ BN gives
+π◦
+⟳(z,λ) (β) = u ∑
+γ∈L
+�
+π◦
+⟳(z,λ) (A, γ) + π◦
+⟳(z,λ) (B, γ)
+� �
+δβ,λ − qγβ
+�
++ ∑
+γ∈L
+π◦
+⟳(z,λ) (γ) qγβ.
+(SI.17)
+Differentiating this equation with respect to u at u = 0 and using Lemma 1 yields Equation SI.16.
+Since the state of the process consists of both a configuration of traits and a network structure,
+the next result gives a recurrence for calculating a modified version of ⟨·⟩◦
+(z,λ), using conditioning
+on the network structure. In particular, for a function ϕ : BN → R defined on just configura-
+tions, we let ⟨ϕ | β⟩◦
+(z,λ) =
+d
+du
+���
+u=0E◦
+⟳(z,λ) [ϕ | β]. This quantity can be calculated as follows:
+31
+
+Proposition 1. For every function ϕ : BN → R, we have
+υ (β) ⟨ϕ | β⟩◦
+(z,λ) = δλ,β (ϕ (z) − ρ◦
+A (z, λ) ϕ (A) − ρ◦
+B (z, λ) ϕ (B))
++ ∑
+γ∈L
+υ (γ) ∑
+(R,α)
+p◦
+(R,α) (γ) qγβ ⟨ϕ�α | γ⟩◦
+(z,λ) ,
+(SI.18)
+where, for �α : N → N , ϕ�α : BN → R is the map defined by ϕ�α (x) = ϕ (x�α) for x ∈ BN .
+Proof. For x ∈ BN
+⊺ , differentiating both sides of Equation SI.6 with respect to u at u = 0 gives
+d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (x, β)
+= δz,xδλ,β + ∑
+y∈BN
+⊺
+∑
+γ∈L
+�
+d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (y, γ)
+�
+P◦
+(y,γ)→(x,β)
+= δz,xδλ,β + ∑
+y∈BN
+⊺
+∑
+γ∈L
+�
+d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (y, γ)
+�
+∑
+(R,α)
+y�α=x
+p◦
+(R,α) (γ) qγβ.
+(SI.19)
+Doing so for x ∈ {A, B} gives
+d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (A, β) = ∑
+γ∈L
+�
+d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (A, γ)
+�
+qγβ − ρ◦
+A (z, λ) υ (β)
++ ∑
+y∈BN
+⊺
+∑
+γ∈L
+�
+d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (y, γ)
+�
+∑
+(R,α)
+y�α=A
+p◦
+(R,α) (γ) qγβ; (SI.20a)
+d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (B, β) = ∑
+γ∈L
+�
+d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (B, γ)
+�
+qγβ − ρ◦
+B (z, λ) υ (β)
++ ∑
+y∈BN
+⊺
+∑
+γ∈L
+�
+d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (y, γ)
+�
+∑
+(R,α)
+y�α=B
+p◦
+(R,α) (γ) qγβ. (SI.20b)
+If ϕ : BN → R is a fixed function, then, by definition,
+υ (β) ⟨ϕ | β⟩◦
+(z,λ) = ∑
+x∈BN
+υ (β) d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (x, β)
+π◦
+⟳(z,λ) (β) ϕ (x) .
+(SI.21)
+Combining Lemma 2 and Eqs. SI.19–SI.20 with the fact that
+υ (β) d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (x, β)
+π◦
+⟳(z,λ) (β)
+= d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (x, β) − (δA,xρ◦
+A (z, λ) + δB,xρ◦
+B (z, λ)) d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (β) (SI.22)
+then gives Equation SI.18 after some tedious but straightforward simplifications.
+32
+
+Corollary 2. With I ⊆ N and η[β]
+I
+(z, λ) := υ (β)
+�
+∑i∈N π[β]
+i xi − xI | β
+�◦
+(z,λ), we have
+η[β]
+I
+(z, λ) = δλ,β
+�
+∑
+i∈N
+π[β]
+i zi − zI
+�
++ ∑
+γ∈L ∑
+(R,α)
+p◦
+(R,α) (γ) qγβη[γ]
+�α(I) (z, λ) .
+(SI.23)
+Subject to ∑i∈N π[β]
+i η[β]
+i
+(z, λ) = 0 for some β ∈ L, the solution to Equation SI.23 is unique.
+Proof. Setting ϕ (x) = ∑i∈N π[β]
+i xi − xI in Proposition 1 gives Equation SI.23. Conversely,
+we know that η[β]
+I
+(z, λ) := υ (β)
+�
+∑i∈N π[β]
+i xi − xI | β
+�◦
+(z,λ) solves Equation SI.23, so that
+there is at least one solution to Equation SI.23. By the Fixation Axiom, the dimensionality of
+the space of solutions to Equation SI.23 is determined by that of the case |I| = 1. (The reason
+is that all subsets of size greater than one are transient under the ancestral process.) Specifically,
+the recurrence for I = {i} is
+η[β]
+i
+(z, λ) = δλ,β (ρ◦
+A (z, λ) − zi) + ∑
+γ∈L ∑
+j∈N
+e◦
+ji (γ) qγβη[γ]
+j
+(z, λ)
++ ∑
+γ∈L
+�
+1 − ∑
+j∈N
+e◦
+ji (γ)
+�
+qγβη[γ]
+i
+(z, λ) .
+(SI.24)
+If �η (z, λ) is another solution to Equation SI.24, then χ (z, λ) := η (z, λ) − �η (z, λ) satisfies
+χ[β]
+i
+(z, λ) = ∑
+γ∈L ∑
+j∈N
+e◦
+ji (γ) qγβχ[γ]
+j
+(z, λ) + ∑
+γ∈L
+�
+1 − ∑
+j∈N
+e◦
+ji (γ)
+�
+qγβχ[γ]
+i
+(z, λ) . (SI.25)
+Noting that any constant function is a solution to Equation SI.25, and the space of solutions to
+this equation is one-dimensional as a result of the Fixation Axiom, there must exist K ∈ R
+such that η (z, λ) = �η (z, λ) + K. Since the solution η[β]
+i
+(z, λ) = υ (β) ⟨xi | β⟩◦
+(z,λ) satisfies
+∑i∈N π[β]
+i η[β]
+i
+(z, λ) = 0 for all β ∈ L, it follows that K = 0 and η (z, λ) = �η (z, λ) whenever
+�η (z, λ) satisfies Equation SI.23 and ∑i∈N η[β]
+i
+�η[β]
+i
+(z, λ) = 0 for some β ∈ L. We note that
+∑i∈N π[β]
+i η[β]
+i
+(z, λ) = 0 for some β ∈ L ensures that this equation holds for all β ∈ L.
+33
+
+SI.1.5
+Calculating first-order effects of selection on fixation probabilities
+SI.1.5.1
+Fixed initial configurations
+Note that for functions ϕ : BN → R and φ : L → R, we have
+⟨φϕ⟩◦
+(z,λ) = d
+du
+�����
+u=0
+∑
+β∈L
+π◦
+⟳(z,λ) (β) φ (β) E◦
+⟳(z,λ) [ϕ | β]
+= ∑
+β∈L
+υ (β) φ (β) ⟨ϕ | β⟩◦
+(z,λ)
++ (ρ◦
+A (z, λ) ϕ (A) + ρ◦
+B (z, λ) ϕ (B)) ∑
+β∈L
+φ (β) d
+du
+�����
+u=0
+π◦
+⟳(z,λ) (β) .
+(SI.26)
+Therefore, we may rewrite Equation SI.13 as
+d
+dδ
+�����
+δ=0
+ρA (z, λ)
+= ∑
+i,j∈N ∑
+I⊆N
+�
+cji
+I (β) ∑γ∈L qβγπ[γ]
+i
+�
+��
+�
+φ
+�
+xI∪{j} − xI∪{i}
+�
+��
+�
+ϕ
+��◦
+(z,λ)
+= ∑
+i,j∈N ∑
+I⊆N ∑
+β∈L
+υ (β) cji
+I (β) ∑
+γ∈L
+qβγπ[γ]
+i
+��
+xI∪{j} | β
+�◦
+(z,λ) −
+�
+xI∪{i} | β
+�◦
+(z,λ)
+�
+.
+(SI.27)
+Defining η[β]
+I
+(z, λ) := υ (β)
+�
+∑i∈N π[β]
+i xi − xI | β
+�◦
+(z,λ), we then have
+d
+dδ
+�����
+δ=0
+ρA (z, λ) = ∑
+i,j∈N ∑
+I⊆N ∑
+β∈L
+cji
+I (β) ∑
+γ∈L
+qβγπ[γ]
+i
+�
+η[β]
+I∪{i} (z, λ) − η[β]
+I∪{j} (z, λ)
+�
+,
+(SI.28)
+where, by Corollary 2, the terms η are uniquely determined by
+η[β]
+I
+(z, λ) = δλ,β
+�
+∑
+i∈N
+π[β]
+i zi − zI
+�
++ ∑
+γ∈L ∑
+(R,α)
+p◦
+(R,α) (γ) qγβη[γ]
+�α(I) (z, λ) ; (SI.29a)
+∑
+i∈N
+π[β]
+i η[β]
+i
+(z, λ) = 0 for some β ∈ L.
+(SI.29b)
+SI.1.5.2
+Probabilistic initial configurations
+Up until this point, we have focused on fixation probabilities given some fixed initial state,
+(z, λ) ∈ N × L. We now allow mutant types to arise stochastically and consider mean fix-
+34
+
+ation probabilities for both types. For two distributions, µA, µB ∈ ∆
+�
+BN
+⊺ × L
+�
+, we let
+ρA (µA) := E(z,λ)∼µA [ρA (z, λ)] ;
+(SI.30a)
+ρB (µB) := E(z,λ)∼µB [ρB (z, λ)] .
+(SI.30b)
+By the results of §SI.1.5.1, for any µ ∈ ∆
+�
+BN
+⊺ × L
+�
+, we have
+d
+dδ
+�����
+δ=0
+ρA (µ) = ∑
+i,j∈N ∑
+I⊆N ∑
+β∈L
+cji
+I (β) ∑
+γ∈L
+qβγπ[γ]
+i
+�
+η[β]
+I∪{i} (µ) − η[β]
+I∪{j} (µ)
+�
+,
+(SI.31)
+where
+η[β]
+I
+(µ) = E(z,λ)∼µ
+�
+δλ,β
+�
+∑
+i∈N
+π[β]
+i zi − zI
+��
++ ∑
+γ∈L ∑
+(R,α)
+p◦
+(R,α) (γ) qγβη[γ]
+�α(I) (µ) ;
+(SI.32a)
+∑
+i∈N
+π[β]
+i η[β]
+i
+(µ) = 0 for some β ∈ L.
+(SI.32b)
+Letting µ = µA gives the mean fixation probability for type A, while the mean fixation proba-
+bility for type B can be calculated analogously using the equation ρB (µB) = 1 − ρA (µB).
+Although the main focus of our study is on network-transition chains that are both aperiodic
+and irreducible, we do also consider periodic structures. Suppose that among the L networks in
+L, network β transitions deterministically to network β + 1 for β ∈ {1, . . . , L − 1}, and network
+L transitions deterministically to network 1. We can then write Equation SI.32 more explicitly
+as
+η[1]
+I (µ) = E(z,λ)∼µ
+�
+δλ,1
+�
+∑
+i∈N
+π[1]
+i zi − zI
+��
++ ∑
+(R,α)
+p◦
+(R,α) (L) η[L]
+�α(I) (µ) ; (SI.33a)
+η[β]
+I
+(µ) = E(z,λ)∼µ
+�
+δλ,β
+�
+∑
+i∈N
+π[β]
+i zi − zI
+��
++ ∑
+(R,α)
+p◦
+(R,α) (β − 1) η[β−1]
+�α(I) (µ) ;
+(1 < β ⩽ L)
+(SI.33b)
+∑
+i∈N
+π[β]
+i η[β]
+i
+(µ) = 0 for some β ∈ L.
+(SI.33c)
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf,len=1061
+page_content='Strategy evolution on dynamic networks  Qi Su1,2,3, Alex McAvoy1,2, and Joshua B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Plotkin1,2,3  1Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA  2Center for Mathematical Biology, University of Pennsylvania, Philadelphia, PA 19104, USA  3Department of Biology, University of Pennsylvania, Philadelphia, PA 19104, USA  Abstract  Models of strategy evolution on static networks help us understanding how population struc-  ture can promote the spread of traits like cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' One key mechanism is the formation  of altruistic spatial clusters, where neighbors of a cooperative individual are likely to recip-  rocate, which protects prosocial traits from exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' But most real-world interactions  are ephemeral and subject to exogenous restructuring, resulting in dynamic social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Strategic behavior on dynamic networks is difficult to study, and much less is known about  the resulting evolutionary dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Here, we provide an analytical treatment of cooperation  on dynamic networks, allowing for arbitrary spatial and temporal heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We show  that transitions among network structures can favor the spread of cooperation, even if each  individual social network would inhibit cooperation when static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Furthermore, we show that  spatial heterogeneity tends to inhibit cooperation while temporal heterogeneity tends to pro-  mote it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Dynamic networks can therefore have profound effects on the evolution of prosocial  traits, even in cases where individuals have no agency over network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  1  Introduction  The geographic locations of individuals, together with their social or physical connections, con-  strain interactions and shape behavioral evolution in a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' A network is a useful model  of a population’s structure, where nodes represent individuals and edges capture interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  How network structure affects evolutionary dynamics has been extensively investigated over the  last several decades, using techniques including computer simulations, mathematical analysis,  and experimental studies with human subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' A well-known and illustrative finding1 is that  population structure can favor cooperation provided the ratio of the benefit from cooperative be-  havior, b, to its cost, c, exceeds the average number of neighbors, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The mechanism underlying  this cooperation-promoting effect is that spatial structure enables the formation of cooperative  clusters of individuals, who have high payoffs and are capable of resisting invasion by defectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Most existing studies are based on a static network, where the duration and intensity of inter-  actions remain unchanged throughout the evolutionary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In contrast, empirical networks  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='11982v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='soc-ph]  27 Jan 2023  frequently vary over time2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Representative examples include communication networks involv-  ing telephone calls or emails3,4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' networks of physical proximity, where individuals encounter  different people as they move through space5,6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' and ecological networks that change with the  seasons as organisms go through different phases of their life cycles7–9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Temporal features can  even reverse the evolutionary outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For example, whether an idea or information diffuses  throughout a society depends not only on the structure of the network guiding interactions but  also on the timing of those interactions, as the the coexistence of individuals with different active  timing maximizes diffusion10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In the context of epidemics, high concurrency (the number of  neighbors of a node) leads to a lower epidemic threshold under susceptible-infected-susceptible  dynamics, while low concurrency can suppress epidemics11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Despite the attention that other dynamical processes have received on time-varying networks,  the evolution of cooperation in this setting remains much less studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' One reason to discount  any positive effect of dynamic structures comes from intuition on static networks: since coop-  erators spread via clusters, network transitions will tend break up these clusters, likely leading  to diminished reciprocity and exploitation by defectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Another impediment to undertaking  research in this area is the lack of mathematical tools for analyzing strategic interactions on dy-  namic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In static networks, mathematical approaches provide general conditions for how  structure affects evolutionary dynamics12,13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' They also allow for extensive, efficient numerical  explorations into example networks, both artificial and empirical14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Whether these approaches  can be extended to dynamic networks remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Endogenous network transitions often produce predictable results for the evolution of coopera-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For example, if cooperators can selectively seek out new connections with other cooperators  (“cooperation begets friends”) and sever ties with defectors, then it is not surprising to find that  these endogenous networks changes favor the spread cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' But it is much less clear how  exogenous transitions in network structure will affect the evolution of cooperation, and so this is  the main focus of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' There is also substantial evidence for the prevalence of exogenous  network transitions in nature, ranging from weather fluctuations to human-induced changes to  ecosystems15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The scope of models with dynamic networks is broad and can include environ-  mental feedback and ecosystem engineering16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' And even when an organism has some agency  over the structure of their environment, the behavioral trait of interest might be unrelated to these  changes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' movement between cities need not be tied to altruistic tendencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Finally, exoge-  nous network transitions that are not dependent on individual behavior provide the most natural  point of comparison to static structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  In this paper, we study the evolution of strategic behavior in a population whose structure of  social interactions change over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' At any point in time, the population structure is described by  a network whose nodes represent individuals and edges represent interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Individuals may  change their strategies over time, imitating neighbors who have higher payoffs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' and the network  of interactions itself may also change over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The interaction network changes at random  times, unrelated to the current composition of strategies in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We derive general  mathematical results for when cooperative behavior is favored, which apply to any stochastic  transition pattern among any number of networks, each with arbitrary structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Surprisingly, we  2  find that in a large class of networks, stochastic transitions among networks can strongly promote  cooperation, even though they tend to disrupt cooperative clusters in each network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In fact, even  if each individual static network would disfavor cooperation, transitions among them can rescue  cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We conclude by analyzing spatial and temporal burstiness, which we show have  opposite effects on the evolution of cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  2  Model  Our model consists of a finite population of size N, with individuals engaged in pairwise social  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The structure of the population varies over time, and at each discrete time it is  represented by one of L weighted networks, each with N nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For network β ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , L},  we let w[β]  ij denote the weight of the edge between nodes i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We assume that all networks  are undirected, meaning w[β]  ij = w[β]  ji for all i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , N} and β ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Each individual in the population can adopt one of two types, or strategies: “cooperator” (C) or  “defector” (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Individuals interact in pairwise donation games, with cooperators paying a cost c  to generate benefit b for their co-player.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Defectors pay no costs and generate no benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In each  time step, everyone plays a donation game with each of their neighbors in the current network,  β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We denote the state of the population by x, where xi ∈ {0, 1} indicates the type of individual  i, with 0 and 1 representing types D and C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The accumulated payoff to individual i  in network β is then  ui (x, β) =  N  ∑  j=1  w[β]  ij  �−cxi + bxj  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (1)  In other words, individual i receives a benefit w[β]  ij b from of each of its neighbors j who are  cooperators (xj = 1), and i pays a cost w[β]  ij c to each j if i is itself a cooperator (xi = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' An  individual’s accumulated payoff in network β is transformed into fecundity, which represents i’s  propensity to reproduce or, equivalently, to be imitated by another individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The fecundity is  given by Fi (x, β) = 1 + δui (x, β), where δ is called the selection intensity, which we assume  to be small (δ ≪ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' This assumption, called “weak selection,” is common in the literature and  it aims to capture scenarios in which the social trait (C or D) has a small effect on reproductive  success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  After all pairwise games are played in network β and individuals accumulate payoffs, a random  individual i is selected uniformly from the population to update his or her strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' This individ-  ual then imitates the type of a neighbor, j, with probability proportional to j’s fecundity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In other  words, in network β, the probability that i copies j’s type is  eji (x, β) = 1  N  Fj (x, β) w[β]  ji  ∑N  k=1 Fk (x, β) w[β]  ki  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (2)  3  a  Network 1  Network 2  b  c  Strategy updating  d  Network 1  Network 2  Network updating  e  C  C  C  C  D  D  D  C  C  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  C  D  D  D  C  C  D  C  D  D  D  1  2  6  4  7  5  3  1  2  6  4  7  5  3  1  2  6  4  7  5  3  2c  2b  2b  b-2c  2b-4c  2b  b-2c  b-2c  b-2c  2b  0  2c  b  b  1  2  6  4  7  5  3  q11  q12  q21  q22  Interaction at time n+1  Interaction at time n  Figure 1: Evolutionary games on dynamic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' a, The population structure at any time is described by a  network, which may change from one time point to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' (The figure illustrates an example with two possible  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=') b, Each individual (node) in the population adopts the strategy cooperate (C) or defect (D) in games  played with each neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Each individual i accumulates a total payoff ui across pairwise interactions with neigh-  bors, which determine their reproductive rate Fi = 1 + δui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' c, An individual (marked by “?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=') is selected uniformly  at random to update its strategy, and all neighboring individuals, indicated by black circles, compete to be imitated  by the focal node, with probability proportional to reproductive rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' d, After an individual updates its strategy,  the population structure itself either changes (from network 1 to network 2 with probability q12, or from network  2 to network 1 with probability q21) or remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' e, Social interactions and strategy updates repeat on the  population structure at the next time step, n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Here, the factor of 1/N represents the probability that i is chosen to update in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  After each strategic update, the population structure itself then undergoes a transition step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The  probability of moving from network β to network γ is independent of the strategic composition  of the population, and it depends only on the current network state, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The stochastic process  governing these transitions is described by an L × L matrix Q =  �  qβγ  �  , where qβγ is the prob-  ability of transitioning from network β to network γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Note that there may be (and we often  assume) a positive chance that the network will remain unchanged at the transition stage, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  qββ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The pairwise social interactions, strategic update, and network transition, which com-  prise a single time step, are depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  4  3  Results  Without mutation, the population must eventually reach a monomorphic strategic state in which  all individuals have the same type, either cooperate or defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The duration that the population  spends in each network is proportional to the corresponding value in stationary distribution υ,  which is determined by the network transition matrix Q (see Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We assume that a mutant  appears in network β with probability υ (β), and it is located at a node chosen uniformly at  random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We let ρC denote the probability that a single cooperator mutant eventually takes over  a resident population of defectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Likewise, we let ρD be the probability that a single defector  mutant takes over a resident population of cooperators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We use the condition ρC > ρD to  measure whether selection favors cooperation relative to defection17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1  Selection condition for the evolution of cooperation  We first derive a general result applicable to almost any transition pattern, Q, among any finite  number of networks, each with arbitrary spatial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' This result combines several different  quantities describing the dynamics under neutral drift (δ = 0), together with the payoffs for the  game13,18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Let p[β]  ij  := w[β]  ij / ∑N  k=1 w[β]  ik be the one-step random-walk probability of moving from i to j on  network β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' This quantity can be interpreted as the probability that i imitates the strategy of j  under neutral drift, conditioned on i being chosen for an update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In other words, p can be seen  as defining an ancestral process, tracking replacement backwards in time under neutral drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  The most fundamental neutral quantity is the reproductive value of individual i in network β,  which can be interpreted as the probability that a mutant introduced at node i in network β  generates a lineage that eventually takes over the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' This quantity, denoted by π[β]  i  is independent of the payoffs and thus independent of the particular mutant that arises in the  population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The version of reproductive value that we use is a generalization of Fisher’s clas-  sical notion19,20 that also takes into account environmental changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' It can be calculated using  Equation 5 in Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Another neutral quantity we use is related to coalescence times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Under neutral drift, we can look  backward in time and ask how long it takes, on average, before two or more lineages meet at a  common ancestor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Starting in network β, let T[β] be the expected number of steps to the most  recent common ancestor of the entire population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' If τ[β]  ij  is the expected time to the most recent  common ancestor of i and j, then the mean amount of time that i and j are identical by descent is  T[β] − τ[β]  ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The pairwise times to a common ancestor, τ, can be calculated using Equation 8 in  Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  In terms of the neutral quantities π, τ, and T, the general condition for cooperation to be favored  5  over defection under weak selection is given by  N  ∑  i,j=1  L  ∑  β=1  υ (β)  �  L  ∑  γ=1  qβγπ[γ]  i  �  p[β]  ij  N  ∑  ℓ=1  �  −  �  T[β]−τ[β]  jj  �  w[β]  jℓ c  +  �  T[β]−τ[β]  jℓ  �  w[β]  ℓj b  �  >  N  ∑  i,j,k=1  L  ∑  β=1  υ (β)  �  L  ∑  γ=1  qβγπ[γ]  i  �  p[β]  ij p[β]  ik  N  ∑  ℓ=1  �  −  �  T[β]−τ[β]  jk  �  w[β]  kℓ c  +  �  T[β]−τ[β]  jℓ  �  w[β]  ℓk b  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (3)  Broadly speaking, what Equation 3 says is that an individual i is chosen, a cooperator is placed  at a neighbor j of i, and another neighbor k of i is chosen to compare its (weighted) payoff with  that of the cooperator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' If j’s weighted payoff exceeds that of k, then selection favors the evolution  of cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  The condition above reflects a similar intuition behind the corresponding condition for static  networks (L = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' see Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='14 or Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' 1 of McAvoy & Wakeley21), but there are a few  notable effects of network transitions in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The first effect is that the network β is chosen  with probability υ (β), where υ is the stationary distribution of the structure-transition chain  defined by Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Moreover, whereas individual i is chosen with probability based on reproductive  value πi on a static network, here i is chosen based on reproductive value in the next network  following imitation, ∑L  γ=1 qβγπ[γ]  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The reason for this is natural, because once an individual  replaces i in network β, the network immediately transitions to network γ, and so the resulting  reproductive value of i must be understood within the context of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Once β and i are chosen,  the probabilities of choosing neighbors j and k are p[β]  ij and p[β]  ik , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Moreover, if j is a  cooperator, then individual k is also a cooperator for T[β] − τ[β]  jk time steps, and during each such  step k pays cw[β]  kℓ to provide ℓ with a benefit of bw[β]  kℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' This property accounts for the weighting  of benefits and costs in Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Note that the term T[β] cancels out in Equation 3, and  so although this quantity is helpful for gathering intuition, it is not strictly needed to evaluate  whether cooperators are favored by selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Given the vast number of networks with N nodes, as well as the vast space of possible transitions  among them, we focus most of our analysis on transitions between a pair of networks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' L =  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For a given network transition matrix Q, the value 1/q12 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' 1/q21) gives the expected  time during which the population remains in network 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' network 2) before transitioning  to network 2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' network 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We denote 1/q12 and 1/q21 by t1N and t2N, respectively, so  that t1 and t2 correspond to the expected number of times each individual updates prior to a  transition to a different network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Small values of t1 and t2 correspond to frequent changes in the  population structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Sufficiently large values of t1 and t2 indicate that the population structure is  nearly fixed, so that the population will reach an absorbing strategic state (all C or all D) before  the network transitions to a different state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The regime t1 = 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' t2 = 1) means that, on  average, each individual updates their strategy once in network 1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' network 2) before the  network structure changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  6  a  Network 1  Network 2  5  6  7  8  9  10  11  Benefit, b  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='680  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='080  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='040  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='020  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='040  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='060  Rescaled fixation probability, N(  C-  D)  b  //  //  αN  (1−α)N  Dynamic (α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5)  Static network 1 or 2  (α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5)  Dynamic (α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='8)  Static network 1  (α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='8)  Static network 2  (α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='8)  tN  1  tN  1  tN  1  1−  tN  1  1−  Figure 2: Transitions between networks that contain dense and sparse cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We consider dynamic transitions  between two networks, each of which is comprised of two cliques containing aN and (1 − a) N nodes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  a, Each network has a star graph comprising one clique and a complete graph comprising the other clique, with  a single edge connecting the two cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' When network 1 transitions to network 2, the star clique becomes the  complete clique and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' b, The fixation probability of cooperation versus defection, ρC − ρD, as a function  of the benefit b in the donation game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Selection favors cooperation over defection if ρC − ρD exceeds the horizontal  line, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=', ρC > ρD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Dots indicate the results of Monte Carlo simulations on dynamic networks (solid dots) and  on a static network (open dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The vertical lines correspond to analytical predictions for the critical benefit-to-  cost ratio (b/c)∗ on dynamic networks, above which we predict cooperation will be favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The results show  that cooperation is always disfavored in both static network 1 and static network 2, but dynamic transition between  these networks can favor cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Here, we show two examples with different clique sizes, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5 (blue) and  a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='7 (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The beneficial effect of structure transitions is strongest when cliques have equal size (a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  see Supplementary Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Parameter values: N = 40, t = 1, and c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Fixation probabilities are computed  across an ensemble of 107 runs with selection intensity δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='2  Dynamic networks with dense and sparse cliques  We begin by studying dynamic transitions between a pair of networks where each network is  comprised of two cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' One clique is a star graph, which is sparse, and the other clique is  a complete graph, which is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In each network, the two cliques are connected by a single  edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' When the population transitions from one network to another, the star clique becomes the  complete clique and vice versa (see Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' This kind of dynamic network models a situation  in which a portion of the population is densely connected while the remainder of the population  is connected to only a single node;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' and which portion is dense versus sparse changes over time,  as the state transitions between the two networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  When the population evolves on either network 1 or network 2 alone, the fixation probability  of cooperators is always lower than that of defectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' ρC < ρD,meaning that cooperation  is disfavored by selection regardless of the benefit-to-cost ratio b/c (Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Nonetheless,  when the population transitions dynamically between networks 1 and 2, cooperation is favored  7  provided the benefit-to-cost ratio b/c exceeds the critical value (b/c)∗ ≈ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' As a result, we  see that dynamic population structures can favor cooperation, even when all networks involved  would each individually suppress cooperation were they static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Dynamic population structure facilities cooperation across a wide range of population sizes for  the pair of networks shown in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' When t = 1, which means that individuals each  update their strategy once, on average, before the network changes, cooperation can be favored  by selection regardless of network size N (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' By contrast, if the network is static,  then cooperation is favored only when the population size is very small (N < 17)–and, even  then, only if the benefit-to-cost ratio is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For larger population sizes, N ⩾ 17, the critical  benefit-to-cost ratio is negative on a static network, (b/c)∗ < 0, which means that selection  actually favors the evolution of spite, a behavior in which individuals pay a cost c to decrease  the fitness of their opponent by b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For this static network we can prove that (b/c)∗ ≈ −N/2 in  large populations (see Methods), compared to (b/c)∗ ≈ 7 for any population size in a dynamic  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Consequently, we see that the effects of dynamic population structures are dramatic,  capable of converting a spiteful outcome into a cooperative one, and they persist across a wide  range of population sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Dynamic networks also facilitate cooperation across a wide range of structural transition rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  For a sufficiently large population size, N, on a single static network of the type shown in Fig-  ure 2a, the critical benefit-to-cost ratio is negative ((b/c)∗ ≈ −N/2), which means that selec-  tion favors the evolution of spite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' By contrast, dynamic transitions between networks 1 and 2  can favor cooperation, especially when they occur rapidly (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' When the transition rate  is very slow–in particular, when t exceeds  �√  2 + 1  �  N–the population stays in one network for  so long that the evolutionary dynamics are similar to those of a static network, and the critical  benefit-to-cost ratio becomes negative (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In the limit of the transition rate approaching  zero (t → ∞), the “dynamic” network is actually static and our dynamic calculations agree with  those of a static network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='3  How dynamic structures can facilitate the spread of cooperation  To further understand how dynamic structures can favor cooperation more than their static coun-  terparts, we inspect evolutionary trajectories on the dense-sparse graph of Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' When the  network is static, the process is depicted in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Starting from a specific configuration  of cooperators in both hubs and two leaf nodes, cooperation will initially tend to spread in the  star clique while shrinking in the complete clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' After cooperation fixes within the star clique,  selection strongly suppresses further spread to the complete clique because the node connected  to the star clique is exploited by multiple defectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' If ever a defector manages to diffuse to the  hub of the star clique, however, defection will then rapidly spread within the star and ultimately  fix in the entire network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  By contrast, if the population undergoes structural transitions between networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' n2 → n3  in Figure 4b), the star clique of network 1 will transition into the complete clique of network  2, which promotes the exploitation of cooperators and allows defectors to spread (n3 → n4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  8  101  102  103  104  Network size, N  106  104  102  0  102  104  106  Critical benefit-to-cost ratio, (b/c)*  a  Dynamic, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' 3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='00  Static network 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' 3  N/2  5  10  15  20  25  103  102  101  0  101  102  103  10-1  100  101  102  103  104  105  106  Rescaled duration, t  106  104  102  0  102  104  106  Critical benefit-to-cost ratio, (b/c)*  b  Dynamic, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' 3  Static network 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' 3  N/2  104  105  106  107  106  105  104  103  t3+10t2+26t+19  t2+4t+3  t2N+tN2  −t2+2tN+N2  Figure 3: Dynamic structures facilitate cooperation for a broad range of population sizes and network transi-  tion rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We consider transitions between the two networks shown in Figure 2a, each composed of a sparse clique  and a dense clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' a, The critical benefit-to-cost ratio required to favor cooperation as a function of population size,  N, for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5 and t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Dynamic networks can favor cooperation for any population size, N, provided b/c > 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  In contrast, the corresponding static networks favor cooperation only in small populations (N < 17), and they favor  the evolution of spite ((b/c)∗ < 0) in larger populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Dots show exact analytical computations for finite N  (Equation 3), and lines show analytical approximations for large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' b, The critical benefit-to-cost ratio as a func-  tion of the mean duration between network transitions, t, for a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5 and N = 10,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Whereas a static network  always disfavors cooperation, dynamic networks can favor cooperation provided they do not transition too slowly  (t <  �√  2 + 1  �  N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Dots show exact analytical computations for arbitrary t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' the blue line shows an analytical  approximation in the regime t ≪ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' and the red line shows an analytical approximation in the regime t = O (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  9  Meanwhile, the complete clique of network 1 transitions into the star clique of network 2, which  stimulates the expansion of cooperators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The rate of cooperator expansion in one clique exceeds  their exploitation in the other clique so that, overall, network transitions facilitate cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='4  Other dynamic structures  The examples of dynamic structure considered so far may seem highly specialized because the  networks each contain two stylized cliques with a single edge between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' But we find similar  results on networks with many cliques and with more complicated connections between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In  Figure 5a,b, we analyze networks comprised of multiple star and complete cliques, connected by  either hub nodes or by leaf nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In both cases, we again find that dynamic transitions between  networks reduce the critical benefit-to-cost ratio for the evolution of cooperation, compared to  any single static network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' This effect is increasingly strong as the network size grows (see  Supplementary Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For the networks in Figure 5a with N = 1,200, for example, the  critical benefit-to-cost ratio to favor cooperation is (b/c)∗ ≈ 188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1 when the network is static,  which is reduced to (b/c)∗ ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='49 when the network is dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  In addition to networks comprised of star and complete cliques, we also investigated networks  with cliques defined by various types of random graphs, such as Erd¨os-R´enyi and scale-free  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In the former case, node degrees within a clique do not vary substantially, while the  latter exhibits large variation in degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For both classes of random networks, we still find that  dynamic transitions between random networks tends to promote cooperation, compared to each  static network (Figure 5c,d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  In all examples of dynamic networks considered thus far, transitions between networks involve  dense regions of a network swapping with sparse regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Regardless of the exact structure of  the cliques, this general feature of structural transitions conforms to the underlying intuition for  why dynamic networks can facilitate cooperation (Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Dynamic structures can still facil-  itate cooperation even when networks differ in only a small fraction of connections, although  the strength of the effect is weakened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Furthermore, these effects also persist (and can be quite  strong) when populations transition between three or more network structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We give illustra-  tions in Supplementary Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5  The probability and time to fixation of cooperation  We have studied dynamic structures by comparing the fixation probability of a cooperator to  that of a defector, and by calculating the critical benefit-to-cost ratio (b/c)∗ that ensures ρC >  ρD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We can also study the fixation probability ρC in absolute terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We find that a dynamic  population structure increases the fixation probability of cooperators, making them more likely  to overtake the population, compared to a static network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Dynamic population structures also  tends to decrease the duration before one type or another fixes (see Supplementary Figure 4),  as well as shorten the mean conditional time until cooperators fix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The underlying intuition for  these results is evident in Figure 4: on a static network, the population will tend to be stuck  at stage n3 for a long time, before defectors eventually diffuse to the sparse clique;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' whereas  10  Evolution on static network  Evolution on dynamic network  Network 1  Network 1  Network 2  Cooperator  Defector  n1  n2  n3  n4  n1  n2  n5  n6  n3  n4  n7  n8  Strategy  updating  Network  updating  a  b  Figure 4: Intuition for how dynamic structures can facilitate cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Starting from a configuration in  which the hub and two leaf nodes are cooperators (time point n1 in a and b), we illustrate how cooperation can be  favored in dynamic structures even when it is inhibited in each static structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Initially, cooperators are expected  to spread in the star clique and shrink in the complete clique, and the rate of spreading exceeds that of shrinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' a,  The evolutionary process on a static network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Cooperators rapidly take over the star clique and nearly die out in the  complete clique (n1 → n3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The system tends to stay in this state until defectors spread throughout the star clique  (n4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' b, The evolutionary process with network transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Initially, cooperators spread in the star clique and shrink  in the complete clique (n1 → n2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' However, when the network changes, the star clique transitions to the complete  clique and vice versa (n2 → n3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' This transition is followed by the rapid spread of cooperators in the star clique  and (relatively slower) shrinking of cooperators in the complete clique (n3 → n4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' From n1 to n5, the frequency of  cooperators increases in both cliques so that, under dynamic structure transitions, the population tends to result in  cooperators being fixed in both cliques (n8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  11  a  Network 1, (b/c)*  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='31  Network 2, (b/c)*  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='31  Dynamic network, (b/c)*  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='92  Multiple cliques connected via hub nodes  b  Network 1, (b/c)*  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='00  Network 2, (b/c)*  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='00  Dynamic network, (b/c)*  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='34  Multiple cliques connected via leaf nodes  c  Network 1, (b/c)*  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='85  Network 2, (b/c)*  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='80  Dynamic network, (b/c)*  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='86  Multiple ER communities  d  Network 1, (b/c)*  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='64  Network 2, (b/c)*  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='07  Dynamic network, (b/c)*  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='33  Multiple GKK communities  Figure 5: Evolution of cooperation on diverse dynamic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' a, Each individual network comprises four  star cliques and four complete cliques, where each star clique in one network corresponds to a complete clique  in the other network, and clique hubs are fully connected to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' b is similar to a, but cliques are now  sparsely connected via leaf nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Network transitions facilitate cooperation compared to a static structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' c, Each  individual network comprises two sparse and two dense cliques of Erd¨os-R´enyi (ER) random networks22, with  cliques connected by random nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' d, Each individual network comprises two sparse and two dense cliques of  Goh-Kahng-Kim scale-free networks (GKK)23 with exponent 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5, with cliques connected by nodes of the highest  degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In all these examples, network transitions reduce the benefit-to-cost ratio (b/c)∗ required for cooperation  compared to each static network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Parameters: t = 1 and N = 64 for a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For panels c and d, in network 1,  the two sparse cliques have 30 nodes and average degree 4, and the two dense cliques have 40 nodes and average  degree 30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' in network 2, the two sparse cliques have 40 nodes and average degree 4, and the two dense cliques have  30 nodes and average degree 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  12  on dynamic networks, cooperators spread rapidly by selection in both cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Thus, dynamic  networks increase the likelihood that cooperators sweep the population as well as the rate at  which they do so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='6  Spatial and temporal burstiness  We can adapt our method of analysis to study the effects of spatial and temporal burstiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  For dynamically changing networks, spatial burstiness arises when there is temporal variation in  the density of network edges (node degree), whereas temporal burstiness arises when there are  periods of rapidly changing network structures along with periods in which structures change  more slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Empirical networks of both human and non-human (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' honeybee) interactions  are known to exhibit both spatial and temporal burstiness10,24, but the effects of these two forms  of over-dispersion for behavior remains an active area of current research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  To study spatial burstiness, we consider the following minimal model of dynamically varying  networks that differ in their average node degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We construct a pair of networks as follows  (see Figure 6a): (i) we first generate a single network with N nodes and E edges drawn from one  of several classical families of networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Erdos-Reyni random networks22, Watts-Strogatz  small-world networks25, Barab´asi-Albert scale free networks26, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' (ii) we decompose this  network into two networks, by randomly selecting a fraction ε ∈ [0, 1/2] of the edges for net-  work 1 and using the remaining (1 − ε) E edges for network 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' If ε = 1/2 then the resulting  networks 1 and 2 have the same density of interactions, and there is no spatial burstiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For all  other values of ε ̸= 1/2, the network exhibits spatial burstiness, and we study a simple stochastic  transition pattern between these networks, with t1 = t2 = 1 so that each individual updates his  strategy once, on average, before the network switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  We find that spatial burstiness tends to inhibit the evolution of cooperation, whereas spatial reg-  ularity (equal network densities) is more beneficial for cooperation (Figure 6c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In particular,  regardless of the class of network from which networks 1 and 2 are derived, the critical ratio  (b/c)∗ required to favor cooperation is substantially increased (roughly by a factor of two) in  the regime ε → 0 compared to the spatially homogeneous regime ε = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  We also study the effects of temporal burstiness, in which case networks 1 and 2 are chosen to  have the same edge density (ε = 1/2), but there are periods of rapid transitions between the  two networks, punctuated by periods of slow transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' To construct this scenario, instead of  having a single transition matrix, Q, we consider two such matrices, Q f and Qs, corresponding  to fast and slow epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' At any time, the population is either in hidden state f, so that network  transitions occur according to Q f , or alternatively in hidden state s, so that network transitions  occur according to Qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Whenever the population transitions to a new network, the hidden state  is drawn uniformly-at-random from { f, s} (see Figure 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' (Note that the hidden state s or f is  re-sampled only when the network changes, from 1 to 2 or from 2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=')  The speed of network transitions in each hidden state, s and f, is governed by a parameter  ¯t ∈ [0, 1], so that transitions are fast in state f and slow in state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' When the population enters  13  a  b  Hidden  state s  Hidden  state f  on transition to  new structure  Network 1  (edge fraction, ε)  Network 2  (edge fraction, 1−ε)  ε  High spatial  burstiness  Low spatial  burstiness  High temporal  burstiness  Low temporal  burstiness  Figure 6: Effects of spatial and temporal burstiness on cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We consider transitions between two  networks, with either a, spatial burstiness (different edge densities) or b, temporal burstiness (periods of both rapid  and slow transitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' c, The critical benefit-to-cost ratio (b/c)∗ as a function of spatial heterogeneity, ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' When  the two networks have the same edge density, ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5, cooperation is most readily favored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' When the networks  that differ in their edge densities (ε ≪ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5), much larger values of b/c are required to support cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' d The  critical benefit-to-cost ratio (b/c)∗ required to favor cooperation as a function of temporal heterogeneity, ¯t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The  case ¯t = 1 means that networks transition at the same rate, regardless of the hidden state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' When ¯t < 1, the networks  transition more rapidly in state f than in state s, so that there is temporal burstiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Results on spatial and temporal  burtiness are shown for six classes of networks: random regular networks (RR), Erd¨os-R´enyi networks (ER)22,  Watts-Strogatz small-world networks (SW)25 with rewiring probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1, Barab´asi-Albert scale-free networks  (BA)26, Goh-Kahng-Kim scale-free networks (GKK)23 with exponent 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5, and Holme–Kim scale-free networks  (HK)27 with triad formation probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For each such class, we generate 2,000 networks, each with 100 nodes  and average degree 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We take ¯t = 1 in c and ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5 in d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  14  100  30  RR  ER  Critical benefit-to-cost ratio, (b/c)*  sw  80  BA  ratio,  GKK  HK  60  40  RR  ER  sw  20  0008880888088809  BA  GKK ·  HK  10  0L  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='0  Edge fraction, E  Rescaled duration, tstate f, the expected duration before a network transition is small, namely ¯tN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Whereas when  the population enters state s the expected duration of the current network is longer, (2 − ¯t) N  (see Figure 6b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The case ¯t = 1 means that the current network has the same expected duration,  regardless of the hidden state, and there is no temporal burstiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' When ¯t < 1, the networks  transition more quickly in state f than they do in state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Regardless of the value of ¯t, how-  ever, the total accumulated time spent in network 1 is the same as in network 2, throughout the  evolutionary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Temporal burstiness tends to facilitate cooperation, regardless of the overall structure of underly-  ing networks (Figure 6d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In particular, the critical benefit-to-cost required to favor cooperation  is largest when temporal burstiness is absent (¯t = 1), and it is reduced (typically by 20%)  when temporal burstiness is large (¯t = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Therefore, even when two networks have the same  edge density (ε = 1/2) and the accumulated time is spent on each network is the same, tem-  poral burstiness facilitates the spread of cooperation, in stark contrast to our findings for spatial  burstiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  4  Discussion  Many real-world interactions are ephemeral, and the entire network of social interactions may  be subject to exogenous changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Seasonal changes in a species’ environment, for example, can  lead to active and dormant periods, as can diurnal cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Such periodic transitions are widely  used to model temporal networks28–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Stochastic transitions in social structures can arise from  the effects of weather, animal migration and movement, and role reversal32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Motivated by the  ubiquity of structural variation in nature, we provide a treatment of dynamic social networks that  allows for arbitrary stochastic transitions between structures, with arbitrary networks within each  time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Our main mathematical result (Equation 3) predicts when cooperation will evolve on dynamic  networks, under weak selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The population structure in every time step need not be con-  nected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' all that we require is that the population satisfy a coherence condition so that it does not  become fragmented into multiple sub-populations (see §SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1 in Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  In addition to probabilistic transitions, our analysis also extends to deterministic and periodic net-  work transitions (see Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='33 in Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Our work can also cover  other scenarios for changing structures, such as when the direction of public goods or informa-  tion flow changes over time33;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' the number of active nodes or edges varies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' or the population  size fluctuates (in fact, the results in Supplementary Information allow for arbitrary patterns of  replacement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Although prosocial behaviors in different strategic domains may manifest in dif-  ferent ways, such as trust games or dictator games, the desire to pay costs to benefit others has a  substantial degree of domain generality34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Our conclusions, based on donation games, are thus  indicative of how dynamic networks may broadly impact prosocial behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  In the donation game, we have seen that changing social structures can promote cooperation,  and that these effects can be dramatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Even if every network individually disfavors cooperators,  15  transitions between them can facilitate the evolution of cooperation – a result that is reminiscent  of Parrondo’s paradox35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Figure 4 illustrates the mechanism for how this phenomenon arises,  as transitions move individuals between regions of the network that are dense to those that are  sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' These types of changing social structures are common in real-world settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Groups and  communities are more likely to form among people with close geographical locations and similar  religion, culture, and affiliations36,37;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' but connection density will be altered when individuals  migrate or change social groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Changes in connection densities in different communities may  alternatively result from a phase difference, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' in online social networks across different time  zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Spatio-temporal heterogeneity of interaction density within a community also leads to  time-varying connection densities, from sparse to dense and vice versa2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We find that each kind  of burstiness has a clear effect on cooperation, either hindering it in the case of spatial burstiness  or promoting it in the case of temporal burstiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Broadly speaking, our work highlights the  significance of integrating multiple communities into one system, since treating communities  individually and independently may lead to erroneous conclusions about behavioral dynamics38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  All of our results are based on exogenous network transitions, which means that individuals can-  not selectively engineer their neighborhoods based on the traits of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' There are, of course,  many interesting models involving endogenous transitions, in which cooperators can selectively  form links with other cooperators and break links with defectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In such models cooperation  can flourish when structure transitions are rapid enough39–51, for the simple reason that this en-  dogenous dynamic establishes cooperative clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Such “form follows function” models are  frequently aimed at answering the question: what kinds of networks arise from certain traits, and  how do these networks serve the greater good?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' By contrast, our focus is not the coevolutionary  dynamics of trait and structure, but on a different question altogether: what is the impact of ex-  ogenous structural changes for the evolution of behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' This approach is more closely related  to classical studies of network effects on cooperation: given a (dynamic) network, what behav-  ioral traits evolve?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Since exogenous structural changes do not provide any explicit advantage or  disadvantage to cooperators relative to defectors, the resulting evolutionary dynamics of social  traits are all the more intriguing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  We have aimed for generality in framing our mathematical results, but a natural limitation of  our study is the scope of networks we have analysed, compared to the vast space of possible  population structures and transitions among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For this reason, even static structures are still  an active topic of current research in evolutionary game theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We have therefore chosen to  consider a limited number of representative examples of dynamic networks, which showcase the  interesting effects they can have on the evolution of cooperation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Areas for future investigation  include the effects of fluctuating resources on cooperation, alternative evolutionary update rules,  stronger selection, and environments that involve both endogenous and exogenous transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In  fact, although we use cooperation as an example, our analysis is framed quite generally to allow  the study of other traits on dynamic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' To the best of our knowledge, our analytical  findings constitute the first general results for behavioral evolution on dynamic networks, and we  hope that they will be valuable tools in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  16  5  Methods  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1  Analysis of weak selection  Here, we outline a derivation of the critical benefit-to-cost ratio (b/c)∗ for selection to favor co-  operation, based on an extension of the methods of McAvoy & Allen18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Complete mathematical  details are provided in Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  For i, j ∈ N , let w[β]  ij be the weight of edge between nodes i and j in network β ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  We assume that the network is undirected, meaning w[β]  ij  = w[β]  ji  for all i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , N} and  β ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' If i and j share an edge, then they interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The class of models we are interested  in here involve social goods52 in which, on network β, an individual of type A at i pays a cost of  C[β]  ij to donate B[β]  ij to the individual at j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In state (x, β), the total payoff to the individual at i is  ui (x, β) =  N  ∑  j=1  �  −xiC[β]  ij + xjB[β]  ji  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (4)  This net payoff is converted to reproductive rate via the formula Fk (x, β) = eδuk(x,β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' If the pop-  ulation structure is β, then a node in β is first selected uniformly-at-random to die.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Subsequently,  all neighboring nodes in β compete to produce an offspring to fill the vacancy at node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The  probability that j replaces i in state (x, β) is given by Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Let p[β]  ij := w[β]  ij / ∑N  k=1 w[β]  ik be the probability of moving from i to j in one step of a random walk  on network β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Under neutral drift, the probability π[β]  i  that, starting in network β, i generates a  lineage that takes over the population (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' the reproductive value of i in β) satisfies  π[β]  i  = 1  N  N  ∑  j=1  p[β]  ji  L  ∑  γ=1  qβγπ[γ]  j  +  �  1 − 1  N  N  ∑  j=1  p[β]  ij  �  L  ∑  γ=1  qβγπ[γ]  i  ,  (5)  subject to the constraint ∑N  i=1 π[β]  i  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' π is thus determined by a linear system of size O (LN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  For the initial state, we choose the network according to the stationary distribution of the network-  transition chain, and a mutant appears uniformly-at-random within that network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' There are two  mutant-appearance distributions overall, one for C arising after the all-D state (denoted µC) and  one for D arising after the all-C state (denoted µD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Associated to each µ ∈ {µC, µD} is a  quantity η[β]  I  (µ) related to the co-occurrence of a trait in β among the nodes in I ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , N},  which is defined formally in §SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5 of Supplementary Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For our purposes, we need  17  η[β]  I  (µ) only for I containing one or two nodes, in which case η[β]  I  (µC) = η[β]  I  (µD) and  η[β]  ij =  �  �  �  �  �  �  �  �  �  �  �  0  i = j,  1  N υ (β) +  L  ∑  γ=1  qγβ  �  1  N  N  ∑  k=1  p[γ]  ik η[γ]  kj + 1  N  N  ∑  k=1  p[γ]  jk η[γ]  ik +  �  1 − 2  N  �  η[γ]  ij  �  i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (6)  We refer the reader to Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='32 in Supplementary Information for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  It turns out that a scaled version of η, namely τ[β]  ij  := η[β]  ij /υ (β), allows for a more intuitive  interpretation of the selection condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Consider the time-reversed structure transition chain  defined by  �qβγ := υ (γ)  υ (β)qγβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (7)  Using this time-reversed chain in conjunction with Equation 6, we see that  τ[β]  ij  =  �  �  �  �  �  �  �  �  �  �  �  0  i = j,  1  N +  L  ∑  γ=1  �qβγ  �  1  N  N  ∑  k=1  p[γ]  ik τ[γ]  kj + 1  N  N  ∑  k=1  p[γ]  jk τ[γ]  ik +  �  1 − 2  N  �  τ[γ]  ij  �  i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (8)  In the ancestral process, looking backward in time under neutral drift, Nτ[β]  ij  has the interpre-  tation as the expected number of update steps until i and j coalesce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Equivalently, since one of  N individuals is updated in each time step, τ[β]  ij  can be seen as the mean number of generations  needed for i and j to coalesce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' If, conditioned on the population being in state β, T[β] is the mean  time to reach the most recent common ancestor going backward in time, then the mean time that  i and j spend identical by descent is T[β] − τ[β]  ij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Finding τ for all structures and pairs of individ-  uals involves solving a linear system of size O  �  LN2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' (Although T aids in the interpretation of  τ as determining identity by descent, it does not need to be calculated in order to understand the  first-order effects of selection on fixation probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=')  We now have all of the neutral quantities we need to state the selection condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The final piece  is the connection between the payoffs and the replacement probabilities under weak selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' A  straightforward calculation gives eji (x, β) = 1  N p[β]  ij + δ ∑N  k=1 cji  k (β) xk + O  �  δ2�  , where  cji  k (β) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  1  N p[β]  ij  �  −  N  ∑  ℓ=1  C[β]  jℓ + B[β]  jj + p[β]  ij  N  ∑  ℓ=1  C[β]  jℓ −  N  ∑  ℓ=1  p[β]  iℓ B[β]  jℓ  �  k = j,  1  N p[β]  ij  �  B[β]  kj + p[β]  ik  N  ∑  ℓ=1  C[β]  kℓ −  N  ∑  ℓ=1  p[β]  iℓ B[β]  kℓ  �  k ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (9)  18  Putting everything together using Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='31 in Supplementary Information, we see that  d  dδ  �����  δ=0  ρC = 1  N  N  ∑  i,j=1  L  ∑  β=1  υ (β)  �  L  ∑  γ=1  qβγπ[γ]  i  �  p[β]  ij  N  ∑  ℓ=1  �  −  �  T[β]−τ[β]  jj  �  C[β]  jℓ  +  �  T[β]−τ[β]  jℓ  �  B[β]  ℓj  �  − 1  N  N  ∑  i,j,k=1  L  ∑  β=1  υ (β)  �  L  ∑  γ=1  qβγπ[γ]  i  �  p[β]  ij p[β]  ik  N  ∑  ℓ=1  �  −  �  T[β]−τ[β]  jk  �  C[β]  kℓ  +  �  T[β]−τ[β]  jℓ  �  B[β]  ℓk  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (10)  Moreover, an analogous calculation for D gives d  dδ  ���  δ=0ρD = − d  dδ  ���  δ=0ρC, which means that the  condition d  dδ  ���  δ=0ρC > 0 is equivalent to d  dδ  ���  δ=0ρC > d  dδ  ���  δ=0ρD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  In the donation game, we have B[β]  ij = w[β]  ij b and C[β]  ij = w[β]  ij c, and Equation 10 gives  d  dδ  �����  δ=0  ρC > d  dδ  �����  δ=0  ρD ⇐⇒ bµ2 − cν2 > bµ0 − cν0,  (11)  where  µ0 = 1  N  N  ∑  i,j=1  L  ∑  β=1  υ (β)  �  L  ∑  γ=1  qβγπ[γ]  i  �  p[β]  ij  N  ∑  ℓ=1  w[β]  ℓj τ[β]  jℓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (12a)  ν0 = 1  N  N  ∑  i,j=1  L  ∑  β=1  υ (β)  �  L  ∑  γ=1  qβγπ[γ]  i  �  p[β]  ij  N  ∑  ℓ=1  w[β]  jℓ τ[β]  jj ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (12b)  µ2 = 1  N  N  ∑  i,j,k=1  L  ∑  β=1  υ (β)  �  L  ∑  γ=1  qβγπ[γ]  i  �  p[β]  ij p[β]  ik  N  ∑  ℓ=1  w[β]  ℓk τ[β]  jℓ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (12c)  ν2 = 1  N  N  ∑  i,j,k=1  L  ∑  β=1  υ (β)  �  L  ∑  γ=1  qβγπ[γ]  i  �  p[β]  ij p[β]  ik  N  ∑  ℓ=1  w[β]  kℓ τ[β]  jk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (12d)  The critical benefit-to-cost ratio is therefore (b/c)∗ = (ν2 − ν0) / (µ2 − µ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Note that, for simplicity, we have assumed that any node can be selected for death in a given  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In reality, this assumption might not hold because each individual network need not  be connected, which can lead to isolated nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' If an isolated node is chosen for death, then  the individual at this node cannot be immediately replaced by the offspring of a neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' All  of our calculations can be modified to allow for only non-isolated nodes to be chosen for death,  although in practice we do not need to do so in any of our examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  19  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='2  Specific examples  We study the transition between two networks, with transition probabilities given by  qβγ =  �  �  �  �  �  �  �  �  �  1 − p  β = 1, γ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  p  β = 1, γ = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  q  β = 2, γ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  1 − q  β = 2, γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (13)  The expected durations in networks 1 and 2 are q/ (p + q) and p/ (p + q), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  We study evolution on dynamic two-clique networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The two-clique network is made up of a  star clique and a complete clique, with the hubs connected (see Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Let n and m denote  the numbers of nodes in the star and complete cliques, respectively, so that n + m = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We  denote by 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , n the nodes in the star clique and by n + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' n + m the nodes in the complete  clique, where n is the hub of the star and n + m is the node of the complete clique connected  to the hub of the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The other network is obtained by swapping the star and complete cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  The adjacency matrix for the first network satisfies w[1]  ij = 1 only if (i) i = n and j < n, or i < n  and j = n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' (ii) i = n and j = n + m, or i = n + m and j = n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' or (iii) i, j ⩾ n + 1 and i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  The adjacency for the second network satisfies w[2]  ij = 1 only if (i) i = n + 1 and j > n + 1, or  i > n + 1 and j = n + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' (ii) i = n and j = n + m, or i = n + m and j = n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' or (iii) i, j ⩽ n  and i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Using the results of the previous section, we can directly calculate π and τ and calculate the  critical benefit-to-cost ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Here, we provide explicit mathematical results for representative  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Assuming p = q = 1/ (tN) and letting a := n/ (n + m),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' we find that  �b  c  �∗  =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  (2a2−2a+1)t3+(8a2−8a+7)t2+(8a2−8a+15)t+2a2−2a+10  2a(1−a)(t2+4t+3)  N → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  t3+10t2+26t+19  t2+4t+3  N → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' a = 1  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  20a2−20a+33  16a(1−a)  N → ∞ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' t = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  ¯t(¯t+1)  −2¯t2+2¯t+1N  N → ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' t/N = ¯t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' a = 1  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  �  �  �  210N16−520N15−1034N14+1770N13  +14028N12−93440N11+300848N10−330944N9  −663040N8+2230528N7−1096448N6−4570112N5  +10000384N4−9265152N3+4259840N2−786432N  �  �  �  �  �  �  30N16−79N15+225N14+1756N13  −15088N12−13128N11+247296N10−365152N9  −849344N8+2987392N7−1801984N6−5024768N5  +11302912N4−9949184N3+3940352N2−327680N−131072  �  �  �  a = 1  2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (14)  20  In particular, when N → ∞ and a = 1/2, (b/c)∗ is a monotonically increasing function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  We can compare this critical ratio to that of just a single network, which is the same for either  network 1 or network 2 and satisfies  �b  c  �∗  =  �  �  �  �  �  − (1 − a) N  N → ∞,  −3N9−40N8−204N7−848N6−2464N5+1920N4+15872N3−40960N2+24576N  6N8−64N7−520N6−1232N5+3872N4+6272N3−24320N2+22528N+4096  a = 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (15)  21  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='9  1  106  104  102  0  102  104  106  Supplementary Figure 1: The cooperation-promoting effects of structure transitions as the sizes of the two  cliques vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The dynamic network is illustrated in Figure 2a, with a fraction a (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' 1 − a) of nodes in the top  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' bottom) clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The critical benefit-to-cost ratio, (b/c)∗, is shown as a function of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The dots are the results  of numerical calculations with N = 10,000 and the lines are analytical approximations for sufficiently large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The  rescaled duration is t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  22  50  100  150  200  250  300  350  400  0  10  20  30  40  50  a  Network 1  Dynamic  10-1  100  101  102  103  104  0  5  10  15  20  25  b  50  100  150  200  250  300  350  400  0  10  20  30  40  50  60  c  10-1  100  101  102  103  104  0  5  10  15  20  25  30  d  Supplementary Figure 2: Cooperation-promoting effects of dynamic multi-clique networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We consider  networks made up of eight cliques connected via hub nodes (see Figure 5a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' panels a and b here) and via leaf nodes  (see Figure 5b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' panels c and d here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' a,c, The critical ratio (b/c)∗ as a function of population size N, for the rescaled  duration t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' b,d, The critical ratio (b/c)∗ as a function of the rescaled duration t, for N = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  23  a  b  Supplementary Figure 3: Cooperation-promoting effects of structure transitions among more than two net-  works, and when networks differ in a small fraction of connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' a, Structure transitions among three net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Every network transitions to another network with probability 1/ (2tN) and remains unchanged otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  b, Structure transitions between multi-clique networks in which the two networks differ in only two cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We  take N = 150 in a and N = 64 in b, and the rescaled duration is t = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  24  10-1  100  101  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='05  a  10-1  100  101  0  50  100  150  200  250  300  350  400  b  Supplementary Figure 4: Dynamic networks promote and accelerate the fixation of cooperators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We consider  the network with a star clique and a complete-graph clique with N = 16 and a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5 (see Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' a, Fixation  probability of cooperators as a function of the rescaled duration, t, in network 1 and in the dynamic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The  dynamic network leads to the larger fixation probability of cooperators than in network 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' b, Conditional and  unconditional fixation times as functions of the rescaled duration, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Both the conditional and unconditional times  in the dynamic networks are smaller than in network 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='We take selection intensity δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  25  Supplementary Information  SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1  Modeling evolution on dynamic networks  SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1  Assumptions, definitions, and notation  We consider a population of N individuals (labeled N = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , N}), residing at any point in  time on one of L structures (labeled L = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , L}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Implicitly, this means that each of these  L structures is a network on N nodes, although each network need not be connected and some  nodes can be isolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Each individual has type A or B, and the state of population is tracked by  a pair (x, β) ∈ {0, 1}N × L, where xi = 1 means i has type A and xi = 0 means i has type B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  At each time step, a set of individuals to be replaced, R ⊆ N , is chosen, together with an  offspring-to-parent map, α : R → N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Let p(R,α) (x, β) denote the probability of replacement  event (R, α) in state (x, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Once (R, α) is chosen, the type configuration, x, is updated to y,  where yi = xα(i) if i ∈ R and yi = xi if i ̸∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' This update can be specified more succinctly  using an extended mapping �α : N → N defined by �α (j) = α (j) if j ∈ R and �α (j) = j if  j ̸∈ R, which leads to the updated state x�α, where (x�α)i = x�α(i) for i ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The network, β, is  updated via a transition matrix, Q =  �  qβγ  �  β,γ∈L, where qβγ is the probability of transitioning  from network β to network γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' An important feature of the model is that network transitions are  independent of x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' thus, the population structure is exogenous and not influenced by traits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We  assume that Q is irreducible, which guarantees that it has a unique stationary distribution, υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  We assume that for each replacement event, (R, α), type configuration, x, and network, β, the  probability p(R,α) (x, β) is a smooth function of a selection intensity parameter, δ ⩾ 0, in a small  neighborhood of δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Moreover, when δ = 0 (“neutral drift”), we assume that p(R,α) (x, β) is  independent of x (but it can depend on β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We denote by p◦  (R,α) (β) the probability of choosing  (R, α) under neutral drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The chain defined by Q does not depend on the selection intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  We also make the following assumption, which ensures that for every starting configuration  and network, there exists at least one individual whose lineage can take over the population:  Fixation Axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For all network structures β0 ∈ L, there exists a location i ∈ N , an integer  m ⩾ 1, and sequences of replacement events {(Rk, αk)}m  k=1 and networks {βk}m−1  k=1 for which  (i) p(Rk,αk) (x, βk−1) > 0 for every k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , m} and x ∈ {0, 1}N ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (ii) qβk−1βk > 0 for every k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , m − 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (iii) i ∈ Rk for some k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , m};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (iv) �α1 ◦ �α2 ◦ · · · ◦ �αm (j) = i for all locations j ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  These conditions are similar to those used by Allen & McAvoy13 and McAvoy & Allen18,  except here it is modified to account for dynamic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Informally, it guarantees that no  individual lives forever and that the process eventually reaches a state in which all individuals  are identical by descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We note that here it does not require each network to be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  26  Since there is no mutation of traits, all individuals must have the same type when they  are identical by descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The configurations A := (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , 1) and B := (0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , 0) are  the only absorbing configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' (Note that while the configuration of types cannot leave A  or B, the state itself, which includes the network structure, can still change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=') We denote by  BN the set of all configurations, {0, 1}N , and by BN  ⊺ the set of all transient configurations,  {0, 1}N − {A, B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' From the Fixation Axiom, we see that given any starting configuration-  network pair, (x, β) ∈ BN × L, there is a well-defined probability, ρA (x, β) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' ρB (x, β)),  that the population eventually reaches the monomorphic state A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The behavior of these  fixation probabilities (under weak selection, meaning δ ≪ 1) is the main focus of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  We follow the workflow proposed by McAvoy & Allen18 for analyzing mutation-free evo-  lutionary dynamics under weak selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We first study the assortment of traits under neutral  drift (δ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Subsequently, we link these findings to the game using a martingale perturba-  tion argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We avoid reproducing the entire derivation in18;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' instead, we highlight the main  modifications to those arguments necessary to accommodate stochastic network transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='2  Network-mediated reproductive value  With the main assumptions in place, we now introduce some derived, demographic quantities  that we will refer to throughout the analysis of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' If the population is in state (x, β), then  the marginal probability that i produces an offspring that replaces j in the next update is  eij (x, β) :=  ∑  (R,α)  j∈R, α(j)=i  p(R,α) (x, β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1)  The expected change in the abundance of A in state (x, β) can be expressed as  ∆ (x, β) := ∑  i∈N  xi ∑  j∈N  eij (x, β) + ∑  i∈N  xi  �  1 − ∑  j∈N  eji (x, β)  �  − ∑  i∈N  xi  = ∑  i,j∈N  eji (x, β)  �  xj − xi  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='2)  One inconvenient aspect of dealing with the true abundance of A is that it is generally not  a martingale under neutral drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' This property is well-known even in models without dynamic  structure13 and it necessitates working with a weighted frequency instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The notion of repro-  ductive value, which can be (informally) interpreted as the expected contribution of an individual  to future generations, turns out to give the proper weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For our purposes, we interpret the  reproductive value of i ∈ N as the probability that, under neutral drift, i generates a lineage  that eventually takes over the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Because our interest is in fixation probabilities in the  first place, it is not surprising that such a quantity should appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' This quantity depends on the  network structure, but it is independent of the type configuration due to the drift assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Formally, we define the reproductive value of i in network β, denoted π[β]  i , to be the proba-  bility that under neutral drift and starting in structure β, a mutant in node i eventually takes over  the whole population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Let e◦  ij (β) denote the probability, that under neutral drift and in structure  27  β, individual i spreads her strategy to j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' A one-step analysis of the neutral Markov chain gives  π[β]  i  = ∑  j∈N  e◦  ij (β) ∑  γ∈L  qβγπ[γ]  j  +  �  1 − ∑  j∈N  e◦  ji (β)  �  ∑  γ∈L  qβγπ[γ]  i  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='3a)  ∑  i∈N  π[β]  i  = 1  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='3b)  for all i ∈ N and β ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' There is one point of subtlety in relation to reproductive value on  static networks, which relates to the normalization condition ∑i∈N π[β]  i  = 1 for all β ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  The Fixation Axiom guarantees that there is a unique π satisfying Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='3a up to a  scalar multiple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In this case, for any fixed C ∈ R, requiring ∑i∈N ∑β∈L π[β]  i  = C yields  a unique solution to Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Summing both sides of Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='3a over i ∈ N  yields ∑i∈N π[β]  i  = ∑γ∈L qβγ ∑i∈N π[γ]  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Since the chain Q is irreducible, it follows that  ∑i∈N π[β]  i  is independent of β ∈ L, and thus it must be equal to C/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Therefore, asserting  that ∑i∈N ∑β∈L π[β]  i  = L is equivalent to the requirement that ∑i∈N π[β]  i  = 1 for all β ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  As a result, π, which we refer to as network-mediated reproductive value due to its dependence  on network transitions, is uniquely defined by Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Finally, the change in ∑i∈N π[β]  i xi, the π-weighted abundance of A, is  �∆ (x, β) = ∑  i∈N  xi ∑  j∈N  eij (x, β) ∑  γ∈L  qβγπ[γ]  j  + ∑  i∈N  xi  �  1 − ∑  j∈N  eji (x, β)  �  ∑  γ∈L  qβγπ[γ]  i  − ∑  i∈N  π[β]  i xi  = ∑  i,j∈N  eji (x, β) ∑  γ∈L  qβγπ[γ]  i  �  xj − xi  � + ∑  i∈N  xi  �  ∑  γ∈L  qβγπ[γ]  i  − π[β]  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='4)  It follows from Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='3 that, under neutral drift, �∆◦ (x, β) = 0, for all x ∈ BN and  β ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' This property will play a key role in our subsequent weak-selection analysis of the  process (Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='3  A mutation-modified evolutionary process  The process under consideration is mutation-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' However, following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='18, in order to get  an idea of the assortment of types prior to hitting an absorbing configuration, it is convenient  to introduce an artificial mutation that makes the chain ergodic and gives it a unique stationary  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The idea is to choose a state (z, λ) with z ∈ BN  ⊺ , and let mutations bring absorbing  configurations into (z, λ) with some small probability u > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' If P(x,β)→(y,γ) denotes the proba-  bility of transitioning from (x, β) to (y, γ) in the original (mutation-free) chain over the course  28  of one time step, then the transition probabilities for the mutation-modified chain are given by  P⟳(z,λ)  (x,β)→(y,γ) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  u  x ∈ {A, B} , (y, γ) = (z, λ) ,  (1 − u) P(x,β)→(y,γ)  x ∈ {A, B} , (y, γ) ̸= (z, λ) ,  P(x,β)→(y,γ)  x ̸∈ {A, B} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5)  As a result of the Fixation Axiom, there is a unique stationary distribution, π⟳(z,λ), such that  π◦  ⟳(z,λ) (x, β) = ∑  γ∈L  �  π◦  ⟳(z,λ) (A, γ) P⟳(z,λ)  (A,γ)→(x,β) + π◦  ⟳(z,λ) (B, γ) P⟳(z,λ)  (B,γ)→(x,β)  �  + ∑  y∈BN  ⊺  ∑  γ∈L  π◦  ⟳(z,λ) (y, γ) P⟳(z,λ)  (y,γ)→(x,β)  = ∑  γ∈L  π◦  ⟳(z,λ) (A, γ)  �  uδz,xδλ,β + (1 − u) δA,xqγβ  �  + ∑  γ∈L  π◦  ⟳(z,λ) (B, γ)  �  uδz,xδλ,β + (1 − u) δB,xqγβ  �  + ∑  y∈BN  ⊺  ∑  γ∈L  π◦  ⟳(z,λ) (y, γ) P(y,γ)→(x,β)  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='6)  for all x ∈ B and β ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  In one step after state (x, β), the expected change in the π-weighted abundance of A is  �∆⟳(z,λ) (x, β) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  −u  �  1 − ∑i∈N π[λ]  i  zi  �  x = A,  u ∑i∈N π[λ]  i  zi  x = B,  �∆ (x, β)  x ̸∈ {A, B} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='7)  Averaging this expected change over the stationary distribution of the modified chain gives  0 = E⟳(z,λ)  �  �∆⟳(z,λ)  �  = E⟳(z,λ)  �  �∆  �  − u ∑  β∈L  π⟳(z,λ) (A, β)  �  1 − ∑  i∈N  π[λ]  i  zi  �  + u ∑  β∈L  π⟳(z,λ) (B, β) ∑  i∈N  π[λ]  i  zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='8)  Owing to a result of Fudenberg & Imhof53, we know that, in the low-mutation limit,  lim  u→0 ∑  β∈L  π⟳(z,λ) (A, β) = ρA (z, λ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='9a)  lim  u→0 ∑  β∈L  π⟳(z,λ) (B, β) = ρB (z, λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='9b)  29  Therefore, taking the derivative of both sides of Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='8 with respect to u at u = 0 gives  ρA (z, λ) = ∑  i∈N  π[λ]  i  zi + d  du  �����  u=0  E⟳(z,λ)  �  �∆  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='10)  Let ⟨·⟩(z,λ) :=  d  du  ���  u=0E⟳(z,λ) [·].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' By the argument given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='18 Corollary 1, we see that  for any function ϕ : BN × L → R satisfying ϕ (A, β) = ϕ (B, β) = 0 for all β ∈ L,  ⟨ϕ⟩(z,λ) =  ∞  ∑  t=0  E  �  ϕ  �  xt, βt� |  �  x0, β0�  = (z, λ)  �  ,  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='11)  where the summation on the right-hand side converges absolutely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In particular, this equation  holds for the expected change in the π-weighted abundance of A, ϕ = �∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Since we also have  d  dδ  �����  δ=0  eij (x, β) = ∑  I⊆N  cij  I (β) xI  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='12)  for unique coefficients cij  I (β), where xI := ∏i∈I xi, it follows that  d  dδ  �����  δ=0  ρA (z, λ) = d  dδ  �����  δ=0  �  �∆  �  (z,λ)  =  �  d  dδ  �����  δ=0  �∆  �◦  (z,λ)  =  �  d  dδ  �����  δ=0  ∑  i,j∈N  eji (x, β) ∑  γ∈L  qβγπ[γ]  i  �  xj − xi  �  �◦  (z,λ)  = ∑  i,j∈N ∑  I⊆N  �  cji  I (β) ∑  γ∈L  qβγπ[γ]  i  �  xI∪{j} − xI∪{i}  ��◦  (z,λ)  ,  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='13)  where the interchange of the two limits is possible due to Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='11 and the absolute con-  vergence of its summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The second line of Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='13 is where we use the fact that  �∆0 (x, β) = 0 for all x ∈ BN and β ∈ L, highlighting the importance of network-mediated  reproductive value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  As a result of these calculations, what remains in order to understand the first-order effects  of selection on a mutant type’s fixation probability is an analysis of the neutral operator ⟨·⟩◦  (z,λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='4  Analysis of neutral drift  Throughout this section, we denote the stationary distribution of the structure-transition chain,  Q, by υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We also suppress either the configuration or the network when we marginalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For ex-  ample, we write π⟳(z,λ) (x) for ∑β∈L π⟳(z,λ) (x, β) and π⟳(z,λ) (β) for ∑x∈BN π⟳(z,λ) (x, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  In the limit of low mutation, we know π◦  ⟳(z,λ) (A) converges to ρ◦  A (z, λ) and π◦  ⟳(z,λ) (B)  converges to ρ◦  B (z, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The following lemma is a slightly stronger version of this result:  30  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For all networks β ∈ L,  lim  u→0 π◦  ⟳(z,λ) (A, β) = ρ◦  A (z, λ) υ (β) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='14a)  lim  u→0 π◦  ⟳(z,λ) (B, β) = ρ◦  B (z, λ) υ (β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='14b)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Letting x = A in Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='6 and taking u → 0 gives  lim  u→0 π◦  ⟳(z,λ) (A, β) = ∑  γ∈L  �  lim  u→0 π◦  ⟳(z,λ) (A, γ)  �  qγβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='15)  It follows that limu→0 π◦  ⟳(z,λ) (A, β) is proportional to υ (β), for all β ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The constant of  proportionality must be ρ◦  A (z, λ) due to the fact that limu→0 π◦  ⟳(z,λ) (A) = ρ◦  A (z, λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' The  result for limu→0 π◦  ⟳(z,λ) (B, β) follows from analogous reasoning and is omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Neutral fixation probabilities, ρ◦  A (z, λ) and ρ◦  B (z, λ), can be calculated using re-  productive values and the identities ρ◦  A (z, λ) = ∑i∈N π[λ]  i  zi and ρ◦  B (z, λ) = 1 − ∑i∈N π[λ]  i  zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  The following is an immediate consequence of Lemma 1:  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' limu→0 π◦  ⟳(z,λ) (β) = υ (β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  The next lemma establishes a recurrence for d  du  ���  u=0π◦  ⟳(z,λ) (β):  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For every β, we have  d  du  �����  u=0  π◦  ⟳(z,λ) (β) = δβ,λ − υ (β) + ∑  γ∈L  �  d  du  �����  u=0  π◦  ⟳(z,λ) (γ)  �  qγβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='16)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Summing both sides of Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='6 over all x ∈ BN gives  π◦  ⟳(z,λ) (β) = u ∑  γ∈L  �  π◦  ⟳(z,λ) (A, γ) + π◦  ⟳(z,λ) (B, γ)  � �  δβ,λ − qγβ  �  + ∑  γ∈L  π◦  ⟳(z,λ) (γ) qγβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='17)  Differentiating this equation with respect to u at u = 0 and using Lemma 1 yields Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Since the state of the process consists of both a configuration of traits and a network structure,  the next result gives a recurrence for calculating a modified version of ⟨·⟩◦  (z,λ), using conditioning  on the network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' In particular, for a function ϕ : BN → R defined on just configura-  tions, we let ⟨ϕ | β⟩◦  (z,λ) =  d  du  ���  u=0E◦  ⟳(z,λ) [ϕ | β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' This quantity can be calculated as follows:  31  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For every function ϕ : BN → R, we have  υ (β) ⟨ϕ | β⟩◦  (z,λ) = δλ,β (ϕ (z) − ρ◦  A (z, λ) ϕ (A) − ρ◦  B (z, λ) ϕ (B))  + ∑  γ∈L  υ (γ) ∑  (R,α)  p◦  (R,α) (γ) qγβ ⟨ϕ�α | γ⟩◦  (z,λ) ,  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='18)  where, for �α : N → N , ϕ�α : BN → R is the map defined by ϕ�α (x) = ϕ (x�α) for x ∈ BN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For x ∈ BN  ⊺ , differentiating both sides of Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='6 with respect to u at u = 0 gives  d  du  �����  u=0  π◦  ⟳(z,λ) (x, β)  = δz,xδλ,β + ∑  y∈BN  ⊺  ∑  γ∈L  �  d  du  �����  u=0  π◦  ⟳(z,λ) (y, γ)  �  P◦  (y,γ)→(x,β)  = δz,xδλ,β + ∑  y∈BN  ⊺  ∑  γ∈L  �  d  du  �����  u=0  π◦  ⟳(z,λ) (y, γ)  �  ∑  (R,α)  y�α=x  p◦  (R,α) (γ) qγβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='19)  Doing so for x ∈ {A, B} gives  d  du  �����  u=0  π◦  ⟳(z,λ) (A, β) = ∑  γ∈L  �  d  du  �����  u=0  π◦  ⟳(z,λ) (A, γ)  �  qγβ − ρ◦  A (z, λ) υ (β)  + ∑  y∈BN  ⊺  ∑  γ∈L  �  d  du  �����  u=0  π◦  ⟳(z,λ) (y, γ)  �  ∑  (R,α)  y�α=A  p◦  (R,α) (γ) qγβ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='20a)  d  du  �����  u=0  π◦  ⟳(z,λ) (B, β) = ∑  γ∈L  �  d  du  �����  u=0  π◦  ⟳(z,λ) (B, γ)  �  qγβ − ρ◦  B (z, λ) υ (β)  + ∑  y∈BN  ⊺  ∑  γ∈L  �  d  du  �����  u=0  π◦  ⟳(z,λ) (y, γ)  �  ∑  (R,α)  y�α=B  p◦  (R,α) (γ) qγβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='20b)  If ϕ : BN → R is a fixed function, then, by definition,  υ (β) ⟨ϕ | β⟩◦  (z,λ) = ∑  x∈BN  υ (β) d  du  �����  u=0  π◦  ⟳(z,λ) (x, β)  π◦  ⟳(z,λ) (β) ϕ (x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='21)  Combining Lemma 2 and Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='19–SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='20 with the fact that  υ (β) d  du  �����  u=0  π◦  ⟳(z,λ) (x, β)  π◦  ⟳(z,λ) (β)  = d  du  �����  u=0  π◦  ⟳(z,λ) (x, β) − (δA,xρ◦  A (z, λ) + δB,xρ◦  B (z, λ)) d  du  �����  u=0  π◦  ⟳(z,λ) (β) (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='22)  then gives Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='18 after some tedious but straightforward simplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  32  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' With I ⊆ N and η[β]  I  (z, λ) := υ (β)  �  ∑i∈N π[β]  i xi − xI | β  �◦  (z,λ), we have  η[β]  I  (z, λ) = δλ,β  �  ∑  i∈N  π[β]  i zi − zI  �  + ∑  γ∈L ∑  (R,α)  p◦  (R,α) (γ) qγβη[γ]  �α(I) (z, λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='23)  Subject to ∑i∈N π[β]  i η[β]  i  (z, λ) = 0 for some β ∈ L, the solution to Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='23 is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Setting ϕ (x) = ∑i∈N π[β]  i xi − xI in Proposition 1 gives Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Conversely,  we know that η[β]  I  (z, λ) := υ (β)  �  ∑i∈N π[β]  i xi − xI | β  �◦  (z,λ) solves Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='23, so that  there is at least one solution to Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' By the Fixation Axiom, the dimensionality of  the space of solutions to Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='23 is determined by that of the case |I| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' (The reason  is that all subsets of size greater than one are transient under the ancestral process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=') Specifically,  the recurrence for I = {i} is  η[β]  i  (z, λ) = δλ,β (ρ◦  A (z, λ) − zi) + ∑  γ∈L ∑  j∈N  e◦  ji (γ) qγβη[γ]  j  (z, λ)  + ∑  γ∈L  �  1 − ∑  j∈N  e◦  ji (γ)  �  qγβη[γ]  i  (z, λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='24)  If �η (z, λ) is another solution to Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='24, then χ (z, λ) := η (z, λ) − �η (z, λ) satisfies  χ[β]  i  (z, λ) = ∑  γ∈L ∑  j∈N  e◦  ji (γ) qγβχ[γ]  j  (z, λ) + ∑  γ∈L  �  1 − ∑  j∈N  e◦  ji (γ)  �  qγβχ[γ]  i  (z, λ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='25)  Noting that any constant function is a solution to Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='25, and the space of solutions to  this equation is one-dimensional as a result of the Fixation Axiom, there must exist K ∈ R  such that η (z, λ) = �η (z, λ) + K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Since the solution η[β]  i  (z, λ) = υ (β) ⟨xi | β⟩◦  (z,λ) satisfies  ∑i∈N π[β]  i η[β]  i  (z, λ) = 0 for all β ∈ L, it follows that K = 0 and η (z, λ) = �η (z, λ) whenever  �η (z, λ) satisfies Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='23 and ∑i∈N η[β]  i  �η[β]  i  (z, λ) = 0 for some β ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We note that  ∑i∈N π[β]  i η[β]  i  (z, λ) = 0 for some β ∈ L ensures that this equation holds for all β ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  33  SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5  Calculating first-order effects of selection on fixation probabilities  SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1  Fixed initial configurations  Note that for functions ϕ : BN → R and φ : L → R, we have  ⟨φϕ⟩◦  (z,λ) = d  du  �����  u=0  ∑  β∈L  π◦  ⟳(z,λ) (β) φ (β) E◦  ⟳(z,λ) [ϕ | β]  = ∑  β∈L  υ (β) φ (β) ⟨ϕ | β⟩◦  (z,λ)  + (ρ◦  A (z, λ) ϕ (A) + ρ◦  B (z, λ) ϕ (B)) ∑  β∈L  φ (β) d  du  �����  u=0  π◦  ⟳(z,λ) (β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='26)  Therefore, we may rewrite Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='13 as  d  dδ  �����  δ=0  ρA (z, λ)  = ∑  i,j∈N ∑  I⊆N  �  cji  I (β) ∑γ∈L qβγπ[γ]  i  �  ��  �  φ  �  xI∪{j} − xI∪{i}  �  ��  �  ϕ  ��◦  (z,λ)  = ∑  i,j∈N ∑  I⊆N ∑  β∈L  υ (β) cji  I (β) ∑  γ∈L  qβγπ[γ]  i  ��  xI∪{j} | β  �◦  (z,λ) −  �  xI∪{i} | β  �◦  (z,λ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='27)  Defining η[β]  I  (z, λ) := υ (β)  �  ∑i∈N π[β]  i xi − xI | β  �◦  (z,λ), we then have  d  dδ  �����  δ=0  ρA (z, λ) = ∑  i,j∈N ∑  I⊆N ∑  β∈L  cji  I (β) ∑  γ∈L  qβγπ[γ]  i  �  η[β]  I∪{i} (z, λ) − η[β]  I∪{j} (z, λ)  �  ,  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='28)  where, by Corollary 2, the terms η are uniquely determined by  η[β]  I  (z, λ) = δλ,β  �  ∑  i∈N  π[β]  i zi − zI  �  + ∑  γ∈L ∑  (R,α)  p◦  (R,α) (γ) qγβη[γ]  �α(I) (z, λ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='29a)  ∑  i∈N  π[β]  i η[β]  i  (z, λ) = 0 for some β ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='29b)  SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='2  Probabilistic initial configurations  Up until this point, we have focused on fixation probabilities given some fixed initial state,  (z, λ) ∈ N × L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We now allow mutant types to arise stochastically and consider mean fix-  34  ation probabilities for both types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' For two distributions, µA, µB ∈ ∆  �  BN  ⊺ × L  �  , we let  ρA (µA) := E(z,λ)∼µA [ρA (z, λ)] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='30a)  ρB (µB) := E(z,λ)∼µB [ρB (z, λ)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='30b)  By the results of §SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='1, for any µ ∈ ∆  �  BN  ⊺ × L  �  , we have  d  dδ  �����  δ=0  ρA (µ) = ∑  i,j∈N ∑  I⊆N ∑  β∈L  cji  I (β) ∑  γ∈L  qβγπ[γ]  i  �  η[β]  I∪{i} (µ) − η[β]  I∪{j} (µ)  �  ,  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='31)  where  η[β]  I  (µ) = E(z,λ)∼µ  �  δλ,β  �  ∑  i∈N  π[β]  i zi − zI  ��  + ∑  γ∈L ∑  (R,α)  p◦  (R,α) (γ) qγβη[γ]  �α(I) (µ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='32a)  ∑  i∈N  π[β]  i η[β]  i  (µ) = 0 for some β ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='32b)  Letting µ = µA gives the mean fixation probability for type A, while the mean fixation proba-  bility for type B can be calculated analogously using the equation ρB (µB) = 1 − ρA (µB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  Although the main focus of our study is on network-transition chains that are both aperiodic  and irreducible, we do also consider periodic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Suppose that among the L networks in  L, network β transitions deterministically to network β + 1 for β ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' , L − 1}, and network  L transitions deterministically to network 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' We can then write Equation SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='32 more explicitly  as  η[1]  I (µ) = E(z,λ)∼µ  �  δλ,1  �  ∑  i∈N  π[1]  i zi − zI  ��  + ∑  (R,α)  p◦  (R,α) (L) η[L]  �α(I) (µ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='33a)  η[β]  I  (µ) = E(z,λ)∼µ  �  δλ,β  �  ∑  i∈N  π[β]  i zi − zI  ��  + ∑  (R,α)  p◦  (R,α) (β − 1) η[β−1]  �α(I) (µ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (1 < β ⩽ L)  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='33b)  ∑  i∈N  π[β]  i η[β]  i  (µ) = 0 for some β ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  (SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='33c)  References  [1] Ohtsuki, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=', Hauert, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=', Lieberman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' & Nowak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' A simple rule for the evolution of  cooperation on graphs and social networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Nature 441, 502–505 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content='  [2] Holme, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' & Saram¨aki, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Temporal networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
+page_content=' Physics Reports 519, 97–125 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
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+page_content=', R´acz, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tFLT4oBgHgl3EQfEC7f/content/2301.11982v1.pdf'}
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+Submitted to Transportation Science
+manuscript 1
+Authors are encouraged to submit new papers to INFORMS journals by means of a style file template,
+which includes the journal title. However, use of a template does not certify that the paper has been
+accepted for publication in the named journal. INFORMS journal templates are for the exclusive
+purpose of submitting to an INFORMS journal and should not be used to distribute the papers in print
+or online or to submit the papers to another publication.
+Multimodal Transportation Alliance Design with
+Endogenous Demand: Large-Scale Optimization for Rapid
+Gains
+Kayla Cummings*, Vikrant Vaze†, ¨Ozlem Ergun‡, Cynthia Barnhart*
+Transit agencies have the opportunity to outsource certain services to well-established platform-based Mobility on
+Demand (MOD) providers. Such alliances can improve service quality, coverage, and ridership; reduce public sector
+costs and vehicular emissions; and integrate the passenger experience. To amplify the effectiveness of such alliances,
+we develop a fare-setting model that jointly optimizes discounted fares across a multimodal network. We capture com-
+muters’ travel choices with a discrete choice model, resulting in a large-scale, mixed-integer, non-convex optimization
+problem. To solve this challenging problem, we develop a two-stage decomposition with the pricing decisions in the first
+stage and a mixed-integer linear optimization problem optimizing fare discounts and the induced passenger behaviors
+in the second stage. To solve the decomposition, we develop a new solution approach combining tailored coordinate
+descent, parsimonious second-stage evaluations, and interpolations using special ordered sets. This approach, enhanced
+by acceleration techniques based on slanted traversal, randomization and warm-start, significantly improves system-wide
+practical outcomes over algorithmic benchmarks. Different alliance priorities result in qualitatively different fare designs:
+flat fares decrease the total vehicle-miles traveled, while geographically-informed discounts improve passenger happi-
+ness. The model responds appropriately to equity-oriented and passenger-centric priorities, improving system utilization
+and lowering prices for low-income residents and long-distance commuters. Finally, our revenue allocation mechanism
+improves outcomes for both operators, thus incentivizing profit-oriented MOD operators to adopt transit priorities.
+Key words: Public transit, transportation pricing, alliance design, mixed-integer non-convex optimization
+1.
+Introduction
+Cities face critical challenges in the quest to improve urban mobility. Prior to the pandemic, congestion was
+steadily rising, translating to $160 billion annual costs to U.S. cities and record-breaking contributions to
+greenhouse gas emissions (Schrank et al. 2015). Recent declines in transit ridership demonstrate the inabil-
+ity of transit’s static infrastructure to accommodate rapidly evolving commuting patterns (The Economist
+2018). Private ride-sharing apps from Transportation Network Companies (TNCs) like Uber and Lyft have
+* Massachusetts Institute of Technology
+† Dartmouth College
+‡ Northeastern University
+1
+arXiv:2301.03414v1  [math.OC]  9 Jan 2023
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+2
+Article submitted to Transportation Science; manuscript no. 1
+challenged this fixed-infrastructure status quo. TNCs transported 2.6 billion passengers in 2017, more than
+doubling the ride-sharing market since 2012 (Schaller 2018). The majority of urban TNC patrons admit that
+they would have otherwise walked, biked, taken public transit, or not made the trip, coinciding with tens
+of millions in annual transit revenue losses, worsening congestion, higher emissions, lower navigability of
+cities, and reduced accessibility to affordable public options (Gehrke and Reardon 2018, Schaller 2018).
+Mobility-on-demand (MOD) services have the potential to service transit deserts—low-density areas
+disconnected from public transit. However, cost presents a key barrier: while all public transit modes operate
+at a loss, MOD services administered by transit agencies incur the highest average per-trip costs ($23.10 vs.
+the next-highest $11.19 for commuter rail) (Kane, Tomer, and Puentes 2016). High labor needs, outdated
+technology, and coordination difficulties lead to inefficient, expensive operations. Notably, average TNC
+trip costs $13, a full $10 less than agency-sponsored MOD trips. Outsourcing all 223 million on-demand
+transit trips to TNCs could hypothetically save billions of dollars for US transit agencies (Kane, Tomer, and
+Puentes 2016). Thus, pricing alliances between TNCs and transit agencies have the potential to improve
+service quality and coverage, while reducing costs and decreasing citywide vehicle-miles travelled (VMT).
+1.1.
+Pricing Alliances in the Real World
+TNCs have the infrastructure to provide more cost-effective MOD services supplementing fixed-route tran-
+sit. Microtransit platforms like BRIDJ and Via—differentiated from TNCs due to fleets comprising mini-
+vans or shuttles as opposed to sedans—have oriented their business model toward complementing transit
+(Via Transp. 2023, BRIDJ 2023). The Federal Transit Administration’s MOD Sandbox program has pro-
+vided millions in funding to transit agencies in US cities such as Dallas, San Francisco, and Los Angeles
+to develop on-demand pilots that fill service gaps in their service regions (Federal Transp. Administration
+2016). Rather than designing a complementary MOD system from scratch and incurring high fixed costs,
+transit agencies could outsource MOD services to TNC platforms that are well-established, highly con-
+nected, and widely trusted; or to microtransit platforms that more closely align with transit agency goals.
+This section formalizes such alliances within a rigorous conceptual framework. We define a pricing
+alliance as a cooperative pricing scheme between a transit agency and an MOD operator with independently
+operated infrastructures serving overlapping or adjacent regions. A pricing alliance seeks to improve each
+operator’s own prioritized metrics whilst also improving system-wide benefits through integration. Indeed,
+real-world pricing alliance pilots have shown great promise in improving regional mobility, providing alter-
+natives with higher service levels, lower fares and increased ridership (The Boston Globe 2022, Mag 2021).
+Pricing alliances are characterized by the intended service populations and the relationship of the MOD
+operator’s system to the fixed-route transit network. Service population may be a targeted demographic,
+e.g. persons with limited mobility, low-income people, or senior citizens; the service population might also
+constitute residents of a particular geographic area, e.g. residents of a transit desert or people traveling
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+3
+(a) GoLink in Dallas, TX (Plano region) utilizes a zone-
+based route structure.
+(b) The NewMo Pilot in Newton, MA utilized a hub-based route
+structure.
+Figure 1
+Route structures of recent pricing alliances. In Plano (Dallas Area Rapid Transit 2020), passengers
+spend up to $3 to travel anywhere within a color-blocked region. In the NewMo Pilot (City of New-
+ton, MA 2021), passengers could travel anywhere within town limits for $2, as long as either trip
+endpoint was one of seven hubs. NewMo now allows passengers to travel anywhere within Newton.
+within a given radius of a transit hub. The MOD infrastructure may complement, substitute, or extend fixed-
+route options. Once participating operators establish the nature of the pricing alliance, the alliance can select
+a joint pricing scheme. Carefully designed fares influence passenger behavior, incentivizing choices that
+benefit the entire system. Table 1 surveys these traits of recent pricing alliances:
+• MOD operator: This can be a TNC like Uber or Lyft, or a microtransit platform like Via.
+• Service population: Recent alliances have served people with limited mobility, seniors, essential work-
+ers, or simply everyone.
+• Fare structure: For passengers traveling within the system, some alliances charge a flat fare and/or
+a variable fare based on distance travelled. Selectively applied, interpretable discount structures for
+jointly offered routes can encourage multimodal travel and engineer outcomes desired by the operators.
+• Route structure: Alliances often require specific trip geography: point-to-point (PTP) (trips must occur
+within a given geographic region), zone-based (partitions a larger region into small zones and requires
+intra-zonal trips), and hub-based (at least one trip endpoint must be anchored at specified locations).
+Figure 1 illustrates zone-based and hub-based route structures.
+• Integration into public transit network: The MOD portion of the network might integrate into the
+transit network in several ways: complementary (provides another mode option to improve service
+quality), substitutive (replaces existing fixed-route transit), first-/last-mile (FLM) (connects travelers
+to the fixed-route network), and extension (serves transit desert regions).
+1.2.
+Literature Review
+This work sits at the intersection of literature on FLM system design and operations management, integrated
+multimodal transport system design, and horizontal cooperation among competing transportation operators.
+
+MobilityOptionsinPlano
+OpcionesdemovilidadenlaszonasdePlano
+CISEMONTC
+WLIS
+CHASEOUKS BIVD
+NWPLANO
+PARK&RIDE
+NONEN
+Mapnottoscale
+Elm30ano
+representa
+laestala
+PLRER
+Travel between zones is not permitted.
+N
+PARKERRD
+STATION
+LegacyWestServiceArea Zone
+Noesta permitidoviajardirectoentrezonas.
+PARXBVO
+LegacyWest Zona del area de servicio
+LegacyWestserves Northwest
+FarNorth Plano serves Parker Road
+North Central Plano/Chase Oaks
+Far North Plano Service Area Zone
+PlanoPark&Ride,forconnections
+Station,forconnectionstoDART rail
+servesParker RoadStation,for
+FarNorthPlano Zonadelareadeservicio
+to DART buses.
+and buses.
+connections to DART rail and buses.
+Legacy Westsinve la Northwest Plano
+Far North Plano sirve la Parkor Road
+NorthCentral Plano/Chase Oaks sinve la
+NorthCentral Plano/Chase Oaks ServiceAreaZone
+Park&Ride,para conexionesalos
+Station,paraconexiones alostrenesy
+ParkerRoadStation,para conexionesalos
+NorthCentral Plano/ChaseOaksZonadelareadesenvicig
+sutobusos de DART.
+autobuses de DART.
+trenesy autobusos de DART.
+DARTTransitFacility
+Terminal de DART
+DARTRed&OrangeLines
+DART lineas Rojas yNaranjas
+ CollinCollege-SpringCreekCampusWaltham
+Watertown
+Eligibledestinations:
+City centers:
+NewtonCentre
+NeedhamStreet
+Newton
+WellsAve/Mt.ldaCampus
+ofUMassAmherst
+Transit hubs:
+NewtonvilleCommuterRail
+ChestnutHillGreenLine
+Newton Highlands
+Green Line
+Needham Heights
+CommuterRail
+NeedhamCummings et al.: Transportation Alliance Design with Endogenous Demand
+4
+Article submitted to Transportation Science; manuscript no. 1
+Table 1
+Survey of recent pricing alliances. PTP: point-to-point. Flat fare: same price for every passenger.
+Mode: fares vary by travel mode. Distance: fares increase with distance traveled. (Dallas Area Rapid Transit
+2020, Regional Transp. Commission of Southern Nevada 2021, City of Seattle, WA 2021, MBTA 2021, City of
+Newton, MA 2021, City of Jersey City, NJ 2021, St. Louis Metro 2021, MARTA 2021, Indianapolis Public Transp.
+Corporation 2021) ∗ Not operated by a transit agency. ∗∗ Specially marketed for senior citizens. ∗∗∗ Available to
+everyone, but only in case of transit closures. ∗∗∗∗ Transported essential workers at the beginning of the
+COVID-19 pandemic.
+Program
+City
+Transit agency
+MOD Op. Service population
+Fares
+Routes Integration
+GoLink
+Dallas, TX
+DART
+Via
+Everyone
+Mode
+Zone
+Complementary, FLM
+RTC On-demand Pilot Las Vegas, NV
+RTC
+Lyft
+Paratransit
+Distance PTP
+Substitutive
+Via to Transit
+Seattle, WA
+King County Metro Via
+Everyone
+Flat fare
+Hub
+Complementary, FLM
+The RIDE Flex
+Boston, MA
+MBTA
+Uber, Lyft
+Paratransit
+Flat fare
+PTP
+Substitutive
+NewMo Pilot
+Newton, MA
+City of Newton∗
+Via
+Everyone, seniors∗∗
+Flat fare
+Hub
+Extension, FLM
+No program name
+Jersey City, NJ
+NJ TRANSIT
+Via
+Everyone
+Distance Zone
+Complementary, extension
+No program name
+St. Louis, MO
+St. Louis Metro
+Lyft
+Everyone
+Distance Hub
+Complementary, FLM
+MARTAConnect
+Atlanta, GA
+MARTA
+Uber, Lyft
+Everyone (closures)∗∗∗ Distance PTP
+Extension
+IndyGo + Uber
+Indianapolis, IN IndyGo
+Uber
+Essential workers∗∗∗∗
+Flat fare
+PTP
+Substitutive
+FLM system design and operations: Research on demand responsive connector (DRC) systems devel-
+ops analytical models to evaluate service quality and determine first-mile system parameters. In particu-
+lar, such work specifies optimal zone size and headways, identifies transition points between regions best
+serviced by fixed-route vs. flexible services, and establishes best practices for inter-zone transfer coordina-
+tion (Chandra and Quadrifoglio 2013, Kim, Levy, and Schonfeld 2019, Kim and Schonfeld 2014, Lee and
+Savelsbergh 2017, Li and Quadrifoglio 2010, Lu, Quadrifoglio, and Petrelli 2017, Lu, Shen, and Quadri-
+foglio 2014). The tactical question of how to operate a first-mile system is also well-studied. The Dial-A-
+Ride Problem (DARP) encompasses the vehicle routing problem faced by transit agencies, given a set of
+trip requests and a vehicle fleet (Ho et al. 2018, Molenbruch, Braekers, and Caris 2017). The Integrated
+DARP (IDARP) designs vehicle routes and schedules to meet trip requests, allowing transfers with fixed-
+route timetabled service (Posada, Andersson, and Hall 2017). Closely related to IDARP is the problem
+of matching individual carpoolers and integrating their trips with transit timetables (Stiglic et al. 2018).
+Finally, many studies design strategies for routing and scheduling (Wang 2019), pricing (Chen and Wang
+2018), and trip request acceptance (Agussurja, Cheng, and Lau 2019) for FLM transportation systems.
+Multimodal network optimization with endogenous demand: Our work is related to literature on opti-
+mal design and operation of transportation systems that acknowledges and leverages endogenous demand.
+Past research has modeled decision-making travelers with preferences. One-to-one and many-to-one assign-
+ment problems among travelers and suppliers have been addressed with preference-based stable matchings
+to prevent participants from leaving ride-sharing systems (Wang, Agatz, and Erera 2018) or transit systems
+(Rasulkhani and Chow 2019). Passenger decisions are also often captured by discrete choice models. Bert-
+simas, Ng, and Yan (2020) jointly determine frequencies and prices for multimodal transit to minimize wait
+times, subject to passenger mode and route choices. Cadarso et al. (2017) optimize airline scheduling, fleet
+assignment, and fares while capturing the effects of competing high-speed rail service, taking passengers’
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+5
+mode choices into consideration. Wei, Vaze, and Jacquillat (2020) develop fixed-route transit timetables
+to maximize welfare, subject to competition with ride-sourcing companies, and congestion effects from
+passengers’ mode switching. Wang, Jacquillat, and Vaze (2022) optimize a network of vertiports for sup-
+porting urban aerial mobility, with passenger mode choices described by two alternative models, including
+a multinomial logit model. Banerjee et al. (2021) tackle a welfare-maximizing system design and pricing
+problem for centrally coordinated multimodal transport networks with price-dependent demand, and for-
+mulate it using mixed-integer convex optimization. In contrast, we tackle a multi-objective pricing alliance
+design problem with a practically suitable pricing scheme that enables transparent price communication to
+passengers, but also prevents its convexification and, in turn, heightens the computational challenge.
+Horizontal Cooperation: Finally, we review cooperation models among competing operators. Litera-
+ture on horizontal cooperation in logistics and airline scheduling is particularly mature (Cruijssen, Dullaert,
+and Fleuren 2007, Guajardo and R¨onnqvist 2016, Wright, Groenevelt, and Shumsky 2010, Hu, Caldentey,
+and Vulcano 2013). Chun, Kleywegt, and Shapiro (2017) design a liner shipping alliance with endogenous
+linear demand for a homogeneous product; shipping companies first trade physical capacity on respective
+networks, and then compete to sell substitutable products in an overlapping market. Our work also involves
+joint products over a shared network subject to endogenous demand, but the allied operators offer those
+products together rather than exchanging capacity to compete. Algaba et al. (2019) formulate an urban trans-
+portation network flow game, using exogenous passenger and cost information to coordinate a single-fare
+payment among competing operators. Bian and Liu (2019a,b) design mechanisms for the first-mile problem
+incorporating personalized passenger requirements. Siddiq, Tang, and Zhang (2021) investigate incentive
+mechanisms to inspire commuters to use public transportation, modeling commuters, transit agency, ride-
+sharing platform, municipal government, and local private enterprises as stakeholders.
+Liu and Chow (2022) investigate whether competing transit agencies can share data to improve selfish
+outcomes when setting frequencies, subject to user equilibrium passenger flows. Policymakers can lever-
+age results of their Bayesian game and coalition formation model to inform decisions about establishing
+mandatory data-sharing amongst transit operators, but the model is not amenable to large-scale operations
+management. The most similar study to ours in this branch of literature is by Schlicher and Lurkin (2022),
+who formulate a transport choice game in which operators cooperatively price their pooled resources, sub-
+ject to passengers making travel choices according to a multinomial logit model. They design a market share
+exchange allocation rule that ensures a stable grand coalition. Their study differs from ours in that each
+operator offers homogeneous products with a single price to travelers with unspecified origins and destina-
+tions, thus entirely ignoring network effects. In summary, most existing studies individually model either
+operator or passenger incentives when designing integrated, multimodal urban transportation systems; to
+our knowledge, studies incorporating both strategic operators and passengers provide only general high-
+level intervention recommendations and rules of thumb. Our work differs in that we provide a prescriptive
+and strategic design framework to build pricing alliances at scale and in full operational detail.
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+6
+Article submitted to Transportation Science; manuscript no. 1
+1.3.
+Contributions
+We propose a prescriptive pricing alliance to enable incentive-aligned collaboration between transit agencies
+and established ride-sharing operators. A fare-setting model is formulated to maximize total system-wide
+benefits across the integrated network. Our framework helps operators navigate competing alliance objec-
+tives: (1) enhancing access to high-quality public transportation options for underserved populations, (2)
+lowering vehicle emissions and congestion from single-occupancy vehicle trips, and (3) maintaining the
+financial well-being of participating operators to ensure that the profit-oriented operators are incentivized to
+participate. A key technical challenge when optimizing these objectives lies in capturing interdependencies
+between fares and commuters’ travel choices. In response, our model integrates a discrete choice model of
+passengers’ route and mode decisions based on prices and non-pricing attributes like travel times.
+From a technical standpoint, our fare-setting model is a large-scale, mixed-integer, non-convex opti-
+mization problem—a challenging class of problems. Our first technical contribution is to design a two-
+stage decomposition in which the first-stage pricing decisions parameterize second-stage fare discounts
+and the induced passenger behaviors. The second stage becomes a more tractable mixed-integer linear
+optimization problem that can be solved with commercial solvers. To solve the full model, we develop a
+new solution approach combining tailored coordinate descent, parsimonious second-stage evaluations, and
+interpolations using Special Ordered Sets of type 2 (SOS2) (Misener and Floudas 2010). We also develop
+acceleration techniques based on slanted coordinate traversal and search direction randomization. This solu-
+tion approach—our second technical contribution—is applicable to any two-stage formulation with a low-
+dimensional, convex, continuous first-stage and any computationally expensive black-box second stage.
+This solution approach is found to significantly improve outcomes, for passengers and operators, compared
+to those obtained with state-of-the-art benchmarks based on Bayesian Optimization (Mockus 2012).
+From a practical standpoint, we design a large-scale case study focused on the morning commute in the
+Greater Boston Area. We find that our model sets fares that are in realistic ranges and have interpretable
+connections to alliance goals. For example, an alliance with a greater focus on minimizing total VMT
+prefers flat rather than distance-varying fares to increase system utilization by long-distance commuters.
+On the other hand, alliances with a greater emphasis on increasing transit access will set discounts with
+greater geographic variation to make alliance routes more attractive to heterogeneous populations. The clear
+alignment between operator goals and passenger choices achieved by our fare structures illustrates the value
+of modeling endogenous demand. Moreover, analysis of our results shows that the model is appropriately
+responsive to equity-oriented objectives: it enables the alliance to lower fares for, and increase utilization
+by, low-income and long-distance commuters. Finally, when compared to non-cooperative pricing, our fares
+and our tailored revenue allocation mechanism together incentivize revenue-oriented MOD operators not
+only to participate in the alliance but also to adopt the transit operator’s priorities.
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+7
+Section 2 presents the allied fare-setting model formulation, its two-stage decomposition enabling
+tractable solutions, as well as our revenue allocation mechanism. Section 3 describes our parsimonious
+SOS2-based coordinate descent approach, whose computational performance is compared against bench-
+marks in Section 4. We present our practical insights in Section 5 and conclude in Section 6.
+2.
+Pricing Alliance Design Problem
+We now present our design pipeline for the Pricing Alliance Design Problem (PADP). In the PADP, the
+alliance—i.e., the jointly acting operators—sets a fare structure that optimizes joint operator priorities over
+the integrated multimodal network, subject to the passengers’ endogenous route choice decisions. The indi-
+vidual operators must then decide whether or not to participate in the alliance they have designed based on
+the optimized fares and a revenue allocation mechanism.
+2.1.
+Assumptions
+Before formulating the allied fare-setting model, we specify our characterization of system-wide benefits,
+our model of passenger decision-making, and our assumptions about static fares.
+System-wide benefits. The alliance cooperatively set fares over an integrated network with the objective
+of maximizing overall benefits to society, including travelers, operators, and the rest of society (Daganzo
+2012). Consistent with the motivation of this work, we assume that transit agency’s own objective is identi-
+cal to that of the alliance. We characterize an operator’s benefits as its fare revenue and a passenger’s benefits
+as its average utility across all available travel options. High passenger utility corresponds to the availability
+of many high-quality travel options. Finally, there are many ways to capture the system’s impact on the
+rest of society, defined as everyone except the travelers and operators. Most people who take alternative
+travel options choose to drive personal vehicles, contributing to negative externalities, such as air pollution.
+Because a pricing alliance involves no change in permanent infrastructure but rather better utilization of the
+existing infrastructure, the key benefits of the alliance to the rest of society are likely to come from single-
+occupancy VMT reduction under the allied pricing regime. We ultimately compute system-wide benefits
+as a weighted sum of operator revenue, passenger utility, and a penalty for the outside-option VMT. The
+weights are determined by the alliance’s relative priorities and can be varied to evaluate trade-offs.
+Passenger discrete choice model. We model travelers as rational agents making travel decisions accord-
+ing to a multinomial logit (MNL) discrete choice model. In an MNL model, the choice probabilities are
+proportional to each option’s exponentiated utility, also known as its attractiveness (McFadden 1974). The
+MNL choice model allows us to embed a closed form of the passengers’ decision-making process in the
+alliance fare-setting model, but it also presents limitations related to the independence of irrelevant alter-
+natives (IIA) property. Some have circumvented such inaccuracies by using the general attraction model
+(GAM), of which the MNL model is a special case (Gallego, Ratliff, and Shebalov 2015). The GAM for-
+mulates each choice probability as a function not only of the available options’ attractiveness, but also the
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+8
+Article submitted to Transportation Science; manuscript no. 1
+shadow attractiveness of unavailable options. In practice, researchers have set the shadow attractiveness
+values to zero, in the absence of reliable data to estimate these parameters (Wei, Vaze, and Jacquillat 2020).
+Others leverage the nested MNL (Williams 1977), of which the MNL is also a special case. Lo, Yip, and
+Wan (2004) and Bertsimas, Ng, and Yan (2020) assume that passengers select travel mode in the first level,
+and then they select a route under that mode in the second level. In our work, we populate passengers’ route
+choice sets with the fastest route from each available travel mode (transit-only, MOD-only, or transit-MOD
+hybrid), including the option to drive, referred to in the literature as the outside option or the no purchase
+alternative. Thus, our simplified choice model framework is equivalent to a GAM with zero shadow attrac-
+tiveness values, or to a mode-route nested MNL with second-level choice sets containing one route each.
+Fare-setting. The alliance sets fares over the integrated network. Some MOD operators might set time-
+varying fares on their independently operated network. In particular, TNCs may implement fare multipliers
+to manage two-sided markets between drivers and riders (Castillo, Knoepfle, and Weyl 2017). In a pricing
+alliance, however, the MOD operator is a contractor to the transit agency and consequently agrees to set
+time- and demand-homogeneous fares over allied network. This agreement facilitates transparent communi-
+cation with passengers who can easily anticipate public sector prices, and it also allows the transit operator
+to set a budget for the alliance with higher confidence. We also assume that each operator is capable of
+serving all demand redistribution that occurs as a result of the newly set fares, and that capacity reallocation
+is therefore unnecessary to consider in the pricing alliance design process. Schlicher and Lurkin (2022)
+make a similar assumption: they assume a constant marginal cost due to market shares that do not change
+significantly. From a transit perspective, this implies that the fixed-route options (e.g., buses) have low load
+factors to start with, especially in and near transit deserts; on the MOD side, it implies that the operators
+have large driver pools driven by private market dynamics. In summary, we assume that the load difference
+on the integrated network when transitioning from non-cooperative to allied fare-setting will not impose a
+large enough change in network utilization to necessitate consideration of the associated resource allocation
+decisions. In the practical case study of Section 5, we validate this assumption by showing that the potential
+pricing alliances indeed do not pose a risk of over-saturating the integrated infrastructure.
+2.2.
+Exact Formulation
+We now provide notation for formulating our allied fare-setting model with endogenous demand. Passengers
+select from a set of routes, R, serviced by a set of operators, O, which includes a public transit operator
+and an MOD operator, so that |O| = 2. A route is a sequence of trip legs, each served by some operator’s
+infrastructure. To capture flat and distance-based fares, we define non-discounted price of route r ∈ R as:
+�
+k∈Or
+(β0
+k + ∆rk · β∆
+k )
+(1)
+where Or ⊆ O is the set of operators serving route r; ∆rk is the distance of route r covered by operator
+k; and β0
+k and β∆
+k , respectively, are the base fare, and markup per unit distance traveled, for operator k’s
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+9
+sub-network. We collectively refer to the base fares and distance-based markups of all operators as the
+fare parameters (β), which are decision variables in our model. Fare parameters are constrained by (non-
+negative) upper and lower bounds ((β0
+min,β0
+max), (β∆
+min,β∆
+max)) determined by local legislative or operational
+requirements.
+In addition to the fare parameters, the operators jointly select a set of discounted routes. Only a subset of
+routes, RDE ⊆ R, may be discount-eligible (DE). Rather than deciding whether or not each individual route
+should receive a discount, the discount-eligible routes may be grouped into discount activation categories.
+Routes in the same discount activation category may share common geographic components specified by
+the alliance. By grouping routes into categories, passengers can easily interpret which routes are discounted
+from a map or a simple set of rules. Example definitions for discount activation categories might include all
+routes anchored on a particular hub location, or all routes whose origins and destinations are contained in
+specified regions. Let Ra ⊂ R be the set of routes corresponding to discount activation category a ∈ A. The
+sets Ra partition RDE, i.e., ∪a∈ARa = RDE and Ra ∩ Rb = ∅ for a ̸= b ∈ A. Let xa ∈ {0,1} denote the
+decision variable that activates discounts on all routes in Ra. This assumption is not restrictive; absence of
+activation categories can be handled easily by putting each route in its own category: |Ra| = 1,∀a ∈ A. We
+note that the relaxation of this assumption might result in discount rules that are difficult to communicate to
+passengers in large-scale systems.
+Λ is the discount multiplier for the routes selected to receive a discount, with an allowable range of
+[Λmin,Λmax] : 0 ≤ Λmin ≤ Λmax ≤ 1. The customer-facing price, pr, of route r ∈ R, is given as:
+pr =
+�
+(1 − Λ · xa) ·
+��
+k∈Or(β0
+k + ∆rk · β∆
+k )
+�
+if r ∈ Ra,a ∈ A
+�
+k∈Or(β0
+k + ∆rk · β∆
+k )
+if r ∈ R \ RDE
+(2)
+We consider a set N of passenger types. Each passenger type i ∈ N is identified by a unique combination
+of origin, destination, and preference profile as described by their route choice utility coefficients. Ri is the
+set of routes available to passengers of type i ∈ N. Some passengers are more averse to expensive travel
+options, whereas others are more sensitive to travel time, constituting different preference profiles. There
+are Ni passengers of type i ∈ N. We denote the utility to a passenger of type i ∈ N of route r ∈ Ri as
+uir + αi · pr, where uir is the utility from non-monetary route attributes and αi ≤ 0 is the utility per unit
+price. The market share sir of route r ∈ Ri for passenger type i ∈ N is computed according to MNL as:
+sir =
+exp(uir + αi · pr)
+exp(ui0) + �
+s∈Ri exp(uis + αi · ps),
+(3)
+where the outside option—not in set R = �
+i∈N Ri—has a utility ui0 and a market share computed as:
+si0 =
+exp(ui0)
+exp(ui0) + �
+s∈Ri exp(uis + αi · ps).
+(4)
+Finally, the operators’ relative priorities over the system-wide performance metrics are captured by non-
+negative objective function weights: µP AX,µREV ,µV MT, respectively, corresponding to passenger benefits,
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+10
+Article submitted to Transportation Science; manuscript no. 1
+operator benefits, and the benefits from negative externality reduction. Table 2 summarizes all notation.
+Model PADP-FS (5)-(13) provides the exact formulation for the PADP fare-setting model. It jointly sets
+fares and discounts to maximize system-wide benefits across the integrated network (objective function (5)).
+Discounts are applied on selected routes (Constraints (6) and (7)) and utility-maximizing passengers make
+route selections according to an MNL (Constraints (8) and (9)). Fare parameters and the discount multipliers
+obey bounds (Constraints (10)-(12)). Discount activation decisions are binary (Constraints (13)).
+Table 2
+Notation.
+Component
+Type
+Description
+A
+Set
+Discount activation categories
+N
+Set
+Passenger types
+O
+Set
+Operators
+R
+Set
+Intrasystem routes, not including the outside option
+Or
+Set
+Operators who help service route r ∈ R
+Ri
+Set
+Route options available to passengers of type i ∈ N
+Ra
+Set
+Routes in discount activation category a ∈ A
+RDE
+Set
+Discount-eligible routes, i.e., �
+a∈A Ra
+Ni
+Param. Number of passengers of type i ∈ N
+∆i0
+Param. Driving distance for a passenger of type i ∈ N
+∆rk
+Param
+Distance the passenger travels with operator k ∈ O on route r ∈ R
+uir
+Param. Non-monetary utility accrued by a passenger of type i ∈ N on route r ∈ Ri
+ui0
+Param. Utility accrued by a passenger of type i ∈ N by driving
+αi
+Param. Utility per unit price to a passenger of type i ∈ N
+β0
+min,β0
+max
+Param. Minimum and maximum allowable base fares
+β∆
+min,β∆
+max
+Param. Minimum and maximum allowable distance-based markups
+Λmin,Λmax
+Param. Minimum and maximum allowable values of discount multipliers
+µP AX,µREV ,µV MT
+Param. Relative priority weights of system-wide performance metrics
+xa
+Var.
+Binary. Whether to activate discount option a ∈ A
+β0
+k,β∆
+k
+Var.
+Continuous. Base fare and markup of operator k ∈ O
+pr
+Var.
+Continuous. Customer-facing price of route r ∈ R
+Λ
+Var.
+Continuous. Discount multiplier applied to routes with activated discounts
+sir
+Var.
+Continuous. Proportion of passengers of type i ∈ N who choose route r ∈ Ri
+si0
+Var.
+Continuous. Proportion of passengers of type i ∈ N who choose the outside option
+(PADP-FS)
+max
+�
+i∈N
+Ni·
+�
+µP AX ·
+�
+ui0+
+�
+r∈Ri
+(uir+αi·pr)
+�
++µREV ·
+�
+r∈Ri
+pr·sir−µV MT ·(∆i0·si0)
+�
+(5)
+s.t.
+pr =
+�
+k∈Or
+(β0
+k + ∆rk · β∆
+k )
+r ∈ R \ RDE
+(6)
+pr = (1 − Λ · xa) ·
+� �
+k∈Or
+(β0
+k + ∆rk · β∆
+k )
+�
+a ∈ A,r ∈ Ra
+(7)
+sir =
+exp(uir + αi · pr)
+exp(ui0) + �
+s∈Ri exp(uis + αi · ps)
+i ∈ N,r ∈ Ri
+(8)
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+11
+si0 =
+exp(ui0)
+exp(ui0) + �
+s∈Ri exp(uis + αi · ps)
+i ∈ N
+(9)
+β0
+min ≤ β0
+k ≤ β0
+max
+k ∈ O
+(10)
+β∆
+min ≤ β∆
+k ≤ β∆
+max
+k ∈ O
+(11)
+Λmin ≤ Λ ≤ Λmax
+(12)
+xa ∈ {0,1}
+a ∈ A
+(13)
+2.3.
+Two-stage Decomposition
+The PADP-FS model is a non-convex mixed-integer nonlinear optimization problem (MINLOP). There
+are no commercial solvers that accommodate non-convex MINLOPs, and no open-source solvers accept
+non-convex MINLOPs at practically large scale. Therefore, we propose a different solution approach. We
+decompose the formulation to tractably obtain high-quality solutions for practically sized problems (tens
+of thousands of variables and hundreds of thousands of constraints in our case study). By letting first-
+stage pricing decisions parameterize second-stage discount activations and induced passenger behaviors,
+the second stage can be formulated as a more tractable mixed integer linear optimization problem (MILOP).
+Let B := [β0
+min,β0
+max]2 × [β∆
+min,β∆
+max]2 and L := [Λmin,Λmax] respectively be the sets of allowable fare
+parameters and discount multipliers. We parameterize the second-stage problem by ( �β, �Λ) ∈ B × L and
+define S( �β, �Λ) as the feasible region parameterized by ( �β, �Λ). We utilize the sales-based linear optimization
+formulation by Gallego, Ratliff, and Shebalov (2015) to reformulate the choice model constraints. The
+premise of the reformulation rests on proportionality constraints. Let γir = exp(ui0)/exp(uir + αi · pr) be
+the ratio of the attractiveness values of the outside option and route r ∈ Ri. Since the fare parameters are
+determined in the first stage, γir is a constant in the second stage formulation for the non-discount-eligible
+routes. Then constraints (8) and (9) are reformulated as follows:
+si0 = γir · sir
+i ∈ N,r ∈ Ri
+(14)
+si0 +
+�
+r∈Ri
+sir = 1
+i ∈ N
+(15)
+si0 ≥ 0
+i ∈ N
+(16)
+sir ≥ 0
+i ∈ N,r ∈ Ri
+(17)
+Equation (14) ensures that the market share of each route is proportional to its attractiveness. Constraints
+(15), (16), and (17) ensure that the market shares are non-negative and sum to 1. Note that Equation (14) still
+includes bilinearities for discount-eligible routes r ∈ Ri ∩ RDE. For a type i passenger, γir is either equal
+to the discounted price (γir = γir( �β, �Λ)) or full price (γir = γir( �β,0)), depending on whether the model
+selects the discount for route r. Let Na ⊂ N be the set of passenger types with at least one route option
+corresponding to discount activation category a ∈ A. We linearize constraint (14) as (18) using big-M
+constraints, letting M s
+ir = γir( �β,0) ≥ 0.
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+12
+Article submitted to Transportation Science; manuscript no. 1
+�
+�
+�
+�
+�
+�
+�
+�
+�
+si0 ≤ γir( �β,0) · sir
+si0 ≥ γir( �β,0) · sir − M s
+ir · xa
+si0 ≤ γir( �β, �Λ) · sir + M s
+ir · (1 − xa)
+si0 ≥ γir( �β, �Λ) · sir
+a ∈ A,i ∈ Na,r ∈ Ri ∩ Ra
+(18)
+We similarly handle the bilinearities presented by the revenue terms in the objective function. We define a
+new decision variable wir = pr · sir for passenger types i ∈ N with discount-eligible routes r ∈ Ri ∩ RDE.
+The linearized constraints (19) set the value of wir with M w
+ir = �Λ ·
+��
+k∈Or(�β0
+k + ∆rk · �β∆
+k )
+�
+≥ 0.
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+wir ≤
+��
+k∈Or(�β0
+k + ∆rk · �β∆
+k )
+�
+· sir
+wir ≥
+��
+k∈Or(�β0
+k + ∆rk · �β∆
+k )
+�
+· sir − M w
+ir · xa
+wir ≤ (1 − �Λ) ·
+��
+k∈Or(�β0
+k + ∆rk · �β∆
+k )
+�
+· sir + M w
+ir · (1 − xa)
+wir ≥ (1 − �Λ) ·
+��
+k∈Or(�β0
+k + ∆rk · �β∆
+k )
+�
+· sir
+a ∈ A,i ∈ Na,r ∈ Ri ∩ Ra
+(19)
+Table 3 summarizes the additional and modified notation for the second-stage model.
+Table 3
+Additional and modified notation for second-stage model of the decomposition framework.
+Component Type
+Description
+B
+Set
+Allowable fare parameter values
+L
+Set
+Allowable percent discount values
+Na
+Set
+Passenger types with access to at least one discount-eligible route in
+discount activation category a ∈ A, i.e. Ri ∩ Ra ̸= ∅
+γir(β,Λ)
+Parameter Ratio of outside option attractiveness to attractiveness of route r ∈ Ri
+with price parameters β and discount Λ for passenger type i ∈ N
+M w
+ir,M s
+ir
+Parameter Big-M parameters for each i ∈ N,r ∈ Ri
+wir
+Variable
+Continuous. Equivalent to pr · sir for discount-eligible routes r ∈ Ri ∩ RDE
+In response to fare parameters set in the first stage, the second-stage problem activates discounts that
+optimize system-wide performance metrics, subject to induced passenger decisions. The optimal value of
+the second-stage problem is denoted by W in equation (20).
+W( �β, �Λ) :=
+max
+(x,s,w,p)∈S( �
+β,�Λ)
+�
+i∈N
+Ni ·
+�
+µP AX ·
+�
+ui0 +
+�
+r∈Ri
+(uir +αi ·pr)
+�
++µREV ·
+�
+r∈Ri
+wir −µV MT ·(∆i0 ·si0)
+�
+, (20)
+where S( �β, �Λ) is given as follows:
+Then we define the PADP-FS2SD model, the two-stage decomposition of the PADP-FS model.
+(PADP-FS2SD)
+max
+W(β,Λ)
+(26)
+s.t.
+β ∈ B
+(27)
+Λ ∈ L
+(28)
+LEMMA 1. Formulations PADP-FS and PADP-FS2SD are equivalent.
+The proof of Lemma 1 is in Appendix A.
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+13
+S( �β, �Λ) ≡
+�
+(x,s,w,p) ∈ {0,1}|A| × R
+�
+i∈N (|Ri|+1)
++
+× R
+�
+i∈N |Ri∩RDE| × R|R| :
+Constraints (18) - (19)
+si0 +
+�
+r∈Ri
+sir = 1
+i ∈ N
+(21)
+si0 = γir( �β,0) · sir
+i ∈ N,r ∈ Ri \ RDE
+(22)
+wir =
+� �
+k∈Or
+(�β0
+k + ∆rk · �β∆
+k )
+�
+· sir
+i ∈ N,r ∈ Ri \ RDE
+(23)
+pr =
+�
+k∈Or
+(�β0
+k + ∆rk · �β∆
+k )
+r ∈ R \ RDE
+(24)
+pr = (1 − �Λ · xa) ·
+� �
+k∈Or
+(�β0
+k + ∆rk · �β∆
+k )
+�
+a ∈ A,r ∈ Ra
+�
+(25)
+2.4.
+Revenue Allocation Mechanism
+When considering a pricing alliance, an operator assesses whether the cooperative regime would improve its
+prioritized system-wide metrics over the non-cooperative regime. A non-cooperative fare-setting game and
+a solution approach for it are presented in Appendix B. The MOD operator is solely revenue-maximizing,
+while the transit agency maximizes a linear combination of multiple system-wide metrics. By entering a
+pricing alliance, the transit agency (denoted as TR) is guaranteed to fare no worse than that under the non-
+cooperative regime. However, it remains to ensure that the revenue-maximizing MOD operator will not lose
+revenue by cooperating with the transit agency, which would ensure the MOD operator’s participation.
+We now design a revenue allocation mechanism that guarantees the MOD operator’s alliance participa-
+tion. Let βnc and (βa,Λa) respectively denote the non-cooperative equilibrium fare parameters and allied
+optimal fare parameters. Let fk(βnc) denote the revenue of operator k ∈ O in the non-cooperative regime,
+and f(βa,Λa) denote the combined revenue of both operators in the alliance.
+LEMMA 2. Let δ = f(βa,Λa) − �
+k∈O fk(βnc) be the surplus allied revenue compared to the total
+non-cooperative revenue. Define Φk : R2
++ × R+ → R+ to be the revenue allocation to operator k ∈ O :=
+{TR,MOD}:
+ΦT R((fk(βnc))k∈O,f(βa,Λa)) =
+�
+fT R(βnc) + δ
+2·
+if δ ≥ 0,
+f(βa,Λa) − fMOD(βnc)
+otherwise.
+(29)
+ΦMOD((fk(βnc))k∈O,f(βa,Λa)) = fMOD(βnc) +
+� δ
+2
+�+
+(30)
+(a) The MOD operator will enter the pricing alliance with payment rule Φ.
+(b) When δ ≥ 0, the mechanism satisfies Pareto efficiency, symmetry, the core property, scale invariance,
+and independence of irrelevant alternatives.
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+14
+Article submitted to Transportation Science; manuscript no. 1
+The proof of Lemma 2 is in Appendix C. Because the alliance’s priorities may also include benefits to
+passengers and/or benefits to the rest of the society in the form of reduced VMT, the alliance may earn
+less revenue than the operators’ combined revenue in the non-cooperative regime. Despite this, the transit
+operator can choose to guarantee that, by cooperating, the revenue-oriented MOD operator earns at least as
+much as it would have earned otherwise. We assume that the MOD operator will participate in the alliance if
+its non-cooperative and allied revenues are equal. In the event that the alliance accrues strictly more revenue
+than that in the non-cooperative regime, the operators split the surplus evenly.
+3.
+Solution Approach
+3.1.
+Motivation
+Section 2.3 presented a two-stage decomposition of the allied fare-setting formulation, with the second
+stage characterized as a mixed-integer linear optimization problem and the first stage as a low-dimensional
+decision problem over a convex space. Without an analytic closed-form of W, the function’s gradients are
+inaccessible, eliminating the possibility of using any gradient-based approaches. Bayesian Optimization is
+applicable and has been leveraged in recent urban transportation studies focusing on MOD systems (Liu
+et al. 2019), but it does not provide clear convergence criteria. PADP-2SD is also not amenable to Benders
+decomposition due to the nonlinear interdependencies between first- and second-stage decisions. Our prob-
+lem’s incompatibility with the simpler centralized welfare-maximization structure of the problem tackled
+by Banerjee et al. (2021) implies that their convexification strategy cannot be applied either.
+Our solution strategy approximates gradient descent for solving the first-stage problem. Because the first-
+stage feasible space is low-dimensional and convex, we begin with a coordinate descent framework, which
+takes turns fixing all fare parameters except one and greedily optimizing along the free dimension. Even
+one-dimensional search is difficult because the search space is a continuous spectrum of optimal MILOP
+solutions. While a solution of the second-stage problem is fast enough to be a useful tool (see Section 4:
+needing at most 5 seconds on average), it is also slow enough to warrant a judicious selection of first-stage
+valuation points. Thus, our tailored coordinate descent approach scans each search direction by solving a
+model that approximates the best solution along that search direction. After evaluating a few points along the
+free search direction with the second-stage MILOP, an auxiliary model interpolates intermediate solutions
+along that search direction with Special Ordered Sets of type 2 (SOS2) (Misener and Floudas 2010). The
+process terminates when no improvements are found along any coordinate direction. We define this as the
+basic SOS2 Coordinate Descent (SOS2-CD) approach in Section 3.2.
+To further improve final solution quality given a computational budget, we develop three acceleration
+strategies that build upon SOS2-CD. First, rather than using an arbitrary search direction sequence, we intro-
+duce more opportunities to escape local optima by randomizing search direction order. Second, we exploit
+the fact that the SOS2 approximation model is valid along any search direction through the first-stage prob-
+lem’s search space, and not just those parallel to coordinate axes. Natural search direction candidates are
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+15
+those where each operator’s base fare and markup are jointly varied while holding all other parameters con-
+stant. By considering SOS2 coordinate descent over such slanted directions, we unlock directions navigating
+trade-offs between high base fares and low markups vs. low base fares and high markups, which would
+be unavailable with single-coordinate search directions. Finally, we show how to mitigate the SOS2-CD’s
+sensitivity to random initializations by leveraging warm-start solutions. 3.3 describes the final algorithm,
+including acceleration strategies, initialization procedures, and the incorporation of time limits. Computa-
+tional results in Section 4 demonstrate the effectiveness of SOS2-CD and all acceleration strategies.
+3.2.
+SOS2 Coordinate Descent
+Let Y := B × L denote the search space over which to optimize W, and y := (β,λ) ∈ Y denote a solution.
+Let Yi(y) = {z ∈ R : (y1,··· ,yi−1,z,yi+1,··· ,yn) ∈ Y} be the subset of the feasible space with all dimen-
+sions other than the ith fixed to those of solution y ∈ Y. We define S(y) and S∗(y) ⊆ S(y), respectively,
+to be the set of feasible and optimal second-stage decisions given fare parameters y := (β,Λ) ∈ Y:
+S∗(y) := arg
+max
+(x,s,w,p)∈S(y)
+�
+i∈N
+Ni ·
+�
+µP AX ·
+�
+ui0 +
+�
+r∈Ri
+(uir + αi · pr)
+�
++ µREV ·
+�
+r∈Ri
+wir − µV MT · (∆i0 · si0)
+�
+.
+Basic Coordinate Descent. Coordinate descent is a greedy method that successively optimizes a mul-
+tivariate function along coordinate axes. Starting from an initial point, it cyclically optimizes along every
+coordinate direction holding all other dimensions fixed. Algorithm 1 presents basic coordinate descent to
+solve the PADP. Optimizing even a single dimension of W is hard, because it entails navigating a contin-
+uous spectrum of optimal solutions to MILOPs, which does not have closed analytic form. Therefore, we
+will propose a rigorous method for tractably modifying Step 7 of Algorithm 1 using SOS2 interpolation.
+Algorithm 1 Coordinate Descent for maximization of W
+1: ARG y0 : Initial solution in Y
+2: procedure COORDINATE DESCENT(y0)
+3:
+objPrev ← −∞; objCur ← W(y0); k ← 0; n ← dim(y0)
+4:
+while objCur − objPrev > ϵ do
+5:
+k ← k + 1;objPrev ← objCur;yk ← yk−1
+6:
+for i ∈ {1,...,n} do
+7:
+yk
+i = arg maxz∈Yi(yk) W(yk
+1,··· ,yk
+i−1,z,yk
+i+1,··· ,yk
+n)
+8:
+objCur ← W(yk)
+9:
+return yk
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+16
+Article submitted to Transportation Science; manuscript no. 1
+SOS2 Interpolation. Our SOS2 interpolation procedure performs approximate local search along a spec-
+ified direction to produce the next candidate solution. First, we solve D second-stage models at evenly
+spaced points along the search direction, obtaining a sequence of fare parameter values acting as anchors
+for the SOS2 interpolation. We denote the anchors by yd := (βd,Λd) and their corresponding solutions by
+(xd,sd,wd,pd) ∈ S∗(yd),∀d ∈ {1,...,D}. Let Ω = {yd : d ∈ {1,...,D}} be the ordered set of anchors.
+Larger D values interpolate more accurately, but the solution is also more computationally expensive.
+Figure 2 visualizes the selection of the next candidate solution using SOS2 variables. W is exactly eval-
+uated at every anchor and approximated between the anchors using the interpolated anchor solutions. The
+next candidate solution is selected where the approximation of W is maximized. Rather than directly inter-
+polating W, or wir variables linearizing prsir terms, we interpolate the price and market share variables, p
+and s. Otherwise, the interpolated values of W will be convex combinations of anchor valuations (straight
+line segments connecting consecutive anchors in Figure 2), eliminating any chance of selecting fare param-
+eters between anchors. Moreover, since the objective function’s nonlinearities are quadratic in nature due to
+the multiplicative revenue terms, we capture them with this SOS2 approximation.
+Figure 2
+SOS2 interpolation of W value and the selection of next candidate first-stage solution y∗.
+For a given number of anchor points D, the SOS2 model (Misener and Floudas 2010) is algebraically
+specified as SOS2(D) ≡
+�
+(λ,z) ∈ RD
++ × {0,1}D−1 : �D
+d=1 λd = 1,�D−1
+d=1 zd = 1,λ1 ≤ z1,λd ≤ zd−1 +
+zd ∀d ∈ {2,...,D},λD ≤ zD−1
+�
+. Here, the λ variables are the convex combination weights for outputs
+at fare parameters (βd,Λd), and each binary zd variable indicates whether to select the segment between
+anchors d and d + 1. Now we use the SOS2 variables to approximate W along a given coordinate axis.
+Expression (31) presents the set of fare parameters, SOS2∗(Ω), that optimize approximated W given the
+ordered anchor set Ω. Expression (32) denotes optimal solutions at all anchors. The optimal SOS2 vari-
+ables are selected to maximize the approximated W function in Constraint (33). Finally, the approximately
+optimal fares are interpolated in equation (34). The approximated objective function is quadratic, making
+
+M
+Wevaluatedatanchor
+SOS2 interpolation of W
+TruevalueofW
+Search direction
+True optimumCummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+17
+(33) a mixed-integer quadratic optimization problem. Fortunately, it can be solved almost instantly to global
+optimality with commercial solvers, because D is small by design.
+SOS2∗(Ω) :=
+(31)
+�
+y : (xd,sd,wd,pd) ∈ S∗(yd),
+∀yd ∈ Ω
+(32)
+(λ∗,z∗) ∈
+arg max
+(λ,z)∈SOS2(D)
+�
+i∈N
+Ni ·
+�
+µP AX ·
+�
+ui0 +
+�
+r∈Ri
+(uir + αi ·
+�
+d∈D
+pd
+r · λd)
+�
++
+µREV ·
+�
+r∈Ri
+��
+d∈D
+(pd
+r · λd) ·
+�
+d∈D
+(sd
+ir · λd)
+�
+− µV MT · ∆i0 ·
+�
+d∈D
+(sd
+i0 · λd)
+�
+(33)
+y =
+�
+yd∈Ω
+λ∗
+dyd
+�
+(34)
+Summary of SOS2 Coordinate Descent. We present SOS2-CD in Algorithm 2, which replaces the
+one-dimensional optimization in Step 7 of Algorithm 1 with the SOS2-based approximation. The sub-
+routine SEARCH DIRECTIONS provides a comprehensive ordered list of search directions that can poten-
+tially be multidimensional and/or randomized (options further discussed in the next subsection); the default
+is to cycle through coordinate axes, i.e. to return searchDirections = {1,...,dim(y0)} when random
+and multidim are both set to false. The subroutine GENERATE ANCHORS returns evenly spaced SOS2
+anchors along the specified search direction. The current solution is included in the anchor set to ensure
+that the new solution is at least as good as the previous. Appendix D presents subroutines SEARCH DIREC-
+TIONS and GENERATE ANCHORS in full detail. After generating the anchors in Step 7, Step 8 computes an
+optimal solution for each anchor, uses these anchor solutions for interpolation, and picks the solution that
+maximizes approximated W over the given search direction. Step 9 computes true value of W at the new
+candidate solution and updates the current solution if necessary. The algorithm iterates until convergence.
+3.3.
+Final Algorithm
+We now present three strategies that provide SOS2-CD with additional opportunities to escape local optima
+and thus improve solution quality. The first strategy relaxes the assumption of deterministic search direc-
+tion order. Because the order of search direction is arbitrary, we can randomize it after each iteration. We
+can select this strategy by setting the random argument to TRUE. Second, since the SOS2 approximation
+model is valid along any search direction intersecting the current solution, not just the coordinate axes,
+an operator’s fare parameter pair (base fare and markup) defines a natural subset of dimensions to search
+simultaneously. Given a pair of dimensions, this strategy randomly selects the spanning dimension, and then
+selects the line’s slope in this 2D plane uniformly at random from the set of affine lines that intersect the
+current solution and span the selected dimensions. Finally, it drops anchors at evenly spaced points along the
+sampled line and obtains the next candidate solution maximizing approximated value of W. While there are
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+18
+Article submitted to Transportation Science; manuscript no. 1
+Algorithm 2 SOS2 Coordinate Descent for maximization of W
+1: ARGS y0 : Initial solution in Y; D: Number of SOS2 anchors; random: Boolean, whether to randomize
+search directions; multidim: Boolean, whether to use multidimensional slanted search directions
+2: procedure SOS2 COORDINATE DESCENT(y0,D,random,multidim)
+3:
+objPrev ← −∞; objCur ← W(y0); k ← 0
+4:
+while objCur − objPrev > ϵ do
+5:
+k ← k + 1;objPrev ← objCur;yk ← yk−1
+6:
+for i ∈ SEARCH DIRECTIONS (random, multidim) do
+7:
+Ω ← GENERATE ANCHORS(yk,i,D)
+8:
+Interpolate an optimal solution: draw some y∗ from SOS2∗(Ω)
+9:
+yk ← y∗ if W(y∗) > objCur else yk
+10:
+objCur ← W(yk)
+11:
+return yk
+many possibilities for multidimensional search directions, we limit to each operator’s fare parameter pair.
+Thus, when the algorithm’s multidim argument is set to TRUE, the list of search directions contains three
+items: (1) transit parameters, (2) MOD parameters, and (3) the discount multiplier. Whenever an operator’s
+fare parameters are selected as the search direction, we sample a new affine line with the aforementioned
+procedure. Appendix D fully specifies the subroutine SEARCH DIRECTIONS.
+The last acceleration strategy incorporates a timed warm-start procedure. The outcome of a single round
+of SOS2-CD may depend on the initial solution. From a random set of initial solutions, the basic implemen-
+tation repeats SOS2-CD until a computational time budget limit has elapsed. Each repetition of SOS2-CD is
+called a trajectory. The best fare parameters found across all trajectories are returned. Convergence to higher
+quality solutions may be more likely given intelligent initializations. We can warm-start the algorithm by
+first obtaining a few samples in the region with a specified warmStartProcedure, and selecting the best
+starting points from them. The warmStartProcedure might simply be uniform sampling from the region,
+or it can consist of searching the space in a more principled way, such as with Bayesian Optimization.
+Algorithm 3 presents the overall solution algorithm. τ W S and τ each define the time limits devoted
+to the warm-start and SOS2-CD procedures, respectively. The arguments random and multidim are
+Booleans indicating whether randomized and/or multidimensional search directions will be used. The
+warmStartProcedure specifies the procedure for generating informed initializations.
+4.
+Computational Results
+We now discuss the accuracy and tractability of our approach through several computational experiments
+using a large-scale profit maximization case study of the Greater Boston Area (see Section 5.1 for details).
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+19
+Algorithm 3 Timed SOS2-CD with warm-start initialization
+1: ARGS τ W S: Warm-start time limit (seconds); τ: SOS2-CD time limit (seconds); random: Boolean,
+whether to randomize search directions; multidim: Boolean, whether to use multidimensional slanted
+search directions; warmStartProcedure: Initialization procedure; D: Number of SOS2 anchors
+2: procedure TIMED SOS2-CD(τ W S, τ, random, multidim, warmStartPocedure, D)
+3:
+objCur ← −∞; Y0 ← ∅; T W S ← τ W S; T ← τ; draw y∗ ∈ Y uniformly at random
+4:
+while T W S > 0 do
+// generate warm-start solutions
+5:
+Draw y ∈ Y with warmStartPocedure
+6:
+Subtract from T W S the time to run warmStartPocedure and to compute W(y)
+7:
+if T W S ≥ 0 then: Insert (y,W(y)) into set Y0
+8:
+while T > 0 do
+// execute SOS2-CD
+9:
+if Y0 ̸= ∅ then: y0 ← arg max{W(y) : (y,W(y)) ∈ Y0}; Remove (y0,W(y0)) from Y0
+10:
+else: Draw y0 from Y uniformly at random
+11:
+�y ← SOS2 COORDINATE DESCENT (y0,D,random,multidim)
+12:
+Subtract from T the time to run SOS2 COORDINATE DESCENT and to compute W(�y)
+13:
+if W(�y) > objCur and T ≥ 0 then: y∗ ← �y; objCur ← W(�y)
+14:
+return y∗
+All optimization models are solved with Gurobi v9.0 and the JuMP package in Julia v1.4 (Dunning,
+Huchette, and Lubin 2017).
+4.1.
+Comparisons under 1-Hour Computational Time Budget
+We now demonstrate the superior computational performance of our approach (Algorithm 3). Table 4 com-
+pares different versions of our approach, with different combinations of acceleration strategies, including
+multidimensional search (SOS2-CD-MD), randomized search directions (SOS2-CD-R), both (SOS2-CD-
+MD-R) and neither (SOS2-CD). None of these four approaches use intelligent warm-starts. We establish
+two algorithmic benchmarks against which to compare our computational results. The first benchmark is
+Brute-Force Coordinate Descent (BF-CD). BF-CD differs from SOS2-CD in the way it conducts each iter-
+ation of coordinate descent. BF-CD uses a much higher number of “anchors” along the search direction
+and solves a second-stage model at each anchor. Instead of the SOS2-based interpolation, it just selects the
+anchor with the highest value of the second-stage objective function as the new candidate solution. The
+trade-off at each iteration is a drastic computation time increase for a more accurate evaluation of the points
+along the search direction. We implement BF-CD using a 1% granularity for the discount multiplier and a
+$0.01 granularity for both base fares and markups. The second benchmark is Bayesian Optimization (BO)—
+a global optimization method for black-box functions that are computationally expensive to evaluate and
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+20
+Article submitted to Transportation Science; manuscript no. 1
+may not have gradients (Mockus 2012). Our black-box function is W. BO imposes upon W a prior belief
+about the space of possible objective values based on the candidate solutions considered so far. The poste-
+rior distribution decides which candidate solution to evaluate next, so that our sequential search successfully
+explores unseen regions in the decision space and exploits regions that are more likely to host global optima
+based on prior beliefs. Appendix F includes a detailed account of BO, including all hyperparameter settings.
+We also tested time-limited SOS2-CD-MD-R with BO warm-starts, with varying time limit allocations
+to the warm-starts. In other words, we execute Algorithm 3 where the warmStartProcedure is Bayesian
+Optimization. The warm-start trials have names ending in BO-TL, where TL is the BO warm-start time
+limit in minutes. Table 4 presents the performances statistics across 50 trials each with a 1-hour limit. All
+outcomes are expressed in surplus USD over the average 1-hour BO benchmark performance.
+First, we observe that all four variations of our approach, even without warm-starts, significantly out-
+perform the BO benchmark in terms of the average (by $13.5K-$19.3K) and best-case (by $5.1K-$5.8K)
+performance. Moreover, our approaches with multidimensional search (SOS2-CD-MD-R and SOS2-CD-
+MD), beat the BO benchmark also on the worst-case performance across the 50 trials (by $7.5K-$22.3K).
+Note that the average-case as well as the worst-case performance of the approaches with either acceleration
+strategy (MD or R or both) were superior to those of the basic SOS2-CD approach. The BF-CD approach
+never terminated within the one-hour time limit; in fact, BF-CD could not even evaluate one full set of
+anchors in all but 7 cases. Furthermore, our approaches with warm-starts perform even better than those
+without. In particular, a 40-minute BO warm-start drastically outperforms the benchmarks in the worst case
+and provides the best average-case performance, while 20 minute BO warm-start provides the strongest
+best-case performance. In summary, all our approaches significantly beat benchmarks, and all three accel-
+eration strategies (random search, slanted search and warm-start) were found to enhance the performance
+of our basic SOS2-CD solution approach.
+Table 4
+Objective function statistics with 1-hour time limits and 50 trials each, expressed in terms of
+surplus compared to average 1-hour BO performance. ∗ Average BO performance = $3,634,074.
+Objective (Thousand $)
+Algorithm
+random multidim warmStartProcedure τ W S
+τ
+Min
+Avg.
+Max
+BO
+-
+-
+-
+-
+-
+−28.0
+0.0∗
+18.6
+SOS2-CD-MD-R-BO-50
+Yes
+Yes
+BO
+50
+10
+−11.9
+15.4
+24.1
+SOS2-CD-MD-R-BO-40
+Yes
+Yes
+BO
+40
+20
+12.6
+20.3
+24.1
+SOS2-CD-MD-R-BO-30
+Yes
+Yes
+BO
+30
+30
+0.0
+19.7
+24.0
+SOS2-CD-MD-R-BO-20
+Yes
+Yes
+BO
+20
+40
+−4.3
+19.1
+24.6
+SOS2-CD-MD-R-BO-10
+Yes
+Yes
+BO
+10
+50
+−9.8
+20.0
+24.2
+SOS2-CD-MD-R
+Yes
+Yes
+-
+-
+60
+−20.5
+19.0
+23.7
+BF-CD
+-
+-
+-
+-
+60
+-
+-
+4.8
+SOS2-CD
+No
+No
+-
+-
+60 −103.8
+13.5
+24.1
+SOS2-CD-MD
+No
+Yes
+-
+-
+60
+−5.7
+19.3
+24.1
+SOS2-CD-R
+Yes
+No
+-
+-
+60
+−77.6
+17.1
+24.4
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+21
+4.2.
+Comparisons under Higher Computational Time Budgets
+All comparisons in the previous subsection assumed a 1-hour computational time budget and showed the
+significant superiority of our basic approach over the benchmarks, as well as the value of our acceleration
+strategies. A question emerges as to whether these findings hold when longer computational budgets are
+available. To answer this question, Table 5 compares performances under three time budgets - 1 hour, 6
+hours, and 12 hours. We additionally provide statistics on the number of trajectories. All versions of our
+approach under all time budgets outperform the BO benchmark on average. This is especially true for the
+versions of SOS2-CD with acceleration strategies. The larger time budgets allow accelerated SOS2-CD to
+offer robust performance. In general, the trajectories of SOS2-CD with multidimensional search (that is,
+SOS2-CD-MD and SOS2-CD-MD-R) converge more quickly, allowing more trajectories to be computed
+within a given time limit. BF-CD is extremely slow and did not terminate before the 12-hour time limit in
+any of our runs. We report the performance statistics for BF-CD corresponding to the best solutions found
+within the computational time budgets, prior to termination. While the best-case runs of BF-CD provide a
+slight edge over all benchmarks (of merely $200 USD), the average-case and the worst-case performance is
+significantly worse than our methods. While BF-CD is more thorough for a single random initialization, it
+is too computationally intensive to properly explore the search region. Note that warm-starts did not provide
+much additional value for longer time budgets and hence warm-start approaches are omitted from Table 5.
+For additional analyses of the solution times and SOS2 optimality gaps, see Appendix E.
+Table 5
+Objective function statistics with varying time budgets and 50 trials each, expressed in terms of
+surplus compared to 1-hour BO performance. ∗ Average BO performance = $3,634,074.
+Time
+Algorithm
+Trajectories
+Objective (Thousand USD)
+limit
+Min
+Avg.
+Max
+Min
+Avg.
+Max
+1 hour
+BO
+-
+-
+-
+−28.0
+0∗
+18.6
+BF-CD
+0
+0
+0
+-
+-
+4.8
+SOS2-CD
+2
+3.8
+5
+−103.8
+13.5
+24.1
+SOS2-CD-MD
+2
+4.0
+7
+−5.7
+19.3
+24.1
+SOS2-CD-R
+2
+3.8
+5
+−77.6
+17.1
+24.4
+SOS2-CD-MD-R
+2
+4.2
+6
+−20.5
+19.0
+23.7
+6 hours
+BO
+-
+-
+-
+7.7
+17.9
+23.9
+BF-CD
+0
+0
+0
+−2132.7 −791.9
+24.6
+SOS2-CD
+16
+20.1
+25
+−2.7
+22.3
+24.3
+SOS2-CD-MD
+15
+21.3
+28
+22.0
+23.3
+24.1
+SOS2-CD-R
+13
+19.7
+24
+22.4
+23.8
+24.4
+SOS2-CD-MD-R
+15
+21.4
+26
+21.1
+23.3
+24.3
+12 hours
+BO
+-
+-
+-
+18.7
+21.9
+23.9
+BF-CD
+0
+0
+0
+−2003.4 −112.7
+24.6
+SOS2-CD
+32
+39.6
+50
+22.0
+23.7
+24.3
+SOS2-CD-MD
+32
+42.4
+54
+22.8
+23.7
+24.2
+SOS2-CD-R
+26
+38.2
+47
+23.5
+24.0
+24.4
+SOS2-CD-MD-R
+31
+42.7
+52
+22.2
+23.7
+24.4
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+22
+Article submitted to Transportation Science; manuscript no. 1
+5.
+Insights from Practical Case Study
+To inform the real-world policymakers’ decisions, we obtained practical results with the PADP model over
+a Greater Boston Area case study described in Section 5.1. Section 5.2 confirms that our model yields
+interpretable outputs with prices in realistic ranges. An equity-oriented case study in Section 5.3 underlines
+the value of accurately capturing passenger preferences. Section 5.4 demonstrates the value of cooperative
+pricing. All results are obtained with the SOS2-CD-MD-R approach and a 12-hour time limit.
+5.1.
+Greater Boston Area Case Study
+We model a potential pricing alliance between the Massachusetts Bay Transit Authority (MBTA) and a TNC
+like Uber or Lyft, in the Greater Boston Area. We use Lyft data for generating case study inputs because
+of fare parameter data availability (Lyft Inc. 2020). MBTA subsidizes Uber and Lyft trips as part of their
+on-demand paratransit program called The RIDE Flex (MBTA 2021). We model an alliance with a wider
+passenger scope that aligns with MBTA goals outlined in a recent report (MDOT 2019). In this report,
+the MBTA identified 14 towns (called “urban gateways.”) adjacent to the commuter rail network whose
+residents had the greatest likelihood of utilizing—and benefiting from—targeted transit expansion efforts.
+We identify these 14 towns as the service region of the potential pricing alliance, as depicted in Figure 3.
+Figure 3
+Partial map of urban gateways, as identified by the MBTA (MDOT 2019). The regions are identi-
+fied with solid-line bounding boxes. The dashed-line bounding box demarcates the region that we
+identified as the inner city. (Urban gateway not depicted: Brockton, located south of the inner city.)
+Our case study integrates many datasets describing travel characteristics in the Greater Boston Area
+during the weekday morning commute (6-10 am). We consider passenger travel patterns for those who
+
+Newburyport
+HAVERHILL
+WestNewbury
+Newbury
+URBANGATEWAYS
+Groveland
+Methuen
+Georgetown
+Rowley
+Dracut
+LAWRENCE
+Boxford
+Ipswich
+Dunstable
+Tyngsboroug
+NorthAndover
+OMELL
+Andoyer
+Topsfield
+Hamilton
+Groton
+Tewksbury
+Chelmsford
+Middleton
+Wenham
+Westford
+NorthReading
+Danvers
+Ayer
+Beverly
+Billerica
+Wilnt
+ngtor
+Lynnfield
+Littleton
+READING
+Carlisle
+Peabod
+BURLINGTPN
+WAKEFIELD
+SALEM
+Bedford
+/Marblehead
+Harvard
+Acton
+WOBURN
+Saugus
+LYNN
+MELROSE
+Wampscott
+Concord
+STONEHAM
+Lexington
+Maynard
+Malden
+Stow
+Lincoln
+Arlington
+Medford
+Nahant
+Reve
+Everett
+elmontr
+WALTHAM
+samerville
+ne
+Sudbury
+Winthrop
+atertowncambr.Ige
+Bosto
+Wayiand
+Westo
+Newton
+Brookline
+FRAMINGHAM
+Wellesley
+Bostor
+Natick
+Needham
+HUHCummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+23
+commute from the service region to the inner city (Boston and Cambridge), or those who commute locally
+within the service region. We define a local commute as either working in the town of residence or in an
+adjacent town that is also part of the service region. For example, commutes between Salem and Lynn or
+between Burlington and Melrose are considered local. We use Origin-Destination Employment Statistics
+from the Longitudinal Employer-Household Dynamics (LODES) datasets provided by the U.S. Census
+Bureau to approximate the commuting population at a census tract level (U.S. Census Bureau 2017).
+We obtain MBTA’s commuter rail network data using MBTA General Transit Specification Feed data
+(MBTA 2018), while the MOD operator corresponds to all potential direct travel options and first-mile
+connections in the service region. We construct route choice sets for each passenger type by first executing
+Yen’s k-shortest paths algorithm (Yen 1970). We then include in each passenger type’s route choice set their
+fastest option of each mode: transit-only, MOD only, hybrid (MOD first mile to transit), and driving (which
+corresponds to the outside option). We represent the utility of each route option as a linear combination
+of travel and wait time; incurred costs including fare, gasoline, and parking fees as appropriate; and mode
+discomfort relative to the convenience of driving. The discount activation categories correspond to town
+pairs. In other words, a discount might be activated from any town in the service region to the inner city, to
+an adjacent town that is also part of the service region, or to itself. In total, there are 77 discount activation
+categories in the case study. We allow each operator to set fares up to a maximum of $10 for base fares, $5
+per mile for distance-based markups, and a maximum 0.5 for discount multiplier. Transit-only routes are
+not eligible for discount, while all others are. Appendix G provides more details about the case study.
+5.2.
+Model Validation
+First, we will demonstrate that the allied fare-setting model sets route prices in realistic and reasonable
+ranges from a practical standpoint. Further, we find that the optimal fares intuitively reflect various port-
+folios of alliance priorities. We vary the objective function coefficients µ, i.e., relative weights among the
+three performance metrics: revenue, passenger utility, and VMT. In particular, we focus on regimes with
+varying combinations of priorities between revenue and passenger utility (i.e., setting µV MT = 0), as well as
+between revenue and VMT (i.e., setting µP AX = 0). We do not emphasize regimes that completely exclude
+revenue as a priority, because they intuitively result in zero fares and are not interesting from an analysis
+standpoint. Thus, all experiments have µREV > 0. We also do not analyze regimes that vary all three metrics
+for reasons explained later in this section. Table 6 presents summary statistics about route prices, system uti-
+lization, and system-wide performance metrics across the tested priority regimes. Figure 4 depicts optimal
+fare parameters and discount multipliers, demonstrating the different fare-setting strategies of each regime.
+We extract a few representative solutions from Table 6 and present them in Table 7 alongside real-world
+fares, and the corresponding ridership values obtained by our model for the real-world fares. We compute
+the MOD base fare by combining Lyft’s published minimum fare and service fee, and we compute their
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+24
+Article submitted to Transportation Science; manuscript no. 1
+Table 6
+Aggregate metrics for different operator priority regimes. System-wide performance metrics are
+normalized against best possible values. System utilization (util.) is the alliance’s total market share, i.e. the
+percentage of travelers electing to travel on a transit, MOD, or hybrid option instead of driving a
+single-occupancy vehicle.
+Objective weights
+Route price ($)
+Performance
+System
+µP AX
+µREV
+µV MT
+Min.
+Mean
+Max.
+PAX
+REV
+VMT
+util. %
+1.0
+0
+0
+$0.00
+$0.00
+$0.00
+100.00%
+0.00%
+100.00%
+50.32%
+1.0
+0.2
+0
+$0.00
+$0.04
+$0.54
+99.26%
+3.19%
+100.59%
+50.32%
+1.0
+0.4
+0
+$0.18
+$4.24
+$12.29
+80.15%
+61.62%
+113.01%
+45.02%
+1.0
+0.6
+0
+$3.63
+$7.88
+$23.62
+70.50%
+78.38%
+117.89%
+40.20%
+1.0
+0.8
+0
+$6.63
+$10.43
+$25.89
+64.61%
+85.66%
+120.39%
+37.33%
+1.0
+1.0
+0
+$7.04
+$12.00
+$28.14
+59.99%
+90.04%
+122.35%
+35.93%
+0.8
+1.0
+0
+$7.17
+$13.14
+$29.66
+56.35%
+92.79%
+123.81%
+34.95%
+0.6
+1.0
+0
+$7.47
+$15.20
+$31.64
+51.94%
+95.39%
+125.33%
+32.95%
+0.4
+1.0
+0
+$6.91
+$16.27
+$38.89
+46.81%
+97.57%
+127.13%
+31.82%
+0.2
+1.0
+0
+$8.59
+$18.59
+$45.27
+40.03%
+99.33%
+129.17%
+29.95%
+0
+1.0
+0
+$10.00 $21.58
+$60.37
+30.91%
+100.00%
+131.65%
+27.65%
+0
+1.0
+0.2
+$9.71
+$18.28
+$42.20
+45.90%
+97.77%
+127.10%
+29.95%
+0
+1.0
+0.4
+$8.52
+$16.41
+$36.63
+59.59%
+89.37%
+121.46%
+30.97%
+0
+1.0
+0.6
+$8.50
+$13.02
+$25.58
+70.54%
+76.54%
+116.32%
+33.99%
+0
+1.0
+0.8
+$5.22
+$11.13
+$21.65
+80.62%
+56.33%
+110.38%
+35.17%
+0
+1.0
+1.0
+$1.83
+$8.39
+$14.59
+90.11%
+29.53%
+104.39%
+37.83%
+0
+0.8
+1.0
+$0.00
+$6.77
+$10.00
+95.40%
+10.83%
+100.87%
+39.48%
+0
+0.6
+1.0
+$0.00
+$6.67
+$9.96
+95.58%
+10.45%
+100.81%
+39.54%
+0
+0.4
+1.0
+$0.00
+$3.70
+$6.00
+97.63%
+6.53%
+100.43%
+44.04%
+0
+0.2
+1.0
+$0.00
+$0.00
+$0.00
+100.00%
+0.00%
+100.00%
+50.32%
+0
+0
+1.0
+$0.00
+$0.00
+$0.00
+100.00%
+0.00%
+100.00%
+50.32%
+(a) Optimal fare parameter values.
+(b) Optimal discount characteristics.
+Figure 4
+Optimal fares across varying alliance priority regimes.
+
+MOD base fare
+Fare value
+10.0
+Transit base fare
+7.5
+5.0
+MOD markup
+2.5
+Transit markup
+0.0
+(1,0,0)
+(0.1.0)
+(0.1.1
+(0,0,1)
+(PAX weight, REV weight, VMT weight)roportion of discounts activated
+0.6
+Percent discount
+50%
+40%
+0.2
+30%
+0.0
+(1,0,0)
+(1,1,0)
+(0,0,1)
+0.1.0
+(PAX weight, REV weight, VMT weight)Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+25
+markup by combining the published markups per unit distance and time, assuming an average vehicle speed
+of 25 mph (Lyft Inc. 2020). We ignore fare multipliers utilized to manage the two-sided market, since they
+are outside the scope of this work; note that this may lead to slight undercounting of real MOD fares and
+slight overcounting of the real ridership. The real-world MBTA commuter rail base fare and markup are
+interpolated from its zone-based pricing structure, which assigns higher prices to farther zones (MBTA
+2020). While all regimes have slightly lower ridership than that under real fares, all benchmarks achieve
+non-negligible improvements in system-wide metrics. In particular, the REV, REV+PAX, and REV+VMT
+regimes respectively achieve objective value increases of 47.4%, 5.6%, and 1.8% respectively.
+Table 7
+Summary statistics of representative priority regimes. REV, REV+PAX, and REV+VMT regimes
+respectively have objective weights (µREV ,µP AX,µV MT ) equal to (1,0,0),(1,1,0), and (1,0,1). “% routes
+discounted” provides the proportion of routes in the system with activated discounts. “Number of travelers”
+is the total alliance passenger count originating within the alliance service region.
+Priority regime
+Base fares ($)
+Markups ($/mile)
+Discount
+% routes
+Number of
+MOD
+Transit MOD
+Transit
+Multiplier
+discounted
+travelers
+Real fares
+$4.53
+$4.50
+$1.07
+$0.16
+0%
+0%
+25,816
+REV
+$10.00
+$10.00
+$2.37
+$0.63
+50%
+31.20%
+18,309
+REV+PAX
+$10.00
+$8.50
+$0.56
+$0.25
+31%
+70.13%
+23,793
+REV+VMT
+$10.00
+$1.83
+$0.16
+$0.00
+50%
+6.50%
+25,051
+As shown in Table 6, each set of priorities induces interpretable optimal prices and passenger deci-
+sions. The minimum, mean, and maximum real-world route prices in the service region are respectively
+$4, $10.23, and $54.46, while those given by our model are in the range $0-$60.37; thus optimal fares
+are set at the correct order of magnitude across all regimes. Intuitively, route prices are the highest for the
+revenue maximizing regime, and they decrease gradually as the importance of VMT or passenger utility
+increases. System-wide performance metrics are normalized against the best possible values across tested
+regimes, naturally achieved by each metric’s corresponding single-objective optimization. The lowest rev-
+enue is achieved in regimes that solely maximize passenger utility or minimize VMT, because very low
+prices achieve very low revenues, but increase passenger happiness and entice more passengers away from
+single-occupancy vehicles. Analogously, the highest fares achieve the highest revenue, with more passen-
+gers electing to travel outside the system, and lowering overall passenger utility. System-wide outcomes
+vary smoothly with gradually changing alliance priorities. Note that even under single-objective revenue
+maximization,, route prices remain in the ballpark of real-world fares. While both base fares and discount
+multiplier reach their upper limits, the both optimal markups stay in the interior of the allowable range.
+We attribute the model’s realism to the incorporation of the endogenous passenger choice model into the
+fare-setting model, as opposed to modeling exogenous demand parametrically. These intuitive observations
+confirm that our fare-setting model is suitable for generating trustworthy qualitative insights.
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+26
+Article submitted to Transportation Science; manuscript no. 1
+At a quick glance, minimizing VMT and maximizing passenger utility seem to achieve similar outcomes
+in Table 6—lower prices and higher system utilization—raising the question of why it is worth modeling
+them separately. We turn to Figure 4 to illustrate how each objective yields qualitatively very different
+designs. As VMT minimization increases in importance (moving from the middle towards the right in Fig-
+ure 4a), the markup is zeroed out, equalizing fares across longer and shorter routes. The elimination of a
+distance-based markup entices more longer-distance commuters to travel on the allied network, thus lower-
+ing VMT. When the system maximizes passenger utility, a more nuanced fare structure emerges to address
+heterogeneous passenger preferences. All pricing levers are employed: base fares, markups, discount multi-
+pliers, and discount activations. Higher markups and base fares are coupled with more numerous discounts
+across hybrid options, illustrated in the left halves of both Figures 4a and 4b. Thus, prioritization of each
+objective (PAX versus VMT) results in similar system-wide performance metrics by qualitatively different
+means. To extract the corresponding fare designs, we consider case studies prioritizing at most one of PAX
+and VMT at a time, with varying weights for revenue. Section 5.3 further investigates geographic factors.
+Finally, note that the variation in system utilization due to allied fare-setting is small compared to the
+integrated network’s total loads. Before the pandemic, MBTA’s bus load factor during peak hours was
+already below 75% (Hicks 2017). MBTA commuter rail transported around 120k passengers on an average
+weekday in 2018 (MBTA 2022), with 81.2% of inbound ridership on peak trains (BRMPO 2012), yielding
+approximately 49k travelers on MBTA commuter rail during the AM rush. Moreover, approximately 116k
+daily TNC rides were destined for Boston in 2018, while another 15.3k originated in the alliance service
+region every day (MDPU 2018). In contrast, the ridership numbers in Table 7 show that the alliance’s
+ridership under real fares is a small proportion of the entire integrated network and that the system is capable
+of accommodating all demand redistribution as a result of allied fare-setting. In fact, Table 7 shows that
+the aggregate alliance ridership is slightly lower than under real fares. Thus, drops on other routes will
+compensate for the slight ridership increases that may happen on certain routes under our proposed pricing
+alliance, ensuring that the system-wide transit load factors and MOD detour times are expected to remain
+largely unchanged as a result of the pricing alliance. We conclude that the linked resource reallocation
+problem need not be considered, when looking for rapid gains through pricing alliance formation.
+5.3.
+Ensuring Equitable Access through a Refined Income-Aware Model Specification
+Our fare-setting model captures passenger’s travel preferences and travel decisions when designing fares, a
+critical step to satisfying passenger needs. However, different groups of passengers may have very different
+preferences. Ignoring such differences can lead to inequitable and socially undesirable outcomes. After all,
+equity is a key driver for integrating on-demand services into public transportation options. Acknowledging
+this challenge, in this section we further refine our choice model and quantify the impacts of this nuanced
+model specification on system-wide metrics compared to an aggregated, average-case choice model.
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+27
+To this end, we augment our case study. Towns targeted for transit expansion by the MBTA in our case
+study have wide-ranging median household incomes, translating into varying price sensitivities. Affluent
+travelers’ route choice decisions are less susceptible to changing fares than those of low-income travelers.
+To partly account for such passenger heterogeneity, we compute a ratio of each town’s median household
+income to the average of the median household incomes across the entire service region (U.S. Census
+Bureau 2018). We use this income ratio to scale passengers’ price sensitivities. See Appendix G for details.
+Table 8
+Allied system’s daily morning rush ridership with (Refined) and without (Base) the choice model
+refinement, their percentage difference (Diff.), real-world ridership (Real), median household income (HHI), and
+average distance (Dist.) of alliance routes originating in the corresponding town and ending in the inner city.
+HHI
+Dist.
+REV+PAX
+REV
+REV+VMT
+Town
+$K
+Miles Base Refined
+Diff.
+Base Refined
+Diff.
+Base Refined
+Diff.
+Real
+Lawrence
+41.6
+32.4
+1585
+1982
+125%
+918
+1881
+205% 1331
+2065
+155% 1566
+Lowell
+52.0
+30.8
+2000
+2525
+126% 1285
+1565
+122% 1852
+2633
+142% 2115
+Lynn
+54.6
+13.5
+3539
+3769
+107% 2283
+2560
+112% 3615
+3188
+88%
+4023
+Brockton
+55.1
+24.6
+3498
+3709
+106% 1949
+3364
+173% 3416
+3919
+115% 3872
+Salem
+65.6
+18.9
+1885
+2043
+108% 1189
+1382
+116% 1961
+1786
+91%
+2171
+Haverhill
+67.6
+39.1
+1605
+1746
+109% 1068
+1214
+114% 1630
+1809
+111% 1801
+Framingham
+79.1
+26.1
+1203
+1327
+110%
+780
+936
+120% 1093
+1117
+102% 1198
+Waltham
+85.7
+11.8
+1974
+1873
+95%
+1597
+1713
+107% 2101
+1946
+93%
+2249
+Woburn
+88.7
+14.3
+1795
+1828
+102% 1479
+1633
+110% 2047
+1910
+93%
+2199
+Stoneham
+94.8
+11.6
+761
+785
+103%
+610
+691
+113%
+880
+830
+94%
+948
+Wakefield
+95.3
+13.3
+1004
+1007
+100%
+783
+889
+114% 1140
+1076
+94%
+1228
+Melrose
+103.7
+9.8
+810
+821
+101%
+637
+711
+112%
+889
+842
+95%
+949
+Burlington
+105.4
+17.6
+1091
+1144
+105%
+975
+1053
+108% 1264
+1193
+94%
+1340
+Reading
+112.6
+16.6
+846
+871
+103%
+703
+787
+112%
+972
+927
+95%
+1038
+Winchester
+159.5
+10.8
+197
+207
+105%
+193
+198
+103%
+218
+210
+96%
+227
+We compare the allied system ridership across priority regimes and towns under fares corresponding to
+base (i.e., income-agnostic) as well as refined (i.e., income-aware) choice models. We first calculate fares
+using the fare-setting model incorporating income-agnostic and income-aware choice models separately,
+and then evaluate both fare designs by calculating (and reporting in Table 8) ridership using only the income-
+aware choice model. Due to the use of the income-aware model, the system ridership in the REV+PAX
+regime increases by 13% on average for the towns with below-average median HHI compared to a less
+than 2% average increase for the towns with above-average median HHI. While ridership increases across
+the board due to generally lower fares, the greatest increases occur in Lawrence and Lowell, which are the
+two towns with the lowest median household incomes. The lower middle-income bracket (Lynn, Brockton,
+Salem, Haverhill, and Framingham) sees the next-highest ridership increase. Similar trends are observed in
+the REV regime: ridership increases across all towns due to the fact that higher revenue can be achieved with
+lower fares and higher volume. The above-average income towns gain ridership by only 10% on average,
+while the below-average income towns have a 37% gain.
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+28
+Article submitted to Transportation Science; manuscript no. 1
+In contrast to the 7% and 23% average ridership gains in REV+PAX and REV regimes, average ridership
+grows by less than 4% in the REV+VMT regime. But interestingly, this regime has significantly larger
+ridership increases for longer distance passengers. In particular, the five towns that are farthest from the
+inner city area—Lawrence, Lowell, Brockton, Haverhill, and Framingham—are the exact five towns with
+ridership increases. Their average increase is about 25% while the remaining 10 towns see an average 7%
+drop in ridership. Thus, the REV+VMT regime increases access to the allied network for commuters who
+are farther from the inner city. Overall, the prioritized system-wide objectives improved by 19.16%, 0.35%
+and 0.99% respectively for REV+PAX, REV, and REV+VMT regimes compared to the real-world fares.
+These results underline the importance of capturing passenger preference heterogeneity to amplify our
+model’s practical impact. A choice model that reflects passenger preferences more accurately improves
+system-wide outcomes and especially maximizes passenger benefits. We conclude that our income-aware
+refined model improves transportation equity for passengers—as compared to the income-agnostic aggre-
+gate model—making it a valuable tool for transit agencies to incorporate into strategic decision-making.
+5.4.
+Quantifying the Value of Cooperation
+To quantify the value of operator cooperation for operators and passengers, we solve the non-cooperative
+fare-setting model for all allied priorities depicted in Table 6. In each experiment, the transit operator’s
+priorities are identical to alliance priorities, whereas the MOD operator always maximizes revenue. Thus,
+our experiment reflects an assumption that the prospective alliance will adopt the transit operator’s priorities.
+Figure 5 depicts the non-cooperative equilibrium fares. Discount multipliers are not included since they
+are not applicable in the non-cooperative setting. Table 9 compares allied outcomes to corresponding non-
+cooperative outcomes. In particular, we report the percentage increase in the alliance objective compared
+to that computed under the non-cooperative setting (transit obj. % inc.), and the percentage increase in
+MOD operator revenue due to the alliance (MOD rev. % inc.). We also provide the revenue allocations
+to both operators as determined by the revenue allocation mechanism in Section 2.4. The non-cooperative
+system utilization and non-cooperative average route prices are also provided for each experiment. Figure
+6 illustrates passenger mode choices across all tested regimes for both allied and non-cooperative fares.
+Figure 5
+Non-cooperative fare parameters for different transit operator priorities.
+
+MOD base fare
+Fare value
+10.0
+Transit base fare
+7.5
+5.0
+MOD markup
+2.5
+0.0
+Transit markup
+(1,0,0)
+(1,1,0)
+(0,1,0)
+(0,1,1)
+(0,0,1)
+(PAX weight, REV weight, VMT weight)Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+29
+Table 9
+Allied vs. non-cooperative outcomes. Transit operator’s priorities represented as µT R.
+Transit obj. weights
+Transit obj.
+MOD rev.
+MOD allied
+Transit allied
+System
+Route price
+µP AX
+T R
+µREV
+T R
+µV MT
+T R
+% inc.
+% inc.
+rev. alloc.
+rev. alloc.
+util. %
+($) Mean
+1.0
+0
+0
+9.60%
+0.00%
+$501.90K
+-$501.90K
+35.01%
+$12.83
+1.0
+0.2
+0
+6.87%
+0.00%
+$501.90K
+-$501.90K
+35.01%
+$12.83
+1.0
+0.4
+0
+4.93%
+0.00%
+$487.91K
+$1848.89K
+32.41%
+$13.45
+1.0
+0.6
+0
+4.80%
+0.00%
+$479.04K
+$2337.68K
+30.46%
+$14.78
+1.0
+0.8
+0
+5.45%
+0.00%
+$473.18K
+$2624.27K
+29.21%
+$15.83
+1.0
+1.0
+0
+7.57%
+0.00%
+$468.14K
+$2824.63K
+28.16%
+$16.96
+0.8
+1.0
+0
+27.71%
+0.00%
+$465.06K
+$2940.94K
+27.39%
+$17.77
+0.6
+1.0
+0
+5.82%
+0.00%
+$462.29K
+$3022.50K
+26.70%
+$18.54
+0.4
+1.0
+0
+1.31%
+0.00%
+$460.45K
+$3101.99K
+26.43%
+$18.66
+0.2
+1.0
+0
+0.54%
+1.04%
+$463.61K
+$3173.57K
+26.00%
+$18.97
+0
+1.0
+0
+0.33%
+1.33%
+$462.79K
+$3195.02K
+25.65%
+$19.24
+0
+1.0
+0.2
+1.55%
+4.42%
+$481.71K
+$3108.48K
+26.66%
+$18.50
+0
+1.0
+0.4
+0.52%
+0.00%
+$465.54K
+$2839.52K
+27.50%
+$18.07
+0
+1.0
+0.6
+0.37%
+4.34%
+$492.25K
+$2376.74K
+28.67%
+$17.40
+0
+1.0
+0.8
+0.51%
+0.00%
+$482.97K
+$1573.28K
+31.13%
+$15.33
+0
+1.0
+1.0
+0.54%
+0.00%
+$494.38K
+$591.27K
+33.48%
+$13.76
+0
+0.8
+1.0
+0.60%
+0.00%
+$501.90K
+-$105.59K
+35.01%
+$12.83
+0
+0.6
+1.0
+0.73%
+0.00%
+$501.90K
+-$119.73K
+35.01%
+$12.83
+0
+0.4
+1.0
+0.92%
+0.00%
+$501.90K
+-$262.97K
+35.01%
+$12.83
+0
+0.2
+1.0
+1.36%
+0.00%
+$501.90K
+-$501.90K
+35.01%
+$12.83
+0
+0
+1.0
+1.90%
+0.00%
+$501.90K
+-$501.90K
+35.01%
+$12.83
+Figure 6
+Market shares by mode across priority regimes for allied and non-cooperative fare-setting models.
+Figure 5 illustrates that the MOD operator’s revenue-maximizing strategy remains relatively constant,
+regardless of the transit operator’s priorities. Still, when the transit operator prioritizes VMT minimization or
+passenger benefits maximization, the MOD operator selects higher fare parameters (mainly through higher
+markups) than in the corresponding allied settings. Thus, the average route price (Route price ($) Mean)
+columns in Tables 6 and 9 show that the non-cooperative average route prices are higher than average allied
+route prices in every regime except for the one where the transit operator only prioritizes revenue.
+In scenarios that partially maximize passenger benefits or minimize VMT, the alliance sets MOD fare
+parameters lower than in the non-cooperative regime (Figure 4a vs. 5). Figure 6 shows that MOD-only mar-
+ket shares decrease and hybrid market shares increase in the non-cooperative regime as passenger benefits
+or VMT are increasingly prioritized. We can conclude that, although fewer passengers utilize MOD-only
+
+30
+Mode
+Transit
+MOD
+Hybrid
+Fare model
+10
+Allied
+Non-coop
+0
+(1,0,0)
+(1,1,0)
+(0,1,0)
+(0,1,1)
+(0,0,1)
+(PAX weight, REV weight, VMT weight)Cummings et al.: Transportation Alliance Design with Endogenous Demand
+30
+Article submitted to Transportation Science; manuscript no. 1
+options in non-cooperative scenarios where the transit operator is VMT- or passenger-oriented, a higher
+volume of passengers selects hybrid options due to the very low (or free) transit fares observed in Figure
+5. Thus, the MOD operator earns more revenue in those non-cooperative scenarios in which the transit
+operator is more altruistic. This is reflected in the revenue allocation mechanism: observe in Table 9 that
+the MOD operator earns strictly more revenue in almost all regimes where the transit operator is not solely
+a revenue maximizer, even though the system as a whole generates strictly less total revenue, as seen in
+comparison with Table 6. The revenue allocation mechanism ensures that the MOD operator receives their
+non-cooperative earnings, despite the lower MOD fares that the alliance sets to achieve lower VMT or
+higher passenger benefits. This in turn reduces the transit’s revenue allocation as high VMT or low pas-
+senger benefits are increasingly penalized. On the other hand, the transit operator always strictly improves
+its objective of optimizing total system-wide performance, however it chooses to define it. As a result, the
+MOD operator interestingly finds it in its interest to adopt transit’s priorities as transit increasingly diverges
+from revenue maximization. In other words, the revenue-maximizing MOD operator would not prefer a
+revenue-maximizing alliance. In fact, the MOD operator would benefit most from total altruism on transit’s
+side (an exclusive focus on either passenger utility or VMT reduction). Passengers win due to the strictly
+lower prices and higher system utilization that result from such alliances.
+The transit operator must ultimately set the ceiling in terms of the price they are willing to pay for
+the alliance benefits. Table 9 shows that the transit agency runs a deficit to appease the MOD operator if
+its revenue emphasis is too low. A deficit alone might not be enough to dissuade the transit agency from
+participating in the alliance: as we have noted, every public transit mode operates at a loss, especially on-
+demand options (Kane, Tomer, and Puentes 2016). To weigh the financial implications of an alliance, the
+agency may compare the magnitude of the loss to the cost of the analogous MOD system operated by
+transit on their own in the absence of outsourcing through an alliance. Oftentimes, transit agencies also
+receive grants to fund pricing alliances (Federal Transp. Administration 2016), and the daily deficit rate
+can be compared to the grant award amount and intended duration. To enable a daily deficit rate that keeps
+pace with financial resources over time, the transit agency may propose to reset overall performance metric
+goals to induce different optimal fares, or to adjust the geographic and/or temporal scope of the alliance.
+In the end, each transit agency is expected to choose the trade-off point between its financial, passenger-
+focused, and environmental goals that most closely aligns with their overall policy and various practical
+and financial constraints. Regardless, our allied and non-cooperative fare-setting models together with our
+revenue allocation mechanism jointly provide a toolkit usable by transit agencies to weigh these trade-offs
+as they evaluate a potential pricing alliance.
+6.
+Conclusions, Limitations and Future Directions
+We contribute a pricing alliance design framework to enable incentive-aligned collaboration between transit
+agencies and MOD operators. Our allied fare-setting model captures the interdependent decisions of pas-
+sengers and operators, allowing the alliance to maximize benefits for all stakeholders across the integrated
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+31
+network. The prescriptive pricing framework can be generalized to different types of large-scale alliances
+with varying MOD operators, service populations, fare structures, service goals, and network configurations.
+We accomplish large scale by developing a tractable two-stage fare-setting formulation equivalent to the
+original mixed-integer non-convex optimization problem, which we then solve with a tailored SOS2 coor-
+dinate descent approach. From a technical standpoint, our approach selects consistently and significantly
+higher quality solutions than benchmarks based on Bayesian Optimization, enabling additional system-
+wide benefits worth tens of thousands dollars per day over the service region. Practically speaking, the high
+quality solutions from our allied fare-setting model together with our dedicated revenue allocation mech-
+anism work together to align revenue-oriented MOD operators with transit goals of passenger utility and
+single-occupancy VMT reduction. In other words, cooperative pricing results in win-win-win outcomes for
+passengers, MOD operators, and transit agencies. Finally, the income-aware nuanced version of our fare-
+setting model helps enhance passenger equity-related goals: by tuning passenger route choice models, the
+alliance can prioritize lower fares and higher utilization for low-income or long-distance commuters.
+Our analysis is based on a few assumptions which can be relaxed in future work. We assumed that the
+MOD operator serves as a contractor to the transit agency and agrees to set static fares for trips in the inte-
+grated system. While this is one good way to ensure public sector prices are transparently communicated,
+it may also be possible to set fare schemes that allow MOD operators, especially TNCs, to maintain their
+dynamic prices, perhaps through the transit operator subsidizing the cost of a passenger’s trip up to a fixed
+dollar amount. Additionally, we model average-case travel times over a fixed set of route options for the
+MOD operator’s portion of the network to represent average-case operations, which simplifies their typi-
+cally dynamic routing scheme. In other words, we consider the case where the alliance sets one permanent
+fare scheme that is optimized for average-case performance. Future research may consider optimizing for
+the worst-case performance, and/or integrating dynamic routing into the fare-setting model if the alliance
+prefers dynamic fares, requiring the integration of two complex problem classes. Finally, we did not incor-
+porate joint resource reallocation into the pricing scheme, due to the observation that the system is capable
+of accommodating all demand re-distributions attributed to changing prices. This assumption works well
+for our case study over the Greater Boston Area because the allied system’s ridership is a small fraction of
+the larger service region with a high-capacity existing network. Future research could consider relaxing this
+assumption to generalize the analysis by jointly modeling the capacity allocation and pricing decisions.
+Acknowledgments
+This material is based upon work supported by the National Science Foundation under Grant no. 1122374.
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+37
+Appendix A:
+Proof of Lemma 1: Two-stage Decomposition is Equivalent to Full Formulation
+Given an optimal solution from each formulation, we construct a feasible solution for the other with the same objective
+value. In this section, we introduce the following notation for a non-discounted route price r ∈ R, for expositional
+brevity:
+σr(β) =
+�
+k∈Or
+(β0
+k + ∆rk · β∆
+k ).
+(35)
+PADP-FS2SD → PADP-FS. Let (β2SD,Λ2SD,x2SD,s2SD,w2SD,p2SD) be an optimal solution to the PADP-
+FS2SD model with objective value z2SD. We construct the following solution to the PADP-FS model.
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+βF S = β2SD
+ΛF S = Λ2SD
+xF S = x2SD
+sF S = s2SD
+pF S = p2SD
+(36)
+We demonstrate feasibility of (36) to the PADP-FS model.
+• xF S are binary by construction.
+• Constraints (6) and (7) hold by construction.
+• Constraints (8) and (9) are feasible due to constraints (18), (21), and (22). For a given i ∈ N, constraints (18)
+and (22) ensure that
+sF S
+ir ∝ exp(uir + αi · pF S
+r )
+∀r ∈ Ri, and
+sF S
+i0 ∝ exp(ui0).
+Constraint (21) forces each sir and si0 to be normalized by exp(ui0) + �
+r∈Ri exp(uir + αi · pF S
+r ). The result
+follows.
+• Constraints (10) - (12) hold by definition of B and L.
+Finally, to show that solution (36) has the same objective value as z2SD, we must only prove that pF S
+r
+· sF S
+ir = w2SD
+ir
+for all i ∈ N and r ∈ Ri, which follows directly from Constraints (19) and (23).
+PADP-FS → PADP-FS2SD. Let (βF S,ΛF S,pF S,sF S,xF S) be an optimal solution to the PADP-FS model with
+objective value zF S. We construct the following solution to the PADP-FS2SD model.
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+β2SD = βF S
+Λ2SD = ΛF S
+x2SD = xF S
+s2SD = sF S
+w2SD
+ir
+= pF S
+r
+· sF S
+ir
+∀i ∈ N,r ∈ Ri
+p2SD = pF S
+(37)
+First we demonstrate the feasibility of (37) to PADP-FS2SD.
+• s2SD, w2SD and p2SD are non-negative and x2SD are binary by construction.
+• Constraints (18) and (22): Consider any i ∈ N. For r ∈ Ri \ RDE, and by expressions (8) and (9),
+γir(β2SD,0) · s2SD
+ir
+= γir(βF S,0) · sF S
+ir
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+38
+Article submitted to Transportation Science; manuscript no. 1
+=
+exp(ui0)
+exp(uir + αi · pF S
+r ) ·
+exp(uir + αi · pF S
+r )
+ui0 + �
+s∈Ri exp(uis + αi · pF S
+s )
+=
+exp(ui0)
+ui0 + �
+s∈Ri exp(uis + αi · pF S
+s )
+= sF S
+i0 = s2SD
+i0
+.
+Thus constraints (22) are satisfied. Now, consider any a ∈ A with r ∈ Ria. If x2SD
+a
+= 0, then we can use the
+exact same argument as above to confirm that γir(β2SD,0) · s2SD
+ir
+= s2SD
+i0
+again holds. Since
+γir(β2SD,Λ2SD) · s2SD
+ir
+≤ γir(β2SD,0) · s2SD
+ir
+= s2SD
+i0
+≤ γi0(β2SD,0)
+≤ γi0(β2SD,0) + γir(β2SD,Λ2SD) · s2SD
+ir
+= γir(β2SD,Λ2SD) · s2SD
+ir
++ M s
+ir · (1 − x2SD
+a
+),
+the constraints hold. A similar argument applies when x2SD
+a
+= 1.
+• Constraints (19) and (23): For any i ∈ N, Constraints (23) hold by construction for all r ∈ Ri \ RDE. For a ∈ A
+with r ∈ Ria, we have w2SD
+ir
+= p2SD
+r
+· s2SD
+ir
+regardless of the value of x2SD
+a
+. If x2SD
+a
+= 0, then
+(1 − Λ2SD) · σr(β2SD) · s2SD
+ir
+≤ w2SD
+ir
+= σr(β2SD) · s2SD
+ir
+= (1 − Λ2SD) · σr(β2SD) · s2SD
+ir
++ Λ2SD · σr(β2SD) · s2SD
+ir
+≤ (1 − Λ2SD) · σr(β2SD) · s2SD
+ir
++ Λ2SD · σr(β2SD)
+= (1 − Λ2SD) · σr(β2SD) · s2SD
+ir
++ M w
+ir · (1 − x2SD
+a
+),
+and hence constraints (19) hold. A similar argument applies when x2SD
+a
+= 1.
+• Constraints (21): Consider any i ∈ N. By expressions (8) and (9),
+s2SD
+i0
++
+�
+r∈Ri
+s2SD
+ir
+= sF S
+i0 +
+�
+r∈Ri
+sF S
+ir
+=
+exp(ui0)
+exp(ui0) + �
+s∈Ri exp(uis + αi · pF S
+s )+
+�
+r∈Ri
+exp(uir + αi · pF S
+r )
+exp(ui0) + �
+s∈Ri exp(uis + αi · pF S
+s )
+= 1.
+• Constraints (24) and (25) hold by construction.
+• Constraints β ∈ B and Λ ∈ L hold by construction.
+Now we show that solution (37) to Formulation PADP-FS2SD has the same objective value of zF S.
+z2SD =
+�
+i∈N
+Ni ·
+�
+µP AX · (ui0 +
+�
+r∈Ri
+(uir + αi · p2SD
+r
+) + µREV ·
+�
+r∈Ri
+w2SD
+ir
+− µV MT · (∆i0 · s2SD
+i0
+)
+�
+=
+�
+i∈N
+Ni ·
+�
+µP AX · (ui0 +
+�
+r∈Ri
+(uir + αi · p2SD
+r
+) + µREV ·
+�
+r∈Ri
+p2SD
+r
+· s2SD
+ir
+− µV MT · (∆i0 · s2SD
+i0
+)
+�
+=
+�
+i∈N
+Ni ·
+�
+µP AX · (ui0 +
+�
+r∈Ri
+(uir + αi · pF S
+r ) + µREV ·
+�
+r∈Ri
+pF S
+r
+· sF S
+ir − µV MT · (∆i0 · sF S
+i0 )
+�
+= zF S
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+39
+Appendix B:
+Non-cooperative Fare-setting Game and Iterated Best Response Algorithm
+To benchmark the solutions of our allied fare-setting model, we formulate the non-cooperative fare-setting problem
+between a transit operator and an MOD operator. When they are not allied, each operator can only make fare-setting
+decisions over their portion of the integrated network. The discount activation decision is not applicable in this context,
+and operators set their own fare parameters so as to optimize their prioritized performance metrics over the integrated
+network, subject to endogenous passenger decisions.
+First we introduce new notation. Let µk = (µP AX
+k
+,µREV
+k
+,µV MT
+k
+) be the relative priority weights of operator k ∈ O
+for each system-wide performance metric. For a given k ∈ O, let −k indicate the other operator, i.e. the singleton
+−k ∈ O \ {k}. Each operator sets fare parameters βk = (β0
+k,β∆
+k ). Let Bk := [β0
+min,β0
+max] × [β∆
+min,β∆
+max] be the set
+of valid fare parameters that operator k can set. Let si0(β) and sir(β) respectively denote the market shares of the
+outside option and of route r ∈ Ri for passenger type i ∈ N, as functions by both operators’ fare parameters β.
+sir(β) =
+exp(uir + αi · σr(β))
+exp(ui0) + �
+s∈Ri exp(uis + αi · σs(β))
+i ∈ N,r ∈ Ri
+si0(β) =
+exp(ui0)
+exp(ui0) + �
+s∈Ri exp(uis + αi · σs(β))
+i ∈ N
+Let ˆβ−k denote the fixed fare parameter decisions of operator −k. Then σr(βk; ˆβ−k) denotes the non-discounted price
+of route r ∈ R as a function of the fare parameter decisions of operator k, parameterized by the fixed decisions of oper-
+ator −k, where σr is originally defined in Equation (35). Similarly, sir(βk; ˆβ−k) and si0(βk; ˆβ−k) represent market
+shares as functions of the fare parameter decisions of operator k, given fixed decisions of operator −k. Formulation
+PADP-NC is the non-cooperative fare setting model of operator k.
+(PADP-NC)
+max
+βk∈Bk Wk(βk; ˆβ−k)
+≡
+max
+βk∈Bk
+�
+i∈N
+Ni ·
+�
+µP AX
+k
+·
+�
+ui0 +
+�
+r∈Ri
+(uir + αi · σr(βk; ˆβ−k))
+�
++
+µREV
+k
+·
+�
+r∈Ri
+σr(βk; ˆβ−k) · sir(βk; ˆβ−k)−
+µV MT
+k
+· (∆i0 · si0(βk; ˆβ−k))
+�
+An operator’s non-cooperative fare-setting model is a simplified version of the allied fare-setting model, with more
+restricted decisions. Operator k sets their fare parameters within allowable bounds to maximize system-wide benefits
+over the integrated network, subject to the induced passenger decisions and for given fare parameters set by the other
+operator. Each operator’s model has two decision variables: their base fare and their distance-based markup. Fare
+parameters β must satisfy condition (38) below to be considered a solution to the non-cooperative fare setting problem.
+βk = arg max
+β′
+k∈Bk Wk(β′
+k;β−k)
+k ∈ O
+(38)
+When fare parameters β satisfy condition (38), they define a Nash equilibrium (NE), or more precisely, a pure strategy
+Nash equilibrium. Fare parameters form an NE when neither operator can unilaterally change their pricing strategy to
+improve their own objective, given the fare parameters set by the other operator. We compute an NE using an iterated
+best response algorithm, which takes turns alternating between each operator’s optimal fare parameters computation
+in response to their competitor’s fixed fare parameters, until convergence.
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+40
+Article submitted to Transportation Science; manuscript no. 1
+A pure strategy Nash equilibrium to the non-cooperative fare-setting game is a set of fares such that each opera-
+tor’s decision is the optimal response given the other operator’s fare parameters. An iterated best response algorithm
+randomly initializes the fare parameters and lets each operator take turns setting their best response. The algorithm
+converges when neither operator can gain by changing their response (Fudenberg and Tirole 1991).
+Algorithm 4 Iterated best response for non-cooperative fare setting
+1: procedure ITERATED BEST RESPONSE(ϵ)
+2:
+βk ← uniform draw from Bk for each k ∈ O
+3:
+objPrevk ← −∞ for each k ∈ O
+4:
+objCurk ← Wk(β) for each k ∈ O
+5:
+while maxk∈O(objCurk − objPrevk) > ϵ do
+6:
+for k ∈ O do
+7:
+objPrevk ← objCurk
+8:
+obj ← maxβ′
+k∈Bk Wk(β′
+k;β−k)
+9:
+if obj > objCurk then
+10:
+objCurk ← obj
+11:
+βk ← arg maxβ′
+k∈Bk Wk(β′
+k;β−k)
+12:
+return β
+In general, without proof of existence of a Nash equilibrium, Algorithm 4 is not guaranteed to converge, let alone
+to the same β in the face of random initializations. To prevent infinite oscillation, a maximum iteration limit can
+potentially be implemented. Yet, in all runs of our computational experiments performed when generating Table 9, a
+unique NE was always obtained across the 10 random initializations in a given row of the table.
+Appendix C:
+Proof of Lemma 2: Revenue Allocation Mechanism
+Proof of Lemma (2a): The MOD operator participates in the alliance whenever they can earn at least as much
+revenue by cooperating with the transit agency as they can otherwise. By construction,
+ΦMOD((fk(βnc))k∈O,f(βa,Λa)) = fMOD(βnc) +
+� δ
+2
+�+ ≥ fMOD(βnc),
+so the result follows.
+Proof of Lemma (2b): In the case that allied revenue exceeds combined non-cooperative revenue, we show that the
+allocation is a Nash bargaining solution, which is a classic payment rule guaranteeing properties of Pareto efficiency,
+symmetry, scale invariance, and independence of irrelevant alternatives.
+We frame the alliance revenue allocation problem as a collective bargaining problem. Let the disagreement outcome
+be the operator revenues resulting from non-cooperation, d = (fk(βnc))k∈O. Let X be the set of all revenue allocations
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+41
+such that each operator receives at least their disagreement outcome, and such that the sum of revenue allocations does
+not exceed allied revenue p = f(βa,Λa).
+X(d,p) :=
+��
+ak
+�
+k∈O : ak ≥ dk,∀k ∈ O;
+�
+k∈O
+ak ≤ p
+�
+Let F be the set of all such allocation problems, with each (d,p) ∈ F corresponding to a different allocation problem
+(i.e., a different potential alliance). We seek an allocation solution function Φ : F → X that allocates the available profit
+according to the axioms of Pareto efficiency, symmetry, scale invariance, and independence of irrelevant alternatives.
+It is known that Nash bargaining solutions – which satisfy the above axioms– exactly coincide with optimal solutions
+to optimization problem (39).
+max
+a∈X(d,p)
+�
+k∈O
+(ak − dk)
+(39)
+Expression (40) clearly solves (39).
+dk + p − �
+k∈O dk
+2
+= fk(βnc) + δ
+2 ≡ Φk(d,p),
+∀k ∈ O
+(40)
+To establish the core property, we point to Lemma (2a), as well as the fact that ΦT R(d,p) ≥ fT R(βnc) in the case that
+δ ≥ 0. This proves Lemma (2a).
+Appendix D:
+SOS2 Coordinate Descent Subroutines
+This section specifies subroutines of SOS2 Coordinate Descent, presented in Algorithm 2. The subroutine SEARCH
+DIRECTIONS provides a comprehensive ordered list of search directions that can be multidimensional and/or random-
+ized or neither. The subroutine GENERATE ANCHORS computes an ordered set of evenly spaced SOS2 anchors along
+the specified search direction. In case of a multidimensional search direction, we determine the range of slopes that
+will ensure that the line spans the selected dimension, and sample uniformly from that range.
+Figure 7 visualizes the computation of the slope range in lines (16)-(21) of the GENERATE ANCHORS procedure.
+The most negative slope of a line passing through the current solution is determined by the maximum of the slopes
+of the two line segments connecting the current solution to the upper left and bottom right corners of the diagram.
+Similarly, the most positive slope is determined by the minimum of the slopes of the two line segments connecting the
+current solution to the lower left and upper right corners.
+Appendix E:
+Performance of Algorithm Components
+In this section, we examine the performance of individual components of the algorithm.
+Figure 8 illustrates the distribution of solution times in seconds for the second-stage model. Each calculation of W,
+which requires solving the second-stage model once, needs on average 5.7 seconds of CPU time, which is fast enough
+to be useful, but slow enough to warrant judicious selection of candidate first-stage solution points to evaluate. Each
+of the 31,553 observations was obtained in under one minute. The observations were aggregated across all solutions
+of the second-stage model represented in the paper, including runs of SOS2-CD, BF-CD, and BO.
+Figure 9 depicts the accuracy of the SOS2 model with anchors placed at 10% intervals for the discount multiplier
+axis and at $1 intervals for the rest of the fare parameters. In each trial, a fare parameter combination and a search
+dimension were selected uniformly at random. The “true optimal” fare parameters for each trial (i.e., each combination
+of current solution and search direction) were computed using a brute-force approach, through exhaustive enumeration
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+42
+Article submitted to Transportation Science; manuscript no. 1
+Algorithm 5 Subroutines for SOS2 Coordinate Descent
+1: procedure SEARCH DIRECTIONS(random, multidim)
+2:
+if multidim then
+3:
+searchDirs ← {(β0
+T R,β∆
+T R), (β0
+MOD,β∆
+MOD), Λ}
+// index names of y
+4:
+else
+5:
+searchDirs ← {β0
+T R, β∆
+T R, β0
+MOD, β∆
+MOD, Λ}
+6:
+if random then
+7:
+searchDirs ← SHUFFLE(searchDirs)
+// randomly permutes the set searchDirs
+8:
+return searchDirs
+9: procedure GENERATE ANCHORS((β,Λ), searchDir, D)
+10:
+if searchDir ∈ {(β0
+k′,β∆
+k′) : k′ ∈ O} then
+// if searchDir is multidimensional
+11:
+β0
+k,β∆
+k ← searchDir
+12:
+if Unif(0,1) < 0.5 then
+// select spanning dimension
+13:
+(x,xmin,xmax),(y,ymin,ymax) ← (β0
+k,β0
+min,β0
+max),(β∆
+k ,β∆
+min,β∆
+max)
+14:
+else
+15:
+(x,xmin,xmax),(y,ymin,ymax) ← (β∆
+k ,β∆
+min,β∆
+max),(β0
+k,β0
+min,β0
+max)
+16:
+if x == 0 then
+// determine valid slopes for spanning affine lines
+17:
+mmin ← ymin−y
+xmax−x; mmax ← ymax−y
+xmax−x
+18:
+else if x == xmax then
+19:
+mmin ← ymax−y
+xmin−x; mmax ← ymin−y
+xmin−x
+20:
+else
+21:
+mmin ← max{ ymax−y
+xmin−x, ymin−y
+xmax−x}; mmax ← min{ ymin−y
+xmin−x, ymax−y
+xmax−x}
+// see Figure 7
+22:
+m ← Unif(mmin,mmax); b ← y − m · x; δx = xmax − xmin
+// slope of spanning affine line
+23:
+anchors ← {(xmin + i−1
+D−1 · δx,b + m · (xmin + i−1
+D−1 · δx),β0
+−k,β∆
+−k,Λ) : i ∈ {1,...,D}}
+24:
+else
+25:
+if searchDir == Λ then
+26:
+maxV al ← Λmax; minV al ← Λmin
+27:
+else if searchDir in {β0
+k : k ∈ O} then
+28:
+maxV al ← β0
+max; minV al ← β0
+min
+29:
+else
+30:
+maxV al ← β∆
+max; minV al ← β∆
+min
+31:
+anchors ← {(minV al + i−1
+D−1 · (maxV al − minV al),(β,Λ) \ {searchDir}) : i ∈ {1,...,D}}
+32:
+Insert (β,Λ) into the ordered set anchors
+33:
+return anchors
+of every solution along that affine line at a 1% granularity for discount multiplier and a $0.10 granularity for the other
+four fare parameters. Then the W obtained through SOS2∗ procedure was evaluated and compared with this “true
+optimal” solution, to compute the SOS2 optimality gap. The mean optimality gap was 0.25%, and it was 0% in 29.8%
+of the instances. This establishes the trustworthiness of the SOS2 model outputs, especially given the drastic CPU time
+reduction they provide. We repeated this procedure for 10 hours, resulting in 927 solutions.
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+43
+Figure 7
+Computing the range of slopes such that the selected dimension is spanned.
+Figure 8
+Distribution of 31,553 second-stage model solution times. Mean solution time = 5.7s.
+Figure 9
+Distribution of SOS2 model optimality gaps. Mean gap = 0.25%, across 927 solutions. Exact solu-
+tions (with 0% gaps) obtained in 276 of the 927 cases.
+Figure 10 shows the distribution of BF-CD solution times for a single trajectory across 50 trials. BF-CD never
+terminates before an hour elapses.
+Appendix F:
+Bayesian Optimization Benchmark
+Bayesian Optimization (BO) is a sequential search strategy for optimizing low-dimensional black-box functions
+(Mockus 2012). Typically, the function being optimized takes a long time to evaluate and has no analytical form, pre-
+cluding access to gradients. We first provide a high-level overview of BO, and then we summarize the design choices
+that we use in this paper. In particular, we adapt the setup of a recent dedicated study by Liu et al. (2019) on using
+Bayesian Optimization to select MOD system service parameters subject to passenger mode choice. Interested readers
+
+0.3
+0.2
+Density
+0.1
+0.0
+10
+20
+30
+Solution time (s)1000
+750
+Density
+500
+250
+0 -
+0.00
+0.01
+0.02
+0.03
+SOs2 model optimality gapYmax
+Ymax - y
+Ymax - y
+Xmin - X
+X - xewx
+Other dimension
+βcur
+Ymin - y
+Ymin - y
+Xmin - X
+Xmax - X
+Ymin
+Xmin
+x
+xewx
+Spanning dimensionCummings et al.: Transportation Alliance Design with Endogenous Demand
+44
+Article submitted to Transportation Science; manuscript no. 1
+Figure 10
+BF-CD single-trajectory solution times across 50 trials.
+are referred to Liu et al. (2019) for a more detailed description of BO in this application context. Readers interested in
+a general BO tutorial are referred to Brochu, Cora, and de Freitas (2010).
+In the absence of a closed-form representation of our black-box function f : x → R, the BO framework first imposes
+a prior belief upon f via a probabilistic surrogate model. Given this surrogate model, BO iteratively (i) updates
+the likelihood of historical observations with new evaluations of f to obtain a more informative posterior, and (ii)
+queries an acquisition function that uses the updated posterior to recommend the next value of x that should be
+evaluated. A very common surrogate model for the black-box function is called a Gaussian Process (GP), which is a
+stochastic process in which any finite set of random variables follows a multivariate Gaussian distribution (Mockus
+2012). After building the GP with historical observations, the GP maps a given point in the search space to a univariate
+Gaussian distribution. We interpret this output distribution as a set of potential values of f(x), accounting for noise.
+The acquisition function is a BO design choice intended to help in navigating the trade-off between exploration and
+exploitation, i.e. exploring more of the search space versus exploiting regions where a globally optimal solution is
+suspected to exist. As in Liu et al. (2019), we use a GP upper confidence bound as our acquisition function, which
+characterizes the BO optimization process as a multi-armed bandit problem.
+Appendix G:
+Greater Boston Area Case Study
+First we describe the primary case study. Then we explain the income-aware modification to the choice model.
+Primary case study. Our case study is based on several datasets describing travel characteristics in the Greater
+Boston Area. We consider a pricing alliance for the morning commute, restricting our time window to 6-10AM on a
+typical weekday in Fall 2018. Our data sources and their purposes are as follows:
+• Massachusetts Bay Transit Authority (MBTA) Focus40 report: We consider MBTA “urban gateways,” character-
+ized as regions located beyond the rapid transit network but in close proximity to the commuter rail network.
+More formally, an urban gateway is a town with high potential to be receptive to additional development of pub-
+lic transportation options, especially if they connect to commuter rail hubs (MDOT 2019). These towns define
+the service region of our case study.
+• MBTA Fall 2018 General Transit Feed Specification (GTFS): We use the MBTA GTFS to extract rapid transit
+and commuter rail stations and edges. We also collect valid transfer edges from those enumerated by the feed
+(MBTA 2018).
+• 2017 Longitudinal Employer-Household Dynamics (LEHD) Origin-Destination Employment Statistics
+(LODES): This data specifies the number of jobs corresponding to each origin-destination census tract pair,
+where the origin represents the employee’s home tract and the destination represents their work tract. We used
+this data as a proxy for daily morning commute demand (U.S. Census Bureau 2017).
+
+0.100
+0.075
+Density
+0.050
+0.025
+0.000
+15
+20
+25
+Solution time (h)Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+45
+• U.S. Census Bureau American Community Survey, 2014-2018: We extracted town-level data on the modes of
+transportation to work, median household income, and total population for towns in selected priority places (U.S.
+Census Bureau 2018). This data was primarily used to calibrate the passenger utility parameters.
+• TIGER/Line shapefiles: We used this data to assign network components to census tracts. We use the centroids
+of census tracts in priority places to define passenger origins and destinations (U.S. Census Bureau 2019).
+• Uber Movement travel times: We used travel times aggregated over the 4th quarter of 2018 in the Greater Boston
+Area to approximate travel times of first- and last-mile edges and outside options (Uber Technologies Inc. 2020).
+• Lyft fare parameters: We calculate the real-world prices of the MOD operator’s services using Lyft’s base fares,
+distance-based markups, and time-based markups (Lyft Inc. 2020). We also use this data to calibrate the passen-
+ger utility parameters.
+• MBTA commuter rail fares: We characterize the real-world prices of the transit operator’s services using the
+MBTA commuter rail fares. We interpolate the base fares and distance-based markups from the discretized,
+zone-based fare structure together with the distances between transit stations in each zone (MBTA 2020).
+We define the transit network using commuter rail and rapid transit edges. Each passenger’s origin and destination
+are census tract centroids from the LODES dataset. We built MOD operator network’s edges by connecting centroids
+to transit stations and to each other within each priority place. We filtered out unnecessary edges that corresponded to
+very low demand; implementing a 90% service goal enabled us to reduce the MOD operator’s edge set by 50%. We
+only consider those commuters who work locally (within their home priority place) or in the inner city (the dashed
+bounding box in Figure 3). Commuters to the inner city were assumed to walk the last leg of their trip, from their final
+transit station to their destination centroid. Because the rapid transit network is so well connected within the inner city,
+approximately 90% of all inner city destinations in the data were within a 0.5 mile walk of at least one transit station.
+To generate the route choice sets, we first built a routing network. This routing network is a transformation of the
+physical network, consisting of MOD, transit, waiting, walking, and transfer edges. The cost of each edge was the
+time to “traverse”, whether that traversal entailed travel on the transit and/or MOD system, travel by foot, or stationary
+waiting. We filtered out the trips with more than one transfer, by performing a transformation on the relevant transit
+stations by replicating those nodes—each transfer station had one node for incoming transfers and one for outgoing
+transfers. A path through this network from a passenger’s origin to destination captures their total travel time along
+that path. Over this routing network, we implemented Yen’s algorithm for finding the k-shortest loopless paths over a
+directed graph with non-negative edge costs (Yen 1970). We first obtained up to 10 shortest paths for each commuter to
+the inner city. Some paths were effectively duplicates, in that every aspect of the path was the same except for the final
+transit station. We identified “unique” routes by their starting and transfer locations. With this definition of uniqueness,
+we selected the shortest path for each mode: direct via transit (with a potential walking, driving, or local bus option
+for the first mile, selecting whichever was the least costly from each passenger’s available options), direct via MOD
+operator, or hybrid (MOD first mile transferring to transit, and potentially an MOD last mile for local commuters).
+We selected a route choice model specification representative of the factors influencing passenger decisions in our
+model: monetary costs (i.e., fares) and travel time (both within and outside the system), and alternative-specific con-
+stants (ASC) for each represented mode (driving, transit, MOD, and hybrid). To calibrate choice model parameters,
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+46
+Article submitted to Transportation Science; manuscript no. 1
+we used a simplified version of the non-cooperative pricing game. We fixed fare parameters to their real-world values
+and selected model coefficients that make real-world fares correspond to a Nash equilibrium. This simplified game
+for the calibration purposes included only three passenger types: local commuters, commuters to inner city, and those
+traveling on the transit network outside of the alliance region. Each passenger had a simplified choice set represent-
+ing average-case travel times and distances. We solved the simplified game by enumerating solutions over a coarse
+grid of candidate choice model coefficients values. Finally, all passengers were grouped into 3 sensitivity profiles:
+time-sensitive, price sensitive, and intermediate. Intermediate category uses the calibrated parameters as is. We char-
+acterize the time-sensitive passengers by doubling the time-sensitivity coefficients and halving the price-sensitivity
+coefficients; and vice versa for the price-sensitive passengers. Table 10 shows the final calibrated route choice model
+coefficients. All signs and relative magnitudes are as expected intuitively, with ASCs indicating that all else equal,
+people prefer driving the most and transit the least. Price and travel time impact passenger utility negatively. The
+calibrated parameters thus clearly pass the common sense test.
+Table 10
+Route choice model’s calibrated coefficients.
+MOD ASC Transit ASC Hybrid ASC
+Driving ASC
+Price (USD)
+Travel time (min)
+-0.75
+-1.125
+-0.9375
+0
+-0.05
+-0.0075
+Modifications to Make the Model Income-aware. In the modified case study in Section (5.3), rather than uni-
+formly dividing the Greater Boston Area population into the three sensitivity profiles, we normalized time sensitivity
+and calibrated price sensitivity to the relative median household income (HHI) of each town, as described in Section
+5.3. The price-sensitivity coefficient for each town was obtained by dividing the primary case study’s coefficient in
+Table 10 by that town’s income ratio. This scaling method leads to the towns with the lower values of median HHI
+(i.e., the towns with a lower income ratio) appropriately correspond to higher price sensitivity values.
+For each town, Table 11 displays its price sensitivity coefficient and income ratio—i.e., a ratio of that town’s median
+HHI to the average of the median HHIs across the service region (which was $92,618.17). Absolute values of the
+median HHIs by town are available in Table 8. Note that the average of the median HHIs for the rest of the transit
+service population outside of the alliance service region is 9% higher than that within the service region, underlining
+that the proposed alliance particularly aims to serve lower income populations.
+
+Cummings et al.: Transportation Alliance Design with Endogenous Demand
+Article submitted to Transportation Science; manuscript no. 1
+47
+Table 11
+Price sensitivity coefficients in the passengers’ route choice model and the income ratios by town
+for the income-aware case study.
+Town
+Price Sensitivity Coefficient Income Ratio
+Outside
+-0.04592
+1.0888
+Brockton
+-0.0840
+0.5953
+Framingham
+-0.0585
+0.8544
+Waltham
+-0.0541
+0.9251
+Lynn
+-0.0848
+0.5895
+Salem
+-0.0706
+0.7079
+Lowell
+-0.0891
+0.5613
+Lawrence
+-0.1114
+0.4490
+Haverhill
+-0.0685
+0.7297
+Burlington
+-0.0439
+1.1385
+Reading
+-0.0411
+1.2161
+Wakefield
+-0.0486
+1.0290
+Stoneham
+-0.0488
+1.0239
+Melrose
+-0.0446
+1.1201
+Woburn
+-0.0522
+0.9582
+Winchester
+-0.0293
+1.7225
+
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+page_content='Submitted to Transportation Science  manuscript 1  Authors are encouraged to submit new papers to INFORMS journals by means of a style file template,  which includes the journal title.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' However, use of a template does not certify that the paper has been  accepted for publication in the named journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' INFORMS journal templates are for the exclusive  purpose of submitting to an INFORMS journal and should not be used to distribute the papers in print  or online or to submit the papers to another publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Multimodal Transportation Alliance Design with  Endogenous Demand: Large-Scale Optimization for Rapid  Gains  Kayla Cummings*, Vikrant Vaze†, ¨Ozlem Ergun‡, Cynthia Barnhart*  Transit agencies have the opportunity to outsource certain services to well-established platform-based Mobility on  Demand (MOD) providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Such alliances can improve service quality, coverage, and ridership;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' reduce public sector  costs and vehicular emissions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' and integrate the passenger experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' To amplify the effectiveness of such alliances,  we develop a fare-setting model that jointly optimizes discounted fares across a multimodal network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We capture com-  muters’ travel choices with a discrete choice model, resulting in a large-scale, mixed-integer, non-convex optimization  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' To solve this challenging problem, we develop a two-stage decomposition with the pricing decisions in the first  stage and a mixed-integer linear optimization problem optimizing fare discounts and the induced passenger behaviors  in the second stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' To solve the decomposition, we develop a new solution approach combining tailored coordinate  descent, parsimonious second-stage evaluations, and interpolations using special ordered sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' This approach, enhanced  by acceleration techniques based on slanted traversal, randomization and warm-start, significantly improves system-wide  practical outcomes over algorithmic benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Different alliance priorities result in qualitatively different fare designs:  flat fares decrease the total vehicle-miles traveled, while geographically-informed discounts improve passenger happi-  ness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The model responds appropriately to equity-oriented and passenger-centric priorities, improving system utilization  and lowering prices for low-income residents and long-distance commuters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Finally, our revenue allocation mechanism  improves outcomes for both operators, thus incentivizing profit-oriented MOD operators to adopt transit priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Key words: Public transit, transportation pricing, alliance design, mixed-integer non-convex optimization  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Introduction  Cities face critical challenges in the quest to improve urban mobility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Prior to the pandemic, congestion was  steadily rising, translating to $160 billion annual costs to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' cities and record-breaking contributions to  greenhouse gas emissions (Schrank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Recent declines in transit ridership demonstrate the inabil-  ity of transit’s static infrastructure to accommodate rapidly evolving commuting patterns (The Economist  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Private ride-sharing apps from Transportation Network Companies (TNCs) like Uber and Lyft have  Massachusetts Institute of Technology  † Dartmouth College  ‡ Northeastern University  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='03414v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='OC]  9 Jan 2023  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  2  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  challenged this fixed-infrastructure status quo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' TNCs transported 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6 billion passengers in 2017, more than  doubling the ride-sharing market since 2012 (Schaller 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The majority of urban TNC patrons admit that  they would have otherwise walked, biked, taken public transit, or not made the trip, coinciding with tens  of millions in annual transit revenue losses, worsening congestion, higher emissions, lower navigability of  cities, and reduced accessibility to affordable public options (Gehrke and Reardon 2018, Schaller 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Mobility-on-demand (MOD) services have the potential to service transit deserts—low-density areas  disconnected from public transit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' However, cost presents a key barrier: while all public transit modes operate  at a loss, MOD services administered by transit agencies incur the highest average per-trip costs ($23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='10 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  the next-highest $11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='19 for commuter rail) (Kane, Tomer, and Puentes 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' High labor needs, outdated  technology, and coordination difficulties lead to inefficient, expensive operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Notably, average TNC  trip costs $13, a full $10 less than agency-sponsored MOD trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Outsourcing all 223 million on-demand  transit trips to TNCs could hypothetically save billions of dollars for US transit agencies (Kane, Tomer, and  Puentes 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Thus, pricing alliances between TNCs and transit agencies have the potential to improve  service quality and coverage, while reducing costs and decreasing citywide vehicle-miles travelled (VMT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Pricing Alliances in the Real World  TNCs have the infrastructure to provide more cost-effective MOD services supplementing fixed-route tran-  sit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Microtransit platforms like BRIDJ and Via—differentiated from TNCs due to fleets comprising mini-  vans or shuttles as opposed to sedans—have oriented their business model toward complementing transit  (Via Transp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 2023, BRIDJ 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The Federal Transit Administration’s MOD Sandbox program has pro-  vided millions in funding to transit agencies in US cities such as Dallas, San Francisco, and Los Angeles  to develop on-demand pilots that fill service gaps in their service regions (Federal Transp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Administration  2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Rather than designing a complementary MOD system from scratch and incurring high fixed costs,  transit agencies could outsource MOD services to TNC platforms that are well-established, highly con-  nected, and widely trusted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' or to microtransit platforms that more closely align with transit agency goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  This section formalizes such alliances within a rigorous conceptual framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We define a pricing  alliance as a cooperative pricing scheme between a transit agency and an MOD operator with independently  operated infrastructures serving overlapping or adjacent regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' A pricing alliance seeks to improve each  operator’s own prioritized metrics whilst also improving system-wide benefits through integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Indeed,  real-world pricing alliance pilots have shown great promise in improving regional mobility, providing alter-  natives with higher service levels, lower fares and increased ridership (The Boston Globe 2022, Mag 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Pricing alliances are characterized by the intended service populations and the relationship of the MOD  operator’s system to the fixed-route transit network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Service population may be a targeted demographic,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' persons with limited mobility, low-income people, or senior citizens;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' the service population might also  constitute residents of a particular geographic area, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' residents of a transit desert or people traveling  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  3  (a) GoLink in Dallas, TX (Plano region) utilizes a zone-  based route structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  (b) The NewMo Pilot in Newton, MA utilized a hub-based route  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Figure 1  Route structures of recent pricing alliances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In Plano (Dallas Area Rapid Transit 2020), passengers  spend up to $3 to travel anywhere within a color-blocked region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In the NewMo Pilot (City of New-  ton, MA 2021), passengers could travel anywhere within town limits for $2, as long as either trip  endpoint was one of seven hubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' NewMo now allows passengers to travel anywhere within Newton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  within a given radius of a transit hub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The MOD infrastructure may complement, substitute, or extend fixed-  route options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Once participating operators establish the nature of the pricing alliance, the alliance can select  a joint pricing scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Carefully designed fares influence passenger behavior, incentivizing choices that  benefit the entire system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Table 1 surveys these traits of recent pricing alliances:  MOD operator: This can be a TNC like Uber or Lyft, or a microtransit platform like Via.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Service population: Recent alliances have served people with limited mobility, seniors, essential work-  ers, or simply everyone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Fare structure: For passengers traveling within the system, some alliances charge a flat fare and/or  a variable fare based on distance travelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Selectively applied, interpretable discount structures for  jointly offered routes can encourage multimodal travel and engineer outcomes desired by the operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Route structure: Alliances often require specific trip geography: point-to-point (PTP) (trips must occur  within a given geographic region), zone-based (partitions a larger region into small zones and requires  intra-zonal trips), and hub-based (at least one trip endpoint must be anchored at specified locations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Figure 1 illustrates zone-based and hub-based route structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Integration into public transit network: The MOD portion of the network might integrate into the  transit network in several ways: complementary (provides another mode option to improve service  quality), substitutive (replaces existing fixed-route transit), first-/last-mile (FLM) (connects travelers  to the fixed-route network), and extension (serves transit desert regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Literature Review  This work sits at the intersection of literature on FLM system design and operations management, integrated  multimodal transport system design, and horizontal cooperation among competing transportation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  MobilityOptionsinPlano  OpcionesdemovilidadenlaszonasdePlano  CISEMONTC  WLIS  CHASEOUKS BIVD  NWPLANO  PARK&RIDE  NONEN  Mapnottoscale  Elm30ano  representa  laestala  PLRER  Travel between zones is not permitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  N  PARKERRD  STATION  LegacyWestServiceArea Zone  Noesta permitidoviajardirectoentrezonas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  PARXBVO  LegacyWest Zona del area de servicio  LegacyWestserves Northwest  FarNorth Plano serves Parker Road  North Central Plano/Chase Oaks  Far North Plano Service Area Zone  PlanoPark&Ride,forconnections  Station,forconnectionstoDART rail  servesParker RoadStation,for  FarNorthPlano Zonadelareadeservicio  to DART buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  and buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  connections to DART rail and buses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Legacy Westsinve la Northwest Plano  Far North Plano sirve la Parkor Road  NorthCentral Plano/Chase Oaks sinve la  NorthCentral Plano/Chase Oaks ServiceAreaZone  Park&Ride,para conexionesalos  Station,paraconexiones alostrenesy  ParkerRoadStation,para conexionesalos  NorthCentral Plano/ChaseOaksZonadelareadesenvicig  sutobusos de DART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  autobuses de DART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  trenesy autobusos de DART.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  DARTTransitFacility  Terminal de DART  DARTRed&OrangeLines  DART lineas Rojas yNaranjas  CollinCollege-SpringCreekCampusWaltham  Watertown  Eligibledestinations:  City centers:  NewtonCentre  NeedhamStreet  Newton  WellsAve/Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='ldaCampus  ofUMassAmherst  Transit hubs:  NewtonvilleCommuterRail  ChestnutHillGreenLine  Newton Highlands  Green Line  Needham Heights  CommuterRail  NeedhamCummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  4  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  Table 1  Survey of recent pricing alliances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' PTP: point-to-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Flat fare: same price for every passenger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Mode: fares vary by travel mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Distance: fares increase with distance traveled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' (Dallas Area Rapid Transit  2020, Regional Transp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Commission of Southern Nevada 2021, City of Seattle, WA 2021, MBTA 2021, City of  Newton, MA 2021, City of Jersey City, NJ 2021, St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Louis Metro 2021, MARTA 2021, Indianapolis Public Transp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Corporation 2021) ∗ Not operated by a transit agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' ∗∗ Specially marketed for senior citizens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' ∗∗∗ Available to  everyone, but only in case of transit closures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' ∗∗∗∗ Transported essential workers at the beginning of the  COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Program  City  Transit agency  MOD Op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Service population  Fares  Routes Integration  GoLink  Dallas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' TX  DART  Via  Everyone  Mode  Zone  Complementary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' FLM  RTC On-demand Pilot Las Vegas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' NV  RTC  Lyft  Paratransit  Distance PTP  Substitutive  Via to Transit  Seattle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' WA  King County Metro Via  Everyone  Flat fare  Hub  Complementary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' FLM  The RIDE Flex  Boston,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' MA  MBTA  Uber,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Lyft  Paratransit  Flat fare  PTP  Substitutive  NewMo Pilot  Newton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' MA  City of Newton∗  Via  Everyone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' seniors∗∗  Flat fare  Hub  Extension,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' FLM  No program name  Jersey City,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' NJ  NJ TRANSIT  Via  Everyone  Distance Zone  Complementary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' extension  No program name  St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Louis, MO  St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Louis Metro  Lyft  Everyone  Distance Hub  Complementary, FLM  MARTAConnect  Atlanta, GA  MARTA  Uber, Lyft  Everyone (closures)∗∗∗ Distance PTP  Extension  IndyGo + Uber  Indianapolis, IN IndyGo  Uber  Essential workers∗∗∗∗  Flat fare  PTP  Substitutive  FLM system design and operations: Research on demand responsive connector (DRC) systems devel-  ops analytical models to evaluate service quality and determine first-mile system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In particu-  lar, such work specifies optimal zone size and headways, identifies transition points between regions best  serviced by fixed-route vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' flexible services, and establishes best practices for inter-zone transfer coordina-  tion (Chandra and Quadrifoglio 2013, Kim, Levy, and Schonfeld 2019, Kim and Schonfeld 2014, Lee and  Savelsbergh 2017, Li and Quadrifoglio 2010, Lu, Quadrifoglio, and Petrelli 2017, Lu, Shen, and Quadri-  foglio 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The tactical question of how to operate a first-mile system is also well-studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The Dial-A-  Ride Problem (DARP) encompasses the vehicle routing problem faced by transit agencies, given a set of  trip requests and a vehicle fleet (Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 2018, Molenbruch, Braekers, and Caris 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The Integrated  DARP (IDARP) designs vehicle routes and schedules to meet trip requests, allowing transfers with fixed-  route timetabled service (Posada, Andersson, and Hall 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Closely related to IDARP is the problem  of matching individual carpoolers and integrating their trips with transit timetables (Stiglic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Finally, many studies design strategies for routing and scheduling (Wang 2019), pricing (Chen and Wang  2018), and trip request acceptance (Agussurja, Cheng, and Lau 2019) for FLM transportation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Multimodal network optimization with endogenous demand: Our work is related to literature on opti-  mal design and operation of transportation systems that acknowledges and leverages endogenous demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Past research has modeled decision-making travelers with preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' One-to-one and many-to-one assign-  ment problems among travelers and suppliers have been addressed with preference-based stable matchings  to prevent participants from leaving ride-sharing systems (Wang, Agatz, and Erera 2018) or transit systems  (Rasulkhani and Chow 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Passenger decisions are also often captured by discrete choice models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Bert-  simas, Ng, and Yan (2020) jointly determine frequencies and prices for multimodal transit to minimize wait  times, subject to passenger mode and route choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Cadarso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' (2017) optimize airline scheduling, fleet  assignment, and fares while capturing the effects of competing high-speed rail service, taking passengers’  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  5  mode choices into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Wei, Vaze, and Jacquillat (2020) develop fixed-route transit timetables  to maximize welfare, subject to competition with ride-sourcing companies, and congestion effects from  passengers’ mode switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Wang, Jacquillat, and Vaze (2022) optimize a network of vertiports for sup-  porting urban aerial mobility, with passenger mode choices described by two alternative models, including  a multinomial logit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' (2021) tackle a welfare-maximizing system design and pricing  problem for centrally coordinated multimodal transport networks with price-dependent demand, and for-  mulate it using mixed-integer convex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In contrast, we tackle a multi-objective pricing alliance  design problem with a practically suitable pricing scheme that enables transparent price communication to  passengers, but also prevents its convexification and, in turn, heightens the computational challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Horizontal Cooperation: Finally, we review cooperation models among competing operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Litera-  ture on horizontal cooperation in logistics and airline scheduling is particularly mature (Cruijssen, Dullaert,  and Fleuren 2007, Guajardo and R¨onnqvist 2016, Wright, Groenevelt, and Shumsky 2010, Hu, Caldentey,  and Vulcano 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Chun, Kleywegt, and Shapiro (2017) design a liner shipping alliance with endogenous  linear demand for a homogeneous product;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' shipping companies first trade physical capacity on respective  networks, and then compete to sell substitutable products in an overlapping market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Our work also involves  joint products over a shared network subject to endogenous demand, but the allied operators offer those  products together rather than exchanging capacity to compete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Algaba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' (2019) formulate an urban trans-  portation network flow game, using exogenous passenger and cost information to coordinate a single-fare  payment among competing operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Bian and Liu (2019a,b) design mechanisms for the first-mile problem  incorporating personalized passenger requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Siddiq, Tang, and Zhang (2021) investigate incentive  mechanisms to inspire commuters to use public transportation, modeling commuters, transit agency, ride-  sharing platform, municipal government, and local private enterprises as stakeholders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Liu and Chow (2022) investigate whether competing transit agencies can share data to improve selfish  outcomes when setting frequencies, subject to user equilibrium passenger flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Policymakers can lever-  age results of their Bayesian game and coalition formation model to inform decisions about establishing  mandatory data-sharing amongst transit operators, but the model is not amenable to large-scale operations  management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The most similar study to ours in this branch of literature is by Schlicher and Lurkin (2022),  who formulate a transport choice game in which operators cooperatively price their pooled resources, sub-  ject to passengers making travel choices according to a multinomial logit model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' They design a market share  exchange allocation rule that ensures a stable grand coalition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Their study differs from ours in that each  operator offers homogeneous products with a single price to travelers with unspecified origins and destina-  tions, thus entirely ignoring network effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In summary, most existing studies individually model either  operator or passenger incentives when designing integrated, multimodal urban transportation systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' to  our knowledge, studies incorporating both strategic operators and passengers provide only general high-  level intervention recommendations and rules of thumb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Our work differs in that we provide a prescriptive  and strategic design framework to build pricing alliances at scale and in full operational detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  6  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Contributions  We propose a prescriptive pricing alliance to enable incentive-aligned collaboration between transit agencies  and established ride-sharing operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' A fare-setting model is formulated to maximize total system-wide  benefits across the integrated network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Our framework helps operators navigate competing alliance objec-  tives: (1) enhancing access to high-quality public transportation options for underserved populations, (2)  lowering vehicle emissions and congestion from single-occupancy vehicle trips, and (3) maintaining the  financial well-being of participating operators to ensure that the profit-oriented operators are incentivized to  participate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' A key technical challenge when optimizing these objectives lies in capturing interdependencies  between fares and commuters’ travel choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In response, our model integrates a discrete choice model of  passengers’ route and mode decisions based on prices and non-pricing attributes like travel times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  From a technical standpoint, our fare-setting model is a large-scale, mixed-integer, non-convex opti-  mization problem—a challenging class of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Our first technical contribution is to design a two-  stage decomposition in which the first-stage pricing decisions parameterize second-stage fare discounts  and the induced passenger behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The second stage becomes a more tractable mixed-integer linear  optimization problem that can be solved with commercial solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' To solve the full model, we develop a  new solution approach combining tailored coordinate descent, parsimonious second-stage evaluations, and  interpolations using Special Ordered Sets of type 2 (SOS2) (Misener and Floudas 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We also develop  acceleration techniques based on slanted coordinate traversal and search direction randomization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' This solu-  tion approach—our second technical contribution—is applicable to any two-stage formulation with a low-  dimensional, convex, continuous first-stage and any computationally expensive black-box second stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  This solution approach is found to significantly improve outcomes, for passengers and operators, compared  to those obtained with state-of-the-art benchmarks based on Bayesian Optimization (Mockus 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  From a practical standpoint, we design a large-scale case study focused on the morning commute in the  Greater Boston Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We find that our model sets fares that are in realistic ranges and have interpretable  connections to alliance goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' For example, an alliance with a greater focus on minimizing total VMT  prefers flat rather than distance-varying fares to increase system utilization by long-distance commuters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  On the other hand, alliances with a greater emphasis on increasing transit access will set discounts with  greater geographic variation to make alliance routes more attractive to heterogeneous populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The clear  alignment between operator goals and passenger choices achieved by our fare structures illustrates the value  of modeling endogenous demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Moreover, analysis of our results shows that the model is appropriately  responsive to equity-oriented objectives: it enables the alliance to lower fares for, and increase utilization  by, low-income and long-distance commuters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Finally, when compared to non-cooperative pricing, our fares  and our tailored revenue allocation mechanism together incentivize revenue-oriented MOD operators not  only to participate in the alliance but also to adopt the transit operator’s priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  7  Section 2 presents the allied fare-setting model formulation, its two-stage decomposition enabling  tractable solutions, as well as our revenue allocation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Section 3 describes our parsimonious  SOS2-based coordinate descent approach, whose computational performance is compared against bench-  marks in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We present our practical insights in Section 5 and conclude in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Pricing Alliance Design Problem  We now present our design pipeline for the Pricing Alliance Design Problem (PADP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In the PADP, the  alliance—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=', the jointly acting operators—sets a fare structure that optimizes joint operator priorities over  the integrated multimodal network, subject to the passengers’ endogenous route choice decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The indi-  vidual operators must then decide whether or not to participate in the alliance they have designed based on  the optimized fares and a revenue allocation mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Assumptions  Before formulating the allied fare-setting model, we specify our characterization of system-wide benefits,  our model of passenger decision-making, and our assumptions about static fares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  System-wide benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The alliance cooperatively set fares over an integrated network with the objective  of maximizing overall benefits to society, including travelers, operators, and the rest of society (Daganzo  2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Consistent with the motivation of this work, we assume that transit agency’s own objective is identi-  cal to that of the alliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We characterize an operator’s benefits as its fare revenue and a passenger’s benefits  as its average utility across all available travel options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' High passenger utility corresponds to the availability  of many high-quality travel options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Finally, there are many ways to capture the system’s impact on the  rest of society, defined as everyone except the travelers and operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Most people who take alternative  travel options choose to drive personal vehicles, contributing to negative externalities, such as air pollution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Because a pricing alliance involves no change in permanent infrastructure but rather better utilization of the  existing infrastructure, the key benefits of the alliance to the rest of society are likely to come from single-  occupancy VMT reduction under the allied pricing regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We ultimately compute system-wide benefits  as a weighted sum of operator revenue, passenger utility, and a penalty for the outside-option VMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The  weights are determined by the alliance’s relative priorities and can be varied to evaluate trade-offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Passenger discrete choice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We model travelers as rational agents making travel decisions accord-  ing to a multinomial logit (MNL) discrete choice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In an MNL model, the choice probabilities are  proportional to each option’s exponentiated utility, also known as its attractiveness (McFadden 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The  MNL choice model allows us to embed a closed form of the passengers’ decision-making process in the  alliance fare-setting model, but it also presents limitations related to the independence of irrelevant alter-  natives (IIA) property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Some have circumvented such inaccuracies by using the general attraction model  (GAM), of which the MNL model is a special case (Gallego, Ratliff, and Shebalov 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The GAM for-  mulates each choice probability as a function not only of the available options’ attractiveness, but also the  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  8  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  shadow attractiveness of unavailable options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In practice, researchers have set the shadow attractiveness  values to zero, in the absence of reliable data to estimate these parameters (Wei, Vaze, and Jacquillat 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Others leverage the nested MNL (Williams 1977), of which the MNL is also a special case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Lo, Yip, and  Wan (2004) and Bertsimas, Ng, and Yan (2020) assume that passengers select travel mode in the first level,  and then they select a route under that mode in the second level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In our work, we populate passengers’ route  choice sets with the fastest route from each available travel mode (transit-only, MOD-only, or transit-MOD  hybrid), including the option to drive, referred to in the literature as the outside option or the no purchase  alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Thus, our simplified choice model framework is equivalent to a GAM with zero shadow attrac-  tiveness values, or to a mode-route nested MNL with second-level choice sets containing one route each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Fare-setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The alliance sets fares over the integrated network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Some MOD operators might set time-  varying fares on their independently operated network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In particular, TNCs may implement fare multipliers  to manage two-sided markets between drivers and riders (Castillo, Knoepfle, and Weyl 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In a pricing  alliance, however, the MOD operator is a contractor to the transit agency and consequently agrees to set  time- and demand-homogeneous fares over allied network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' This agreement facilitates transparent communi-  cation with passengers who can easily anticipate public sector prices, and it also allows the transit operator  to set a budget for the alliance with higher confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We also assume that each operator is capable of  serving all demand redistribution that occurs as a result of the newly set fares, and that capacity reallocation  is therefore unnecessary to consider in the pricing alliance design process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Schlicher and Lurkin (2022)  make a similar assumption: they assume a constant marginal cost due to market shares that do not change  significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' From a transit perspective, this implies that the fixed-route options (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=', buses) have low load  factors to start with, especially in and near transit deserts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' on the MOD side, it implies that the operators  have large driver pools driven by private market dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In summary, we assume that the load difference  on the integrated network when transitioning from non-cooperative to allied fare-setting will not impose a  large enough change in network utilization to necessitate consideration of the associated resource allocation  decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In the practical case study of Section 5, we validate this assumption by showing that the potential  pricing alliances indeed do not pose a risk of over-saturating the integrated infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Exact Formulation  We now provide notation for formulating our allied fare-setting model with endogenous demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Passengers  select from a set of routes, R, serviced by a set of operators, O, which includes a public transit operator  and an MOD operator, so that |O| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' A route is a sequence of trip legs, each served by some operator’s  infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' To capture flat and distance-based fares, we define non-discounted price of route r ∈ R as:  �  k∈Or  (β0  k + ∆rk · β∆  k )  (1)  where Or ⊆ O is the set of operators serving route r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' ∆rk is the distance of route r covered by operator  k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' and β0  k and β∆  k , respectively, are the base fare, and markup per unit distance traveled, for operator k’s  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  9  sub-network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We collectively refer to the base fares and distance-based markups of all operators as the  fare parameters (β), which are decision variables in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Fare parameters are constrained by (non-  negative) upper and lower bounds ((β0  min,β0  max), (β∆  min,β∆  max)) determined by local legislative or operational  requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  In addition to the fare parameters, the operators jointly select a set of discounted routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Only a subset of  routes, RDE ⊆ R, may be discount-eligible (DE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Rather than deciding whether or not each individual route  should receive a discount, the discount-eligible routes may be grouped into discount activation categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Routes in the same discount activation category may share common geographic components specified by  the alliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' By grouping routes into categories, passengers can easily interpret which routes are discounted  from a map or a simple set of rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Example definitions for discount activation categories might include all  routes anchored on a particular hub location, or all routes whose origins and destinations are contained in  specified regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Let Ra ⊂ R be the set of routes corresponding to discount activation category a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The  sets Ra partition RDE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=', ∪a∈ARa = RDE and Ra ∩ Rb = ∅ for a ̸= b ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Let xa ∈ {0,1} denote the  decision variable that activates discounts on all routes in Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' This assumption is not restrictive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' absence of  activation categories can be handled easily by putting each route in its own category: |Ra| = 1,∀a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We  note that the relaxation of this assumption might result in discount rules that are difficult to communicate to  passengers in large-scale systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Λ is the discount multiplier for the routes selected to receive a discount, with an allowable range of  [Λmin,Λmax] : 0 ≤ Λmin ≤ Λmax ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The customer-facing price, pr, of route r ∈ R, is given as:  pr =  �  (1 − Λ · xa) ·  ��  k∈Or(β0  k + ∆rk · β∆  k )  �  if r ∈ Ra,a ∈ A  �  k∈Or(β0  k + ∆rk · β∆  k )  if r ∈ R \\ RDE  (2)  We consider a set N of passenger types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Each passenger type i ∈ N is identified by a unique combination  of origin, destination, and preference profile as described by their route choice utility coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Ri is the  set of routes available to passengers of type i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Some passengers are more averse to expensive travel  options, whereas others are more sensitive to travel time, constituting different preference profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' There  are Ni passengers of type i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We denote the utility to a passenger of type i ∈ N of route r ∈ Ri as  uir + αi · pr, where uir is the utility from non-monetary route attributes and αi ≤ 0 is the utility per unit  price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The market share sir of route r ∈ Ri for passenger type i ∈ N is computed according to MNL as:  sir =  exp(uir + αi · pr)  exp(ui0) + �  s∈Ri exp(uis + αi · ps),  (3)  where the outside option—not in set R = �  i∈N Ri—has a utility ui0 and a market share computed as:  si0 =  exp(ui0)  exp(ui0) + �  s∈Ri exp(uis + αi · ps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  (4)  Finally, the operators’ relative priorities over the system-wide performance metrics are captured by non-  negative objective function weights: µP AX,µREV ,µV MT, respectively, corresponding to passenger benefits,  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  10  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  operator benefits, and the benefits from negative externality reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Table 2 summarizes all notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Model PADP-FS (5)-(13) provides the exact formulation for the PADP fare-setting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' It jointly sets  fares and discounts to maximize system-wide benefits across the integrated network (objective function (5)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Discounts are applied on selected routes (Constraints (6) and (7)) and utility-maximizing passengers make  route selections according to an MNL (Constraints (8) and (9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Fare parameters and the discount multipliers  obey bounds (Constraints (10)-(12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Discount activation decisions are binary (Constraints (13)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Table 2  Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Component  Type  Description  A  Set  Discount activation categories  N  Set  Passenger types  O  Set  Operators  R  Set  Intrasystem routes, not including the outside option  Or  Set  Operators who help service route r ∈ R  Ri  Set  Route options available to passengers of type i ∈ N  Ra  Set  Routes in discount activation category a ∈ A  RDE  Set  Discount-eligible routes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=', �  a∈A Ra  Ni  Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Number of passengers of type i ∈ N  ∆i0  Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Driving distance for a passenger of type i ∈ N  ∆rk  Param  Distance the passenger travels with operator k ∈ O on route r ∈ R  uir  Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Non-monetary utility accrued by a passenger of type i ∈ N on route r ∈ Ri  ui0  Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Utility accrued by a passenger of type i ∈ N by driving  αi  Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Utility per unit price to a passenger of type i ∈ N  β0  min,β0  max  Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Minimum and maximum allowable base fares  β∆  min,β∆  max  Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Minimum and maximum allowable distance-based markups  Λmin,Λmax  Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Minimum and maximum allowable values of discount multipliers  µP AX,µREV ,µV MT  Param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Relative priority weights of system-wide performance metrics  xa  Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Whether to activate discount option a ∈ A  β0  k,β∆  k  Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Base fare and markup of operator k ∈ O  pr  Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Customer-facing price of route r ∈ R  Λ  Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Discount multiplier applied to routes with activated discounts  sir  Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Proportion of passengers of type i ∈ N who choose route r ∈ Ri  si0  Var.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Proportion of passengers of type i ∈ N who choose the outside option  (PADP-FS)  max  �  i∈N  Ni·  �  µP AX ·  �  ui0+  �  r∈Ri  (uir+αi·pr)  �  +µREV ·  �  r∈Ri  pr·sir−µV MT ·(∆i0·si0)  �  (5)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  pr =  �  k∈Or  (β0  k + ∆rk · β∆  k )  r ∈ R \\ RDE  (6)  pr = (1 − Λ · xa) ·  � �  k∈Or  (β0  k + ∆rk · β∆  k )  �  a ∈ A,r ∈ Ra  (7)  sir =  exp(uir + αi · pr)  exp(ui0) + �  s∈Ri exp(uis + αi · ps)  i ∈ N,r ∈ Ri  (8)  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  11  si0 =  exp(ui0)  exp(ui0) + �  s∈Ri exp(uis + αi · ps)  i ∈ N  (9)  β0  min ≤ β0  k ≤ β0  max  k ∈ O  (10)  β∆  min ≤ β∆  k ≤ β∆  max  k ∈ O  (11)  Λmin ≤ Λ ≤ Λmax  (12)  xa ∈ {0,1}  a ∈ A  (13)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Two-stage Decomposition  The PADP-FS model is a non-convex mixed-integer nonlinear optimization problem (MINLOP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' There  are no commercial solvers that accommodate non-convex MINLOPs, and no open-source solvers accept  non-convex MINLOPs at practically large scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Therefore, we propose a different solution approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We  decompose the formulation to tractably obtain high-quality solutions for practically sized problems (tens  of thousands of variables and hundreds of thousands of constraints in our case study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' By letting first-  stage pricing decisions parameterize second-stage discount activations and induced passenger behaviors,  the second stage can be formulated as a more tractable mixed integer linear optimization problem (MILOP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Let B := [β0  min,β0  max]2 × [β∆  min,β∆  max]2 and L := [Λmin,Λmax] respectively be the sets of allowable fare  parameters and discount multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We parameterize the second-stage problem by ( �β, �Λ) ∈ B × L and  define S( �β, �Λ) as the feasible region parameterized by ( �β, �Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We utilize the sales-based linear optimization  formulation by Gallego, Ratliff, and Shebalov (2015) to reformulate the choice model constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The  premise of the reformulation rests on proportionality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Let γir = exp(ui0)/exp(uir + αi · pr) be  the ratio of the attractiveness values of the outside option and route r ∈ Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Since the fare parameters are  determined in the first stage, γir is a constant in the second stage formulation for the non-discount-eligible  routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Then constraints (8) and (9) are reformulated as follows:  si0 = γir · sir  i ∈ N,r ∈ Ri  (14)  si0 +  �  r∈Ri  sir = 1  i ∈ N  (15)  si0 ≥ 0  i ∈ N  (16)  sir ≥ 0  i ∈ N,r ∈ Ri  (17)  Equation (14) ensures that the market share of each route is proportional to its attractiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Constraints  (15), (16), and (17) ensure that the market shares are non-negative and sum to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Note that Equation (14) still  includes bilinearities for discount-eligible routes r ∈ Ri ∩ RDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' For a type i passenger, γir is either equal  to the discounted price (γir = γir( �β, �Λ)) or full price (γir = γir( �β,0)), depending on whether the model  selects the discount for route r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Let Na ⊂ N be the set of passenger types with at least one route option  corresponding to discount activation category a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We linearize constraint (14) as (18) using big-M  constraints, letting M s  ir = γir( �β,0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  12  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  �  �  �  �  �  �  �  �  �  si0 ≤ γir( �β,0) · sir  si0 ≥ γir( �β,0) · sir − M s  ir · xa  si0 ≤ γir( �β, �Λ) · sir + M s  ir · (1 − xa)  si0 ≥ γir( �β, �Λ) · sir  a ∈ A,i ∈ Na,r ∈ Ri ∩ Ra  (18)  We similarly handle the bilinearities presented by the revenue terms in the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We define a  new decision variable wir = pr · sir for passenger types i ∈ N with discount-eligible routes r ∈ Ri ∩ RDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  The linearized constraints (19) set the value of wir with M w  ir = �Λ ·  ��  k∈Or(�β0  k + ∆rk · �β∆  k )  �  ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  wir ≤  ��  k∈Or(�β0  k + ∆rk · �β∆  k )  �  sir  wir ≥  ��  k∈Or(�β0  k + ∆rk · �β∆  k )  �  sir − M w  ir · xa  wir ≤ (1 − �Λ) ·  ��  k∈Or(�β0  k + ∆rk · �β∆  k )  �  sir + M w  ir · (1 − xa)  wir ≥ (1 − �Λ) ·  ��  k∈Or(�β0  k + ∆rk · �β∆  k )  �  sir  a ∈ A,i ∈ Na,r ∈ Ri ∩ Ra  (19)  Table 3 summarizes the additional and modified notation for the second-stage model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Table 3  Additional and modified notation for second-stage model of the decomposition framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Component Type  Description  B  Set  Allowable fare parameter values  L  Set  Allowable percent discount values  Na  Set  Passenger types with access to at least one discount-eligible route in  discount activation category a ∈ A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Ri ∩ Ra ̸= ∅  γir(β,Λ)  Parameter Ratio of outside option attractiveness to attractiveness of route r ∈ Ri  with price parameters β and discount Λ for passenger type i ∈ N  M w  ir,M s  ir  Parameter Big-M parameters for each i ∈ N,r ∈ Ri  wir  Variable  Continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Equivalent to pr · sir for discount-eligible routes r ∈ Ri ∩ RDE  In response to fare parameters set in the first stage, the second-stage problem activates discounts that  optimize system-wide performance metrics, subject to induced passenger decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The optimal value of  the second-stage problem is denoted by W in equation (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  W( �β, �Λ) :=  max  (x,s,w,p)∈S( �  β,�Λ)  �  i∈N  Ni ·  �  µP AX ·  �  ui0 +  �  r∈Ri  (uir +αi ·pr)  �  +µREV ·  �  r∈Ri  wir −µV MT ·(∆i0 ·si0)  �  , (20)  where S( �β, �Λ) is given as follows:  Then we define the PADP-FS2SD model, the two-stage decomposition of the PADP-FS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  (PADP-FS2SD)  max  W(β,Λ)  (26)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  β ∈ B  (27)  Λ ∈ L  (28)  LEMMA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Formulations PADP-FS and PADP-FS2SD are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  The proof of Lemma 1 is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  13  S( �β, �Λ) ≡  �  (x,s,w,p) ∈ {0,1}|A| × R  �  i∈N (|Ri|+1)  +  × R  �  i∈N |Ri∩RDE| × R|R| :  Constraints (18) - (19)  si0 +  �  r∈Ri  sir = 1  i ∈ N  (21)  si0 = γir( �β,0) · sir  i ∈ N,r ∈ Ri \\ RDE  (22)  wir =  � �  k∈Or  (�β0  k + ∆rk · �β∆  k )  �  sir  i ∈ N,r ∈ Ri \\ RDE  (23)  pr =  �  k∈Or  (�β0  k + ∆rk · �β∆  k )  r ∈ R \\ RDE  (24)  pr = (1 − �Λ · xa) ·  � �  k∈Or  (�β0  k + ∆rk · �β∆  k )  �  a ∈ A,r ∈ Ra  �  (25)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Revenue Allocation Mechanism  When considering a pricing alliance, an operator assesses whether the cooperative regime would improve its  prioritized system-wide metrics over the non-cooperative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' A non-cooperative fare-setting game and  a solution approach for it are presented in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The MOD operator is solely revenue-maximizing,  while the transit agency maximizes a linear combination of multiple system-wide metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' By entering a  pricing alliance, the transit agency (denoted as TR) is guaranteed to fare no worse than that under the non-  cooperative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' However, it remains to ensure that the revenue-maximizing MOD operator will not lose  revenue by cooperating with the transit agency, which would ensure the MOD operator’s participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  We now design a revenue allocation mechanism that guarantees the MOD operator’s alliance participa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Let βnc and (βa,Λa) respectively denote the non-cooperative equilibrium fare parameters and allied  optimal fare parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Let fk(βnc) denote the revenue of operator k ∈ O in the non-cooperative regime,  and f(βa,Λa) denote the combined revenue of both operators in the alliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  LEMMA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Let δ = f(βa,Λa) − �  k∈O fk(βnc) be the surplus allied revenue compared to the total  non-cooperative revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Define Φk : R2  + × R+ → R+ to be the revenue allocation to operator k ∈ O :=  {TR,MOD}:  ΦT R((fk(βnc))k∈O,f(βa,Λa)) =  �  fT R(βnc) + δ  2·  if δ ≥ 0,  f(βa,Λa) − fMOD(βnc)  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  (29)  ΦMOD((fk(βnc))k∈O,f(βa,Λa)) = fMOD(βnc) +  � δ  2  �+  (30)  (a) The MOD operator will enter the pricing alliance with payment rule Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  (b) When δ ≥ 0, the mechanism satisfies Pareto efficiency, symmetry, the core property, scale invariance,  and independence of irrelevant alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  14  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  The proof of Lemma 2 is in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Because the alliance’s priorities may also include benefits to  passengers and/or benefits to the rest of the society in the form of reduced VMT, the alliance may earn  less revenue than the operators’ combined revenue in the non-cooperative regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Despite this, the transit  operator can choose to guarantee that, by cooperating, the revenue-oriented MOD operator earns at least as  much as it would have earned otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We assume that the MOD operator will participate in the alliance if  its non-cooperative and allied revenues are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In the event that the alliance accrues strictly more revenue  than that in the non-cooperative regime, the operators split the surplus evenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Solution Approach  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Motivation  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3 presented a two-stage decomposition of the allied fare-setting formulation, with the second  stage characterized as a mixed-integer linear optimization problem and the first stage as a low-dimensional  decision problem over a convex space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Without an analytic closed-form of W, the function’s gradients are  inaccessible, eliminating the possibility of using any gradient-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Bayesian Optimization is  applicable and has been leveraged in recent urban transportation studies focusing on MOD systems (Liu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 2019), but it does not provide clear convergence criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' PADP-2SD is also not amenable to Benders  decomposition due to the nonlinear interdependencies between first- and second-stage decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Our prob-  lem’s incompatibility with the simpler centralized welfare-maximization structure of the problem tackled  by Banerjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' (2021) implies that their convexification strategy cannot be applied either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Our solution strategy approximates gradient descent for solving the first-stage problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Because the first-  stage feasible space is low-dimensional and convex, we begin with a coordinate descent framework, which  takes turns fixing all fare parameters except one and greedily optimizing along the free dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Even  one-dimensional search is difficult because the search space is a continuous spectrum of optimal MILOP  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' While a solution of the second-stage problem is fast enough to be a useful tool (see Section 4:  needing at most 5 seconds on average), it is also slow enough to warrant a judicious selection of first-stage  valuation points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Thus, our tailored coordinate descent approach scans each search direction by solving a  model that approximates the best solution along that search direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' After evaluating a few points along the  free search direction with the second-stage MILOP, an auxiliary model interpolates intermediate solutions  along that search direction with Special Ordered Sets of type 2 (SOS2) (Misener and Floudas 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The  process terminates when no improvements are found along any coordinate direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We define this as the  basic SOS2 Coordinate Descent (SOS2-CD) approach in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  To further improve final solution quality given a computational budget, we develop three acceleration  strategies that build upon SOS2-CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' First, rather than using an arbitrary search direction sequence, we intro-  duce more opportunities to escape local optima by randomizing search direction order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Second, we exploit  the fact that the SOS2 approximation model is valid along any search direction through the first-stage prob-  lem’s search space, and not just those parallel to coordinate axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Natural search direction candidates are  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  15  those where each operator’s base fare and markup are jointly varied while holding all other parameters con-  stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' By considering SOS2 coordinate descent over such slanted directions, we unlock directions navigating  trade-offs between high base fares and low markups vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' low base fares and high markups, which would  be unavailable with single-coordinate search directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Finally, we show how to mitigate the SOS2-CD’s  sensitivity to random initializations by leveraging warm-start solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3 describes the final algorithm,  including acceleration strategies, initialization procedures, and the incorporation of time limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Computa-  tional results in Section 4 demonstrate the effectiveness of SOS2-CD and all acceleration strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  SOS2 Coordinate Descent  Let Y := B × L denote the search space over which to optimize W, and y := (β,λ) ∈ Y denote a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Let Yi(y) = {z ∈ R : (y1,··· ,yi−1,z,yi+1,··· ,yn) ∈ Y} be the subset of the feasible space with all dimen-  sions other than the ith fixed to those of solution y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We define S(y) and S∗(y) ⊆ S(y), respectively,  to be the set of feasible and optimal second-stage decisions given fare parameters y := (β,Λ) ∈ Y:  S∗(y) := arg  max  (x,s,w,p)∈S(y)  �  i∈N  Ni ·  �  µP AX ·  �  ui0 +  �  r∈Ri  (uir + αi · pr)  �  + µREV ·  �  r∈Ri  wir − µV MT · (∆i0 · si0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Basic Coordinate Descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Coordinate descent is a greedy method that successively optimizes a mul-  tivariate function along coordinate axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Starting from an initial point, it cyclically optimizes along every  coordinate direction holding all other dimensions fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Algorithm 1 presents basic coordinate descent to  solve the PADP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Optimizing even a single dimension of W is hard, because it entails navigating a contin-  uous spectrum of optimal solutions to MILOPs, which does not have closed analytic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Therefore, we  will propose a rigorous method for tractably modifying Step 7 of Algorithm 1 using SOS2 interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Algorithm 1 Coordinate Descent for maximization of W  1: ARG y0 : Initial solution in Y  2: procedure COORDINATE DESCENT(y0)  3:  objPrev ← −∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' objCur ← W(y0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' k ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' n ← dim(y0)  4:  while objCur − objPrev > ϵ do  5:  k ← k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='objPrev ← objCur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='yk ← yk−1  6:  for i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=',n} do  7:  yk  i = arg maxz∈Yi(yk) W(yk  1,··· ,yk  i−1,z,yk  i+1,··· ,yk  n)  8:  objCur ← W(yk)  9:  return yk  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  16  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  SOS2 Interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Our SOS2 interpolation procedure performs approximate local search along a spec-  ified direction to produce the next candidate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' First, we solve D second-stage models at evenly  spaced points along the search direction, obtaining a sequence of fare parameter values acting as anchors  for the SOS2 interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We denote the anchors by yd := (βd,Λd) and their corresponding solutions by  (xd,sd,wd,pd) ∈ S∗(yd),∀d ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=',D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Let Ω = {yd : d ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=',D}} be the ordered set of anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Larger D values interpolate more accurately, but the solution is also more computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Figure 2 visualizes the selection of the next candidate solution using SOS2 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' W is exactly eval-  uated at every anchor and approximated between the anchors using the interpolated anchor solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The  next candidate solution is selected where the approximation of W is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Rather than directly inter-  polating W, or wir variables linearizing prsir terms, we interpolate the price and market share variables, p  and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Otherwise, the interpolated values of W will be convex combinations of anchor valuations (straight  line segments connecting consecutive anchors in Figure 2), eliminating any chance of selecting fare param-  eters between anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Moreover, since the objective function’s nonlinearities are quadratic in nature due to  the multiplicative revenue terms, we capture them with this SOS2 approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Figure 2  SOS2 interpolation of W value and the selection of next candidate first-stage solution y∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  For a given number of anchor points D, the SOS2 model (Misener and Floudas 2010) is algebraically  specified as SOS2(D) ≡  �  (λ,z) ∈ RD  + × {0,1}D−1 : �D  d=1 λd = 1,�D−1  d=1 zd = 1,λ1 ≤ z1,λd ≤ zd−1 +  zd ∀d ∈ {2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=',D},λD ≤ zD−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Here, the λ variables are the convex combination weights for outputs  at fare parameters (βd,Λd), and each binary zd variable indicates whether to select the segment between  anchors d and d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Now we use the SOS2 variables to approximate W along a given coordinate axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Expression (31) presents the set of fare parameters, SOS2∗(Ω), that optimize approximated W given the  ordered anchor set Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Expression (32) denotes optimal solutions at all anchors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The optimal SOS2 vari-  ables are selected to maximize the approximated W function in Constraint (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Finally, the approximately  optimal fares are interpolated in equation (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The approximated objective function is quadratic, making  M  Wevaluatedatanchor  SOS2 interpolation of W  TruevalueofW  Search direction  True optimumCummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  17  (33) a mixed-integer quadratic optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Fortunately, it can be solved almost instantly to global  optimality with commercial solvers, because D is small by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  SOS2∗(Ω) :=  (31)  �  y : (xd,sd,wd,pd) ∈ S∗(yd),  ∀yd ∈ Ω  (32)  (λ∗,z∗) ∈  arg max  (λ,z)∈SOS2(D)  �  i∈N  Ni ·  �  µP AX ·  �  ui0 +  �  r∈Ri  (uir + αi ·  �  d∈D  pd  r · λd)  �  +  µREV ·  �  r∈Ri  ��  d∈D  (pd  r · λd) ·  �  d∈D  (sd  ir · λd)  �  − µV MT · ∆i0 ·  �  d∈D  (sd  i0 · λd)  �  (33)  y =  �  yd∈Ω  λ∗  dyd  �  (34)  Summary of SOS2 Coordinate Descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We present SOS2-CD in Algorithm 2, which replaces the  one-dimensional optimization in Step 7 of Algorithm 1 with the SOS2-based approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The sub-  routine SEARCH DIRECTIONS provides a comprehensive ordered list of search directions that can poten-  tially be multidimensional and/or randomized (options further discussed in the next subsection);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' the default  is to cycle through coordinate axes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' to return searchDirections = {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=',dim(y0)} when random  and multidim are both set to false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The subroutine GENERATE ANCHORS returns evenly spaced SOS2  anchors along the specified search direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The current solution is included in the anchor set to ensure  that the new solution is at least as good as the previous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Appendix D presents subroutines SEARCH DIREC-  TIONS and GENERATE ANCHORS in full detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' After generating the anchors in Step 7, Step 8 computes an  optimal solution for each anchor, uses these anchor solutions for interpolation, and picks the solution that  maximizes approximated W over the given search direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Step 9 computes true value of W at the new  candidate solution and updates the current solution if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The algorithm iterates until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Final Algorithm  We now present three strategies that provide SOS2-CD with additional opportunities to escape local optima  and thus improve solution quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The first strategy relaxes the assumption of deterministic search direc-  tion order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Because the order of search direction is arbitrary, we can randomize it after each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We  can select this strategy by setting the random argument to TRUE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Second, since the SOS2 approximation  model is valid along any search direction intersecting the current solution, not just the coordinate axes,  an operator’s fare parameter pair (base fare and markup) defines a natural subset of dimensions to search  simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Given a pair of dimensions, this strategy randomly selects the spanning dimension, and then  selects the line’s slope in this 2D plane uniformly at random from the set of affine lines that intersect the  current solution and span the selected dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Finally, it drops anchors at evenly spaced points along the  sampled line and obtains the next candidate solution maximizing approximated value of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' While there are  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  18  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  Algorithm 2 SOS2 Coordinate Descent for maximization of W  1: ARGS y0 : Initial solution in Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' D: Number of SOS2 anchors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' random: Boolean, whether to randomize  search directions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' multidim: Boolean, whether to use multidimensional slanted search directions  2: procedure SOS2 COORDINATE DESCENT(y0,D,random,multidim)  3:  objPrev ← −∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' objCur ← W(y0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' k ← 0  4:  while objCur − objPrev > ϵ do  5:  k ← k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='objPrev ← objCur;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='yk ← yk−1  6:  for i ∈ SEARCH DIRECTIONS (random, multidim) do  7:  Ω ← GENERATE ANCHORS(yk,i,D)  8:  Interpolate an optimal solution: draw some y∗ from SOS2∗(Ω)  9:  yk ← y∗ if W(y∗) > objCur else yk  10:  objCur ← W(yk)  11:  return yk  many possibilities for multidimensional search directions, we limit to each operator’s fare parameter pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Thus, when the algorithm’s multidim argument is set to TRUE, the list of search directions contains three  items: (1) transit parameters, (2) MOD parameters, and (3) the discount multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Whenever an operator’s  fare parameters are selected as the search direction, we sample a new affine line with the aforementioned  procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Appendix D fully specifies the subroutine SEARCH DIRECTIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  The last acceleration strategy incorporates a timed warm-start procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The outcome of a single round  of SOS2-CD may depend on the initial solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' From a random set of initial solutions, the basic implemen-  tation repeats SOS2-CD until a computational time budget limit has elapsed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Each repetition of SOS2-CD is  called a trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The best fare parameters found across all trajectories are returned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Convergence to higher  quality solutions may be more likely given intelligent initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We can warm-start the algorithm by  first obtaining a few samples in the region with a specified warmStartProcedure, and selecting the best  starting points from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The warmStartProcedure might simply be uniform sampling from the region,  or it can consist of searching the space in a more principled way, such as with Bayesian Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Algorithm 3 presents the overall solution algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' τ W S and τ each define the time limits devoted  to the warm-start and SOS2-CD procedures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The arguments random and multidim are  Booleans indicating whether randomized and/or multidimensional search directions will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The  warmStartProcedure specifies the procedure for generating informed initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Computational Results  We now discuss the accuracy and tractability of our approach through several computational experiments  using a large-scale profit maximization case study of the Greater Boston Area (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  19  Algorithm 3 Timed SOS2-CD with warm-start initialization  1: ARGS τ W S: Warm-start time limit (seconds);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' τ: SOS2-CD time limit (seconds);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' random: Boolean,  whether to randomize search directions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' multidim: Boolean, whether to use multidimensional slanted  search directions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' warmStartProcedure: Initialization procedure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' D: Number of SOS2 anchors  2: procedure TIMED SOS2-CD(τ W S, τ, random, multidim, warmStartPocedure, D)  3:  objCur ← −∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Y0 ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' T W S ← τ W S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' T ← τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' draw y∗ ∈ Y uniformly at random  4:  while T W S > 0 do  // generate warm-start solutions  5:  Draw y ∈ Y with warmStartPocedure  6:  Subtract from T W S the time to run warmStartPocedure and to compute W(y)  7:  if T W S ≥ 0 then: Insert (y,W(y)) into set Y0  8:  while T > 0 do  // execute SOS2-CD  9:  if Y0 ̸= ∅ then: y0 ← arg max{W(y) : (y,W(y)) ∈ Y0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Remove (y0,W(y0)) from Y0  10:  else: Draw y0 from Y uniformly at random  11:  �y ← SOS2 COORDINATE DESCENT (y0,D,random,multidim)  12:  Subtract from T the time to run SOS2 COORDINATE DESCENT and to compute W(�y)  13:  if W(�y) > objCur and T ≥ 0 then: y∗ ← �y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' objCur ← W(�y)  14:  return y∗  All optimization models are solved with Gurobi v9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0 and the JuMP package in Julia v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='4 (Dunning,  Huchette, and Lubin 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Comparisons under 1-Hour Computational Time Budget  We now demonstrate the superior computational performance of our approach (Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Table 4 com-  pares different versions of our approach, with different combinations of acceleration strategies, including  multidimensional search (SOS2-CD-MD), randomized search directions (SOS2-CD-R), both (SOS2-CD-  MD-R) and neither (SOS2-CD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' None of these four approaches use intelligent warm-starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We establish  two algorithmic benchmarks against which to compare our computational results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The first benchmark is  Brute-Force Coordinate Descent (BF-CD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' BF-CD differs from SOS2-CD in the way it conducts each iter-  ation of coordinate descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' BF-CD uses a much higher number of “anchors” along the search direction  and solves a second-stage model at each anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Instead of the SOS2-based interpolation, it just selects the  anchor with the highest value of the second-stage objective function as the new candidate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The  trade-off at each iteration is a drastic computation time increase for a more accurate evaluation of the points  along the search direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We implement BF-CD using a 1% granularity for the discount multiplier and a  $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='01 granularity for both base fares and markups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The second benchmark is Bayesian Optimization (BO)—  a global optimization method for black-box functions that are computationally expensive to evaluate and  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  20  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  may not have gradients (Mockus 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Our black-box function is W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' BO imposes upon W a prior belief  about the space of possible objective values based on the candidate solutions considered so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The poste-  rior distribution decides which candidate solution to evaluate next, so that our sequential search successfully  explores unseen regions in the decision space and exploits regions that are more likely to host global optima  based on prior beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Appendix F includes a detailed account of BO, including all hyperparameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  We also tested time-limited SOS2-CD-MD-R with BO warm-starts, with varying time limit allocations  to the warm-starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In other words, we execute Algorithm 3 where the warmStartProcedure is Bayesian  Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The warm-start trials have names ending in BO-TL, where TL is the BO warm-start time  limit in minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Table 4 presents the performances statistics across 50 trials each with a 1-hour limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' All  outcomes are expressed in surplus USD over the average 1-hour BO benchmark performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  First, we observe that all four variations of our approach, even without warm-starts, significantly out-  perform the BO benchmark in terms of the average (by $13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='5K-$19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3K) and best-case (by $5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1K-$5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='8K)  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Moreover, our approaches with multidimensional search (SOS2-CD-MD-R and SOS2-CD-  MD), beat the BO benchmark also on the worst-case performance across the 50 trials (by $7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='5K-$22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Note that the average-case as well as the worst-case performance of the approaches with either acceleration  strategy (MD or R or both) were superior to those of the basic SOS2-CD approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The BF-CD approach  never terminated within the one-hour time limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' in fact, BF-CD could not even evaluate one full set of  anchors in all but 7 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Furthermore, our approaches with warm-starts perform even better than those  without.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In particular, a 40-minute BO warm-start drastically outperforms the benchmarks in the worst case  and provides the best average-case performance, while 20 minute BO warm-start provides the strongest  best-case performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In summary, all our approaches significantly beat benchmarks, and all three accel-  eration strategies (random search, slanted search and warm-start) were found to enhance the performance  of our basic SOS2-CD solution approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Table 4  Objective function statistics with 1-hour time limits and 50 trials each, expressed in terms of  surplus compared to average 1-hour BO performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' ∗ Average BO performance = $3,634,074.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Objective (Thousand $)  Algorithm  random multidim warmStartProcedure τ W S  τ  Min  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Max  BO −28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0∗  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6  SOS2-CD-MD-R-BO-50  Yes  Yes  BO  50  10  −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='9  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='4  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1  SOS2-CD-MD-R-BO-40  Yes  Yes  BO  40  20  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1  SOS2-CD-MD-R-BO-30  Yes  Yes  BO  30  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='7  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  SOS2-CD-MD-R-BO-20  Yes  Yes  BO  20  40  −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6  SOS2-CD-MD-R-BO-10  Yes  Yes  BO  10  50  −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='8  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='2  SOS2-CD-MD-R  Yes  Yes 60  −20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='5  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='7  BF-CD 60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='8  SOS2-CD  No  No 60 −103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='8  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='5  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1  SOS2-CD-MD  No  Yes 60  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='7  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1  SOS2-CD-R  Yes  No 60  −77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='4  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  21  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Comparisons under Higher Computational Time Budgets  All comparisons in the previous subsection assumed a 1-hour computational time budget and showed the  significant superiority of our basic approach over the benchmarks, as well as the value of our acceleration  strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' A question emerges as to whether these findings hold when longer computational budgets are  available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' To answer this question, Table 5 compares performances under three time budgets - 1 hour, 6  hours, and 12 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We additionally provide statistics on the number of trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' All versions of our  approach under all time budgets outperform the BO benchmark on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' This is especially true for the  versions of SOS2-CD with acceleration strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The larger time budgets allow accelerated SOS2-CD to  offer robust performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In general, the trajectories of SOS2-CD with multidimensional search (that is,  SOS2-CD-MD and SOS2-CD-MD-R) converge more quickly, allowing more trajectories to be computed  within a given time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' BF-CD is extremely slow and did not terminate before the 12-hour time limit in  any of our runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We report the performance statistics for BF-CD corresponding to the best solutions found  within the computational time budgets, prior to termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' While the best-case runs of BF-CD provide a  slight edge over all benchmarks (of merely $200 USD), the average-case and the worst-case performance is  significantly worse than our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' While BF-CD is more thorough for a single random initialization, it  is too computationally intensive to properly explore the search region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Note that warm-starts did not provide  much additional value for longer time budgets and hence warm-start approaches are omitted from Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  For additional analyses of the solution times and SOS2 optimality gaps, see Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Table 5  Objective function statistics with varying time budgets and 50 trials each, expressed in terms of  surplus compared to 1-hour BO performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' ∗ Average BO performance = $3,634,074.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Time  Algorithm  Trajectories  Objective (Thousand USD)  limit  Min  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Max  Min  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Max  1 hour  BO −28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  0∗  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6  BF-CD  0  0  0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='8  SOS2-CD  2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='8  5  −103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='8  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='5  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1  SOS2-CD-MD  2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  7  −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='7  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
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+page_content='7  6 hours  BO 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
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+page_content='3  12 hours  BO 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
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+page_content='7  52  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
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+page_content='7  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='4  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  22  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Insights from Practical Case Study  To inform the real-world policymakers’ decisions, we obtained practical results with the PADP model over  a Greater Boston Area case study described in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='2 confirms that our model yields  interpretable outputs with prices in realistic ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' An equity-oriented case study in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3 underlines  the value of accurately capturing passenger preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='4 demonstrates the value of cooperative  pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' All results are obtained with the SOS2-CD-MD-R approach and a 12-hour time limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Greater Boston Area Case Study  We model a potential pricing alliance between the Massachusetts Bay Transit Authority (MBTA) and a TNC  like Uber or Lyft, in the Greater Boston Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We use Lyft data for generating case study inputs because  of fare parameter data availability (Lyft Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' MBTA subsidizes Uber and Lyft trips as part of their  on-demand paratransit program called The RIDE Flex (MBTA 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We model an alliance with a wider  passenger scope that aligns with MBTA goals outlined in a recent report (MDOT 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In this report,  the MBTA identified 14 towns (called “urban gateways.”) adjacent to the commuter rail network whose  residents had the greatest likelihood of utilizing—and benefiting from—targeted transit expansion efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  We identify these 14 towns as the service region of the potential pricing alliance, as depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Figure 3  Partial map of urban gateways, as identified by the MBTA (MDOT 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The regions are identi-  fied with solid-line bounding boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The dashed-line bounding box demarcates the region that we  identified as the inner city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' (Urban gateway not depicted: Brockton, located south of the inner city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=')  Our case study integrates many datasets describing travel characteristics in the Greater Boston Area  during the weekday morning commute (6-10 am).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We consider passenger travel patterns for those who  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Newburyport  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='HAVERHILL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='WestNewbury  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Newbury  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='URBANGATEWAYS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Groveland  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Methuen  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Georgetown  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Rowley  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Dracut  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='LAWRENCE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Boxford  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Ipswich  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Dunstable  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Tyngsboroug  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='NorthAndover  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='OMELL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Andoyer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Topsfield  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Hamilton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Groton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Tewksbury  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Chelmsford  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Middleton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Wenham  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Westford  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='NorthReading  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Danvers  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Ayer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Beverly  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Billerica  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Wilnt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='ngtor  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Lynnfield  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Littleton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
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+page_content='Ige  Bosto  Wayiand  Westo  Newton  Brookline  FRAMINGHAM  Wellesley  Bostor  Natick  Needham  HUHCummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  23  commute from the service region to the inner city (Boston and Cambridge), or those who commute locally  within the service region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We define a local commute as either working in the town of residence or in an  adjacent town that is also part of the service region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' For example, commutes between Salem and Lynn or  between Burlington and Melrose are considered local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We use Origin-Destination Employment Statistics  from the Longitudinal Employer-Household Dynamics (LODES) datasets provided by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Census  Bureau to approximate the commuting population at a census tract level (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Census Bureau 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  We obtain MBTA’s commuter rail network data using MBTA General Transit Specification Feed data  (MBTA 2018), while the MOD operator corresponds to all potential direct travel options and first-mile  connections in the service region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We construct route choice sets for each passenger type by first executing  Yen’s k-shortest paths algorithm (Yen 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We then include in each passenger type’s route choice set their  fastest option of each mode: transit-only, MOD only, hybrid (MOD first mile to transit), and driving (which  corresponds to the outside option).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We represent the utility of each route option as a linear combination  of travel and wait time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' incurred costs including fare, gasoline, and parking fees as appropriate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' and mode  discomfort relative to the convenience of driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The discount activation categories correspond to town  pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In other words, a discount might be activated from any town in the service region to the inner city, to  an adjacent town that is also part of the service region, or to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In total, there are 77 discount activation  categories in the case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We allow each operator to set fares up to a maximum of $10 for base fares, $5  per mile for distance-based markups, and a maximum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='5 for discount multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Transit-only routes are  not eligible for discount, while all others are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Appendix G provides more details about the case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Model Validation  First, we will demonstrate that the allied fare-setting model sets route prices in realistic and reasonable  ranges from a practical standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Further, we find that the optimal fares intuitively reflect various port-  folios of alliance priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We vary the objective function coefficients µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=', relative weights among the  three performance metrics: revenue, passenger utility, and VMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In particular, we focus on regimes with  varying combinations of priorities between revenue and passenger utility (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=', setting µV MT = 0), as well as  between revenue and VMT (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=', setting µP AX = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We do not emphasize regimes that completely exclude  revenue as a priority, because they intuitively result in zero fares and are not interesting from an analysis  standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Thus, all experiments have µREV > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We also do not analyze regimes that vary all three metrics  for reasons explained later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Table 6 presents summary statistics about route prices, system uti-  lization, and system-wide performance metrics across the tested priority regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Figure 4 depicts optimal  fare parameters and discount multipliers, demonstrating the different fare-setting strategies of each regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  We extract a few representative solutions from Table 6 and present them in Table 7 alongside real-world  fares, and the corresponding ridership values obtained by our model for the real-world fares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We compute  the MOD base fare by combining Lyft’s published minimum fare and service fee, and we compute their  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  24  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  Table 6  Aggregate metrics for different operator priority regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' System-wide performance metrics are  normalized against best possible values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' System utilization (util.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=') is the alliance’s total market share, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' the  percentage of travelers electing to travel on a transit, MOD, or hybrid option instead of driving a  single-occupancy vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Objective weights  Route price ($)  Performance  System  µP AX  µREV  µV MT  Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
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+page_content='32%  (a) Optimal fare parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  (b) Optimal discount characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Figure 4  Optimal fares across varying alliance priority regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  MOD base fare  Fare value  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  Transit base fare  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  MOD markup  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='5  Transit markup  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  (1,0,0)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1  (0,0,1)  (PAX weight, REV weight, VMT weight)roportion of discounts activated  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6  Percent discount  50%  40%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='2  30%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  (1,0,0)  (1,1,0)  (0,0,1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  (PAX weight, REV weight, VMT weight)Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  25  markup by combining the published markups per unit distance and time, assuming an average vehicle speed  of 25 mph (Lyft Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We ignore fare multipliers utilized to manage the two-sided market, since they  are outside the scope of this work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' note that this may lead to slight undercounting of real MOD fares and  slight overcounting of the real ridership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The real-world MBTA commuter rail base fare and markup are  interpolated from its zone-based pricing structure, which assigns higher prices to farther zones (MBTA  2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' While all regimes have slightly lower ridership than that under real fares, all benchmarks achieve  non-negligible improvements in system-wide metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In particular, the REV, REV+PAX, and REV+VMT  regimes respectively achieve objective value increases of 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='4%, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6%, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='8% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Table 7  Summary statistics of representative priority regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' REV, REV+PAX, and REV+VMT regimes  respectively have objective weights (µREV ,µP AX,µV MT ) equal to (1,0,0),(1,1,0), and (1,0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' “% routes  discounted” provides the proportion of routes in the system with activated discounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' “Number of travelers”  is the total alliance passenger count originating within the alliance service region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Priority regime  Base fares ($)  Markups ($/mile)  Discount  % routes  Number of  MOD  Transit MOD  Transit  Multiplier  discounted  travelers  Real fares  $4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='53  $4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
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+page_content='16  0%  0%  25,816  REV  $10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='00  $10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='00  $2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='37  $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='63  50%  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='20%  18,309  REV+PAX  $10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='00  $8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
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+page_content='25  31%  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='13%  23,793  REV+VMT  $10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='00  $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='83  $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='16  $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='00  50%  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='50%  25,051  As shown in Table 6, each set of priorities induces interpretable optimal prices and passenger deci-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The minimum, mean, and maximum real-world route prices in the service region are respectively  $4, $10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='23, and $54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='46, while those given by our model are in the range $0-$60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='37;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' thus optimal fares  are set at the correct order of magnitude across all regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Intuitively, route prices are the highest for the  revenue maximizing regime, and they decrease gradually as the importance of VMT or passenger utility  increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' System-wide performance metrics are normalized against the best possible values across tested  regimes, naturally achieved by each metric’s corresponding single-objective optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The lowest rev-  enue is achieved in regimes that solely maximize passenger utility or minimize VMT, because very low  prices achieve very low revenues, but increase passenger happiness and entice more passengers away from  single-occupancy vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Analogously, the highest fares achieve the highest revenue, with more passen-  gers electing to travel outside the system, and lowering overall passenger utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' System-wide outcomes  vary smoothly with gradually changing alliance priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Note that even under single-objective revenue  maximization,, route prices remain in the ballpark of real-world fares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' While both base fares and discount  multiplier reach their upper limits, the both optimal markups stay in the interior of the allowable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  We attribute the model’s realism to the incorporation of the endogenous passenger choice model into the  fare-setting model, as opposed to modeling exogenous demand parametrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' These intuitive observations  confirm that our fare-setting model is suitable for generating trustworthy qualitative insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  26  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  At a quick glance, minimizing VMT and maximizing passenger utility seem to achieve similar outcomes  in Table 6—lower prices and higher system utilization—raising the question of why it is worth modeling  them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We turn to Figure 4 to illustrate how each objective yields qualitatively very different  designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' As VMT minimization increases in importance (moving from the middle towards the right in Fig-  ure 4a), the markup is zeroed out, equalizing fares across longer and shorter routes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The elimination of a  distance-based markup entices more longer-distance commuters to travel on the allied network, thus lower-  ing VMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' When the system maximizes passenger utility, a more nuanced fare structure emerges to address  heterogeneous passenger preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' All pricing levers are employed: base fares, markups, discount multi-  pliers, and discount activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Higher markups and base fares are coupled with more numerous discounts  across hybrid options, illustrated in the left halves of both Figures 4a and 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Thus, prioritization of each  objective (PAX versus VMT) results in similar system-wide performance metrics by qualitatively different  means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' To extract the corresponding fare designs, we consider case studies prioritizing at most one of PAX  and VMT at a time, with varying weights for revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3 further investigates geographic factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Finally, note that the variation in system utilization due to allied fare-setting is small compared to the  integrated network’s total loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Before the pandemic, MBTA’s bus load factor during peak hours was  already below 75% (Hicks 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' MBTA commuter rail transported around 120k passengers on an average  weekday in 2018 (MBTA 2022), with 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='2% of inbound ridership on peak trains (BRMPO 2012), yielding  approximately 49k travelers on MBTA commuter rail during the AM rush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Moreover, approximately 116k  daily TNC rides were destined for Boston in 2018, while another 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3k originated in the alliance service  region every day (MDPU 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In contrast, the ridership numbers in Table 7 show that the alliance’s  ridership under real fares is a small proportion of the entire integrated network and that the system is capable  of accommodating all demand redistribution as a result of allied fare-setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In fact, Table 7 shows that  the aggregate alliance ridership is slightly lower than under real fares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Thus, drops on other routes will  compensate for the slight ridership increases that may happen on certain routes under our proposed pricing  alliance, ensuring that the system-wide transit load factors and MOD detour times are expected to remain  largely unchanged as a result of the pricing alliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We conclude that the linked resource reallocation  problem need not be considered, when looking for rapid gains through pricing alliance formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Ensuring Equitable Access through a Refined Income-Aware Model Specification  Our fare-setting model captures passenger’s travel preferences and travel decisions when designing fares, a  critical step to satisfying passenger needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' However, different groups of passengers may have very different  preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Ignoring such differences can lead to inequitable and socially undesirable outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' After all,  equity is a key driver for integrating on-demand services into public transportation options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Acknowledging  this challenge, in this section we further refine our choice model and quantify the impacts of this nuanced  model specification on system-wide metrics compared to an aggregated, average-case choice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  27  To this end, we augment our case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Towns targeted for transit expansion by the MBTA in our case  study have wide-ranging median household incomes, translating into varying price sensitivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Affluent  travelers’ route choice decisions are less susceptible to changing fares than those of low-income travelers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  To partly account for such passenger heterogeneity, we compute a ratio of each town’s median household  income to the average of the median household incomes across the entire service region (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Census  Bureau 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We use this income ratio to scale passengers’ price sensitivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' See Appendix G for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Table 8  Allied system’s daily morning rush ridership with (Refined) and without (Base) the choice model  refinement, their percentage difference (Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' ), real-world ridership (Real), median household income (HHI), and  average distance (Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=') of alliance routes originating in the corresponding town and ending in the inner city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  HHI  Dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  REV+PAX  REV  REV+VMT  Town  $K  Miles Base Refined  Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Base Refined  Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Base Refined  Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Real  Lawrence  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='4  1585  1982  125%  918  1881  205% 1331  2065  155% 1566  Lowell  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='8  2000  2525  126% 1285  1565  122% 1852  2633  142% 2115  Lynn  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='5  3539  3769  107% 2283  2560  112% 3615  3188  88%  4023  Brockton  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6  3498  3709  106% 1949  3364  173% 3416  3919  115% 3872  Salem  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='9  1885  2043  108% 1189  1382  116% 1961  1786  91%  2171  Haverhill  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1  1605  1746  109% 1068  1214  114% 1630  1809  111% 1801  Framingham  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1  1203  1327  110%  780  936  120% 1093  1117  102% 1198  Waltham  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='7  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='8  1974  1873  95%  1597  1713  107% 2101  1946  93%  2249  Woburn  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='7  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3  1795  1828  102% 1479  1633  110% 2047  1910  93%  2199  Stoneham  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='8  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6  761  785  103%  610  691  113%  880  830  94%  948  Wakefield  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3  1004  1007  100%  783  889  114% 1140  1076  94%  1228  Melrose  103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='7  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='8  810  821  101%  637  711  112%  889  842  95%  949  Burlington  105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='4  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6  1091  1144  105%  975  1053  108% 1264  1193  94%  1340  Reading  112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='6  846  871  103%  703  787  112%  972  927  95%  1038  Winchester  159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='8  197  207  105%  193  198  103%  218  210  96%  227  We compare the allied system ridership across priority regimes and towns under fares corresponding to  base (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=', income-agnostic) as well as refined (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=', income-aware) choice models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We first calculate fares  using the fare-setting model incorporating income-agnostic and income-aware choice models separately,  and then evaluate both fare designs by calculating (and reporting in Table 8) ridership using only the income-  aware choice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Due to the use of the income-aware model, the system ridership in the REV+PAX  regime increases by 13% on average for the towns with below-average median HHI compared to a less  than 2% average increase for the towns with above-average median HHI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' While ridership increases across  the board due to generally lower fares, the greatest increases occur in Lawrence and Lowell, which are the  two towns with the lowest median household incomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The lower middle-income bracket (Lynn, Brockton,  Salem, Haverhill, and Framingham) sees the next-highest ridership increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Similar trends are observed in  the REV regime: ridership increases across all towns due to the fact that higher revenue can be achieved with  lower fares and higher volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The above-average income towns gain ridership by only 10% on average,  while the below-average income towns have a 37% gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  28  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  In contrast to the 7% and 23% average ridership gains in REV+PAX and REV regimes, average ridership  grows by less than 4% in the REV+VMT regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' But interestingly, this regime has significantly larger  ridership increases for longer distance passengers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In particular, the five towns that are farthest from the  inner city area—Lawrence, Lowell, Brockton, Haverhill, and Framingham—are the exact five towns with  ridership increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Their average increase is about 25% while the remaining 10 towns see an average 7%  drop in ridership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Thus, the REV+VMT regime increases access to the allied network for commuters who  are farther from the inner city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Overall, the prioritized system-wide objectives improved by 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='16%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='35%  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='99% respectively for REV+PAX, REV, and REV+VMT regimes compared to the real-world fares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  These results underline the importance of capturing passenger preference heterogeneity to amplify our  model’s practical impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' A choice model that reflects passenger preferences more accurately improves  system-wide outcomes and especially maximizes passenger benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We conclude that our income-aware  refined model improves transportation equity for passengers—as compared to the income-agnostic aggre-  gate model—making it a valuable tool for transit agencies to incorporate into strategic decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Quantifying the Value of Cooperation  To quantify the value of operator cooperation for operators and passengers, we solve the non-cooperative  fare-setting model for all allied priorities depicted in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In each experiment, the transit operator’s  priorities are identical to alliance priorities, whereas the MOD operator always maximizes revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Thus,  our experiment reflects an assumption that the prospective alliance will adopt the transit operator’s priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Figure 5 depicts the non-cooperative equilibrium fares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Discount multipliers are not included since they  are not applicable in the non-cooperative setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Table 9 compares allied outcomes to corresponding non-  cooperative outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In particular, we report the percentage increase in the alliance objective compared  to that computed under the non-cooperative setting (transit obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' % inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='), and the percentage increase in  MOD operator revenue due to the alliance (MOD rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' % inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We also provide the revenue allocations  to both operators as determined by the revenue allocation mechanism in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The non-cooperative  system utilization and non-cooperative average route prices are also provided for each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Figure  6 illustrates passenger mode choices across all tested regimes for both allied and non-cooperative fares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Figure 5  Non-cooperative fare parameters for different transit operator priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  MOD base fare  Fare value  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  Transit base fare  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  MOD markup  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  Transit markup  (1,0,0)  (1,1,0)  (0,1,0)  (0,1,1)  (0,0,1)  (PAX weight, REV weight, VMT weight)Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  29  Table 9  Allied vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' non-cooperative outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Transit operator’s priorities represented as µT R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Transit obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' weights  Transit obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  MOD rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  MOD allied  Transit allied  System  Route price  µP AX  T R  µREV  T R  µV MT  T R  % inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  % inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
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+page_content='83  Figure 6  Market shares by mode across priority regimes for allied and non-cooperative fare-setting models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Figure 5 illustrates that the MOD operator’s revenue-maximizing strategy remains relatively constant,  regardless of the transit operator’s priorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Still, when the transit operator prioritizes VMT minimization or  passenger benefits maximization, the MOD operator selects higher fare parameters (mainly through higher  markups) than in the corresponding allied settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Thus, the average route price (Route price ($) Mean)  columns in Tables 6 and 9 show that the non-cooperative average route prices are higher than average allied  route prices in every regime except for the one where the transit operator only prioritizes revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  In scenarios that partially maximize passenger benefits or minimize VMT, the alliance sets MOD fare  parameters lower than in the non-cooperative regime (Figure 4a vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Figure 6 shows that MOD-only mar-  ket shares decrease and hybrid market shares increase in the non-cooperative regime as passenger benefits  or VMT are increasingly prioritized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We can conclude that, although fewer passengers utilize MOD-only  30  Mode  Transit  MOD  Hybrid  Fare model  10  Allied  Non-coop  0  (1,0,0)  (1,1,0)  (0,1,0)  (0,1,1)  (0,0,1)  (PAX weight, REV weight, VMT weight)Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  30  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  options in non-cooperative scenarios where the transit operator is VMT- or passenger-oriented, a higher  volume of passengers selects hybrid options due to the very low (or free) transit fares observed in Figure  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Thus, the MOD operator earns more revenue in those non-cooperative scenarios in which the transit  operator is more altruistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' This is reflected in the revenue allocation mechanism: observe in Table 9 that  the MOD operator earns strictly more revenue in almost all regimes where the transit operator is not solely  a revenue maximizer, even though the system as a whole generates strictly less total revenue, as seen in  comparison with Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The revenue allocation mechanism ensures that the MOD operator receives their  non-cooperative earnings, despite the lower MOD fares that the alliance sets to achieve lower VMT or  higher passenger benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' This in turn reduces the transit’s revenue allocation as high VMT or low pas-  senger benefits are increasingly penalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' On the other hand, the transit operator always strictly improves  its objective of optimizing total system-wide performance, however it chooses to define it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' As a result, the  MOD operator interestingly finds it in its interest to adopt transit’s priorities as transit increasingly diverges  from revenue maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In other words, the revenue-maximizing MOD operator would not prefer a  revenue-maximizing alliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In fact, the MOD operator would benefit most from total altruism on transit’s  side (an exclusive focus on either passenger utility or VMT reduction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Passengers win due to the strictly  lower prices and higher system utilization that result from such alliances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  The transit operator must ultimately set the ceiling in terms of the price they are willing to pay for  the alliance benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Table 9 shows that the transit agency runs a deficit to appease the MOD operator if  its revenue emphasis is too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' A deficit alone might not be enough to dissuade the transit agency from  participating in the alliance: as we have noted, every public transit mode operates at a loss, especially on-  demand options (Kane, Tomer, and Puentes 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' To weigh the financial implications of an alliance, the  agency may compare the magnitude of the loss to the cost of the analogous MOD system operated by  transit on their own in the absence of outsourcing through an alliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Oftentimes, transit agencies also  receive grants to fund pricing alliances (Federal Transp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Administration 2016), and the daily deficit rate  can be compared to the grant award amount and intended duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' To enable a daily deficit rate that keeps  pace with financial resources over time, the transit agency may propose to reset overall performance metric  goals to induce different optimal fares, or to adjust the geographic and/or temporal scope of the alliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  In the end, each transit agency is expected to choose the trade-off point between its financial, passenger-  focused, and environmental goals that most closely aligns with their overall policy and various practical  and financial constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Regardless, our allied and non-cooperative fare-setting models together with our  revenue allocation mechanism jointly provide a toolkit usable by transit agencies to weigh these trade-offs  as they evaluate a potential pricing alliance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Conclusions, Limitations and Future Directions  We contribute a pricing alliance design framework to enable incentive-aligned collaboration between transit  agencies and MOD operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Our allied fare-setting model captures the interdependent decisions of pas-  sengers and operators, allowing the alliance to maximize benefits for all stakeholders across the integrated  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  31  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The prescriptive pricing framework can be generalized to different types of large-scale alliances  with varying MOD operators, service populations, fare structures, service goals, and network configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  We accomplish large scale by developing a tractable two-stage fare-setting formulation equivalent to the  original mixed-integer non-convex optimization problem, which we then solve with a tailored SOS2 coor-  dinate descent approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' From a technical standpoint, our approach selects consistently and significantly  higher quality solutions than benchmarks based on Bayesian Optimization, enabling additional system-  wide benefits worth tens of thousands dollars per day over the service region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Practically speaking, the high  quality solutions from our allied fare-setting model together with our dedicated revenue allocation mech-  anism work together to align revenue-oriented MOD operators with transit goals of passenger utility and  single-occupancy VMT reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In other words, cooperative pricing results in win-win-win outcomes for  passengers, MOD operators, and transit agencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Finally, the income-aware nuanced version of our fare-  setting model helps enhance passenger equity-related goals: by tuning passenger route choice models, the  alliance can prioritize lower fares and higher utilization for low-income or long-distance commuters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Our analysis is based on a few assumptions which can be relaxed in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We assumed that the  MOD operator serves as a contractor to the transit agency and agrees to set static fares for trips in the inte-  grated system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' While this is one good way to ensure public sector prices are transparently communicated,  it may also be possible to set fare schemes that allow MOD operators, especially TNCs, to maintain their  dynamic prices, perhaps through the transit operator subsidizing the cost of a passenger’s trip up to a fixed  dollar amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Additionally, we model average-case travel times over a fixed set of route options for the  MOD operator’s portion of the network to represent average-case operations, which simplifies their typi-  cally dynamic routing scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In other words, we consider the case where the alliance sets one permanent  fare scheme that is optimized for average-case performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Future research may consider optimizing for  the worst-case performance, and/or integrating dynamic routing into the fare-setting model if the alliance  prefers dynamic fares, requiring the integration of two complex problem classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Finally, we did not incor-  porate joint resource reallocation into the pricing scheme, due to the observation that the system is capable  of accommodating all demand re-distributions attributed to changing prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' This assumption works well  for our case study over the Greater Boston Area because the allied system’s ridership is a small fraction of  the larger service region with a high-capacity existing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Future research could consider relaxing this  assumption to generalize the analysis by jointly modeling the capacity allocation and pricing decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Acknowledgments  This material is based upon work supported by the National Science Foundation under Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1122374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
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+page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
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+page_content=' Plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' A 9(3):285–344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Wright CP, Groenevelt H, Shumsky RA, 2010 Dynamic revenue management in airline alliances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Transp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 44(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  36  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  Yen J, 1970 An algorithm for finding shortest routes from all source nodes to a given destination in general networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  37  Appendix A:  Proof of Lemma 1: Two-stage Decomposition is Equivalent to Full Formulation  Given an optimal solution from each formulation, we construct a feasible solution for the other with the same objective  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In this section, we introduce the following notation for a non-discounted route price r ∈ R, for expositional  brevity:  σr(β) =  �  k∈Or  (β0  k + ∆rk · β∆  k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  (35)  PADP-FS2SD → PADP-FS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Let (β2SD,Λ2SD,x2SD,s2SD,w2SD,p2SD) be an optimal solution to the PADP-  FS2SD model with objective value z2SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We construct the following solution to the PADP-FS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  �  �  �  �  �  �  �  �  �  �  �  �  �  βF S = β2SD  ΛF S = Λ2SD  xF S = x2SD  sF S = s2SD  pF S = p2SD  (36)  We demonstrate feasibility of (36) to the PADP-FS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  xF S are binary by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Constraints (6) and (7) hold by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Constraints (8) and (9) are feasible due to constraints (18), (21), and (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' For a given i ∈ N, constraints (18)  and (22) ensure that  sF S  ir ∝ exp(uir + αi · pF S  r )  ∀r ∈ Ri, and  sF S  i0 ∝ exp(ui0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Constraint (21) forces each sir and si0 to be normalized by exp(ui0) + �  r∈Ri exp(uir + αi · pF S  r ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The result  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Constraints (10) - (12) hold by definition of B and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Finally, to show that solution (36) has the same objective value as z2SD, we must only prove that pF S  r  sF S  ir = w2SD  ir  for all i ∈ N and r ∈ Ri, which follows directly from Constraints (19) and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  PADP-FS → PADP-FS2SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Let (βF S,ΛF S,pF S,sF S,xF S) be an optimal solution to the PADP-FS model with  objective value zF S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We construct the following solution to the PADP-FS2SD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  β2SD = βF S  Λ2SD = ΛF S  x2SD = xF S  s2SD = sF S  w2SD  ir  = pF S  r  sF S  ir  ∀i ∈ N,r ∈ Ri  p2SD = pF S  (37)  First we demonstrate the feasibility of (37) to PADP-FS2SD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  s2SD, w2SD and p2SD are non-negative and x2SD are binary by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Constraints (18) and (22): Consider any i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' For r ∈ Ri \\ RDE, and by expressions (8) and (9),  γir(β2SD,0) · s2SD  ir  = γir(βF S,0) · sF S  ir  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  38  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  =  exp(ui0)  exp(uir + αi · pF S  r ) ·  exp(uir + αi · pF S  r )  ui0 + �  s∈Ri exp(uis + αi · pF S  s )  =  exp(ui0)  ui0 + �  s∈Ri exp(uis + αi · pF S  s )  = sF S  i0 = s2SD  i0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Thus constraints (22) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Now, consider any a ∈ A with r ∈ Ria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' If x2SD  a  = 0, then we can use the  exact same argument as above to confirm that γir(β2SD,0) · s2SD  ir  = s2SD  i0  again holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Since  γir(β2SD,Λ2SD) · s2SD  ir  ≤ γir(β2SD,0) · s2SD  ir  = s2SD  i0  ≤ γi0(β2SD,0)  ≤ γi0(β2SD,0) + γir(β2SD,Λ2SD) · s2SD  ir  = γir(β2SD,Λ2SD) · s2SD  ir  + M s  ir · (1 − x2SD  a  ),  the constraints hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' A similar argument applies when x2SD  a  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Constraints (19) and (23): For any i ∈ N, Constraints (23) hold by construction for all r ∈ Ri \\ RDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' For a ∈ A  with r ∈ Ria, we have w2SD  ir  = p2SD  r  s2SD  ir  regardless of the value of x2SD  a  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' If x2SD  a  = 0, then  (1 − Λ2SD) · σr(β2SD) · s2SD  ir  ≤ w2SD  ir  = σr(β2SD) · s2SD  ir  = (1 − Λ2SD) · σr(β2SD) · s2SD  ir  + Λ2SD · σr(β2SD) · s2SD  ir  ≤ (1 − Λ2SD) · σr(β2SD) · s2SD  ir  + Λ2SD · σr(β2SD)  = (1 − Λ2SD) · σr(β2SD) · s2SD  ir  + M w  ir · (1 − x2SD  a  ),  and hence constraints (19) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' A similar argument applies when x2SD  a  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Constraints (21): Consider any i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' By expressions (8) and (9),  s2SD  i0  +  �  r∈Ri  s2SD  ir  = sF S  i0 +  �  r∈Ri  sF S  ir  =  exp(ui0)  exp(ui0) + �  s∈Ri exp(uis + αi · pF S  s )+  �  r∈Ri  exp(uir + αi · pF S  r )  exp(ui0) + �  s∈Ri exp(uis + αi · pF S  s )  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Constraints (24) and (25) hold by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Constraints β ∈ B and Λ ∈ L hold by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Now we show that solution (37) to Formulation PADP-FS2SD has the same objective value of zF S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  z2SD =  �  i∈N  Ni ·  �  µP AX · (ui0 +  �  r∈Ri  (uir + αi · p2SD  r  ) + µREV ·  �  r∈Ri  w2SD  ir  − µV MT · (∆i0 · s2SD  i0  )  �  =  �  i∈N  Ni ·  �  µP AX · (ui0 +  �  r∈Ri  (uir + αi · p2SD  r  ) + µREV ·  �  r∈Ri  p2SD  r  s2SD  ir  − µV MT · (∆i0 · s2SD  i0  )  �  =  �  i∈N  Ni ·  �  µP AX · (ui0 +  �  r∈Ri  (uir + αi · pF S  r ) + µREV ·  �  r∈Ri  pF S  r  sF S  ir − µV MT · (∆i0 · sF S  i0 )  �  = zF S  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  39  Appendix B:  Non-cooperative Fare-setting Game and Iterated Best Response Algorithm  To benchmark the solutions of our allied fare-setting model, we formulate the non-cooperative fare-setting problem  between a transit operator and an MOD operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' When they are not allied, each operator can only make fare-setting  decisions over their portion of the integrated network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The discount activation decision is not applicable in this context,  and operators set their own fare parameters so as to optimize their prioritized performance metrics over the integrated  network, subject to endogenous passenger decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  First we introduce new notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Let µk = (µP AX  k  ,µREV  k  ,µV MT  k  ) be the relative priority weights of operator k ∈ O  for each system-wide performance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' For a given k ∈ O, let −k indicate the other operator, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' the singleton  −k ∈ O \\ {k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Each operator sets fare parameters βk = (β0  k,β∆  k ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Let Bk := [β0  min,β0  max] × [β∆  min,β∆  max] be the set  of valid fare parameters that operator k can set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Let si0(β) and sir(β) respectively denote the market shares of the  outside option and of route r ∈ Ri for passenger type i ∈ N, as functions by both operators’ fare parameters β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  sir(β) =  exp(uir + αi · σr(β))  exp(ui0) + �  s∈Ri exp(uis + αi · σs(β))  i ∈ N,r ∈ Ri  si0(β) =  exp(ui0)  exp(ui0) + �  s∈Ri exp(uis + αi · σs(β))  i ∈ N  Let ˆβ−k denote the fixed fare parameter decisions of operator −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Then σr(βk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' ˆβ−k) denotes the non-discounted price  of route r ∈ R as a function of the fare parameter decisions of operator k, parameterized by the fixed decisions of oper-  ator −k, where σr is originally defined in Equation (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Similarly, sir(βk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' ˆβ−k) and si0(βk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' ˆβ−k) represent market  shares as functions of the fare parameter decisions of operator k, given fixed decisions of operator −k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Formulation  PADP-NC is the non-cooperative fare setting model of operator k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  (PADP-NC)  max  βk∈Bk Wk(βk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' ˆβ−k)  ≡  max  βk∈Bk  �  i∈N  Ni ·  �  µP AX  k �  ui0 +  �  r∈Ri  (uir + αi · σr(βk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' ˆβ−k))  �  +  µREV  k �  r∈Ri  σr(βk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' ˆβ−k) · sir(βk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' ˆβ−k)−  µV MT  k  (∆i0 · si0(βk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' ˆβ−k))  �  An operator’s non-cooperative fare-setting model is a simplified version of the allied fare-setting model, with more  restricted decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Operator k sets their fare parameters within allowable bounds to maximize system-wide benefits  over the integrated network, subject to the induced passenger decisions and for given fare parameters set by the other  operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Each operator’s model has two decision variables: their base fare and their distance-based markup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Fare  parameters β must satisfy condition (38) below to be considered a solution to the non-cooperative fare setting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  βk = arg max  β′  k∈Bk Wk(β′  k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='β−k)  k ∈ O  (38)  When fare parameters β satisfy condition (38), they define a Nash equilibrium (NE), or more precisely, a pure strategy  Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Fare parameters form an NE when neither operator can unilaterally change their pricing strategy to  improve their own objective, given the fare parameters set by the other operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We compute an NE using an iterated  best response algorithm, which takes turns alternating between each operator’s optimal fare parameters computation  in response to their competitor’s fixed fare parameters, until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  40  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  A pure strategy Nash equilibrium to the non-cooperative fare-setting game is a set of fares such that each opera-  tor’s decision is the optimal response given the other operator’s fare parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' An iterated best response algorithm  randomly initializes the fare parameters and lets each operator take turns setting their best response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The algorithm  converges when neither operator can gain by changing their response (Fudenberg and Tirole 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Algorithm 4 Iterated best response for non-cooperative fare setting  1: procedure ITERATED BEST RESPONSE(ϵ)  2:  βk ← uniform draw from Bk for each k ∈ O  3:  objPrevk ← −∞ for each k ∈ O  4:  objCurk ← Wk(β) for each k ∈ O  5:  while maxk∈O(objCurk − objPrevk) > ϵ do  6:  for k ∈ O do  7:  objPrevk ← objCurk  8:  obj ← maxβ′  k∈Bk Wk(β′  k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='β−k)  9:  if obj > objCurk then  10:  objCurk ← obj  11:  βk ← arg maxβ′  k∈Bk Wk(β′  k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='β−k)  12:  return β  In general, without proof of existence of a Nash equilibrium, Algorithm 4 is not guaranteed to converge, let alone  to the same β in the face of random initializations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' To prevent infinite oscillation, a maximum iteration limit can  potentially be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Yet, in all runs of our computational experiments performed when generating Table 9, a  unique NE was always obtained across the 10 random initializations in a given row of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Appendix C:  Proof of Lemma 2: Revenue Allocation Mechanism  Proof of Lemma (2a): The MOD operator participates in the alliance whenever they can earn at least as much  revenue by cooperating with the transit agency as they can otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' By construction,  ΦMOD((fk(βnc))k∈O,f(βa,Λa)) = fMOD(βnc) +  � δ  2  �+ ≥ fMOD(βnc),  so the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Proof of Lemma (2b): In the case that allied revenue exceeds combined non-cooperative revenue, we show that the  allocation is a Nash bargaining solution, which is a classic payment rule guaranteeing properties of Pareto efficiency,  symmetry, scale invariance, and independence of irrelevant alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  We frame the alliance revenue allocation problem as a collective bargaining problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Let the disagreement outcome  be the operator revenues resulting from non-cooperation, d = (fk(βnc))k∈O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Let X be the set of all revenue allocations  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  41  such that each operator receives at least their disagreement outcome, and such that the sum of revenue allocations does  not exceed allied revenue p = f(βa,Λa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  X(d,p) :=  ��  ak  �  k∈O : ak ≥ dk,∀k ∈ O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  �  k∈O  ak ≤ p  �  Let F be the set of all such allocation problems, with each (d,p) ∈ F corresponding to a different allocation problem  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=', a different potential alliance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We seek an allocation solution function Φ : F → X that allocates the available profit  according to the axioms of Pareto efficiency, symmetry, scale invariance, and independence of irrelevant alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  It is known that Nash bargaining solutions – which satisfy the above axioms– exactly coincide with optimal solutions  to optimization problem (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  max  a∈X(d,p)  �  k∈O  (ak − dk)  (39)  Expression (40) clearly solves (39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  dk + p − �  k∈O dk  2  = fk(βnc) + δ  2 ≡ Φk(d,p),  ∀k ∈ O  (40)  To establish the core property, we point to Lemma (2a), as well as the fact that ΦT R(d,p) ≥ fT R(βnc) in the case that  δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' This proves Lemma (2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Appendix D:  SOS2 Coordinate Descent Subroutines  This section specifies subroutines of SOS2 Coordinate Descent, presented in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The subroutine SEARCH  DIRECTIONS provides a comprehensive ordered list of search directions that can be multidimensional and/or random-  ized or neither.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The subroutine GENERATE ANCHORS computes an ordered set of evenly spaced SOS2 anchors along  the specified search direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In case of a multidimensional search direction, we determine the range of slopes that  will ensure that the line spans the selected dimension, and sample uniformly from that range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Figure 7 visualizes the computation of the slope range in lines (16)-(21) of the GENERATE ANCHORS procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  The most negative slope of a line passing through the current solution is determined by the maximum of the slopes  of the two line segments connecting the current solution to the upper left and bottom right corners of the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Similarly, the most positive slope is determined by the minimum of the slopes of the two line segments connecting the  current solution to the lower left and upper right corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Appendix E:  Performance of Algorithm Components  In this section, we examine the performance of individual components of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Figure 8 illustrates the distribution of solution times in seconds for the second-stage model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Each calculation of W,  which requires solving the second-stage model once, needs on average 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='7 seconds of CPU time, which is fast enough  to be useful, but slow enough to warrant judicious selection of candidate first-stage solution points to evaluate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Each  of the 31,553 observations was obtained in under one minute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The observations were aggregated across all solutions  of the second-stage model represented in the paper, including runs of SOS2-CD, BF-CD, and BO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Figure 9 depicts the accuracy of the SOS2 model with anchors placed at 10% intervals for the discount multiplier  axis and at $1 intervals for the rest of the fare parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In each trial, a fare parameter combination and a search  dimension were selected uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The “true optimal” fare parameters for each trial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=', each combination  of current solution and search direction) were computed using a brute-force approach, through exhaustive enumeration  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  42  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  Algorithm 5 Subroutines for SOS2 Coordinate Descent  1: procedure SEARCH DIRECTIONS(random,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' multidim)  2:  if multidim then  3:  searchDirs ← {(β0  T R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='β∆  T R),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' (β0  MOD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='β∆  MOD),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Λ}  // index names of y  4:  else  5:  searchDirs ← {β0  T R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' β∆  T R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' β0  MOD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' β∆  MOD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Λ}  6:  if random then  7:  searchDirs ← SHUFFLE(searchDirs)  // randomly permutes the set searchDirs  8:  return searchDirs  9: procedure GENERATE ANCHORS((β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='Λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' searchDir,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' D)  10:  if searchDir ∈ {(β0  k′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='β∆  k′) : k′ ∈ O} then  // if searchDir is multidimensional  11:  β0  k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='β∆  k ← searchDir  12:  if Unif(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='5 then  // select spanning dimension  13:  (x,xmin,xmax),(y,ymin,ymax) ← (β0  k,β0  min,β0  max),(β∆  k ,β∆  min,β∆  max)  14:  else  15:  (x,xmin,xmax),(y,ymin,ymax) ← (β∆  k ,β∆  min,β∆  max),(β0  k,β0  min,β0  max)  16:  if x == 0 then  // determine valid slopes for spanning affine lines  17:  mmin ← ymin−y  xmax−x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' mmax ← ymax−y  xmax−x  18:  else if x == xmax then  19:  mmin ← ymax−y  xmin−x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' mmax ← ymin−y  xmin−x  20:  else  21:  mmin ← max{ ymax−y  xmin−x, ymin−y  xmax−x};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' mmax ← min{ ymin−y  xmin−x, ymax−y  xmax−x}  // see Figure 7  22:  m ← Unif(mmin,mmax);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' b ← y − m · x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' δx = xmax − xmin  // slope of spanning affine line  23:  anchors ← {(xmin + i−1  D−1 · δx,b + m · (xmin + i−1  D−1 · δx),β0  −k,β∆  −k,Λ) : i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=',D}}  24:  else  25:  if searchDir == Λ then  26:  maxV al ← Λmax;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' minV al ← Λmin  27:  else if searchDir in {β0  k : k ∈ O} then  28:  maxV al ← β0  max;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' minV al ← β0  min  29:  else  30:  maxV al ← β∆  max;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' minV al ← β∆  min  31:  anchors ← {(minV al + i−1  D−1 · (maxV al − minV al),(β,Λ) \\ {searchDir}) : i ∈ {1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=',D}}  32:  Insert (β,Λ) into the ordered set anchors  33:  return anchors  of every solution along that affine line at a 1% granularity for discount multiplier and a $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='10 granularity for the other  four fare parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Then the W obtained through SOS2∗ procedure was evaluated and compared with this “true  optimal” solution, to compute the SOS2 optimality gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The mean optimality gap was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='25%, and it was 0% in 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='8%  of the instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' This establishes the trustworthiness of the SOS2 model outputs, especially given the drastic CPU time  reduction they provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We repeated this procedure for 10 hours, resulting in 927 solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  43  Figure 7  Computing the range of slopes such that the selected dimension is spanned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Figure 8  Distribution of 31,553 second-stage model solution times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Mean solution time = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='7s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Figure 9  Distribution of SOS2 model optimality gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Mean gap = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='25%, across 927 solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Exact solu-  tions (with 0% gaps) obtained in 276 of the 927 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Figure 10 shows the distribution of BF-CD solution times for a single trajectory across 50 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' BF-CD never  terminates before an hour elapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Appendix F:  Bayesian Optimization Benchmark  Bayesian Optimization (BO) is a sequential search strategy for optimizing low-dimensional black-box functions  (Mockus 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Typically, the function being optimized takes a long time to evaluate and has no analytical form, pre-  cluding access to gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We first provide a high-level overview of BO, and then we summarize the design choices  that we use in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In particular, we adapt the setup of a recent dedicated study by Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' (2019) on using  Bayesian Optimization to select MOD system service parameters subject to passenger mode choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Interested readers  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='2  Density  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0  10  20  30  Solution time (s)1000  750  Density  500  250  0 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='03  SOs2 model optimality gapYmax  Ymax - y  Ymax - y  Xmin - X  X - xewx  Other dimension  βcur  Ymin - y  Ymin - y  Xmin - X  Xmax - X  Ymin  Xmin  x  xewx  Spanning dimensionCummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  44  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  Figure 10  BF-CD single-trajectory solution times across 50 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  are referred to Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' (2019) for a more detailed description of BO in this application context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Readers interested in  a general BO tutorial are referred to Brochu, Cora, and de Freitas (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  In the absence of a closed-form representation of our black-box function f : x → R, the BO framework first imposes  a prior belief upon f via a probabilistic surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Given this surrogate model, BO iteratively (i) updates  the likelihood of historical observations with new evaluations of f to obtain a more informative posterior, and (ii)  queries an acquisition function that uses the updated posterior to recommend the next value of x that should be  evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' A very common surrogate model for the black-box function is called a Gaussian Process (GP), which is a  stochastic process in which any finite set of random variables follows a multivariate Gaussian distribution (Mockus  2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' After building the GP with historical observations, the GP maps a given point in the search space to a univariate  Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We interpret this output distribution as a set of potential values of f(x), accounting for noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  The acquisition function is a BO design choice intended to help in navigating the trade-off between exploration and  exploitation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' exploring more of the search space versus exploiting regions where a globally optimal solution is  suspected to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' As in Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' (2019), we use a GP upper confidence bound as our acquisition function, which  characterizes the BO optimization process as a multi-armed bandit problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Appendix G:  Greater Boston Area Case Study  First we describe the primary case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Then we explain the income-aware modification to the choice model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Primary case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Our case study is based on several datasets describing travel characteristics in the Greater  Boston Area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We consider a pricing alliance for the morning commute, restricting our time window to 6-10AM on a  typical weekday in Fall 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Our data sources and their purposes are as follows:  Massachusetts Bay Transit Authority (MBTA) Focus40 report: We consider MBTA “urban gateways,” character-  ized as regions located beyond the rapid transit network but in close proximity to the commuter rail network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  More formally, an urban gateway is a town with high potential to be receptive to additional development of pub-  lic transportation options, especially if they connect to commuter rail hubs (MDOT 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' These towns define  the service region of our case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  MBTA Fall 2018 General Transit Feed Specification (GTFS): We use the MBTA GTFS to extract rapid transit  and commuter rail stations and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We also collect valid transfer edges from those enumerated by the feed  (MBTA 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  2017 Longitudinal Employer-Household Dynamics (LEHD) Origin-Destination Employment Statistics  (LODES): This data specifies the number of jobs corresponding to each origin-destination census tract pair,  where the origin represents the employee’s home tract and the destination represents their work tract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We used  this data as a proxy for daily morning commute demand (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Census Bureau 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='075  Density  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='000  15  20  25  Solution time (h)Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  45  U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Census Bureau American Community Survey, 2014-2018: We extracted town-level data on the modes of  transportation to work, median household income, and total population for towns in selected priority places (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Census Bureau 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' This data was primarily used to calibrate the passenger utility parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  TIGER/Line shapefiles: We used this data to assign network components to census tracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We use the centroids  of census tracts in priority places to define passenger origins and destinations (U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Census Bureau 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Uber Movement travel times: We used travel times aggregated over the 4th quarter of 2018 in the Greater Boston  Area to approximate travel times of first- and last-mile edges and outside options (Uber Technologies Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Lyft fare parameters: We calculate the real-world prices of the MOD operator’s services using Lyft’s base fares,  distance-based markups, and time-based markups (Lyft Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We also use this data to calibrate the passen-  ger utility parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  MBTA commuter rail fares: We characterize the real-world prices of the transit operator’s services using the  MBTA commuter rail fares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We interpolate the base fares and distance-based markups from the discretized,  zone-based fare structure together with the distances between transit stations in each zone (MBTA 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  We define the transit network using commuter rail and rapid transit edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Each passenger’s origin and destination  are census tract centroids from the LODES dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We built MOD operator network’s edges by connecting centroids  to transit stations and to each other within each priority place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We filtered out unnecessary edges that corresponded to  very low demand;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' implementing a 90% service goal enabled us to reduce the MOD operator’s edge set by 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We  only consider those commuters who work locally (within their home priority place) or in the inner city (the dashed  bounding box in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Commuters to the inner city were assumed to walk the last leg of their trip, from their final  transit station to their destination centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Because the rapid transit network is so well connected within the inner city,  approximately 90% of all inner city destinations in the data were within a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='5 mile walk of at least one transit station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  To generate the route choice sets, we first built a routing network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' This routing network is a transformation of the  physical network, consisting of MOD, transit, waiting, walking, and transfer edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The cost of each edge was the  time to “traverse”, whether that traversal entailed travel on the transit and/or MOD system, travel by foot, or stationary  waiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We filtered out the trips with more than one transfer, by performing a transformation on the relevant transit  stations by replicating those nodes—each transfer station had one node for incoming transfers and one for outgoing  transfers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' A path through this network from a passenger’s origin to destination captures their total travel time along  that path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Over this routing network, we implemented Yen’s algorithm for finding the k-shortest loopless paths over a  directed graph with non-negative edge costs (Yen 1970).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We first obtained up to 10 shortest paths for each commuter to  the inner city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Some paths were effectively duplicates, in that every aspect of the path was the same except for the final  transit station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We identified “unique” routes by their starting and transfer locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' With this definition of uniqueness,  we selected the shortest path for each mode: direct via transit (with a potential walking, driving, or local bus option  for the first mile, selecting whichever was the least costly from each passenger’s available options), direct via MOD  operator, or hybrid (MOD first mile transferring to transit, and potentially an MOD last mile for local commuters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  We selected a route choice model specification representative of the factors influencing passenger decisions in our  model: monetary costs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=', fares) and travel time (both within and outside the system), and alternative-specific con-  stants (ASC) for each represented mode (driving, transit, MOD, and hybrid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' To calibrate choice model parameters,  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  46  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  we used a simplified version of the non-cooperative pricing game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We fixed fare parameters to their real-world values  and selected model coefficients that make real-world fares correspond to a Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' This simplified game  for the calibration purposes included only three passenger types: local commuters, commuters to inner city, and those  traveling on the transit network outside of the alliance region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Each passenger had a simplified choice set represent-  ing average-case travel times and distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We solved the simplified game by enumerating solutions over a coarse  grid of candidate choice model coefficients values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Finally, all passengers were grouped into 3 sensitivity profiles:  time-sensitive, price sensitive, and intermediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Intermediate category uses the calibrated parameters as is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' We char-  acterize the time-sensitive passengers by doubling the time-sensitivity coefficients and halving the price-sensitivity  coefficients;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' and vice versa for the price-sensitive passengers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Table 10 shows the final calibrated route choice model  coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' All signs and relative magnitudes are as expected intuitively, with ASCs indicating that all else equal,  people prefer driving the most and transit the least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Price and travel time impact passenger utility negatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The  calibrated parameters thus clearly pass the common sense test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Table 10  Route choice model’s calibrated coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  MOD ASC Transit ASC Hybrid ASC  Driving ASC  Price (USD)  Travel time (min)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='125  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='9375  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='0075  Modifications to Make the Model Income-aware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' In the modified case study in Section (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3), rather than uni-  formly dividing the Greater Boston Area population into the three sensitivity profiles, we normalized time sensitivity  and calibrated price sensitivity to the relative median household income (HHI) of each town, as described in Section  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' The price-sensitivity coefficient for each town was obtained by dividing the primary case study’s coefficient in  Table 10 by that town’s income ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' This scaling method leads to the towns with the lower values of median HHI  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=', the towns with a lower income ratio) appropriately correspond to higher price sensitivity values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  For each town, Table 11 displays its price sensitivity coefficient and income ratio—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=', a ratio of that town’s median  HHI to the average of the median HHIs across the service region (which was $92,618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Absolute values of the  median HHIs by town are available in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' Note that the average of the median HHIs for the rest of the transit  service population outside of the alliance service region is 9% higher than that within the service region, underlining  that the proposed alliance particularly aims to serve lower income populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' : Transportation Alliance Design with Endogenous Demand  Article submitted to Transportation Science;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content=' 1  47  Table 11  Price sensitivity coefficients in the passengers’ route choice model and the income ratios by town  for the income-aware case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
+page_content='  Town  Price Sensitivity Coefficient Income Ratio  Outside  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNE1T4oBgHgl3EQfwwVc/content/2301.03414v1.pdf'}
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+Differentially Private Confidence
+Intervals for Proportions under
+Stratified Random Sampling∗
+Shurong Lin
+Department of Mathematics and Statistics, Boston University, Boston, MA
+e-mail: shrlin@bu.edu
+Mark Bun and Marco Gaboardi
+Department of Computer Science, Boston University, Boston, MA
+e-mail: mbun@bu.edu; gaboardi@bu.edu
+Eric D. Kolaczyk
+Department of Mathematics and Statistics, McGill University, Canada
+e-mail: eric.kolaczyk@mcgill.ca
+Adam Smith
+Department of Computer Science, Boston University, Boston, MA
+e-mail: ads22@bu.edu
+Abstract: Confidence intervals are a fundamental tool for quantifying the
+uncertainty of parameters of interest. With the increase of data privacy
+awareness, developing a private version of confidence intervals has gained
+growing attention from both statisticians and computer scientists. Differ-
+ential privacy is a state-of-the-art framework for analyzing privacy loss
+when releasing statistics computed from sensitive data. Recent work has
+been done around differentially private confidence intervals, yet to the best
+of our knowledge, rigorous methodologies on differentially private confi-
+dence intervals in the context of survey sampling have not been studied. In
+this paper, we propose three differentially private algorithms for construct-
+ing confidence intervals for proportions under stratified random sampling.
+We articulate two variants of differential privacy that make sense for data
+from stratified sampling designs, analyzing each of our algorithms within
+one of these two variants. We establish analytical privacy guarantees and
+asymptotic properties of the estimators. In addition, we conduct simula-
+tion studies to evaluate the proposed private confidence intervals, and two
+applications to the 1940 Census data are provided.
+MSC2020 subject classifications: Primary 68P27, 62G15; secondary
+62Dxx.
+Keywords and phrases: Differential privacy, confidence intervals, strat-
+ified sampling, population proportion.
+∗The research presented in this paper was supported by the U.S. Census Bureau Cooper-
+ative Agreement CB20ADR0160001.
+1
+arXiv:2301.08324v1  [stat.ME]  19 Jan 2023
+
+S. Lin et al./Differentially Private Confidence Intervals
+2
+Contents
+1
+Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+2
+1.1
+Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2
+Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+4
+2.1
+Confidence Intervals for the Population Proportion . . . . . . . .
+5
+2.2
+Differential Privacy . . . . . . . . . . . . . . . . . . . . . . . . . .
+5
+3
+Methodology
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+7
+3.1
+Estimating with Public Sample Sizes . . . . . . . . . . . . . . . .
+7
+3.1.1
+Adding Noise at the Stratum Level . . . . . . . . . . . . .
+8
+3.1.2
+Adding Noise at the Population Level
+. . . . . . . . . . .
+8
+3.2
+Estimating with Private Sample Sizes
+. . . . . . . . . . . . . . .
+10
+4
+Theoretical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+13
+4.1
+Privacy and Coverage Guarantees . . . . . . . . . . . . . . . . . .
+13
+4.2
+Comparisons of Variances . . . . . . . . . . . . . . . . . . . . . .
+15
+4.2.1
+Extrinsic Variances . . . . . . . . . . . . . . . . . . . . . .
+15
+4.2.2
+Comparing with Non-Private CI: One Stratum Case . . .
+16
+5
+Numerical Results
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+17
+5.1
+Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+17
+5.1.1
+Normality Check . . . . . . . . . . . . . . . . . . . . . . .
+18
+5.1.2
+Varying Key Parameters . . . . . . . . . . . . . . . . . . .
+18
+5.2
+Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+20
+5.2.1
+Confidence Intervals for the Unemployment Rate . . . . .
+21
+5.2.2
+Confidence Intervals for the Difference in Income Level . .
+22
+6
+Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+24
+A Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+25
+A.1 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . .
+25
+A.2 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . .
+29
+A.3 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . . .
+30
+A.4 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . .
+31
+A.5 Proof of Theorem 4.4 . . . . . . . . . . . . . . . . . . . . . . . . .
+32
+A.6 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . .
+33
+Acknowledgments
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+37
+References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+37
+1. Introduction
+With the increase of privacy awareness in the modern information era, estab-
+lishing privacy-preserving methodologies for statistics and machine learning has
+become an active research area. Differential privacy, a state-of-the-art privacy
+protection technique [13], is considered a gold standard for rigorous privacy
+guarantees. Not only has it drawn significant attention in academia [14, 15], but
+also it has been deployed by governments, firms, and other data agencies, such
+
+S. Lin et al./Differentially Private Confidence Intervals
+3
+as the U.S. Census Bureau [1], Google [19], Microsoft [7], and Apple [37]. Re-
+cently, the U.S. Census Bureau released a new demonstration of its differentially
+private Disclosure Avoidance System (DAS) for the 2020 Census [4, 23].
+Simply put, a differentially private mechanism guarantees privacy via care-
+fully injecting random noise (e.g., drawn from the Laplace or Gaussian dis-
+tribution) into the data analysis or modeling procedure. At the intersection
+of differential privacy and statistics, both statisticians and computer scientists
+are working on developing private versions of statistical inference procedures.
+Early work discussing differential privacy in the context of statistics includes
+[16, 12, 41, 36]. More recent work has explored statistical inference and estima-
+tion under the constraint of differential privacy [5, 28, 30].
+As one of the most fundamental tools for statistical inference, confidence in-
+tervals are ubiquitous in quantifying the uncertainty of parameters of interest.
+In this paper, we propose three differentially private algorithms for constructing
+confidence intervals for the population proportion under stratified random sam-
+pling. To the best of our knowledge, our work is the first to establish rigorous
+methodologies on differentially private confidence intervals in the context of sur-
+vey sampling. Survey sampling is an important area in statistics that involves
+selecting a sample of individuals from a target population to conduct a survey.
+It provides timely and cost-efficient estimates of population characteristics of
+interest and is widely used in broad-scale data gatherings, such as the American
+Community Survey (ACS), the Survey of Income and Program Participation
+(SIPP), and the Current Population Survey (CPS).
+This paper provides the first study of differentially private confidence intervals
+for data from stratified sampling designs. Specifically:
+• We articulate two specific variants of differential privacy that are appropri-
+ate for data from stratified sampling designs. In addition to the standard
+notion of differential privacy, we also consider settings in which the sample
+stratum sample sizes are fixed and public. This latter setting allows for
+simpler algorithms and tighter confidence intervals.
+• We give methods to propagate the uncertainty due to the application of
+differentially private mechanisms (adding random noise) into the construc-
+tion of confidence intervals. A necessary bias correction is made to achieve
+(asymptotic) unbiased variance estimates. Central limit theorem (CLT)-
+type statements are provided to guarantee the confidence level asymptot-
+ically.
+• We assess the performance of the proposed algorithms both in theory and
+through simulations. The theoretical analysis comparing the non-private
+and private methods gives practitioners a sense of how the algorithms
+would work prior to applying them to real data.
+• To support the theoretical analysis of one of the algorithms, we study
+the behavior of a reciprocal normal variable in depth. A general form of
+the Taylor expansion (for conditional moments) is obtained to solve the
+problem of the non-existence of moments due to its heavy-tailed nature.
+The paper is organized as follows. We briefly discuss the existing work on
+
+S. Lin et al./Differentially Private Confidence Intervals
+4
+differentially private confidence intervals in Section 1.1. Section 2 provides pre-
+liminaries on confidence intervals of population proportions and differential pri-
+vacy. In Section 3, we discuss the methodology of three differentially private
+algorithms. Section 4 provides theorems on both privacy and asymptotic cov-
+erage guarantees. Numerical experiments, including simulation studies and two
+applications to the 1940 Census data, are conducted in Section 5. Section 6
+discusses the implications of our methods and general research directions on
+differentially private confidence intervals.
+1.1. Related Work
+Differentially private confidence intervals have recently been studied for other
+settings. Some studied differentially private confidence intervals for the popula-
+tion mean of normally distributed data [26, 10, 22] . Other tasks on confidence
+intervals have also been explored. Drechsler et al. designed and evaluated sev-
+eral strategies to obtain differentially private confidence intervals for the median
+[9]. Wang et al. provided confidence intervals for differentially private models
+trained with objective or output perturbation algorithms [40].
+Besides, bootstrapping is a popular technique for constructing more general
+differentially private confidence intervals. Ferrando et al. proposed a general-
+purpose approach to construct confidence intervals for a population parameter
+[20]. A numerical confidence interval for the difference of mean was provided [8].
+The nonparametric bootstrap was considered in [2]. Covington et al. described a
+method to induce distributions of mean and covariance estimates via the bag of
+little bootstraps (BLB), which can further produce private confidence intervals
+[6].
+Our work is the first to study design-based approaches to sampling. In a
+design-based setting, the values of interest are viewed as fixed but unknown con-
+stants. Randomness only comes from the sampling design. The selection proba-
+bilities introduced with the design will be used for estimation. On the contrary,
+in a model-based setting, a parametric model is postulated. Design-based meth-
+ods in sampling can be more reassuring than model-based approaches for the
+reason that in many cases, no accurate prior information about the population
+distribution is available. More discussion of design-based versus model-based
+approaches in sampling can be found in [38].
+2. Preliminaries
+In this section, we provide some preliminaries on population proportion estima-
+tion and differential privacy. We first review the classic Wald confidence interval
+for the population proportion under stratified random sampling. Then we define
+a notion of differential privacy specifically for stratified data. Some properties
+of differential privacy are revisited in preparation for the theoretical analysis in
+Section 4.
+
+S. Lin et al./Differentially Private Confidence Intervals
+5
+2.1. Confidence Intervals for the Population Proportion
+In stratified random sampling, a population of N individuals is partitioned into
+H strata, where stratum h has Nh individuals, and simple random sampling
+of nh individuals is conducted within each stratum. When the objective is to
+estimate the proportion of individuals having some attribute in the population,
+one can estimate it by the sample proportion. Let yhi be the corresponding
+indicator variable: yhi = 1 when the individual i in stratum h has the attribute
+and yhi = 0 otherwise. One can estimate the population proportion
+p = 1
+N
+H
+�
+h=1
+Nh
+�
+i=1
+yhi
+by the sample proportion
+ˆp = 1
+N
+H
+�
+h=1
+Nh
+nh
+nh
+�
+i=1
+yhi =
+H
+�
+h=1
+whˆph
+where wh
+def
+= Nh
+N
+and ˆph
+def
+=
+1
+nh
+nh
+�
+i=1
+yhi. Its variance Var(ˆp) =
+H
+�
+h=1
+w2
+hVar(ˆph),
+where
+Var(ˆph) =
+�Nh − nh
+Nh − 1
+� ph(1 − ph)
+nh
+.
+An unbiased estimator for Var(ˆph) is given by the sample variance in the stratum
+�
+Var(ˆph) =
+�Nh − nh
+Nh
+� ˆph(1 − ˆph)
+nh − 1
+.
+(1)
+Then an unbiased estimator for Var(ˆp) is given by �
+Var(ˆp) =
+H
+�
+h=1
+w2
+h�
+Var(ˆph). An
+approximate 100%(1−α) confidence interval for p based on a normal distribution
+can be constructed:
+ˆp ± z1− α
+2
+�
+�
+Var(ˆp),
+(2)
+where z1− α
+2 denotes the 1 − α
+2 quantile of standard normal distribution. The
+normal approximation is useful when all the sample sizes are moderate to large.
+Otherwise, the t distribution with appropriate degrees of freedom is typically
+used to replace the standard normal distribution. For small sample sizes, various
+specialized confidence intervals have been developed [21].
+2.2. Differential Privacy
+Differential privacy ensures that the output of data analysis or a query does not
+differ much when the data set is changed by one record, such that one can not
+
+S. Lin et al./Differentially Private Confidence Intervals
+6
+infer the presence or absence of any individual. If two data sets x, x′ differ by
+one record, we say that x, x′ are adjacent or neighboring, written as x ∼ x′.
+The definition of differential privacy depends on how we define adjacency. For
+the partitioned data under stratified sampling, there are two ways to change a
+record: (1) one way is to substitute one record within a stratum, with all the
+stratum sample sizes fixed. We refer to this adjacency relation as “substitute-one
+relation within a stratum” and denote it by ∼ss. This relation corresponds to the
+case where the sample sizes are public and fixed; (2) another way to obtain an
+adjacent data set is to remove or add one record from one stratum; we refer to
+the corresponding relation as, which we call “remove/add-one relation”, denoted
+by ∼r. In this case, one of the stratum sample sizes will change by one, as will
+the overall sample size. This relation corresponds to the case where the sample
+sizes are private.
+Under either adjacency relation, we can define zero-concentrated differentially
+private (ρ-zCDP) as in [3]:
+Definition 1 (ρ-zCDP). Let X ∗ denote the space of the input data with an
+arbitrary finite dimension. Under the adjacency relation ∼, a randomized algo-
+rithm M : X ∗ → Y is ρ-zero-concentrated-differentially private (ρ-zCDP) if, for
+every pair of adjacent data sets x ∼ x′ ∈ X ∗, and all α ∈ (1, ∞),
+Dα(M(x)∥M(x′)) ≤ ρα,
+where Dα(M(x)∥M(x′)) is the α-Rényi divergence [39] between the distribution
+of M(x) and the distribution of M(x′).
+The parameter ρ indicates the privacy level. A smaller ρ means a more re-
+strictive distance control between M(x) and M(x′). As a result, the outputs on
+two adjacent data sets are harder to tell apart and the algorithm achieves higher
+privacy. We call ρ the privacy budget when we deliberately design an algorithm
+to satisfy ρ-zCDP.
+Depending on the adjacency notion, there are two types of differential pri-
+vacy: bounded and unbounded differential privacy [27]. Definition 1 under the
+“remove/add-one relation” corresponds to the standard unbounded differential
+privacy. The sample size of the data set changes when one record is added or
+removed to obtain an adjacent data set. With “substitute-one within a stratum”
+relation ∼ss, the resulting notion corresponds to the bounded version of differ-
+ential privacy where the sizes of two adjacent data sets are the same. But it
+is somewhat different from the standard notion of bounded differential privacy
+in that for the latter, substitutions can happen across strata. That is, we can
+change both the record and the stratum it is part of.
+In the literature on differential privacy, (ϵ, δ)-DP ([14] Definition 2.4) is con-
+sidered the classic notion. We consider ρ-zCDP because (1) ρ-zCDP implies
+(ϵ, δ)-DP ([3] Proposition 1.3), (2) the application of the Gaussian mechanism
+to achieve zCDP facilitates the theoretical analyses, and (3) the composition of
+ρ-zCDP is straightforward. The Gaussian mechanism is a prototypical exam-
+ple of a mechanism satisfying zCDP, which perturbs the true values by adding
+
+S. Lin et al./Differentially Private Confidence Intervals
+7
+Gaussian noise. We provide the Gaussian mechanism and the composition and
+post-processing properties of ρ-zCDP in the following propositions. All proposi-
+tions can be found in [3] and will be used in the analyses of privacy guarantees
+in Section 4.
+Definition 2 (Sensitivity). A function q: X ∗ → R has sensitivity ∆ if for all
+pairs of adjacent data sets x ∼ x′ ∈ X ∗, we have |q(x) − q(x′)| ≤ ∆.
+Proposition 1 (Gaussian Mechanism of ρ-zCDP). Let q : X ∗ → R be a
+sensitivity-∆ query. Consider the mechanism M : X ∗ → R that on input x,
+releases a sample from N(q(x), ∆2/(2ρ)). Then, M satisfies ρ-zCDP.
+A smaller budget leads to larger noise added to the query on average. Con-
+sequently, the output is more private.
+Proposition 2 (Composition). Let M : X ∗ → Y and M ′ : X ∗ → Z be two
+randomized algorithms. Suppose M satisfies ρ-zCDP and M ′ satisfies ρ′-zCDP,
+then algorithm M ′′ = (M, M ′) : X ∗ → Y × Z is (ρ + ρ′)-zCDP.
+Proposition 3 (Post-processing). Let M : X ∗ → Y and f : Y → Z be
+randomized algorithms. If M is ρ-zCDP, then so is the composed algorithm
+M ′ = f ◦ M : X ∗ → Z.
+3. Methodology
+Our goal is to release a ρ-zCDP confidence interval for the population proportion
+p under stratified random sampling. To construct a confidence interval as in (2),
+we need to estimate both p and the variance of the estimator privately. Recall
+that the non-private estimator of population proportion is given by the sample
+proportion
+ˆp =
+H
+�
+h=1
+whˆph.
+We assume the stratum sizes Nh are all public and fixed, thus so are wh. To get
+a private estimator for p, denoted by ˜p, we can add noise at the level of either
+the (non-private) estimator ˆp or the estimator ˆph. With ˜p, we further devise a
+private estimator for Var(˜p). Based on this idea, two algorithms for the case of
+public sample sizes are designed by adding noise at the stratum or population
+level in section 3.1. In section 3.2, we additionally propose adding noise at the
+stratum level when sample sizes are private. Throughout the paper, the accents
+ˆ· and ˜· are used to represent non-private and private estimators, respectively.
+3.1. Estimating with Public Sample Sizes
+When sample sizes nh are fixed, there are two natural strategies for perturbing
+ˆp: add Gaussian noise to (1) the stratum-level statistics ˆph’s, or (2) the overall
+statistic ˆp. Adding noise to the ˆph’s has the advantage of producing private
+estimators for stratum-level proportions simultaneously.
+
+S. Lin et al./Differentially Private Confidence Intervals
+8
+3.1.1. Adding Noise at the Stratum Level
+We apply the Gaussian mechanism to each stratum to derive a private estimator
+˜ph
+def
+= ˆph + eh where eh is the Gaussian noise. Then the private estimator for
+the population proportion is
+˜p
+def
+=
+H
+�
+h=1
+wh˜ph.
+As a result, the variance of ˜p consists of both the intrinsic variances of estimating
+ph’s by ˆph’s and the additional variability from added noise:
+Var(˜p) =
+H
+�
+h=1
+w2
+h
+�
+Var(ˆph) + w2
+h Var(eh)
+�
+(3)
+where Var(eh), h = 1, ..., H are public since they do not depend on the data.
+To obtain a private confidence interval for ˆp, we will need to privately estimate
+Var(ˆph). Note that the added noise biases the term ˆph(1− ˆph) in the non-private
+estimate of Var(ˆph) in (1). More specifically, Ee[˜ph(1−˜ph)] = ˆph(1−ˆph)−Var(eh)
+where Ee denotes the expectation taken on the randomness of the added noise.
+Then a private unbiased estimator of Var(ˆph) in the right-hand side in (3) is
+given by
+�
+Var(ˆph)
+def
+=
+�Nh − nh
+Nh
+� ˜ph(1 − ˜ph) + Var(eh)
+nh − 1
+.
+(4)
+To estimate Var(˜p), we set
+�
+Var(˜p)
+def
+=
+H
+�
+h=1
+w2
+h
+�
+�
+Var(ˆph) + Var(eh)
+�
+This approach is formulated in Algorithm 1 which we call StrNz-PubSz (adding
+noise at the stratum level with public sample sizes). The theoretical results re-
+garding privacy level and asymptotic coverage are provided in Theorems 4.1 and
+4.2.
+3.1.2. Adding Noise at the Population Level
+An alternative strategy is to directly add noise to the non-private estimator of
+p, i.e., ˆp. The sensitivity of ˆp is
+∆p = max
+h
+wh
+nh
+.
+Since wh and nh are public, ∆p can be made public. We set ˜p = ˆp + e where
+e is the Gaussian noise with standard deviation proportional to ∆p. Then, the
+variance of ˜p becomes
+Var(˜p) = Var(ˆp) + Var(e).
+(5)
+
+S. Lin et al./Differentially Private Confidence Intervals
+9
+Algorithm 1 Adding noise at the stratum level with public sample sizes, StrNz-
+PubSz
+Input: ˆph, nh, Nh, wh, ρ, α.
+Output: ρ-zCDP (1 − α) CI for the population proportion.
+1: for h = 1 to H do
+2:
+Generate Gaussian noise eh ∼ N(0,
+1
+2ρn2
+h ), and let
+˜ph ← ˆph + eh.
+3:
+Estimate Var(˜ph) by
+�Vh ←
+� Nh − nh
+Nh
+� ˜ph(1 − ˜ph) +
+1
+2ρn2
+h
+nh − 1
++
+1
+2ρn2
+h
+.
+4: end for
+5: Estimate p by ˜p ←
+H
+�
+h=1
+wh ˜ph and Var(˜p) by �V ←
+H
+�
+h=1
+w2
+h �Vh.
+6: Return
+˜p ± z1−α/2
+�
+�V ,
+where z1−α/2 is the (1 − α/2)-quantile of the standard normal distribution.
+Recall that
+�
+Var(ˆp) =
+H
+�
+h=1
+w2
+h
+�Nh − nh
+Nh
+� ˆph(1 − ˆph)
+nh − 1
+is an unbiased estimator for Var(ˆp). To get a private estimator for Var(˜p),
+we again apply the Gaussian mechanism to �
+Var(ˆp) based on the sensitivity
+of Var(ˆp):
+∆V = max
+h
+�Ch
+nh
+�
+1 − 1
+nh
+��
+,
+where Ch = w2
+h
+Nh−nh
+Nh
+1
+nh−1.
+Since we apply the Gaussian mechanism twice, the total privacy budget
+should be divided into two parts: ρ = ρ1 + ρ2 to spend on adding noise to ˆp
+and Var(ˆp), respectively. The composition property (Proposition 2) ensures the
+total privacy level is ρ. The resulting algorithm, PopNz-PubSz, is presented
+in Algorithm 2.
+Remark 1. When there are multiple strata with similar sampling rates, Algo-
+rithm 1 yields a wider confidence interval for p than Algorithm 2 does, given the
+same privacy budget. However, Algorithm 1 additionally produces private confi-
+dence intervals for ˆph which may be of interest for release. In Section 4.2.1, we
+compare the two algorithms quantitatively.
+Remark 2. Proportions are always between 0 and 1. One can post-process
+proportion estimates (˜ph in Algorithm 1 and ˜p in Algorithm 2) by clipping them
+onto interval [0,1] without undermining privacy. When the privacy budget is
+very small, the noisy proportion estimates are likely to lie outside [0,1]. Thus,
+clipping moves the confidence interval toward the truth and a higher coverage
+
+S. Lin et al./Differentially Private Confidence Intervals
+10
+Algorithm 2 Adding noise at the population level with public sample sizes,
+PopNz-PubSz
+Input: ˆp, ˆph, nh, Nh, wh, ρ, α.
+Output: A ρ-zCDP (1 − α) CI for Population Proportion.
+1: Split the budget ρ = ρ1 + ρ2. Let ρ1 = ρ2 if not specified otherwise.
+2: Generate noise e ∼ N(0,
+∆2
+p
+2ρ1 ) where ∆p = maxh
+wh
+nh and let
+˜p ← ˆp + e.
+3: Generate noise eV
+∼
+N(0, ∆2
+V
+2ρ2 ) where ∆V
+=
+maxh
+�
+Ch
+nh
+�
+1 −
+1
+nh
+��
+and Ch
+=
+w2
+h
+Nh−nh
+Nh
+1
+nh−1 . Let
+�V ←
+H
+�
+h=1
+w2
+h
+� Nh − nh
+Nh
+� ˆph(1 − ˆph)
+nh − 1
++ ∆2
+p
+2ρ1
++ eV .
+4: Return
+˜p ± z1−α/2
+�
+�V ,
+where z1−α/2 is the (1 − α/2)-quantile of the standard normal distribution.
+rate will be observed. With a moderate or large budget, clipping does not make
+a noticeable difference.
+Lastly, one can always clip the output confidence intervals onto [0,1] without
+privacy loss.
+3.2. Estimating with Private Sample Sizes
+When sample sizes are public information, keeping the proportions private is es-
+sentially protecting only the numerator (the counts of individuals with y = 1).
+In some cases where subpopulation proportions also need to be estimated, Al-
+gorithms 1 and 2 with public sample sizes can lead to privacy leakage since the
+counts become the denominator. Therefore, a method of constructing confidence
+intervals for proportions to keep both the counts and sample sizes private is nec-
+essary. We protect the sample sizes by adding noise to them. As a result, sample
+sizes are not fixed and therefore we need the unbounded notion of differential
+privacy with the adjacency relation ∼r. In the following, we extend Algorithm
+1 to serve the needs of privacy protection of sample sizes by adding noise at the
+stratum level. (It is not obvious how to extend Algorithm 2, which adds noise
+at the population level. It requires more sophisticated mechanisms; we briefly
+discuss in Section 6.)
+To begin, we first consider the setting of simple random sampling. The idea
+is to add independent Gaussian noise to both the numerator and denominator
+for each stratum. For ease of notation, we first consider a single stratum with
+count c = �n
+i=1 xi. We know
+c ∼ Hypergeometric(N, K, n),
+where K is the total number of individuals with the attribute of interest. The
+count c has mean n K
+N = np and variance n K
+N
+N−K
+N
+N−n
+N−1 = n2 Var(ˆp). By applying
+
+S. Lin et al./Differentially Private Confidence Intervals
+11
+the Gaussian mechanism to c and n with privacy budgets ρ1 and ρ2, respectively,
+we have private count ˜c and sample size ˜n:
+˜c | c ∼ N(c, 1
+2ρ1
+)
+and
+˜n ∼ N(n, 1
+2ρ2
+).
+The unconditional mean and variance for c are
+E(˜c) = E[E(˜c | c)] = E(c) = np
+and
+Var(˜c) = E Var(˜c | c) + Var E(˜c | c) =
+1
+2ρ1
++ n2 Var(ˆp).
+(6)
+By the composition property of zCDP, we get a private estimator for proportion
+p, denoted by ˜p, with privacy level ρ = ρ1 + ρ2. Since ˜c and ˜n are independent
+variables, in principle,
+E(˜p) = E
+� ˜c
+˜n
+�
+= E(˜c)E
+� 1
+˜n
+�
+,
+(7)
+and
+Var(˜p) = E
+� ˜c
+˜n
+�2
+−
+�
+E
+� ˜c
+˜n
+��2
+= E˜c2E
+� 1
+˜n2
+�
+− (E˜c)2
+�
+E 1
+˜n
+�2
+.
+(8)
+However, the moments of 1
+˜n do not exist, thus neither do those of ˜p. Generally
+speaking, the ratio of two independent normal random variables has a heavy-
+tailed distribution with no moments [33, 17]. The shape of the distribution could
+be unimodal, bimodal, symmetric, or asymmetric. It is primarily determined by
+the coefficient of variation of the denominator variable, CV . When CV is suffi-
+ciently small, a normal distribution approximation can be effective. It has been
+shown theoretically that a normal distribution can be arbitrarily close to the
+ratio variable in an interval centered at the ratio of means of two normal ran-
+dom variables [17]. Experiments have provided guidelines for when the normal
+approximation can be used. For example, a simple rule is that the approxima-
+tion is reasonable whenever CV is less than 0.1 [29]. Other practical rules are
+mentioned in [25, 33].
+We take advantage of the normal approximation to construct a ρ-zCDP con-
+fidence interval for the proportion. We present the following estimation strategy
+in Algorithm 3, StrNz-PrivSz. In the algorithm, we clip ˜nh in (9) to ensure the
+denominator is not too small. Otherwise, the ratio can be arbitrarily large. Such
+a post-processing step preserves the same privacy guarantee. For the theoretical
+analysis, we do not clip ˜nh, but instead, we consider the ratio variable ˜ch/˜nh
+given the event Sh = {1 ≤ ˜nh ≤ 2nh − 1} (a symmetric area around the mean
+of ˜nh). It is more convenient for the analysis. The asymptotic behaviors of ˜ph
+
+S. Lin et al./Differentially Private Confidence Intervals
+12
+Algorithm 3 Adding noise at the stratum level with private sample sizes,
+StrNz-PrivSz
+Input: Nh, wh, nh, ch, ρ, α.
+Output: A ρ-zCDP (1 − α) CI for the population proportion.
+1: Split the budget ρ = ρ1 + ρ2. Let ρ1 = ρ2 if not specified otherwise.
+2: for h = 1 to H do
+3:
+Generate e(1)
+h
+∼ N(0,
+1
+2ρ1 ) and e(2)
+h
+∼ N(0,
+1
+2ρ2 ), and let
+�
+˜ch ← ch + e(1)
+h
+˜nh ← max(nh + e(2)
+h , 2)
+(9)
+4:
+Let
+˜ph ← ˜ch
+˜nh
+(10)
+5:
+Let
+�Vh ←
+� Nh − ˜nh
+Nh − 1
+� ˜ph(1 − ˜ph)
+˜nh
++
+1
+2ρ1˜n2
+h
++
+˜p2
+h
+2ρ2˜n2
+h
+.
+(11)
+6: end for
+7: Estimate p by ˜p ←
+H
+�
+h=1
+wh ˜ph and Var(˜p) by �V ←
+H
+�
+h=1
+w2
+h �Vh.
+8: Return
+˜p ± z1−α/2
+�
+�V ,
+where z1−α/2 is the (1 − α/2)-quantile of the standard normal distribution.
+in the algorithm and ˜ch/˜nh | Sh are essentially the same since Pr(˜nh ≥ 2) → 1
+and Pr(Sh) → 1 as n → ∞. We will see the private estimator of the variance of
+˜ph we derive from the analysis of ˜ch/˜nh | Sh works well and the algorithm does
+achieve the desired coverage level.
+We consider the ratio of two independent normal variables. By independence,
+what remains unclear is the behavior of the reciprocal of a normal distribution.
+(We should mention that the Inverse Gaussian distribution is a different dis-
+tribution than the reciprocal distribution we discuss here.) In Theorem 3.1, we
+provide a general form of the Taylor series of conditional mean and variance of
+a reciprocal normal distribution. To our best knowledge, this is the first com-
+plete result of the Taylor series, with the remainder term specified. We prove
+the theorem in the Proofs section. We use k = 2 to derive an estimator for the
+variance of ˜p Algorithm 3, which leads to (11).
+Theorem 3.1 (Conditional mean and variance of a reciprocal normal distribu-
+tion). For random variable X ∼ N(µ, σ2) where µ > 1 and σ2 > 0, given the
+event S = {1 ≤ X ≤ 2µ − 1}, for any integer k > 0, the first two moments of
+1
+X | S have the following expansions:
+E
+� 1
+X | S
+�
+= 1
+µ
+k
+�
+j=0
+(2j − 1)!!σ2j
+µ2j
++ O
+�σ2k+2
+µ2k+2
+�
+(12)
+
+S. Lin et al./Differentially Private Confidence Intervals
+13
+and
+E
+� 1
+X2 | S
+�
+= 1
+µ2
+k
+�
+j=0
+(2j + 1)!!σ2j
+µ2j
++ O
+�σ2k+2
+µ2k+2
+�
+.
+(13)
+4. Theoretical Results
+In this section, we present the theoretical results of both privacy and asymptotic
+coverage guarantees. In addition, comparisons of the three algorithms in terms
+of variance and width ratios are discussed.
+4.1. Privacy and Coverage Guarantees
+Our theoretical results are two-fold. First, the proposed algorithms satisfy the
+desired privacy level under the corresponding adjacency relation, which is pre-
+sented in Theorem 4.1.
+Theorem 4.1 (Privacy Guarantee). Algorithms 1 and 2 satisfy ρ-zCDP under
+the adjacency relation ∼ss; Algorithm 3 satisfies ρ-zCDP under the adjacency
+relation ∼r.
+Proofs are presented in the Proofs section.
+On the other hand, for the confidence intervals to be useful, we provide the-
+orems that guarantee the asymptotic coverage for each algorithm. The central
+limit theorem (CLT) asserts (essentially) that the sample mean is asymptoti-
+cally normally distributed regardless of the original distribution. Therefore, the
+sample mean can be used to construct a confidence interval for the population
+mean. In the finite-population sampling designs we are considering, variants of
+CLTs can be found among [18, 24, 31] and others. We restate a general form
+of the finite-population CLT for simple random sampling in Theorem A.2 and
+provide asymptotic coverage guarantees in the following theorems.
+Theorem 4.2 (Algorithm 1). For a population of size N, let p be the pro-
+portion in the population with the attribute of interest. Under stratified random
+sampling with sample sizes nh within the stratum of size Nh, h = 1, .., H, let
+�V =
+H
+�
+h=1
+w2
+h �Vh where
+�Vh =
+�Nh − nh
+Nh
+� ˜ph(1 − ˜ph) +
+1
+2ρn2
+h
+nh − 1
++
+1
+2ρn2
+h
+.
+(14)
+for ρ > 0 as described in Algorithm 1. If ρ = ω(1/nh) for all h, then as Nh −nh
+and nh both tend to infinity for every stratum,
+(i) �V
+p→ Var(ˆp), and
+
+S. Lin et al./Differentially Private Confidence Intervals
+14
+(ii) for 0 < α < 1,
+Pr
+�
+p ∈
+�
+˜p − z1−α/2
+�
+�V , ˜p + z1−α/2
+�
+�V
+��
+→ 1 − α.
+(15)
+Theorem 4.3 (Algorithm 2). For a population of size N, let p be the proportion
+in the population with the attribute of interest. Under stratified random sampling
+with sample sizes nh within the stratum of size Nh, h = 1, .., H, let
+�V =
+H
+�
+h=1
+w2
+h
+�Nh − nh
+Nh
+� ˆph(1 − ˆph)
+nh − 1
++ ∆2
+p
+2ρ1
++ eV
+(16)
+where eV ∼ N(0, ∆2
+V
+2ρ2 ) for ρ1, ρ2 > 0 as described in Algorithm 2. If ρ1 =
+ω(1/nh) and ρ2 = ω(1/nh) for all h, then as Nh − nh and nh both tend to
+infinity for every stratum,
+(i) �V
+p→ Var(ˆp), and
+(ii) for 0 < α < 1,
+Pr
+�
+p ∈
+�
+˜p − z1−α/2
+�
+�V , ˜p + z1−α/2
+�
+�V
+��
+→ 1 − α.
+(17)
+Proofs of the above theorems use the finite-population CLT and are provided
+in the Proofs section.
+For Algorithm 3, the asymptotic behavior of ˜p is grounded on the normal
+approximation to a ratio variable in addition to the CLT. We revisit the result
+of normal approximation by [17] in Theorem A.4. Based on the approximation,
+we have shown the consistency of ˜p in the case of simple random sampling.
+Theorem 4.4. Under simple random sampling, let c be the count of individuals
+having the attribute of interest and n be the sample size. The true population
+proportion is denoted by p. Let ˜p = ˜c/˜n where ˜c ∼ N(c,
+1
+2ρ1 ) and ˜n ∼ N(n,
+1
+2ρ2 )
+for ρ1, ρ2 > 0. Under the conditions that ρ2 = ω(1/n), ρ1 = ω(1/n), ˜p is a
+consistent estimator for p.
+With the foundation of the above consistency, we establish the asymptotic
+properties:
+Theorem 4.5 (Algorithm 3). For a population of size N, let p be the pro-
+portion in the population with the attribute of interest. Under stratified random
+sampling with sample sizes nh within the stratum of size Nh, h = 1, .., H, let
+�V =
+H
+�
+h=1
+w2
+h �Vh where
+�Vh =
+�Nh − ˜nh
+Nh − 1
+� ˜ph(1 − ˜ph)
+˜nh
++
+1
+2ρ1˜n2
+h
++
+˜p2
+h
+2ρ2˜n2
+h
+(18)
+
+S. Lin et al./Differentially Private Confidence Intervals
+15
+for ρ1, ρ2 > 0 as described in Algorithm 3. If ρ1 = ω(1/nh) and ρ2 = ω(1/nh)
+for all h, then as Nh − nh and nh both tend to infinity for every stratum,
+(i) �V
+p→ Var(ˆp), and
+(ii) for 0 < α < 1,
+Pr
+�
+p ∈
+�
+˜p − z1−α/2
+�
+�V , ˜p + z1−α/2
+�
+�V
+��
+→ 1 − α.
+(19)
+To prove Theorem 4.5, we start with a single stratum. We use a normal dis-
+tribution (denoted by p∗
+h) to approximate that of the proportion estimator ˜ph,
+with the distance between the two distribution vanishing to zero in an interval.
+Then for multiple strata, we show that the linear combination of the normal
+variables (denoted by p∗) is an accurate approximation to ˜p. Last but not least,
+due to the consistency stated in Theorem 4.4, the noisy estimator �V is a consis-
+tent estimator for the variance of p∗. Then, a Wald confidence interval can be
+constructed using ˜p and �V . Details are presented in the Proofs section.
+4.2. Comparisons of Variances
+The theorems presented in Section 4.1 ensure that, under proper conditions, the
+desired coverage is achieved asymptotically. Therefore, to compare the perfor-
+mance of the different proposed confidence intervals, we compare their widths,
+which are determined by their variance estimates. In this section, we will an-
+alyze our variance estimates and compare the resulting widths to that of the
+non-private confidence interval.
+4.2.1. Extrinsic Variances
+To investigate how much additional uncertainty is caused by adding noise, we
+decompose the variances of the private estimators into two parts: (1) the inherent
+component coming from the estimation from the sampling data, i.e, Var(ˆp), and
+(2) the extrinsic component introduced by the added noise, written as
+Vex
+def
+= Var(˜p) − Var(ˆp).
+Table 1 provides the (approximate) variances of ˜p for three algorithms, where
+wh = Nh
+N are the stratum weights. The variances are derived in the proofs of
+Theorems 4.2, 4.3, and 4.5. The additional variance terms, Vex, can be rewritten
+in terms of uh
+def
+=
+Nh
+nh instead of wh, as shown in the second row of the table.
+In fact, uh are called sampling weights in the literature on survey sampling. A
+sample weight is defined as the number of individuals that each respondent in
+the sample is representing in the population. It is the reciprocal of the sampling
+rate
+nh
+Nh and plays an important role in statistical inference for survey data
+[34, 11]. Understanding the relation between sampling weights and the variance
+
+S. Lin et al./Differentially Private Confidence Intervals
+16
+of the noisy estimators is helpful for practitioners to make survey designs and
+the choice of algorithms.
+Table 1
+(Approximate) variances of ˜p.
+Algorithm
+StrNz-PubSz
+PopNz-PubSz
+StrNz-PrivSz (approximate)
+Var(˜p)
+Var(ˆp) +
+1
+2ρ
+�H
+h=1
+w2
+h
+n2
+h
+Var(ˆp) +
+1
+2ρ1 maxh
+w2
+h
+n2
+h
+Var(ˆp) +
+1
+2ρ1
+�H
+h=1
+w2
+h
+n2
+h +
+1
+2ρ2
+�H
+h=1
+w2
+hp2
+h
+n2
+h
+Vex
+1
+2N2
+�H
+h=1
+u2
+h
+ρ
+1
+2N2 maxh
+u2
+h
+ρ1
+1
+2N2
+�H
+h=1 u2
+h( 1
+ρ1 + p2
+h
+ρ2 )
+With a fixed population size N and a chosen privacy level ρ, the extra vari-
+ances Vex induced by the added noise are primarily dictated by uh. In PopNz-
+PubSz where we add noise at the population level, Vex is solely determined by
+the largest sample weight among all strata. If noise is injected into each stra-
+tum, then sampling weights in all strata collectively affect Vex. In particular,
+for StrNz-PrivSz, Vex is impacted by ph additionally. For all three algorithms,
+smaller sampling weights lead to lower extrinsic variance.
+For comparison, we look at the ratio of Vex with the default budgeting ρ1 =
+ρ2 = ρ/2 for PopNz-PubSz and StrNz-PrivSz. The ratio of Vex for StrNz-PubSz
+to PopNz-PubSz is
+�H
+h=1 u2
+h
+2 maxh u2
+h
+.
+(20)
+Roughly speaking, when there are many strata, adding noise at the population
+level gives a smaller variance. To compare StrNz-PrivSz and StrNz-PubSz, the
+ratio of Vex is
+2 �H
+h=1 u2
+h(1 + p2
+h)
+�H
+h=1 u2
+h
+,
+(21)
+which will always be greater than 2 (due to the cost it takes to protect sample
+sizes in StrNz-PrivSz) and at most 4.
+4.2.2. Comparing with Non-Private CI: One Stratum Case
+To assess the width in theory, we also look at the confidence interval width
+ratios by comparing them to the non-private one. Since the parameters Nh, nh,
+ph, ρh come into play together in the stratification setting, it is more practical
+to analyze the special case with one stratum.
+Let the theoretical width ratio (TWR) be
+TWR =
+�
+Var(˜p)
+Var(ˆp).
+In the implementation, the real width ratio (WR), defined as
+�
+�V / Var(ˆp), will
+be very close to TWR in that �V is a consistent estimator for Var(˜p). Table 2
+
+S. Lin et al./Differentially Private Confidence Intervals
+17
+displays some relevant quantities. Note that N−1
+N−n is always less than 1 but tends
+to 1 when the population size is far larger than the sample size.
+Table 2
+Theoretical width ratios and lower bounds.
+Algorithm
+StrNz-PubSz
+PopNz-PubSz
+StrNz-PrivSz
+˜p
+ˆp + N(0,
+1
+2ρn2 )
+ˆp + N(0,
+1
+ρn2 )
+(c + N(0, 1
+ρ))/(n + N(0, 1
+ρ))
+Var(˜p)
+Var(ˆp) +
+1
+2ρn2
+Var(ˆp) +
+1
+ρn2
+Var(ˆp) + 1+p2
+ρn2
+TWR
+�
+1 + N−1
+N−n
+1
+2p(1−p)nρ
+�
+1 + N−1
+N−n
+1
+p(1−p)nρ
+�
+1 + N−1
+N−n
+1+p2
+p(1−p)nρ
+Lower bound of TWR
+�
+1 +
+2
+nρ
+�
+1 +
+4
+nρ
+�
+1 + 2(1+
+√
+2)
+nρ
+We can obtain a lower bound for TWR by dropping the factor
+N−1
+N−n and
+minimizing over p. We can see that the width ratio mainly depends on p and
+the relative magnitude between n and ρ. If p is extreme (tends to 0 or 1), TWR is
+drastically large; when p is around 0.5, TWR is close to the lower bound. Also,
+the added noise induces a term involving ρ. For example, under the regime
+ρ = 1/n, the three algorithms result in an interval of length at least
+√
+3 ≈ 1.73,
+√
+5 ≈ 2.24, and
+�
+3 + 2
+√
+2 ≈ 2.41 as wide, respectively. It is trivial that with
+one stratum, StrNz-PubSz produces a tighter confidence interval than PopNz-
+PubSz does in that the ratio of Vex in (20) is 1/2. However, PopNz-PubSz will
+outperform StrNz-PubSz once there are enough strata such that (20) is greater
+than 1.
+5. Numerical Results
+In this section, we conduct both simulation studies and applications to assess
+and illustrate the numerical performance of the proposed algorithms. We clip
+the proportions ˜ph onto [0, 1] as mentioned in Remark 2.
+5.1. Simulations
+We set up a set of experiments to (1) check the normality of noisy estimators,
+and (2) evaluate the performance of the proposed confidence intervals by varying
+the number of strata H, the true population proportion p, and the privacy level
+ρ. To generate the data, we need to specify the strata sizes Nh and the sampling
+rates rh. The setup of these parameters is presented in Table 3. We generate
+a proportion for each stratum to create heterogeneity across strata. The true
+population proportion is then calculated and reported in each experiment.
+
+S. Lin et al./Differentially Private Confidence Intervals
+18
+Table 3
+Parameter setup. The resulting sample sizes are between 60 and 160.
+Fixed parameter
+Value / Distribution
+Varying parameter
+Value / Distribution
+α
+0.1
+H
+1 or 20
+Nh
+Uniform(1500, 2000)
+ph
+0.5, Uniform(0.4, 0.6) or Uniform(0.05, 0.15)
+rh
+Uniform(0.04, 0.08)
+ρ
+1/ max(nh) or specified in the axis of the plot
+5.1.1. Normality Check
+We first check whether the distributions of ˜p in the three algorithms are reason-
+ably close to the theoretical normal distributions with the corresponding means
+and variances. Figure 1 displays the Q-Q plots of the theoretical distribution of
+˜p versus its sample distribution:
+• Non-private: N(p, Var(ˆp));
+• StrNz-PubSz: N(p, Var(˜p)) as Var(˜p) in (3);
+• PopNz-PubSz: N(p, Var(˜p)) as Var(˜p) in (5);
+• StrNz-PrivSz: N
+�
+p + �H
+h=1
+whph
+2ρ2n2
+h , �H
+h=1 w2
+hVh
+�
+with Vh specified in (51).
+Note that, ˜p in Algorithms StrNz-PubSz and PopNz-PubSz are unbiased for
+p while ˜p in StrNz-PrivSz is not. Nevertheless, under the condition that ρ2 =
+ω(1/nh) in Theorem 3, the bias term �H
+h=1
+whph
+2ρ2n2
+h is negligible and thus we
+do not make a bias correction in Algorithm 3. We observe great alignments
+between the theoretical and experimental distributions, indicating that the pri-
+vate estimators in all three algorithms are indeed highly close to being normally
+distributed.
+5.1.2. Varying Key Parameters
+Assured by the results of the normality check, we experiment with a wide range
+of the privacy budget, different numbers of strata, and true population propor-
+tions.
+We examine the impact of ρ on the performance of the three private esti-
+mators. The simulation is run on 10,000 repetitions and therefore the empirical
+coverage falling into 90% ± 0.006 (departure of two standard deviations) is con-
+sidered appropriate. In Figure 2a, the empirical coverage is reasonable except
+that StrNz-PrivSz gives unnecessarily higher coverage when ρ is smaller than
+around 0.005. This is because the budget is so small for the method that, with
+clipping, it covers the truth more often than needed. In this case, the confidence
+intervals are too wide to be as useful, as shown in Figure 2b. For all three meth-
+ods, the width grows as ρ becomes smaller. However, the rates of width growth
+differ: in the multiple strata case we simulate, the width of PopNz-PubSz grows
+the slowest, StrNz-PrivSz grows the fastest, and StrNz-PubSz is in the middle.
+Thus, the optimal privacy level should be chosen by taking into account the
+method, width, and coverage. For instance, if we want a 90% of confidence level
+and width under 0.1, one can choose the value for ρ as small as (1) 0.001 for
+PopNz-PubSz, (2) 0.003 for StrNz-PubSz, and (3) 0.01 for StrNz-PrivSz.
+
+S. Lin et al./Differentially Private Confidence Intervals
+19
+0.46
+0.48
+0.50
+0.52
+0.54
+Theoretical Quantiles
+0.46
+0.48
+0.50
+0.52
+0.54
+Sample Quantiles
+Non-Private
+0.425
+0.450
+0.475
+0.500
+0.525
+0.550
+0.575
+Theoretical Quantiles
+0.45
+0.50
+0.55
+Sample Quantiles
+StrNz-PubSz
+0.46
+0.48
+0.50
+0.52
+0.54
+0.56
+Theoretical Quantiles
+0.475
+0.500
+0.525
+0.550
+Sample Quantiles
+PopNz-PubSz
+0.40
+0.45
+0.50
+0.55
+0.60
+Theoretical Quantiles
+0.4
+0.5
+0.6
+Sample Quantiles
+StrNz-PrivSz
+Fig 1: Q-Q plots: Theoretical versus sample distributions of ˜p with 20 strata and p = 0.505
+(resulting from ph ∼ Uniform(0.4, 0.6)), based on 10,000 repetitions each.
+In addition, Table 4 shows the numerical results of three experiments with
+different combinations of the numbers of strata and the true population pro-
+portions. The simulation in the middle panel shares the same setting as the
+experiment shown in Figure 2 but has a fixed privacy level: 1/ max(nh). This
+is an analogous regime to ρ = 1/n (for simple random sampling) for multiple
+strata. In the literature on differential privacy, the regime ρ = 1/n for a simple
+random sample is often considered to understand how small ρ can be as the
+sample size increases. Recall that a smaller ρ means a higher privacy level.
+As argued above, clipping ˜ph (or ˜p) onto [0,1] will yield better results in some
+cases. The conclusions coincide with the analyses in Section 4. The empirical
+coverage of the three private ones in all simulations achieves the nominal level of
+90%, as guaranteed by Theorems 4.2, 4.3, and 4.5. The case where StrNz-PrivSz
+gives a 91.9% confidence level in the bottom panel is due to clipping. (When
+the stratum proportions are close to the extreme, clipping is more noticeable.)
+The average width and width ratio (WR) varies. With one single stratum,
+WRs are near the lower bounds of theoretical width ratios (TWR) given in
+Section 4.2.2, which suggests that the lower bounds are almost tight. StrNz-
+PubSz gives a narrower CI than PopNz-PubSz with one stratum. But with
+more strata, PopNz-PubSz outperforms StrNz-PubSz in terms of WR. Having
+more strata means splitting the total privacy budget into smaller portions, which
+leads to adding more noise on the whole. The CI needs to be wider to achieve the
+same confidence level. As for StrNz-PrivSz, however, it always yields the widest
+CI due to the additional price it pays to protect sample sizes simultaneously.
+
+S. Lin et al./Differentially Private Confidence Intervals
+20
+10
+3
+10
+2
+10
+1
+100
+0.80
+0.85
+0.90
+0.95
+1.00
+Empirical coverage
+Non-Private
+PopNz-PubSz
+StrNz-PubSz
+StrNz-PrivSz
+(a) Empirical coverage
+10
+3
+10
+2
+10
+1
+100
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+Width
+0
+2
+4
+8
+16
+Width ratio
+Non-Private
+PopNz-PubSz
+StrNz-PubSz
+StrNz-PrivSz
+(b) Width and ratio
+Fig 2: Setup: 20 strata and p = 0.505 (ph ∼ Uniform(0.4, 0.6)) with 10,000 repetitions.
+Figure (a) is the empirical coverage with the red line indicating the nominal confidence level
+of 90%. The average width and width ratio are displayed in (b) with the non-private as the
+benchmark.
+On the other hand, with the same number of strata (20 here), we see that more
+extreme ph leads to a larger WR than ph in the middle range. This is because
+the factor ph(1 − ph) comes into play as p(1 − p) does in TWR in Table 2 for
+the one stratum case.
+We also provide the sample standard deviation of the widths (width SD). In
+general, the non-private method results in a smaller standard deviation than
+the private ones. In some cases, clipping helps reduce the width SD for the
+private algorithms. With the same privacy level, there is more fluctuation in
+width for PopNz-PubSz compared to StrNz-PubSz. This is because we use one-
+half of the privacy budget and directly add noise to the variance estimate. As
+expected, StrNz-PrivSz has the largest width SD since the magnitude of width
+is the largest and the ratio variable is heavy-tailed by design. Nevertheless,
+compared to the width, the width SD for all methods is so small that it does
+not compromise the effectiveness of the confidence interval.
+5.2. Applications
+In this section, we apply the proposed methods to the 1940 Census full enumer-
+ation from IPUMS USA [35] and evaluate the performance of three differentially
+private confidence intervals. To conduct stratified random sampling on the data
+set, the state-level geographical variable “STATEICP” (49 categories, constitut-
+ing the then-48 states and Washington, D.C.) is used for stratification. Under
+stratified random sampling with H = 49 strata, we estimate the national un-
+employment rate for the first application. In the second application, we are
+interested in studying the discrepancy in income levels between black and white
+
+S. Lin et al./Differentially Private Confidence Intervals
+21
+Table 4
+Simulation results under ρ = 1/n (or ρ = 1/ max(nh)) regime based on 10,000 repetitions.
+The strata sizes and sampling rates are drawn as described in Table 3. For the multiple
+strata case, the resulting sample sizes in nh range from 72 to 152, and ρ is set to be
+1/152 ≈ 6.58 × 10−3. For the one-stratum case, we set the sample size to 152 so that we
+have the same level of privacy.
+Non-Private
+StrNz-PubSz
+PopNz-PubSz
+StrNz-PrivSz
+1 stratum, p = 0.5
+coverage
+0.893
+0.901
+0.894
+0.901
+width
+0.127
+0.228
+0.295
+0.327
+width SD
+5.47 × 10−4
+9.85 × 10−4
+8.89 × 10−3
+3.15 × 10−2
+CI
+(0.436, 0.564)
+(0.386, 0.614)
+(0.352, 0.648)
+(0.34, 0.667)
+WR
+1
+1.786
+2.318
+2.567
+20 strata, p = 0.505 (ph ∼ Uniform(0.4, 0.6))
+coverage
+0.902
+0.895
+0.902
+0.902
+width
+0.035
+0.073
+0.043
+0.111
+width SD
+1.08 × 10−4
+1.58 × 10−4
+5.87 × 10−4
+5.22 × 10−3
+CI
+(0.488, 0.523)
+(0.469, 0.542)
+(0.483, 0.527)
+(0.457, 0.568)
+WR
+1
+2.074
+1.239
+3.168
+20 strata, p = 0.103 (ph ∼ Uniform(0.05, 0.15))
+coverage
+0.902
+0.919
+0.904
+0.899
+width
+0.021
+0.067
+0.033
+0.096
+width SD
+6.17 × 10−4
+5.21 × 10−4
+8.71 × 10−4
+3.94 × 10−3
+CI
+(0.092, 0.113)
+(0.073, 0.143)
+(0.086, 0.119)
+(0.072, 0.168)
+WR
+1
+3.189
+1.571
+4.563
+men.
+5.2.1. Confidence Intervals for the Unemployment Rate
+As an important indicator of the status of the national economy, the unem-
+ployment rate is the percentage of unemployed workers in the total labor force
+consisting of both the employed and unemployed. Thus, we consider all the indi-
+viduals who are either employed or unemployed as the whole population. In the
+1940 Census data set, the binary characteristic “EMPSTAT” represents employ-
+ment status. The full enumeration is considered the truth and the true popula-
+tion proportion is p = 9.346%. To carry out stratified random sampling, sample
+sizes or sampling rates are selected for all 49 strata. For modern relevance, we
+simulate in a manner intended to mimic the canonical design implemented in
+the current American Community Survey (ACS), by choosing a typical range of
+sampling rates used in ACS which is [0.5%, 15%]. See Table 5 for detail.
+
+S. Lin et al./Differentially Private Confidence Intervals
+22
+Table 5
+Sampling rates
+Stratum size
+Sampling rate
+nh ≤ 5 × 104
+15%
+5 × 104 < nh ≤ 105
+10%
+105 < nh ≤ 5 × 105
+5%
+5 × 105 < nh ≤ 106
+2%
+106 < nh ≤ 5 × 106
+1%
+nh > 5 × 106
+0.5%
+To apply and assess the proposed algorithms, we experiment with a wide
+range of small privacy budgets: ρ ∈ [10−6, 10−3]. Each method is repeated 10,000
+times and the empirical coverage, the average CI width, and the average CI
+width ratio (WR) are computed. As shown in Figure 3a, the empirical coverage
+is always around the nominal level which is chosen at the level of 90% for the
+whole range of privacy levels. In Figure 3b, the CI width and CI width ratio
+with the non-private CI as the benchmark, share the same shape. Even when
+the CI given by StrNz-PrivSz is 8 times the non-private CI width, the CI width
+is only 0.01 due to the large sample size. Both CI width and width ratio should
+be taken into account when choosing an optimal privacy level.
+10
+6
+10
+5
+10
+4
+10
+3
+0.80
+0.85
+0.90
+0.95
+1.00
+Empirical coverage
+Non-Private
+PopNz-PubSz
+StrNz-PubSz
+StrNz-PrivSz
+(a) Empirical coverage
+10
+6
+10
+5
+10
+4
+10
+3
+0.00
+0.01
+0.02
+0.03
+Width
+0
+5
+10
+15
+20
+25
+WR
+Non-Private
+PopNz-PubSz
+StrNz-PubSz
+StrNz-PrivSz
+(b) Width and ratio
+Fig 3: The empirical coverage and average width and width ratio of DP-CIs of the unemploy-
+ment rate.
+5.2.2. Confidence Intervals for the Difference in Income Level
+In the second application, we want to investigate whether there was a discrep-
+ancy between the income levels of white males (population 1) and that of black
+males (population 2). Note that only those who had valid income numbers in
+the 1940 Census are considered. Since the poverty thresholds were not devel-
+oped until the 1960s and thus are not available for the 1940 data, the national
+
+S. Lin et al./Differentially Private Confidence Intervals
+23
+income average is used as a threshold instead. We are interested in examining
+the difference in subpopulation proportions of those whose income levels passed
+this threshold.
+The geographic feature “STATEICP” is used for stratification, yielding 49
+strata, with stratum size ranges of (41838, 4621442) for the population of white
+males and (50, 309214) for the population of black males. Sampling rates are
+adaptively chosen based on stratum sizes. For the population of white males, the
+range of sampling rates is also [0.5%, 15%], whereas the range of sampling rates
+is [0.5%, 30%] for the population of black males given its small stratum sizes.
+Additionally, to allow solid approximations based on the asymptotic results, we
+impose that the sample sizes are adjusted to be 50 if the sampling rates give
+smaller sizes than 50. See Table 6 for detail.
+Table 6
+Sampling rates for two populations. Stratum sizes nh ∈ (4.1 × 104, 4.7 × 106) for the
+population of white males and stratum sizes nh ∈ (50, 3.1 × 105) for the population of black
+males. *The sample size will be adjusted to be 50 if the above sampling rate results in a size
+smaller than 50.
+Stratum size Nh of white males
+Sampling rate
+Nh ≤ 5 × 104
+15%
+5 × 104 < Nh ≤ 105
+10%
+105 < Nh ≤ 5 × 105
+5%
+5 × 105 < Nh ≤ 106
+2%
+106 < Nh ≤ 4 × 106
+1%
+Nh > 4 × 106
+0.5%
+Stratum size Nh of black males
+Sampling rate*
+Nh ≤ 500
+30%
+500 < Nh ≤ 5 × 103
+15%
+5 × 103 < Nh ≤ 104
+5%
+104 < Nh ≤ 2 × 104
+2%
+2 × 104 < Nh ≤ 3 × 104
+1%
+nh > 3 × 104
+0.5%
+Let p1 and p2 denote the proportions of eligible individuals who earned more
+than the national average income level $442.12. The true values of proportions
+are p1 = 49.0223% and p2 = 29.5152%. Let pdiff = p1 − p2, then the true
+difference in these two proportions is pdiff = 19.5071%. By the additivity of two
+independent normal distributions, naturally, we use the following differentially
+private CI:
+˜pdiff + z1−α/2
+�
+�V(˜pdiff),
+(22)
+where �V (·) denotes a private estimator of variance, ˜pdiff is defined as ˜p1 − ˜p2
+and
+�V(pdiff) = �V(p1) + �V(p2).
+In Figure 4, similar patterns are observed in this application as in the first.
+All CIs have empirical coverage around/above the nominal confidence level as
+in the simulation study in Section 5.1.2. The phenomenon of higher coverage
+is due to small ρ and effective clipping. When the range of stratum sizes is
+large (it is (50, 309214) in this application), that is, when the stratum sizes are
+very different, a large privacy budget ρ should be chosen. The choice of a small
+ρ harms the estimates of small-sized strata. We advise that the smallest ρ be
+chosen given the tolerance of uncertainty in terms of width and/or width ratio.
+For example, if the accuracy requirement is that the width should be under 0.05
+or WR under 5, then the best choices of ρ among the experiments in Figure 4b
+
+S. Lin et al./Differentially Private Confidence Intervals
+24
+10
+4
+10
+3
+10
+2
+10
+1
+0.80
+0.85
+0.90
+0.95
+1.00
+Empirical coverage
+Non-Private
+PopNz-PubSz
+StrNz-PubSz
+StrNz-PrivSz
+(a) Empirical coverage
+10
+4
+10
+3
+10
+2
+10
+1
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+Width
+0
+5
+10
+15
+20
+25
+30
+WR
+Non-Private
+PopNz-PubSz
+StrNz-PubSz
+StrNz-PrivSz
+(b) Width and ratio
+Fig 4: The empirical coverage and average width and width ratio of DP-CIs of the difference of
+the above-national-income-level proportions between black and white males with valid income
+values.
+are (1) 0.0001 for PopNz-PubSz, (2) 0.0018 for StrNz-PubSz, and (3) 0.0056 for
+StrNz-PrivSz.
+6. Discussion
+We have designed three algorithms to construct confidence intervals for the
+population proportion under stratified random sampling with zero concentrated
+differential privacy guarantees. We consider both the case where the sample sizes
+are public and the case where they are private information. Theoretical results
+including privacy guarantees and asymptotic properties are established. With
+proper conditions on the relation between the privacy budget and sample sizes,
+as stated in the theorems, the resulting confidence intervals will achieve the
+desired coverage asymptotically, and the width tends to be that of a non-private
+confidence interval when the sample sizes go to infinity.
+In the simulation studies and two applications, we have experimented with a
+wide range of privacy budgets under a variety of parameter setups. The three
+algorithms always perform well in terms of empirical coverage. The width and
+width ratio are in a reasonable range even under the strict regime where ρ =
+1/ maxh nh. Typically in practice, a constant between 0.001 to 10 is chosen to
+be the privacy budget. According to our experiments, with the choice of the
+smallest budget in this range, 0.001, the three algorithms still have fairly good
+results even when the smallest stratum has only a size 50 (as demonstrated in
+the second application).
+The comparative analysis of the three algorithms in Section 4.2 gives ac-
+tionable guidance to practitioners. When releasing the population proportion is
+the only goal and there are enough strata (such that Eq.(20) regarding sam-
+
+S. Lin et al./Differentially Private Confidence Intervals
+25
+ple weights is greater than 1), PopNz-PubSz is the better option. However, if
+stratum proportions should also be released or there are just a few strata, StrNz-
+PubSz is preferable. On the other hand, when the population proportion and
+sample sizes must be protected simultaneously, StrNz-PrivSz is the only algo-
+rithm presented in this paper. StrNz-PrivSz, compared to the case with public
+sample sizes, needs a larger budget to meet the same width requirement on
+account of the additional cost of protecting sample sizes.
+There are a few open questions worth considering for future research. In
+this paper, we discuss the case where the number of strata is fixed and the
+sample sizes tend to infinity, we use the finite-population CLTs for each stra-
+tum to derive an aggregated estimator. For the other case where the number
+of stratum tends to infinity instead of sample sizes, one can apply the Linde-
+berg–Lévy–Feller Theorem to obtain the asymptotic normality. More interest-
+ingly, we do not provide ‘PopNz-PrivSz’ – an analogous algorithm to PopNz-
+PubSz for the private sample sizes case. To protect both the population pro-
+portion and the sample sizes, the direct addition of noise to the non-private
+aggregated estimator is not plausible. One should consider more sophisticated
+mechanisms other than directly adding noise to the statistics. If ‘PopNz-PrivSz’
+were proposed, we shall expect it to yield a narrower confidence interval since
+we only need to publish the private population proportion without being able to
+provide private confidence intervals for stratum proportions at the same time.
+Another direction for future research would be optimal budget allocation. We do
+not experiment with different ways to divide the total budget in PopNz-PubSz
+or StrNz-PrivSz but simply split it up evenly. Budgeting for the composed ap-
+plication of the algorithms may also be of interest, like in Section 5.2.2 where
+we apply the algorithms twice for two independent populations. Lastly, one
+broad direction is to develop the differentially private versions for other alter-
+natives to the basic Wald interval, such as the Wilson Interval, Jeffreys interval,
+etc.(see [21] for a comparative summary of seven such types of confidence inter-
+vals for proportions). Many of these latter are specifically designed for the case
+of small sample sizes, which we do not consider here and for which we expect
+fundamentally different approaches to differential privacy likely to be necessary.
+Appendix A: Proofs
+A.1. Proof of Theorem 3.1
+Lemma A.1. Let X ∼ N(µ, σ2) and S = {µ − a ≤ X ≤ µ + a}. For any a > 0
+and an integer k ≥ 1, the conditional even moments
+E[(X − µ)2k | S] = σ2k(2k − 1)!! − O
+�
+e− a2
+2σ2 a2k−1�
+,
+(23)
+where the big-O hides a constant depending on σ and k.
+Proof. Without loss of generality, we assume µ = 0. We prove the lemma by
+
+S. Lin et al./Differentially Private Confidence Intervals
+26
+induction. Set k = 1, integrate by parts,
+E[X2IS] =
+� a
+−a
+x2
+1
+σ
+√
+2π e− x2
+2σ2 dx
+=
+σ
+√
+2π
+�
+−xe− x2
+2σ2 ��a
+−a +
+� a
+−a
+e− x2
+2σ2 dx
+�
+.
+Integrate by substitution, the integral in the second term becomes
+� a
+−a
+e− x2
+2σ2 dx = σ
+√
+2π erf
+�
+a
+σ
+√
+2
+�
+where
+erf(z) = σ
+� z
+0
+e−t2 dt
+is the error function. Then,
+E[X2IS] = σ2 erf
+�
+a
+σ
+√
+2
+�
+− O
+�
+e− a2
+2σ2 a
+�
+.
+Assuming
+E[X2kIS] = σ2k(2k − 1)!! erf
+�
+a
+σ
+√
+2
+�
+− O
+�
+e− a2
+2σ2 a2k−1�
+,
+(24)
+then integrate by parts for the k + 1 case,
+E[X2(k+1)IS] =
+� a
+−a
+x2k+2
+1
+σ
+√
+2π e− x2
+2σ2 dx
+=
+σ
+√
+2π
+� a
+−a
+x2k+1 · x
+σ2 e− x2
+2σ2 dx
+=
+σ
+√
+2π
+�
+−x2k+1e− x2
+2σ2 ��a
+−a + (2k + 1)
+� a
+−a
+x2ke− x2
+2σ2 dx
+�
+= σ2(2k + 1)E[X2kIS] − O
+�
+e− a2
+2σ2 a2k+1�
+.
+Plug in (24), we obtain
+E[X2(k+1)IS] = σ2k+2(2k + 1)!! erf
+�
+a
+σ
+√
+2
+�
+− O
+�
+e− a2
+2σ2 a2k+1�
+.
+So far we have proved (24). Note that
+Pr(S) =
+� a
+−a
+1
+σ
+√
+2π e− x2
+2σ2 dx = erf
+�
+a
+σ
+√
+2
+�
+,
+and that the image of erf(z) is between (−1, 1). Therefore,
+E[X2k | S] = E[X2kIS]/ Pr(S) = σ2k(2k − 1)!! − O
+�
+e− a2
+2σ2 a2k−1�
+.
+
+S. Lin et al./Differentially Private Confidence Intervals
+27
+Proof of (12) in Theorem 3.1. Consider the Taylor series of 1
+x at x = µ:
+1
+x =
+∞
+�
+j=0
+(−(x − µ))j
+µj+1
+= 1
+µ − x − µ
+µ2
++ (x − µ)2
+µ3
+− (x − µ)3
+µ4
++ . . .
+Let ym be the partial sum of the above series, i.e., ym(x) = �m
+k=0
+(−(x−µ))k
+µk+1
+.
+Then ym(x) converges to 1
+x in (0, 2µ) which contains [1, 2µ − 1]. Let
+g(x) =
+∞
+�
+k=0
+|x − µ|k
+µk+1
+=
+�
+1
+x,
+if 1 ≤ x ≤ µ
+1
+2µ−x
+if µ < x ≤ 2µ − 1
+Then g is integrable as
+� 2µ−1
+1
+|g(x)|dν =
+� µ
+1
+1
+xdν +
+� 2µ−1
+µ
+1
+2µ − xdν = 2
+� µ
+1
+1
+xdν < ∞,
+where dν = f(x)dx is induced by N(µ, σ2) conditional on event S. Note also
+that |ym(x)| ≤ g(x) for any naturals m and x ∈ [1, 2µ − 1]. By the dominated
+convergence theorem, the operations of limit and integral are exchangeable for
+ym(x).
+� 2µ−1
+1
+1
+xdν =
+� 2µ−1
+1
+lim
+m→∞ ym(x)dν
+= lim
+m→∞
+� 2µ−1
+1
+ym(x)dν
+= lim
+m→∞
+� 2µ−1
+1
+�
+�
+m
+�
+j=0
+(−(x − µ))j
+µj+1
+�
+� dν
+= lim
+m→∞
+�
+�
+m
+�
+j=0
+� 2µ−1
+1
+(−(x − µ))j
+µj+1
+dν
+�
+�
+(25)
+Then,
+E
+� 1
+X | S
+�
+=
+∞
+�
+j=0
+1
+µj+1 E
+�
+(−(X − µ))j | S
+�
+=
+∞
+�
+j=0
+1
+µ2j+1 E
+�
+(X − µ)2j | S
+�
+=
+k
+�
+j=0
+1
+µ2j+1 E
+�
+(X − µ)2j | S
+�
++ 1
+µ
+∞
+�
+j=k+1
+E
+��X − µ
+µ
+�2j
+| S
+�
+.
+(26)
+The second equality is because the odd moments are zero due to symmetry.
+
+S. Lin et al./Differentially Private Confidence Intervals
+28
+Note that given event S, | X−µ
+µ | ≤ µ−1
+µ
+< 1, then
+E
+��X − µ
+µ
+�2k+2
+| S
+�
+≤
+�µ − 1
+µ
+�2
+E
+��X − µ
+µ
+�2k
+| S
+�
+.
+(27)
+It follows that
+∞
+�
+j=k+1
+E
+��X − µ
+µ
+�2j
+| S
+�
+≤
+∞
+�
+j=0
+�µ − 1
+µ
+�2j
+E
+��X − µ
+µ
+�2k+2
+| S
+�
+=
+µ2
+2µ − 1E
+��X − µ
+µ
+�2k+2
+| S
+�
+= O
+�
+1
+µ2k+1
+�
+· E[(X − µ)2k+2 | S].
+Applying Lemma A.1, by the choice of a = µ − 1, (26) becomes
+E
+� 1
+X | S
+�
+= 1
+µ
+k
+�
+j=0
+(2j − 1)!!σ2j
+µ2j
++ O
+�σ2k+2
+µ2k+2
+�
+.
+Proof of (13) in Theorem 3.1. We conduct a similar procedure for the second
+moment of X | S. Based on the Taylor expansion
+1
+x2 =
+∞
+�
+j=0
+(j + 1)(−(x − µ))j
+µj+2
+= 1
+µ2 − 2(x − µ)
+µ3
++ 3(x − µ)2
+µ4
+− 4(x − µ)3
+µ5
++ · · · ,
+we have
+E
+� 1
+X2 | S
+�
+=
+∞
+�
+j=0
+j + 1
+µj+2 E
+�
+(−(X − µ))j | S
+�
+=
+∞
+�
+j=0
+2j + 1
+µ2j+2 E
+�
+(X − µ)2j | S
+�
+=
+k
+�
+j=0
+2j + 1
+µ2j+2 E
+�
+(X − µ)2j | S
+�
++ 1
+µ2
+∞
+�
+j=k+1
+(2j + 1)E
+��X − µ
+µ
+�2j
+| S
+�
+.
+(28)
+
+S. Lin et al./Differentially Private Confidence Intervals
+29
+Due to (27), it follows that
+∞
+�
+j=k+1
+(2j + 1)E
+��X − µ
+µ
+�2j
+| S
+�
+≤ E
+��X − µ
+µ
+�2k+2
+| S
+�
+·
+∞
+�
+j=0
+(2k + 3 + 2j)
+�µ − 1
+µ
+�2j
+= E
+��X − µ
+µ
+�2k+2
+| S
+�
+·
+�
+�(2k + 3)
+∞
+�
+j=0
+�µ − 1
+µ
+�2j
++ 2
+∞
+�
+j=1
+j
+�µ − 1
+µ
+�2j
+�
+�
+= E
+��X − µ
+µ
+�2k+2
+| S
+�
+·
+�(2k + 3)µ2
+2µ − 1
++ 2µ2(µ − 1)2
+(2µ − 1)2
+�
+= E
+��X − µ
+µ
+�2k+2
+| S
+�
+· O(µ2)
+= O
+� 1
+µ2k
+�
+· E[(X − µ)2k+2 | S],
+(29)
+where the term �∞
+j=1 j
+�
+µ−1
+µ
+�2j
+is a sum of an arithmetic–geometric sequence.
+By Lemma A.1, (28) becomes
+E
+� 1
+X2 | S
+�
+= 1
+µ2
+k
+�
+j=0
+(2j + 1)!!σ2j
+µ2j
++ O
+�σ2k+2
+µ2k+2
+�
+.
+(30)
+A.2. Proof of Theorem 4.1
+Proof for Algorithm 1. Under neighboring relation ∼ss, only one record changes
+within one stratum and sample sizes remain the same. Applying the Gaussian
+mechanism to each stratum at the level of ρ gives ρ-zCDP guarantee. By post-
+processing, the confidence interval is also ρ-zCDP.
+Proof for Algorithm 2. The sensitivities of ˆp and �
+Var(ˆp) are ∆p and ∆V , re-
+spectively. Applying the Gaussian mechanism, it follows that ˜p is ρ1-zCDP and
+�V is ρ2-zCDP. By basic composition, the confidence interval ˜p ± z1− α
+2
+�
+�V is
+(ρ1 + ρ2)-zCDP.
+Proof for Algorithm 3. By the Gaussian mechanism and the basic composition
+property of zCDP, we know that ˜ph is ρ-zCDP. Under neighboring relation
+∼r, only one record changes within one stratum. Then, by post-processing, the
+confidence interval is ρ-zCDP.
+
+S. Lin et al./Differentially Private Confidence Intervals
+30
+A.3. Proof of Theorem 4.2
+Before proving the theorem, we revisit the finite-population CLT first:
+Theorem A.2 (Theorem 1, [32]). Consider a finite population Π = {X1, ..., XN}
+of size N. Let µ be the population mean and ¯Xn be the mean of a simple ran-
+dom sample of size n from Π, and Var( ¯Xn) is the variance of ¯Xn. The finite
+population variance of Π is denoted by
+v =
+1
+N − 1
+N
+�
+i=1
+(Xi − µ)2.
+As N → ∞, if
+1
+min(n, N − n) · max1≤i≤N(Xi − µ)2
+v
+→ 0,
+(31)
+we have
+¯Xn − µ
+�
+Var( ¯Xn)
+d→ N(0, 1).
+(32)
+The variance of ¯Xn is determined by the population variance v which is
+unknown. Nevertheless, the sample variance �
+Var( ¯Xn) can be used to estimate
+v. To make sure the CLT still holds when substituting Var( ¯Xn) by �
+Var( ¯Xn),
+the consistency of �
+Var( ¯Xn) is crucial, as stated in the following lemma.
+Lemma A.3. Let �
+Var( ¯Xn) be the sample variance. �
+Var( ¯Xn) is an unbiased
+estimator for Var( ¯Xn). Moreover, under the condition in Theorem A.2, as N →
+∞,
+�
+Var( ¯Xn)/Var( ¯Xn)
+p→ 1.
+Now we prove Theorem 4.2:
+Proof of Theorem 4.2. It suffices to show
+˜ph−ph
+√
+�Vh
+d→ N(0, 1) for all h. By the
+finite-population CLT in Theorem A.2, we know
+ˆph − ph
+�
+Var(ˆph)
+d→ N(0, 1).
+Since ˜ph = ˆph + eh where eh ∼ N(0,
+1
+2ρn2
+h ), we have
+˜ph − ph
+�
+Var(˜ph)
+d→ N(0, 1)
+(33)
+where
+Var(˜ph) = Var(ˆph) +
+1
+2ρn2
+h
+.
+
+S. Lin et al./Differentially Private Confidence Intervals
+31
+Let
+�Vh =
+�Nh − nh
+Nh
+� ˜ph(1 − ˜ph) +
+1
+2ρn2
+h
+nh − 1
++
+1
+2ρn2
+h
+= �
+Var(ˆph) +
+�Nh − nn
+Nh
+� eh − 2˜pheh − e2
+h +
+1
+2ρn2
+h
+nh − 1
++
+1
+2ρn2
+h
+.
+(34)
+If
+1
+ρnh → 0, we have eh
+p→ 0 and then the second term of (34) is oP ( 1
+nh ).
+Note that �
+Var(ˆph) is of order
+1
+nh , and that by Lemma A.3, �
+Var(ˆph)
+p→ Var(ˆph).
+Therefore, �Vh
+p→ Var(ˆph) +
+1
+2ρn2
+h = Var(˜ph). Combining it with (33), we have
+˜ph − ph
+�
+�Vh
+d→ N(0, 1)
+(35)
+by Slutsky’s Theorem. Then, ˜p−p
+√
+�V
+d→ N(0, 1). Therefore, the confidence interval
+given by p ± z1−α/2
+�
+�V has asymptotic coverage level 1 − α.
+Note that, under the condition
+1
+ρnh → 0, it follows that
+1
+2ρn2
+h = o(Var(ˆph))
+and hence �Vh
+p→ Var(ˆph). Therefore, �V
+p→ Var(ˆp).
+A.4. Proof of Theorem 4.3
+Proof. Since
+ˆp−p
+√
+Var(ˆp)
+d→ N(0, 1) and ˜p = ˆp+N(0, ∆2
+p/2ρ1) with ∆p = maxh
+wh
+nh ,
+it follows that
+˜p − p
+�
+Var(˜p)
+d→ N(0, 1),
+and
+Var(˜p) = Var(ˆp) + ∆2
+p
+2ρ1
+.
+In Algorithm 2, we set
+�V = �
+Var(ˆp) + ∆2
+p
+2ρ1
++ eV ,
+(36)
+where eV ∼ N(0, ∆2
+V
+2ρ2 ) with ∆V = maxh
+�
+Ch
+nh
+�
+1 −
+1
+nh
+��
+and Ch = w2
+h
+Nh−nh
+Nh
+1
+nh−1.
+If
+1
+ρ2nh → 0 for all h, then eV = oP ( 1
+nh ) for all h. Since �
+Var(ˆp)
+d→ Var(ˆp) by
+finite-population CLT, we have �V
+d→ Var(˜p). Therefore, by Slutsky’s Theorem,
+˜p − p
+�
+�V
+d→ N(0, 1).
+(37)
+Then, the confidence interval given by p ± z1−α/2
+�
+�V has the asymptotic cov-
+erage level 1 − α. In fact, �V
+p→ Var(ˆp) if
+1
+ρ1nh → 0 for all h since
+∆2
+p
+2ρ1 =
+o(Var(ˆp)).
+
+S. Lin et al./Differentially Private Confidence Intervals
+32
+A.5. Proof of Theorem 4.4
+Proof. For ˜n ∼ N(n,
+1
+2ρ2 ), by Proposition 3.1, we derive the kth-order Taylor
+series of the conditional expectation of ˜p given S = {1 ≤ ˜n ≤ 2n − 1}:
+E (˜p | S) = p
+k
+�
+j=0
+(2j − 1)!!
+n2j(2ρ2)j + O
+�
+1
+n2k+1ρk+1
+2
+�
+.
+(38)
+For example, when k = 2,
+E (˜p | S) = p
+�
+1 +
+1
+2n2ρ2
++
+3
+4n4ρ2
+2
+�
++ O
+�
+1
+n5ρ3
+2
+�
+.
+(39)
+To obtain a Taylor expansion for the conditional variance, we plug
+E
+� 1
+˜n | S
+�
+= 1
+n
+k
+�
+j=0
+(2j − 1)!!
+n2j(2ρ2)j + O
+�
+1
+n2k+2ρk+1
+2
+�
+and
+E
+� 1
+˜n2 | S
+�
+= 1
+n2
+k
+�
+j=0
+(2j + 1)!!
+n2j(2ρ2)j + O
+�
+1
+n2k+2ρk+1
+2
+�
+into
+Var(˜p | S) = E(˜p2 | S) − (E(˜p | S))2 = E˜c2E
+� 1
+˜n2 | S
+�
+− (E(˜p | S))2,
+by which we derive a general expansion for the conditional variance:
+Var(˜p | S) = Var(ˆp)
+k
+�
+j=0
+(2j + 1)!!
+n2j(2ρ2)j + p2
+�
+�
+�
+k
+�
+j=0
+(2j + 1)!!
+n2j(2ρ2)j −
+�
+�
+k
+�
+j=0
+(2j − 1)!!
+n2j(2ρ2)j
+�
+�
+2�
+�
+�
++
+1
+2ρ1
+k
+�
+j=0
+(2j + 1)!!
+n2j+2(2ρ2)j + O
+�
+1
+n2kρk+1
+2
+�
++ O
+�
+1
+n2k+2ρ1ρk+1
+2
+�
+.
+(40)
+When k = 2,
+Var(˜p | S) = Var(ˆp)
+�
+1 +
+3
+2n2ρ2
++
+15
+4n4ρ2
+2
+�
++ p2
+�
+1
+2n2ρ2
++
+2
+n4ρ2
+2
+−
+6
+8n6ρ3
+2
+−
+9
+16n8ρ4
+2
+�
++
+1
+2ρ1
+� 1
+n2 +
+3
+2n4ρ2
++
+15
+4n6ρ2
+2
+�
++ O
+�
+1
+n4ρ3
+2
+�
++ O
+�
+1
+n6ρ1ρ3
+2
+�
+.
+(41)
+Based on Taylor expansion with k = 2 for both conditional mean and variance
+given in (39) and (41), under the condition
+1
+ρ1n = o(1) and
+1
+ρ2n = o(1), we have
+E(˜p | S) = p + o
+� 1
+n
+�
+
+S. Lin et al./Differentially Private Confidence Intervals
+33
+and
+Var(˜p | S) = Var(ˆp) + o
+� 1
+n
+�
+.
+Then, ˜p | S is asymptotically unbiased. Note that Var(ˆp) is of order 1
+n and thus
+˜p | S has a vanishing variance. Therefore, ˜p | S converges to p in probability.
+Note also that Pr(S) → 1 as n → ∞, then for any ϵ > 0,
+Pr(|˜p − p| > ϵ) = Pr(|˜p − p| > ϵ | S) + Pr(|˜p − p| > ϵ | Sc) → 0.
+That is, ˜p is a consistent estimator for p.
+A.6. Proof of Theorem 4.5
+To prove Theorem 4.5, we need the following theorem and lemmas.
+Theorem A.4 (Theorem 1, [17]). Let X be a normal random variable with
+positive mean µx, variance σ2
+x and coefficient of variation δx = σx/µx such that
+0 < δx < λ ≤ 1, where λ is a known constant. For every ϵ > 0, there exists
+γ(ϵ) ∈ (0,
+�
+λ2 − δ2x) and also a normal random variable Y independent of X,
+with positive mean µy, variance σ2
+y and coefficient of variation δy = σy/µy that
+satisfy the conditions,
+0 < δy ≤ γ(ϵ) ≤
+�
+λ2 − δ2x < λ
+(42)
+for which the following result holds. Any z that belongs to the interval
+I =
+�
+β − σz
+λ , β + σz
+λ
+�
+,
+where β = µx/µy, σz = β
+�
+δ2x + δ2y, satisfies that
+|G(z) − FZ(z)| < ϵ,
+where G(z) is the cumulative distribution function of N(β, σ2
+z), and FZ is that
+of Z = X/Y . Note that once a given Y fulfills the closeness between the cor-
+responding G to FZ , any other random variables with a smaller coefficient of
+variation will satisfy this result too.
+Lemma A.5. For a population of size N, let p be the true proportion in the
+population with the attribute of interest. Consider simple random sampling with
+sample size n. Let Z∗ ∼ N(p, V ) where V =
+�
+N−n
+N−1
+�
+p(1−p)
+n
++
+1
+2ρ1n2 +
+p2
+2ρ2n2 . If
+ρ1 = ω(1/n2) and ρ2 = ω(1/n), as N − n and n both tend to infinity, then for
+any z ∈ (0, 2p),
+|F˜p(z) − FZ∗(z)| → 0.
+(43)
+
+S. Lin et al./Differentially Private Confidence Intervals
+34
+Proof. By the CLT in Theorem A.2, we know that ˆp ∼ AN(p, Var(ˆp)). Recall
+that ˜c = nˆp + N(n,
+1
+2ρ1 ), then ˜c ∼ AN(np, n2 Var(ˆp) +
+1
+2ρ1 ).
+Let ˜X ∼ AN(np, n2 Var(ˆp)+ 1
+2ρ1 ) and X ∼ N(np, n2 Var(ˆp)+ 1
+2ρ1 ). Therefore,
+for any ϵ > 0, there exists some n0 = n0(ϵ) such that for any x and n > n0,
+|F ˜
+X(x) − FX(x)| < ϵ,
+(44)
+where F denotes the cumulative density function. Let Y ∼ N(n,
+1
+2ρ2 ), ˜Z = ˜X/Y
+and Z = X/Y , then
+F ˜
+Z(z) = Pr
+� ˜X
+Y < z
+�
+= Pr( ˜X < Y z) =
+� ∞
+−∞
+F ˜
+X(yz)fy(y)dy,
+where fy(y) is the density function of Y . From (44), |F ˜
+X(yx) − FX(yx)| < ϵ. It
+follows that,
+� ∞
+−∞
+(FX(yx) − ϵ)fy(y)dy <
+� ∞
+−∞
+F ˜
+X(yx)fy(y)dy <
+� ∞
+−∞
+(FX(yx) + ϵ)fy(y)dy,
+which is equivalent to
+����
+� ∞
+−∞
+F ˜
+X(yx)fy(y)dy −
+� ∞
+−∞
+FX(yx)fy(y)dy
+���� < ϵ,
+i.e.,
+|F ˜
+Z(z) − FZ(z)| < ϵ.
+(45)
+Let δx and δy be the coefficients of variation of X and Y , respectively, then
+δ2
+x = (Var(ˆp) +
+1
+2ρ1n2 )/p2 and δ2
+y =
+1
+2ρ2n2 . Under the condition
+1
+ρ1n = o(1),
+we have δ2
+x = O( 1
+n) since Var(ˆp) = O( 1
+n). Under the condition
+1
+ρ2n = o(1),
+we know δ2
+y = o( 1
+n) and then δy = o(δx). When n is sufficiently large, δy is
+sufficiently small. Let λ =
+�
+δ2x + 2δ2y and FZ∗(z) be the distribution function of
+Z∗ ∼ N(p, Var(ˆp)+
+1
+2ρ1n2 +
+p2
+2ρ2n2 ). By Lemma A.4, for a normal random variable
+Y independent of X, with small enough δy, the condition (42) is satisfied and
+we have
+|FZ(z) − FZ∗(z)| < ϵ,
+(46)
+for any z ∈ I =
+�
+p − σz∗
+λ , p + σz∗
+λ
+�
+where σz∗ = p
+�
+δ2x + δ2y. Hence, for z ∈ I,
+|F ˜
+Z(z) − FZ∗(z)| < |F ˜
+Z(z) − FZ(z)| + |FZ(z) − FZ∗(z)| < 2ϵ.
+(47)
+Note also that as n → ∞, σz∗
+λ
+→ p, and the limit of I is (0, 2p).
+So far, we have shown that as n goes to infinity, under the conditions
+1
+ρ1n =
+o(1) and
+1
+ρ2n = o(1), for z ∈ Ih,
+|F ˜
+Z(z) − FZ∗(z)| → 0.
+(48)
+
+S. Lin et al./Differentially Private Confidence Intervals
+35
+Lemma A.6. Let Z1, ..., ZH and Z∗
+1, ..., Z∗
+H be independent continuous random
+variables which depend on n. Let F denote the distribution function. As n → ∞,
+if
+|FZh(z) − FZ∗
+h(z)| → 0
+holds for any h = 1, ..., H and z in an interval (ah, bh) and Pr(Zh ∈ (ah, bh)) →
+1, Pr(Z∗
+h ∈ (ah, bh)) → 1. Then,
+���F�H
+h=1 chZh(z) − F�H
+h=1 chZ∗
+h(z)
+��� → 0
+for any z ∈
+��H
+h=1 chah, �H
+h=1 chbh
+�
+, where ch’s are constants.
+Proof. It suffices to show that, for any z ∈ (a1c1 + a2c2, b1c1 + b2c2),
+��Fc1Z1+c2Z2(z) − Fc1Z∗
+1 +c2Z∗
+2 (z)
+�� → 0
+as n → ∞. We have
+Fc1Z1+c2Z2(z)
+= Pr(c1Z1 + c2Z2 < z)
+= Pr
+�
+Z1 < z − c2Z2
+c1
+�
+=
+�
+R
+FZ1
+�z − c2x
+c1
+�
+fZ2(x)dx
+=
+�
+R
+�
+FZ1
+�z − c2x
+c1
+�
+− FZ∗
+1
+�z − c2x
+c1
+��
+fZ2(x)dx +
+�
+R
+FZ∗
+1
+�z − c2x
+c1
+�
+fZ2(x)dx
+=
+�
+R
+�
+FZ1
+�z − c2x
+c1
+�
+− FZ∗
+1
+�z − c2x
+c1
+��
+fZ2(x)dx + Fc1Z∗
+1 +c2Z2(z).
+(49)
+When a1 < (z − c2x)/c1 < b1, we know
+����FZ1
+�z − c2x
+c1
+�
+− FZ∗
+1
+�z − c2x
+c1
+����� → 0.
+(50)
+Since FZ1(b1) − FZ1(a1) → 1 and FZ∗
+1 (b1) − FZ∗
+1 (a1) → 1, for any a < a1,
+it holds that FZ1(a) → 0 and FZ∗
+1 (a) → 0, and for any b > b1, FZ1(b) → 1
+and FZ∗
+1 (b) → 1. Thus, (50) also holds when (z − c2x)/c1 is outside (a1, b1).
+Therefore, the first term of the right-hand side of (49) converges to 0. Then
+��Fc1Z1+c2Z2(z) − Fc1Z∗
+1 +c2Z2(z)
+�� → 0.
+Similarly, we have
+��Fc1Z∗
+1 +c2Z2(z) − Fc1Z∗
+1 +c2Z∗
+2 (z)
+�� → 0.
+By the triangle inequality,
+��Fc1Z1+c2Z2(z) − Fc1Z∗
+1 +c2Z∗
+2 (z)
+�� → 0.
+
+S. Lin et al./Differentially Private Confidence Intervals
+36
+Proof of Theorem 4.5. By Lemma A.5, for each stratum, under the conditions
+ρ1 = ω(1/nh) and ρ2 = ω(1/nh), the distribution function of ˜ph converges to
+that of N(ph, Vh) in the interval (0, 2ph) where
+Vh =
+�Nh − nh
+Nh − 1
+� ph(1 − ph)
+nh
++
+1
+2ρ1n2
+h
++
+p2
+h
+2ρ2n2
+h
+.
+(51)
+Let p∗ ∼ N(p, V ) where V = �H
+h=1 w2
+hVh. By Lemma A.6, in the interval (0, 2p),
+we have
+|F˜p(z) − Fp∗(z)| → 0.
+(52)
+where F˜p denotes the distribution function of ˜p designed in Algorithm 3 and
+Fp∗ is the distribution function of p∗.
+Let L = p − z1−α/2
+√
+V , U = p + z1−α/2
+√
+V , ˜L = p − z1−α/2
+�
+�V and ˜U =
+p + z1−α/2
+�
+�V . Note that L and U are constants whereas ˜L and ˜U are random
+variables. Provided that nh’s are sufficiently large, U and L lie in the interval
+where the following hold due to (52),
+|F˜p(U) − Fp∗(U)| → 0
+(53)
+and
+|F˜p(L) − Fp∗(L)| → 0.
+(54)
+On the other hand, by Theorems 3.1 and 4.4, we know that ˜ph
+p→ ph and
+1
+˜nh
+p→
+1
+nh under the conditions ρ1 = ω(1/nh) and ρ2 = ω(1/nh). By the continuous
+mapping theorem, �Vh
+p→ Vh as nh → ∞, and, hence, �V
+p→ V . Therefore, ˜U
+p→ U
+and ˜L
+p→ L. Since F˜p is continuous, we have
+|F˜p( ˜U) − F˜p(U)|
+p→ 0
+(55)
+and
+|F˜p(˜L) − F˜p(L)|
+p→ 0.
+(56)
+Therefore,
+Pr
+�
+p ∈
+�
+˜p − z1−α/2
+�
+�V , ˜p + z1−α/2
+�
+�V
+��
+= Pr
+�
+p − z1−α/2
+�
+�V < ˜p < p + z1−α/2
+�
+�V
+�
+=
+�
+F˜p( ˜U) − F˜p(U)
+�
++ (F˜p(U) − Fp∗(U))
+−
+�
+F˜p(˜L) − F˜p(L)
+�
+− (F˜p(L) − Fp∗(L)) + (Fp∗(U) − Fp∗(L)) .
+Putting together (53) through (56) and Fp∗(U) − Fp∗(L) = 1 − α, we have
+lim
+n→∞ Pr
+�
+p ∈
+�
+˜p − z1−α/2
+�
+�V , ˜p + z1−α/2
+�
+�V
+��
+→ 1 − α.
+Since
+1
+˜nh
+p→
+1
+nh , under the conditions ρ1 = ω(1/nh) and ρ2 = ω(1/nh), it
+holds that
+1
+2ρ1˜n2
+h = oP ( 1
+nh ) and
+˜p2
+h
+2ρ2˜n2
+h = oP ( 1
+nh ). Then �Vh
+p→ Var(ˆph). Therefore,
+�V
+p→ Var(ˆp).
+
+S. Lin et al./Differentially Private Confidence Intervals
+37
+Acknowledgments
+We are grateful for helpful conversations with and comments from (in no partic-
+ular order) Rolando Rodriguez, Brian Finley, Jörg Drechsler, Gary Benedetto,
+Michael Freiman, and Justin Doty. This project was funded by the U.S. Census
+Bureau cooperative agreements CB20ADR0160001.
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diff --git a/btE_T4oBgHgl3EQfzhyy/content/tmp_files/load_file.txt b/btE_T4oBgHgl3EQfzhyy/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf,len=2099
+page_content='Differentially Private Confidence  Intervals for Proportions under  Stratified Random Sampling∗  Shurong Lin  Department of Mathematics and Statistics, Boston University, Boston, MA  e-mail: shrlin@bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='edu  Mark Bun and Marco Gaboardi  Department of Computer Science, Boston University, Boston, MA  e-mail: mbun@bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' gaboardi@bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='edu  Eric D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Kolaczyk  Department of Mathematics and Statistics, McGill University, Canada  e-mail: eric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='kolaczyk@mcgill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='ca  Adam Smith  Department of Computer Science, Boston University, Boston, MA  e-mail: ads22@bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='edu  Abstract: Confidence intervals are a fundamental tool for quantifying the  uncertainty of parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' With the increase of data privacy  awareness, developing a private version of confidence intervals has gained  growing attention from both statisticians and computer scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Differ-  ential privacy is a state-of-the-art framework for analyzing privacy loss  when releasing statistics computed from sensitive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Recent work has  been done around differentially private confidence intervals, yet to the best  of our knowledge, rigorous methodologies on differentially private confi-  dence intervals in the context of survey sampling have not been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In  this paper, we propose three differentially private algorithms for construct-  ing confidence intervals for proportions under stratified random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  We articulate two variants of differential privacy that make sense for data  from stratified sampling designs, analyzing each of our algorithms within  one of these two variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We establish analytical privacy guarantees and  asymptotic properties of the estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In addition, we conduct simula-  tion studies to evaluate the proposed private confidence intervals, and two  applications to the 1940 Census data are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  MSC2020 subject classifications: Primary 68P27, 62G15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' secondary  62Dxx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Keywords and phrases: Differential privacy, confidence intervals, strat-  ified sampling, population proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  ∗The research presented in this paper was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Census Bureau Cooper-  ative Agreement CB20ADR0160001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='08324v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='ME]  19 Jan 2023  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  2  Contents  1  Introduction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content='1  Privacy and Coverage Guarantees .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content='1  Extrinsic Variances .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2  Comparing with Non-Private CI: One Stratum Case .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content='  16  5  Numerical Results  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content='1  Simulations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content='2  Varying Key Parameters .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  18  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2  Applications .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  20  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1  Confidence Intervals for the Unemployment Rate .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content='  21  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2  Confidence Intervals for the Difference in Income Level .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  22  6  Discussion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content='  24  A Proofs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content='  25  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content='2 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content='3 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content='4 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  32  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='6 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content='  33  Acknowledgments  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content='  37  References .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content='  37  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Introduction  With the increase of privacy awareness in the modern information era, estab-  lishing privacy-preserving methodologies for statistics and machine learning has  become an active research area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Differential privacy, a state-of-the-art privacy  protection technique [13], is considered a gold standard for rigorous privacy  guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Not only has it drawn significant attention in academia [14, 15], but  also it has been deployed by governments, firms, and other data agencies, such  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  3  as the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Census Bureau [1], Google [19], Microsoft [7], and Apple [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Re-  cently, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Census Bureau released a new demonstration of its differentially  private Disclosure Avoidance System (DAS) for the 2020 Census [4, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Simply put, a differentially private mechanism guarantees privacy via care-  fully injecting random noise (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=', drawn from the Laplace or Gaussian dis-  tribution) into the data analysis or modeling procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' At the intersection  of differential privacy and statistics, both statisticians and computer scientists  are working on developing private versions of statistical inference procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Early work discussing differential privacy in the context of statistics includes  [16, 12, 41, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' More recent work has explored statistical inference and estima-  tion under the constraint of differential privacy [5, 28, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  As one of the most fundamental tools for statistical inference, confidence in-  tervals are ubiquitous in quantifying the uncertainty of parameters of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  In this paper, we propose three differentially private algorithms for constructing  confidence intervals for the population proportion under stratified random sam-  pling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' To the best of our knowledge, our work is the first to establish rigorous  methodologies on differentially private confidence intervals in the context of sur-  vey sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Survey sampling is an important area in statistics that involves  selecting a sample of individuals from a target population to conduct a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  It provides timely and cost-efficient estimates of population characteristics of  interest and is widely used in broad-scale data gatherings, such as the American  Community Survey (ACS), the Survey of Income and Program Participation  (SIPP), and the Current Population Survey (CPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  This paper provides the first study of differentially private confidence intervals  for data from stratified sampling designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Specifically:  We articulate two specific variants of differential privacy that are appropri-  ate for data from stratified sampling designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In addition to the standard  notion of differential privacy, we also consider settings in which the sample  stratum sample sizes are fixed and public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' This latter setting allows for  simpler algorithms and tighter confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  We give methods to propagate the uncertainty due to the application of  differentially private mechanisms (adding random noise) into the construc-  tion of confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' A necessary bias correction is made to achieve  (asymptotic) unbiased variance estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Central limit theorem (CLT)-  type statements are provided to guarantee the confidence level asymptot-  ically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  We assess the performance of the proposed algorithms both in theory and  through simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The theoretical analysis comparing the non-private  and private methods gives practitioners a sense of how the algorithms  would work prior to applying them to real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  To support the theoretical analysis of one of the algorithms, we study  the behavior of a reciprocal normal variable in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' A general form of  the Taylor expansion (for conditional moments) is obtained to solve the  problem of the non-existence of moments due to its heavy-tailed nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We briefly discuss the existing work on  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  4  differentially private confidence intervals in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Section 2 provides pre-  liminaries on confidence intervals of population proportions and differential pri-  vacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In Section 3, we discuss the methodology of three differentially private  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Section 4 provides theorems on both privacy and asymptotic cov-  erage guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Numerical experiments, including simulation studies and two  applications to the 1940 Census data, are conducted in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Section 6  discusses the implications of our methods and general research directions on  differentially private confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Related Work  Differentially private confidence intervals have recently been studied for other  settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Some studied differentially private confidence intervals for the popula-  tion mean of normally distributed data [26, 10, 22] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Other tasks on confidence  intervals have also been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Drechsler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' designed and evaluated sev-  eral strategies to obtain differentially private confidence intervals for the median  [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' provided confidence intervals for differentially private models  trained with objective or output perturbation algorithms [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Besides, bootstrapping is a popular technique for constructing more general  differentially private confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Ferrando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' proposed a general-  purpose approach to construct confidence intervals for a population parameter  [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' A numerical confidence interval for the difference of mean was provided [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  The nonparametric bootstrap was considered in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Covington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' described a  method to induce distributions of mean and covariance estimates via the bag of  little bootstraps (BLB), which can further produce private confidence intervals  [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Our work is the first to study design-based approaches to sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In a  design-based setting, the values of interest are viewed as fixed but unknown con-  stants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Randomness only comes from the sampling design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The selection proba-  bilities introduced with the design will be used for estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' On the contrary,  in a model-based setting, a parametric model is postulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Design-based meth-  ods in sampling can be more reassuring than model-based approaches for the  reason that in many cases, no accurate prior information about the population  distribution is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' More discussion of design-based versus model-based  approaches in sampling can be found in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Preliminaries  In this section, we provide some preliminaries on population proportion estima-  tion and differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We first review the classic Wald confidence interval  for the population proportion under stratified random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Then we define  a notion of differential privacy specifically for stratified data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Some properties  of differential privacy are revisited in preparation for the theoretical analysis in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Confidence Intervals for the Population Proportion  In stratified random sampling, a population of N individuals is partitioned into  H strata, where stratum h has Nh individuals, and simple random sampling  of nh individuals is conducted within each stratum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' When the objective is to  estimate the proportion of individuals having some attribute in the population,  one can estimate it by the sample proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let yhi be the corresponding  indicator variable: yhi = 1 when the individual i in stratum h has the attribute  and yhi = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' One can estimate the population proportion  p = 1  N  H  �  h=1  Nh  �  i=1  yhi  by the sample proportion  ˆp = 1  N  H  �  h=1  Nh  nh  nh  �  i=1  yhi =  H  �  h=1  whˆph  where wh  def  = Nh  N  and ˆph  def  =  1  nh  nh  �  i=1  yhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Its variance Var(ˆp) =  H  �  h=1  w2  hVar(ˆph),  where  Var(ˆph) =  �Nh − nh  Nh − 1  � ph(1 − ph)  nh  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  An unbiased estimator for Var(ˆph) is given by the sample variance in the stratum  �  Var(ˆph) =  �Nh − nh  Nh  � ˆph(1 − ˆph)  nh − 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (1)  Then an unbiased estimator for Var(ˆp) is given by �  Var(ˆp) =  H  �  h=1  w2  h�  Var(ˆph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' An  approximate 100%(1−α) confidence interval for p based on a normal distribution  can be constructed:  ˆp ± z1− α  2  �  �  Var(ˆp),  (2)  where z1− α  2 denotes the 1 − α  2 quantile of standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The  normal approximation is useful when all the sample sizes are moderate to large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Otherwise, the t distribution with appropriate degrees of freedom is typically  used to replace the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For small sample sizes, various  specialized confidence intervals have been developed [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Differential Privacy  Differential privacy ensures that the output of data analysis or a query does not  differ much when the data set is changed by one record, such that one can not  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  6  infer the presence or absence of any individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' If two data sets x, x′ differ by  one record, we say that x, x′ are adjacent or neighboring, written as x ∼ x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  The definition of differential privacy depends on how we define adjacency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For  the partitioned data under stratified sampling, there are two ways to change a  record: (1) one way is to substitute one record within a stratum, with all the  stratum sample sizes fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We refer to this adjacency relation as “substitute-one  relation within a stratum” and denote it by ∼ss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' This relation corresponds to the  case where the sample sizes are public and fixed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' (2) another way to obtain an  adjacent data set is to remove or add one record from one stratum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' we refer to  the corresponding relation as, which we call “remove/add-one relation”, denoted  by ∼r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In this case, one of the stratum sample sizes will change by one, as will  the overall sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' This relation corresponds to the case where the sample  sizes are private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Under either adjacency relation, we can define zero-concentrated differentially  private (ρ-zCDP) as in [3]:  Definition 1 (ρ-zCDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let X ∗ denote the space of the input data with an  arbitrary finite dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Under the adjacency relation ∼, a randomized algo-  rithm M : X ∗ → Y is ρ-zero-concentrated-differentially private (ρ-zCDP) if, for  every pair of adjacent data sets x ∼ x′ ∈ X ∗, and all α ∈ (1, ∞),  Dα(M(x)∥M(x′)) ≤ ρα,  where Dα(M(x)∥M(x′)) is the α-Rényi divergence [39] between the distribution  of M(x) and the distribution of M(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  The parameter ρ indicates the privacy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' A smaller ρ means a more re-  strictive distance control between M(x) and M(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' As a result, the outputs on  two adjacent data sets are harder to tell apart and the algorithm achieves higher  privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We call ρ the privacy budget when we deliberately design an algorithm  to satisfy ρ-zCDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Depending on the adjacency notion, there are two types of differential pri-  vacy: bounded and unbounded differential privacy [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Definition 1 under the  “remove/add-one relation” corresponds to the standard unbounded differential  privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The sample size of the data set changes when one record is added or  removed to obtain an adjacent data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' With “substitute-one within a stratum”  relation ∼ss, the resulting notion corresponds to the bounded version of differ-  ential privacy where the sizes of two adjacent data sets are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' But it  is somewhat different from the standard notion of bounded differential privacy  in that for the latter, substitutions can happen across strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' That is, we can  change both the record and the stratum it is part of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  In the literature on differential privacy, (ϵ, δ)-DP ([14] Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='4) is con-  sidered the classic notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We consider ρ-zCDP because (1) ρ-zCDP implies  (ϵ, δ)-DP ([3] Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='3), (2) the application of the Gaussian mechanism  to achieve zCDP facilitates the theoretical analyses, and (3) the composition of  ρ-zCDP is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The Gaussian mechanism is a prototypical exam-  ple of a mechanism satisfying zCDP, which perturbs the true values by adding  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  7  Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We provide the Gaussian mechanism and the composition and  post-processing properties of ρ-zCDP in the following propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' All proposi-  tions can be found in [3] and will be used in the analyses of privacy guarantees  in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Definition 2 (Sensitivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' A function q: X ∗ → R has sensitivity ∆ if for all  pairs of adjacent data sets x ∼ x′ ∈ X ∗, we have |q(x) − q(x′)| ≤ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Proposition 1 (Gaussian Mechanism of ρ-zCDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let q : X ∗ → R be a  sensitivity-∆ query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Consider the mechanism M : X ∗ → R that on input x,  releases a sample from N(q(x), ∆2/(2ρ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Then, M satisfies ρ-zCDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  A smaller budget leads to larger noise added to the query on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Con-  sequently, the output is more private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Proposition 2 (Composition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let M : X ∗ → Y and M ′ : X ∗ → Z be two  randomized algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Suppose M satisfies ρ-zCDP and M ′ satisfies ρ′-zCDP,  then algorithm M ′′ = (M, M ′) : X ∗ → Y × Z is (ρ + ρ′)-zCDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Proposition 3 (Post-processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let M : X ∗ → Y and f : Y → Z be  randomized algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' If M is ρ-zCDP, then so is the composed algorithm  M ′ = f ◦ M : X ∗ → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Methodology  Our goal is to release a ρ-zCDP confidence interval for the population proportion  p under stratified random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' To construct a confidence interval as in (2),  we need to estimate both p and the variance of the estimator privately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Recall  that the non-private estimator of population proportion is given by the sample  proportion  ˆp =  H  �  h=1  whˆph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  We assume the stratum sizes Nh are all public and fixed, thus so are wh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' To get  a private estimator for p, denoted by ˜p, we can add noise at the level of either  the (non-private) estimator ˆp or the estimator ˆph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' With ˜p, we further devise a  private estimator for Var(˜p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Based on this idea, two algorithms for the case of  public sample sizes are designed by adding noise at the stratum or population  level in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2, we additionally propose adding noise at the  stratum level when sample sizes are private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Throughout the paper, the accents  ˆ· and ˜· are used to represent non-private and private estimators, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Estimating with Public Sample Sizes  When sample sizes nh are fixed, there are two natural strategies for perturbing  ˆp: add Gaussian noise to (1) the stratum-level statistics ˆph’s, or (2) the overall  statistic ˆp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Adding noise to the ˆph’s has the advantage of producing private  estimators for stratum-level proportions simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Adding Noise at the Stratum Level  We apply the Gaussian mechanism to each stratum to derive a private estimator  ˜ph  def  = ˆph + eh where eh is the Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Then the private estimator for  the population proportion is  ˜p  def  =  H  �  h=1  wh˜ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  As a result, the variance of ˜p consists of both the intrinsic variances of estimating  ph’s by ˆph’s and the additional variability from added noise:  Var(˜p) =  H  �  h=1  w2  h  �  Var(ˆph) + w2  h Var(eh)  �  (3)  where Var(eh), h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=', H are public since they do not depend on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  To obtain a private confidence interval for ˆp, we will need to privately estimate  Var(ˆph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Note that the added noise biases the term ˆph(1− ˆph) in the non-private  estimate of Var(ˆph) in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' More specifically, Ee[˜ph(1−˜ph)] = ˆph(1−ˆph)−Var(eh)  where Ee denotes the expectation taken on the randomness of the added noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Then a private unbiased estimator of Var(ˆph) in the right-hand side in (3) is  given by  �  Var(ˆph)  def  =  �Nh − nh  Nh  � ˜ph(1 − ˜ph) + Var(eh)  nh − 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (4)  To estimate Var(˜p), we set  �  Var(˜p)  def  =  H  �  h=1  w2  h  �  �  Var(ˆph) + Var(eh)  �  This approach is formulated in Algorithm 1 which we call StrNz-PubSz (adding  noise at the stratum level with public sample sizes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The theoretical results re-  garding privacy level and asymptotic coverage are provided in Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1 and  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Adding Noise at the Population Level  An alternative strategy is to directly add noise to the non-private estimator of  p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=', ˆp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The sensitivity of ˆp is  ∆p = max  h  wh  nh  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Since wh and nh are public, ∆p can be made public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We set ˜p = ˆp + e where  e is the Gaussian noise with standard deviation proportional to ∆p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Then, the  variance of ˜p becomes  Var(˜p) = Var(ˆp) + Var(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (5)  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  9  Algorithm 1 Adding noise at the stratum level with public sample sizes, StrNz-  PubSz  Input: ˆph, nh, Nh, wh, ρ, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Output: ρ-zCDP (1 − α) CI for the population proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  1: for h = 1 to H do  2:  Generate Gaussian noise eh ∼ N(0,  1  2ρn2  h ), and let  ˜ph ← ˆph + eh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  3:  Estimate Var(˜ph) by  �Vh ←  � Nh − nh  Nh  � ˜ph(1 − ˜ph) +  1  2ρn2  h  nh − 1  +  1  2ρn2  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  4: end for  5: Estimate p by ˜p ←  H  �  h=1  wh ˜ph and Var(˜p) by �V ←  H  �  h=1  w2  h �Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  6: Return  ˜p ± z1−α/2  �  �V ,  where z1−α/2 is the (1 − α/2)-quantile of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Recall that  �  Var(ˆp) =  H  �  h=1  w2  h  �Nh − nh  Nh  � ˆph(1 − ˆph)  nh − 1  is an unbiased estimator for Var(ˆp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' To get a private estimator for Var(˜p),  we again apply the Gaussian mechanism to �  Var(ˆp) based on the sensitivity  of Var(ˆp):  ∆V = max  h  �Ch  nh  �  1 − 1  nh  ��  ,  where Ch = w2  h  Nh−nh  Nh  1  nh−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Since we apply the Gaussian mechanism twice, the total privacy budget  should be divided into two parts: ρ = ρ1 + ρ2 to spend on adding noise to ˆp  and Var(ˆp), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The composition property (Proposition 2) ensures the  total privacy level is ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The resulting algorithm, PopNz-PubSz, is presented  in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' When there are multiple strata with similar sampling rates, Algo-  rithm 1 yields a wider confidence interval for p than Algorithm 2 does, given the  same privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' However, Algorithm 1 additionally produces private confi-  dence intervals for ˆph which may be of interest for release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1, we  compare the two algorithms quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Proportions are always between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' One can post-process  proportion estimates (˜ph in Algorithm 1 and ˜p in Algorithm 2) by clipping them  onto interval [0,1] without undermining privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' When the privacy budget is  very small, the noisy proportion estimates are likely to lie outside [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Thus,  clipping moves the confidence interval toward the truth and a higher coverage  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  10  Algorithm 2 Adding noise at the population level with public sample sizes,  PopNz-PubSz  Input: ˆp, ˆph, nh, Nh, wh, ρ, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Output: A ρ-zCDP (1 − α) CI for Population Proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  1: Split the budget ρ = ρ1 + ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let ρ1 = ρ2 if not specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  2: Generate noise e ∼ N(0,  ∆2  p  2ρ1 ) where ∆p = maxh  wh  nh and let  ˜p ← ˆp + e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  3: Generate noise eV  ∼  N(0, ∆2  V  2ρ2 ) where ∆V  =  maxh  �  Ch  nh  �  1 −  1  nh  ��  and Ch  =  w2  h  Nh−nh  Nh  1  nh−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let  �V ←  H  �  h=1  w2  h  � Nh − nh  Nh  � ˆph(1 − ˆph)  nh − 1  + ∆2  p  2ρ1  + eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  4: Return  ˜p ± z1−α/2  �  �V ,  where z1−α/2 is the (1 − α/2)-quantile of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  rate will be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' With a moderate or large budget, clipping does not make  a noticeable difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Lastly, one can always clip the output confidence intervals onto [0,1] without  privacy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Estimating with Private Sample Sizes  When sample sizes are public information, keeping the proportions private is es-  sentially protecting only the numerator (the counts of individuals with y = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  In some cases where subpopulation proportions also need to be estimated, Al-  gorithms 1 and 2 with public sample sizes can lead to privacy leakage since the  counts become the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Therefore, a method of constructing confidence  intervals for proportions to keep both the counts and sample sizes private is nec-  essary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We protect the sample sizes by adding noise to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' As a result, sample  sizes are not fixed and therefore we need the unbounded notion of differential  privacy with the adjacency relation ∼r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In the following, we extend Algorithm  1 to serve the needs of privacy protection of sample sizes by adding noise at the  stratum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' (It is not obvious how to extend Algorithm 2, which adds noise  at the population level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' It requires more sophisticated mechanisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' we briefly  discuss in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=')  To begin, we first consider the setting of simple random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The idea  is to add independent Gaussian noise to both the numerator and denominator  for each stratum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For ease of notation, we first consider a single stratum with  count c = �n  i=1 xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We know  c ∼ Hypergeometric(N, K, n),  where K is the total number of individuals with the attribute of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The  count c has mean n K  N = np and variance n K  N  N−K  N  N−n  N−1 = n2 Var(ˆp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' By applying  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  11  the Gaussian mechanism to c and n with privacy budgets ρ1 and ρ2, respectively,  we have private count ˜c and sample size ˜n:  ˜c | c ∼ N(c, 1  2ρ1  )  and  ˜n ∼ N(n, 1  2ρ2  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  The unconditional mean and variance for c are  E(˜c) = E[E(˜c | c)] = E(c) = np  and  Var(˜c) = E Var(˜c | c) + Var E(˜c | c) =  1  2ρ1  + n2 Var(ˆp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (6)  By the composition property of zCDP, we get a private estimator for proportion  p, denoted by ˜p, with privacy level ρ = ρ1 + ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Since ˜c and ˜n are independent  variables, in principle,  E(˜p) = E  � ˜c  ˜n  �  = E(˜c)E  � 1  ˜n  �  ,  (7)  and  Var(˜p) = E  � ˜c  ˜n  �2  −  �  E  � ˜c  ˜n  ��2  = E˜c2E  � 1  ˜n2  �  − (E˜c)2  �  E 1  ˜n  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (8)  However, the moments of 1  ˜n do not exist, thus neither do those of ˜p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Generally  speaking, the ratio of two independent normal random variables has a heavy-  tailed distribution with no moments [33, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The shape of the distribution could  be unimodal, bimodal, symmetric, or asymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' It is primarily determined by  the coefficient of variation of the denominator variable, CV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' When CV is suffi-  ciently small, a normal distribution approximation can be effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' It has been  shown theoretically that a normal distribution can be arbitrarily close to the  ratio variable in an interval centered at the ratio of means of two normal ran-  dom variables [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Experiments have provided guidelines for when the normal  approximation can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For example, a simple rule is that the approxima-  tion is reasonable whenever CV is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Other practical rules are  mentioned in [25, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  We take advantage of the normal approximation to construct a ρ-zCDP con-  fidence interval for the proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We present the following estimation strategy  in Algorithm 3, StrNz-PrivSz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In the algorithm, we clip ˜nh in (9) to ensure the  denominator is not too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Otherwise, the ratio can be arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Such  a post-processing step preserves the same privacy guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For the theoretical  analysis, we do not clip ˜nh, but instead, we consider the ratio variable ˜ch/˜nh  given the event Sh = {1 ≤ ˜nh ≤ 2nh − 1} (a symmetric area around the mean  of ˜nh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' It is more convenient for the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The asymptotic behaviors of ˜ph  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  12  Algorithm 3 Adding noise at the stratum level with private sample sizes,  StrNz-PrivSz  Input: Nh, wh, nh, ch, ρ, α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Output: A ρ-zCDP (1 − α) CI for the population proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  1: Split the budget ρ = ρ1 + ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let ρ1 = ρ2 if not specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  2: for h = 1 to H do  3:  Generate e(1)  h  ∼ N(0,  1  2ρ1 ) and e(2)  h  ∼ N(0,  1  2ρ2 ), and let  �  ˜ch ← ch + e(1)  h  ˜nh ← max(nh + e(2)  h , 2)  (9)  4:  Let  ˜ph ← ˜ch  ˜nh  (10)  5:  Let  �Vh ←  � Nh − ˜nh  Nh − 1  � ˜ph(1 − ˜ph)  ˜nh  +  1  2ρ1˜n2  h  +  ˜p2  h  2ρ2˜n2  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (11)  6: end for  7: Estimate p by ˜p ←  H  �  h=1  wh ˜ph and Var(˜p) by �V ←  H  �  h=1  w2  h �Vh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  8: Return  ˜p ± z1−α/2  �  �V ,  where z1−α/2 is the (1 − α/2)-quantile of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  in the algorithm and ˜ch/˜nh | Sh are essentially the same since Pr(˜nh ≥ 2) → 1  and Pr(Sh) → 1 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We will see the private estimator of the variance of  ˜ph we derive from the analysis of ˜ch/˜nh | Sh works well and the algorithm does  achieve the desired coverage level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  We consider the ratio of two independent normal variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' By independence,  what remains unclear is the behavior of the reciprocal of a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (We should mention that the Inverse Gaussian distribution is a different dis-  tribution than the reciprocal distribution we discuss here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=') In Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1, we  provide a general form of the Taylor series of conditional mean and variance of  a reciprocal normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' To our best knowledge, this is the first com-  plete result of the Taylor series, with the remainder term specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We prove  the theorem in the Proofs section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We use k = 2 to derive an estimator for the  variance of ˜p Algorithm 3, which leads to (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1 (Conditional mean and variance of a reciprocal normal distribu-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For random variable X ∼ N(µ, σ2) where µ > 1 and σ2 > 0, given the  event S = {1 ≤ X ≤ 2µ − 1}, for any integer k > 0, the first two moments of  1  X | S have the following expansions:  E  � 1  X | S  �  = 1  µ  k  �  j=0  (2j − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='σ2j  µ2j  + O  �σ2k+2  µ2k+2  �  (12)  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  13  and  E  � 1  X2 | S  �  = 1  µ2  k  �  j=0  (2j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='σ2j  µ2j  + O  �σ2k+2  µ2k+2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (13)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Theoretical Results  In this section, we present the theoretical results of both privacy and asymptotic  coverage guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In addition, comparisons of the three algorithms in terms  of variance and width ratios are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Privacy and Coverage Guarantees  Our theoretical results are two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' First, the proposed algorithms satisfy the  desired privacy level under the corresponding adjacency relation, which is pre-  sented in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1 (Privacy Guarantee).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Algorithms 1 and 2 satisfy ρ-zCDP under  the adjacency relation ∼ss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Algorithm 3 satisfies ρ-zCDP under the adjacency  relation ∼r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Proofs are presented in the Proofs section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  On the other hand, for the confidence intervals to be useful, we provide the-  orems that guarantee the asymptotic coverage for each algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The central  limit theorem (CLT) asserts (essentially) that the sample mean is asymptoti-  cally normally distributed regardless of the original distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Therefore, the  sample mean can be used to construct a confidence interval for the population  mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In the finite-population sampling designs we are considering, variants of  CLTs can be found among [18, 24, 31] and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We restate a general form  of the finite-population CLT for simple random sampling in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2 and  provide asymptotic coverage guarantees in the following theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2 (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For a population of size N, let p be the pro-  portion in the population with the attribute of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Under stratified random  sampling with sample sizes nh within the stratum of size Nh, h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='., H, let  �V =  H  �  h=1  w2  h �Vh where  �Vh =  �Nh − nh  Nh  � ˜ph(1 − ˜ph) +  1  2ρn2  h  nh − 1  +  1  2ρn2  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (14)  for ρ > 0 as described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' If ρ = ω(1/nh) for all h, then as Nh −nh  and nh both tend to infinity for every stratum,  (i) �V  p→ Var(ˆp), and  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  14  (ii) for 0 < α < 1,  Pr  �  p ∈  �  ˜p − z1−α/2  �  �V , ˜p + z1−α/2  �  �V  ��  → 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (15)  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='3 (Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For a population of size N, let p be the proportion  in the population with the attribute of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Under stratified random sampling  with sample sizes nh within the stratum of size Nh, h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='., H, let  �V =  H  �  h=1  w2  h  �Nh − nh  Nh  � ˆph(1 − ˆph)  nh − 1  + ∆2  p  2ρ1  + eV  (16)  where eV ∼ N(0, ∆2  V  2ρ2 ) for ρ1, ρ2 > 0 as described in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' If ρ1 =  ω(1/nh) and ρ2 = ω(1/nh) for all h, then as Nh − nh and nh both tend to  infinity for every stratum,  (i) �V  p→ Var(ˆp), and  (ii) for 0 < α < 1,  Pr  �  p ∈  �  ˜p − z1−α/2  �  �V , ˜p + z1−α/2  �  �V  ��  → 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (17)  Proofs of the above theorems use the finite-population CLT and are provided  in the Proofs section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  For Algorithm 3, the asymptotic behavior of ˜p is grounded on the normal  approximation to a ratio variable in addition to the CLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We revisit the result  of normal approximation by [17] in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Based on the approximation,  we have shown the consistency of ˜p in the case of simple random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Under simple random sampling, let c be the count of individuals  having the attribute of interest and n be the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The true population  proportion is denoted by p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let ˜p = ˜c/˜n where ˜c ∼ N(c,  1  2ρ1 ) and ˜n ∼ N(n,  1  2ρ2 )  for ρ1, ρ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Under the conditions that ρ2 = ω(1/n), ρ1 = ω(1/n), ˜p is a  consistent estimator for p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  With the foundation of the above consistency, we establish the asymptotic  properties:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5 (Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For a population of size N, let p be the pro-  portion in the population with the attribute of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Under stratified random  sampling with sample sizes nh within the stratum of size Nh, h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='., H, let  �V =  H  �  h=1  w2  h �Vh where  �Vh =  �Nh − ˜nh  Nh − 1  � ˜ph(1 − ˜ph)  ˜nh  +  1  2ρ1˜n2  h  +  ˜p2  h  2ρ2˜n2  h  (18)  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  15  for ρ1, ρ2 > 0 as described in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' If ρ1 = ω(1/nh) and ρ2 = ω(1/nh)  for all h, then as Nh − nh and nh both tend to infinity for every stratum,  (i) �V  p→ Var(ˆp), and  (ii) for 0 < α < 1,  Pr  �  p ∈  �  ˜p − z1−α/2  �  �V , ˜p + z1−α/2  �  �V  ��  → 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (19)  To prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5, we start with a single stratum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We use a normal dis-  tribution (denoted by p∗  h) to approximate that of the proportion estimator ˜ph,  with the distance between the two distribution vanishing to zero in an interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Then for multiple strata, we show that the linear combination of the normal  variables (denoted by p∗) is an accurate approximation to ˜p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Last but not least,  due to the consistency stated in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='4, the noisy estimator �V is a consis-  tent estimator for the variance of p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Then, a Wald confidence interval can be  constructed using ˜p and �V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Details are presented in the Proofs section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Comparisons of Variances  The theorems presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1 ensure that, under proper conditions, the  desired coverage is achieved asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Therefore, to compare the perfor-  mance of the different proposed confidence intervals, we compare their widths,  which are determined by their variance estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In this section, we will an-  alyze our variance estimates and compare the resulting widths to that of the  non-private confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Extrinsic Variances  To investigate how much additional uncertainty is caused by adding noise, we  decompose the variances of the private estimators into two parts: (1) the inherent  component coming from the estimation from the sampling data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='e, Var(ˆp), and  (2) the extrinsic component introduced by the added noise, written as  Vex  def  = Var(˜p) − Var(ˆp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Table 1 provides the (approximate) variances of ˜p for three algorithms, where  wh = Nh  N are the stratum weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The variances are derived in the proofs of  Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The additional variance terms, Vex, can be rewritten  in terms of uh  def  =  Nh  nh instead of wh, as shown in the second row of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  In fact, uh are called sampling weights in the literature on survey sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' A  sample weight is defined as the number of individuals that each respondent in  the sample is representing in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' It is the reciprocal of the sampling  rate  nh  Nh and plays an important role in statistical inference for survey data  [34, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Understanding the relation between sampling weights and the variance  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  16  of the noisy estimators is helpful for practitioners to make survey designs and  the choice of algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Table 1  (Approximate) variances of ˜p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Algorithm  StrNz-PubSz  PopNz-PubSz  StrNz-PrivSz (approximate)  Var(˜p)  Var(ˆp) +  1  2ρ  �H  h=1  w2  h  n2  h  Var(ˆp) +  1  2ρ1 maxh  w2  h  n2  h  Var(ˆp) +  1  2ρ1  �H  h=1  w2  h  n2  h +  1  2ρ2  �H  h=1  w2  hp2  h  n2  h  Vex  1  2N2  �H  h=1  u2  h  ρ  1  2N2 maxh  u2  h  ρ1  1  2N2  �H  h=1 u2  h( 1  ρ1 + p2  h  ρ2 )  With a fixed population size N and a chosen privacy level ρ, the extra vari-  ances Vex induced by the added noise are primarily dictated by uh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In PopNz-  PubSz where we add noise at the population level, Vex is solely determined by  the largest sample weight among all strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' If noise is injected into each stra-  tum, then sampling weights in all strata collectively affect Vex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In particular,  for StrNz-PrivSz, Vex is impacted by ph additionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For all three algorithms,  smaller sampling weights lead to lower extrinsic variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  For comparison, we look at the ratio of Vex with the default budgeting ρ1 =  ρ2 = ρ/2 for PopNz-PubSz and StrNz-PrivSz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The ratio of Vex for StrNz-PubSz  to PopNz-PubSz is  �H  h=1 u2  h  2 maxh u2  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (20)  Roughly speaking, when there are many strata, adding noise at the population  level gives a smaller variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' To compare StrNz-PrivSz and StrNz-PubSz, the  ratio of Vex is  2 �H  h=1 u2  h(1 + p2  h)  �H  h=1 u2  h  ,  (21)  which will always be greater than 2 (due to the cost it takes to protect sample  sizes in StrNz-PrivSz) and at most 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Comparing with Non-Private CI: One Stratum Case  To assess the width in theory, we also look at the confidence interval width  ratios by comparing them to the non-private one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Since the parameters Nh, nh,  ph, ρh come into play together in the stratification setting, it is more practical  to analyze the special case with one stratum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Let the theoretical width ratio (TWR) be  TWR =  �  Var(˜p)  Var(ˆp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  In the implementation, the real width ratio (WR), defined as  �  �V / Var(ˆp), will  be very close to TWR in that �V is a consistent estimator for Var(˜p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Table 2  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  17  displays some relevant quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Note that N−1  N−n is always less than 1 but tends  to 1 when the population size is far larger than the sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Table 2  Theoretical width ratios and lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Algorithm  StrNz-PubSz  PopNz-PubSz  StrNz-PrivSz  ˜p  ˆp + N(0,  1  2ρn2 )  ˆp + N(0,  1  ρn2 )  (c + N(0, 1  ρ))/(n + N(0, 1  ρ))  Var(˜p)  Var(ˆp) +  1  2ρn2  Var(ˆp) +  1  ρn2  Var(ˆp) + 1+p2  ρn2  TWR  �  1 + N−1  N−n  1  2p(1−p)nρ  �  1 + N−1  N−n  1  p(1−p)nρ  �  1 + N−1  N−n  1+p2  p(1−p)nρ  Lower bound of TWR  �  1 +  2  nρ  �  1 +  4  nρ  �  1 + 2(1+  √  2)  nρ  We can obtain a lower bound for TWR by dropping the factor  N−1  N−n and  minimizing over p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We can see that the width ratio mainly depends on p and  the relative magnitude between n and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' If p is extreme (tends to 0 or 1), TWR is  drastically large;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' when p is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5, TWR is close to the lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Also,  the added noise induces a term involving ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For example, under the regime  ρ = 1/n, the three algorithms result in an interval of length at least  √  3 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='73,  √  5 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='24, and  �  3 + 2  √  2 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='41 as wide, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' It is trivial that with  one stratum, StrNz-PubSz produces a tighter confidence interval than PopNz-  PubSz does in that the ratio of Vex in (20) is 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' However, PopNz-PubSz will  outperform StrNz-PubSz once there are enough strata such that (20) is greater  than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Numerical Results  In this section, we conduct both simulation studies and applications to assess  and illustrate the numerical performance of the proposed algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We clip  the proportions ˜ph onto [0, 1] as mentioned in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Simulations  We set up a set of experiments to (1) check the normality of noisy estimators,  and (2) evaluate the performance of the proposed confidence intervals by varying  the number of strata H, the true population proportion p, and the privacy level  ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' To generate the data, we need to specify the strata sizes Nh and the sampling  rates rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The setup of these parameters is presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We generate  a proportion for each stratum to create heterogeneity across strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The true  population proportion is then calculated and reported in each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  18  Table 3  Parameter setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The resulting sample sizes are between 60 and 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Fixed parameter  Value / Distribution  Varying parameter  Value / Distribution  α  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1  H  1 or 20  Nh  Uniform(1500, 2000)  ph  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5, Uniform(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='6) or Uniform(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='15)  rh  Uniform(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='08)  ρ  1/ max(nh) or specified in the axis of the plot  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Normality Check  We first check whether the distributions of ˜p in the three algorithms are reason-  ably close to the theoretical normal distributions with the corresponding means  and variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Figure 1 displays the Q-Q plots of the theoretical distribution of  ˜p versus its sample distribution:  Non-private: N(p, Var(ˆp));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  StrNz-PubSz: N(p, Var(˜p)) as Var(˜p) in (3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  PopNz-PubSz: N(p, Var(˜p)) as Var(˜p) in (5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  StrNz-PrivSz: N  �  p + �H  h=1  whph  2ρ2n2  h , �H  h=1 w2  hVh  �  with Vh specified in (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Note that, ˜p in Algorithms StrNz-PubSz and PopNz-PubSz are unbiased for  p while ˜p in StrNz-PrivSz is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Nevertheless, under the condition that ρ2 =  ω(1/nh) in Theorem 3, the bias term �H  h=1  whph  2ρ2n2  h is negligible and thus we  do not make a bias correction in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We observe great alignments  between the theoretical and experimental distributions, indicating that the pri-  vate estimators in all three algorithms are indeed highly close to being normally  distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Varying Key Parameters  Assured by the results of the normality check, we experiment with a wide range  of the privacy budget, different numbers of strata, and true population propor-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  We examine the impact of ρ on the performance of the three private esti-  mators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The simulation is run on 10,000 repetitions and therefore the empirical  coverage falling into 90% ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='006 (departure of two standard deviations) is con-  sidered appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In Figure 2a, the empirical coverage is reasonable except  that StrNz-PrivSz gives unnecessarily higher coverage when ρ is smaller than  around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' This is because the budget is so small for the method that, with  clipping, it covers the truth more often than needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In this case, the confidence  intervals are too wide to be as useful, as shown in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For all three meth-  ods, the width grows as ρ becomes smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' However, the rates of width growth  differ: in the multiple strata case we simulate, the width of PopNz-PubSz grows  the slowest, StrNz-PrivSz grows the fastest, and StrNz-PubSz is in the middle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Thus, the optimal privacy level should be chosen by taking into account the  method, width, and coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For instance, if we want a 90% of confidence level  and width under 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1, one can choose the value for ρ as small as (1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='001 for  PopNz-PubSz, (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='003 for StrNz-PubSz, and (3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='01 for StrNz-PrivSz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  19  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='54  Theoretical Quantiles  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='54  Sample Quantiles  Non-Private  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='425  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='450  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='475  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='525  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='550  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='575  Theoretical Quantiles  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='55  Sample Quantiles  StrNz-PubSz  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='56  Theoretical Quantiles  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='475  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='525  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='550  Sample Quantiles  PopNz-PubSz  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='60  Theoretical Quantiles  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='6  Sample Quantiles  StrNz-PrivSz  Fig 1: Q-Q plots: Theoretical versus sample distributions of ˜p with 20 strata and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='505  (resulting from ph ∼ Uniform(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='6)), based on 10,000 repetitions each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  In addition, Table 4 shows the numerical results of three experiments with  different combinations of the numbers of strata and the true population pro-  portions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The simulation in the middle panel shares the same setting as the  experiment shown in Figure 2 but has a fixed privacy level: 1/ max(nh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' This  is an analogous regime to ρ = 1/n (for simple random sampling) for multiple  strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In the literature on differential privacy, the regime ρ = 1/n for a simple  random sample is often considered to understand how small ρ can be as the  sample size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Recall that a smaller ρ means a higher privacy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  As argued above, clipping ˜ph (or ˜p) onto [0,1] will yield better results in some  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The conclusions coincide with the analyses in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The empirical  coverage of the three private ones in all simulations achieves the nominal level of  90%, as guaranteed by Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The case where StrNz-PrivSz  gives a 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='9% confidence level in the bottom panel is due to clipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' (When  the stratum proportions are close to the extreme, clipping is more noticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=')  The average width and width ratio (WR) varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' With one single stratum,  WRs are near the lower bounds of theoretical width ratios (TWR) given in  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2, which suggests that the lower bounds are almost tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' StrNz-  PubSz gives a narrower CI than PopNz-PubSz with one stratum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' But with  more strata, PopNz-PubSz outperforms StrNz-PubSz in terms of WR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Having  more strata means splitting the total privacy budget into smaller portions, which  leads to adding more noise on the whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The CI needs to be wider to achieve the  same confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' As for StrNz-PrivSz, however, it always yields the widest  CI due to the additional price it pays to protect sample sizes simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  20  10  3  10  2  10  1  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='00  Empirical coverage  Non-Private  PopNz-PubSz  StrNz-PubSz  StrNz-PrivSz  (a) Empirical coverage  10  3  10  2  10  1  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='6  Width  0  2  4  8  16  Width ratio  Non-Private  PopNz-PubSz  StrNz-PubSz  StrNz-PrivSz  (b) Width and ratio  Fig 2: Setup: 20 strata and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='505 (ph ∼ Uniform(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='6)) with 10,000 repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Figure (a) is the empirical coverage with the red line indicating the nominal confidence level  of 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The average width and width ratio are displayed in (b) with the non-private as the  benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  On the other hand, with the same number of strata (20 here), we see that more  extreme ph leads to a larger WR than ph in the middle range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' This is because  the factor ph(1 − ph) comes into play as p(1 − p) does in TWR in Table 2 for  the one stratum case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  We also provide the sample standard deviation of the widths (width SD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In  general, the non-private method results in a smaller standard deviation than  the private ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In some cases, clipping helps reduce the width SD for the  private algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' With the same privacy level, there is more fluctuation in  width for PopNz-PubSz compared to StrNz-PubSz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' This is because we use one-  half of the privacy budget and directly add noise to the variance estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' As  expected, StrNz-PrivSz has the largest width SD since the magnitude of width  is the largest and the ratio variable is heavy-tailed by design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Nevertheless,  compared to the width, the width SD for all methods is so small that it does  not compromise the effectiveness of the confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Applications  In this section, we apply the proposed methods to the 1940 Census full enumer-  ation from IPUMS USA [35] and evaluate the performance of three differentially  private confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' To conduct stratified random sampling on the data  set, the state-level geographical variable “STATEICP” (49 categories, constitut-  ing the then-48 states and Washington, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=') is used for stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Under  stratified random sampling with H = 49 strata, we estimate the national un-  employment rate for the first application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In the second application, we are  interested in studying the discrepancy in income levels between black and white  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  21  Table 4  Simulation results under ρ = 1/n (or ρ = 1/ max(nh)) regime based on 10,000 repetitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  The strata sizes and sampling rates are drawn as described in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For the multiple  strata case, the resulting sample sizes in nh range from 72 to 152, and ρ is set to be  1/152 ≈ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='58 × 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For the one-stratum case, we set the sample size to 152 so that we  have the same level of privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Non-Private  StrNz-PubSz  PopNz-PubSz  StrNz-PrivSz  1 stratum, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5  coverage  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='893  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='901  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='894  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='901  width  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='127  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='228  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='295  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='327  width SD  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='47 × 10−4  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='85 × 10−4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='89 × 10−3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='15 × 10−2  CI  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='436, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='564)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='386, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='614)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='352, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='648)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='34, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='667)  WR  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='786  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='318  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='567  20 strata, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='505 (ph ∼ Uniform(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='6))  coverage  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='902  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='895  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='902  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='902  width  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='073  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='043  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='111  width SD  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='08 × 10−4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='58 × 10−4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='87 × 10−4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='22 × 10−3  CI  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='488, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='523)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='469, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='542)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='483, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='527)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='457, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='568)  WR  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='074  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='239  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='168  20 strata, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='103 (ph ∼ Uniform(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='15))  coverage  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='902  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='919  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='904  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='899  width  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='021  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='067  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='033  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='096  width SD  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='17 × 10−4  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='21 × 10−4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='71 × 10−4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='94 × 10−3  CI  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='092, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='113)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='073, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='143)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='086, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='119)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='072, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='168)  WR  1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='189  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='571  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='563  men.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Confidence Intervals for the Unemployment Rate  As an important indicator of the status of the national economy, the unem-  ployment rate is the percentage of unemployed workers in the total labor force  consisting of both the employed and unemployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Thus, we consider all the indi-  viduals who are either employed or unemployed as the whole population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In the  1940 Census data set, the binary characteristic “EMPSTAT” represents employ-  ment status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The full enumeration is considered the truth and the true popula-  tion proportion is p = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='346%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' To carry out stratified random sampling, sample  sizes or sampling rates are selected for all 49 strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For modern relevance, we  simulate in a manner intended to mimic the canonical design implemented in  the current American Community Survey (ACS), by choosing a typical range of  sampling rates used in ACS which is [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5%, 15%].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' See Table 5 for detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  22  Table 5  Sampling rates  Stratum size  Sampling rate  nh ≤ 5 × 104  15%  5 × 104 < nh ≤ 105  10%  105 < nh ≤ 5 × 105  5%  5 × 105 < nh ≤ 106  2%  106 < nh ≤ 5 × 106  1%  nh > 5 × 106  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5%  To apply and assess the proposed algorithms, we experiment with a wide  range of small privacy budgets: ρ ∈ [10−6, 10−3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Each method is repeated 10,000  times and the empirical coverage, the average CI width, and the average CI  width ratio (WR) are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' As shown in Figure 3a, the empirical coverage  is always around the nominal level which is chosen at the level of 90% for the  whole range of privacy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In Figure 3b, the CI width and CI width ratio  with the non-private CI as the benchmark, share the same shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Even when  the CI given by StrNz-PrivSz is 8 times the non-private CI width, the CI width  is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='01 due to the large sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Both CI width and width ratio should  be taken into account when choosing an optimal privacy level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  10  6  10  5  10  4  10  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='00  Empirical coverage  Non-Private  PopNz-PubSz  StrNz-PubSz  StrNz-PrivSz  (a) Empirical coverage  10  6  10  5  10  4  10  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='03  Width  0  5  10  15  20  25  WR  Non-Private  PopNz-PubSz  StrNz-PubSz  StrNz-PrivSz  (b) Width and ratio  Fig 3: The empirical coverage and average width and width ratio of DP-CIs of the unemploy-  ment rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Confidence Intervals for the Difference in Income Level  In the second application, we want to investigate whether there was a discrep-  ancy between the income levels of white males (population 1) and that of black  males (population 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Note that only those who had valid income numbers in  the 1940 Census are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Since the poverty thresholds were not devel-  oped until the 1960s and thus are not available for the 1940 data, the national  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  23  income average is used as a threshold instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We are interested in examining  the difference in subpopulation proportions of those whose income levels passed  this threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  The geographic feature “STATEICP” is used for stratification, yielding 49  strata, with stratum size ranges of (41838, 4621442) for the population of white  males and (50, 309214) for the population of black males.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Sampling rates are  adaptively chosen based on stratum sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For the population of white males, the  range of sampling rates is also [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5%, 15%], whereas the range of sampling rates  is [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5%, 30%] for the population of black males given its small stratum sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Additionally, to allow solid approximations based on the asymptotic results, we  impose that the sample sizes are adjusted to be 50 if the sampling rates give  smaller sizes than 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' See Table 6 for detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Table 6  Sampling rates for two populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Stratum sizes nh ∈ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1 × 104, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='7 × 106) for the  population of white males and stratum sizes nh ∈ (50, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1 × 105) for the population of black  males.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' *The sample size will be adjusted to be 50 if the above sampling rate results in a size  smaller than 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Stratum size Nh of white males  Sampling rate  Nh ≤ 5 × 104  15%  5 × 104 < Nh ≤ 105  10%  105 < Nh ≤ 5 × 105  5%  5 × 105 < Nh ≤ 106  2%  106 < Nh ≤ 4 × 106  1%  Nh > 4 × 106  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5%  Stratum size Nh of black males  Sampling rate*  Nh ≤ 500  30%  500 < Nh ≤ 5 × 103  15%  5 × 103 < Nh ≤ 104  5%  104 < Nh ≤ 2 × 104  2%  2 × 104 < Nh ≤ 3 × 104  1%  nh > 3 × 104  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5%  Let p1 and p2 denote the proportions of eligible individuals who earned more  than the national average income level $442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The true values of proportions  are p1 = 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='0223% and p2 = 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5152%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let pdiff = p1 − p2, then the true  difference in these two proportions is pdiff = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5071%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' By the additivity of two  independent normal distributions, naturally, we use the following differentially  private CI:  ˜pdiff + z1−α/2  �  �V(˜pdiff),  (22)  where �V (·) denotes a private estimator of variance, ˜pdiff is defined as ˜p1 − ˜p2  and  �V(pdiff) = �V(p1) + �V(p2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  In Figure 4, similar patterns are observed in this application as in the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  All CIs have empirical coverage around/above the nominal confidence level as  in the simulation study in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The phenomenon of higher coverage  is due to small ρ and effective clipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' When the range of stratum sizes is  large (it is (50, 309214) in this application), that is, when the stratum sizes are  very different, a large privacy budget ρ should be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The choice of a small  ρ harms the estimates of small-sized strata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We advise that the smallest ρ be  chosen given the tolerance of uncertainty in terms of width and/or width ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  For example, if the accuracy requirement is that the width should be under 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='05  or WR under 5, then the best choices of ρ among the experiments in Figure 4b  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  24  10  4  10  3  10  2  10  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='95  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='00  Empirical coverage  Non-Private  PopNz-PubSz  StrNz-PubSz  StrNz-PrivSz  (a) Empirical coverage  10  4  10  3  10  2  10  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='35  Width  0  5  10  15  20  25  30  WR  Non-Private  PopNz-PubSz  StrNz-PubSz  StrNz-PrivSz  (b) Width and ratio  Fig 4: The empirical coverage and average width and width ratio of DP-CIs of the difference of  the above-national-income-level proportions between black and white males with valid income  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  are (1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='0001 for PopNz-PubSz, (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='0018 for StrNz-PubSz, and (3) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='0056 for  StrNz-PrivSz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Discussion  We have designed three algorithms to construct confidence intervals for the  population proportion under stratified random sampling with zero concentrated  differential privacy guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We consider both the case where the sample sizes  are public and the case where they are private information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Theoretical results  including privacy guarantees and asymptotic properties are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' With  proper conditions on the relation between the privacy budget and sample sizes,  as stated in the theorems, the resulting confidence intervals will achieve the  desired coverage asymptotically, and the width tends to be that of a non-private  confidence interval when the sample sizes go to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  In the simulation studies and two applications, we have experimented with a  wide range of privacy budgets under a variety of parameter setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The three  algorithms always perform well in terms of empirical coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The width and  width ratio are in a reasonable range even under the strict regime where ρ =  1/ maxh nh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Typically in practice, a constant between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='001 to 10 is chosen to  be the privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' According to our experiments, with the choice of the  smallest budget in this range, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='001, the three algorithms still have fairly good  results even when the smallest stratum has only a size 50 (as demonstrated in  the second application).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  The comparative analysis of the three algorithms in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2 gives ac-  tionable guidance to practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' When releasing the population proportion is  the only goal and there are enough strata (such that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' (20) regarding sam-  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  25  ple weights is greater than 1), PopNz-PubSz is the better option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' However, if  stratum proportions should also be released or there are just a few strata, StrNz-  PubSz is preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' On the other hand, when the population proportion and  sample sizes must be protected simultaneously, StrNz-PrivSz is the only algo-  rithm presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' StrNz-PrivSz, compared to the case with public  sample sizes, needs a larger budget to meet the same width requirement on  account of the additional cost of protecting sample sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  There are a few open questions worth considering for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In  this paper, we discuss the case where the number of strata is fixed and the  sample sizes tend to infinity, we use the finite-population CLTs for each stra-  tum to derive an aggregated estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For the other case where the number  of stratum tends to infinity instead of sample sizes, one can apply the Linde-  berg–Lévy–Feller Theorem to obtain the asymptotic normality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' More interest-  ingly, we do not provide ‘PopNz-PrivSz’ – an analogous algorithm to PopNz-  PubSz for the private sample sizes case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' To protect both the population pro-  portion and the sample sizes, the direct addition of noise to the non-private  aggregated estimator is not plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' One should consider more sophisticated  mechanisms other than directly adding noise to the statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' If ‘PopNz-PrivSz’  were proposed, we shall expect it to yield a narrower confidence interval since  we only need to publish the private population proportion without being able to  provide private confidence intervals for stratum proportions at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Another direction for future research would be optimal budget allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We do  not experiment with different ways to divide the total budget in PopNz-PubSz  or StrNz-PrivSz but simply split it up evenly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Budgeting for the composed ap-  plication of the algorithms may also be of interest, like in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2 where  we apply the algorithms twice for two independent populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lastly, one  broad direction is to develop the differentially private versions for other alter-  natives to the basic Wald interval, such as the Wilson Interval, Jeffreys interval,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' (see [21] for a comparative summary of seven such types of confidence inter-  vals for proportions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Many of these latter are specifically designed for the case  of small sample sizes, which we do not consider here and for which we expect  fundamentally different approaches to differential privacy likely to be necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Appendix A: Proofs  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let X ∼ N(µ, σ2) and S = {µ − a ≤ X ≤ µ + a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For any a > 0  and an integer k ≥ 1, the conditional even moments  E[(X − µ)2k | S] = σ2k(2k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' − O  �  e− a2  2σ2 a2k−1�  ,  (23)  where the big-O hides a constant depending on σ and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Without loss of generality, we assume µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We prove the lemma by  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  26  induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Set k = 1, integrate by parts,  E[X2IS] =  � a  −a  x2  1  σ  √  2π e− x2  2σ2 dx  =  σ  √  2π  �  −xe− x2  2σ2 ��a  −a +  � a  −a  e− x2  2σ2 dx  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Integrate by substitution, the integral in the second term becomes  � a  −a  e− x2  2σ2 dx = σ  √  2π erf  �  a  σ  √  2  �  where  erf(z) = σ  � z  0  e−t2 dt  is the error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Then,  E[X2IS] = σ2 erf  �  a  σ  √  2  �  − O  �  e− a2  2σ2 a  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Assuming  E[X2kIS] = σ2k(2k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' erf  �  a  σ  √  2  �  − O  �  e− a2  2σ2 a2k−1�  ,  (24)  then integrate by parts for the k + 1 case,  E[X2(k+1)IS] =  � a  −a  x2k+2  1  σ  √  2π e− x2  2σ2 dx  =  σ  √  2π  � a  −a  x2k+1 · x  σ2 e− x2  2σ2 dx  =  σ  √  2π  �  −x2k+1e− x2  2σ2 ��a  −a + (2k + 1)  � a  −a  x2ke− x2  2σ2 dx  �  = σ2(2k + 1)E[X2kIS] − O  �  e− a2  2σ2 a2k+1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Plug in (24), we obtain  E[X2(k+1)IS] = σ2k+2(2k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' erf  �  a  σ  √  2  �  − O  �  e− a2  2σ2 a2k+1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  So far we have proved (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Note that  Pr(S) =  � a  −a  1  σ  √  2π e− x2  2σ2 dx = erf  �  a  σ  √  2  �  ,  and that the image of erf(z) is between (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Therefore,  E[X2k | S] = E[X2kIS]/ Pr(S) = σ2k(2k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' − O  �  e− a2  2σ2 a2k−1�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  27  Proof of (12) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Consider the Taylor series of 1  x at x = µ:  1  x =  ∞  �  j=0  (−(x − µ))j  µj+1  = 1  µ − x − µ  µ2  + (x − µ)2  µ3  − (x − µ)3  µ4  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Let ym be the partial sum of the above series, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=', ym(x) = �m  k=0  (−(x−µ))k  µk+1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Then ym(x) converges to 1  x in (0, 2µ) which contains [1, 2µ − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let  g(x) =  ∞  �  k=0  |x − µ|k  µk+1  =  �  1  x,  if 1 ≤ x ≤ µ  1  2µ−x  if µ < x ≤ 2µ − 1  Then g is integrable as  � 2µ−1  1  |g(x)|dν =  � µ  1  1  xdν +  � 2µ−1  µ  1  2µ − xdν = 2  � µ  1  1  xdν < ∞,  where dν = f(x)dx is induced by N(µ, σ2) conditional on event S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Note also  that |ym(x)| ≤ g(x) for any naturals m and x ∈ [1, 2µ − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' By the dominated  convergence theorem, the operations of limit and integral are exchangeable for  ym(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  � 2µ−1  1  1  xdν =  � 2µ−1  1  lim  m→∞ ym(x)dν  = lim  m→∞  � 2µ−1  1  ym(x)dν  = lim  m→∞  � 2µ−1  1  �  �  m  �  j=0  (−(x − µ))j  µj+1  �  � dν  = lim  m→∞  �  �  m  �  j=0  � 2µ−1  1  (−(x − µ))j  µj+1  dν  �  �  (25)  Then,  E  � 1  X | S  �  =  ∞  �  j=0  1  µj+1 E  �  (−(X − µ))j | S  �  =  ∞  �  j=0  1  µ2j+1 E  �  (X − µ)2j | S  �  =  k  �  j=0  1  µ2j+1 E  �  (X − µ)2j | S  �  + 1  µ  ∞  �  j=k+1  E  ��X − µ  µ  �2j  | S  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (26)  The second equality is because the odd moments are zero due to symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  28  Note that given event S, | X−µ  µ | ≤ µ−1  µ  < 1, then  E  ��X − µ  µ  �2k+2  | S  �  ≤  �µ − 1  µ  �2  E  ��X − µ  µ  �2k  | S  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (27)  It follows that  ∞  �  j=k+1  E  ��X − µ  µ  �2j  | S  �  ≤  ∞  �  j=0  �µ − 1  µ  �2j  E  ��X − µ  µ  �2k+2  | S  �  =  µ2  2µ − 1E  ��X − µ  µ  �2k+2  | S  �  = O  �  1  µ2k+1  �  E[(X − µ)2k+2 | S].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Applying Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1, by the choice of a = µ − 1, (26) becomes  E  � 1  X | S  �  = 1  µ  k  �  j=0  (2j − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='σ2j  µ2j  + O  �σ2k+2  µ2k+2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Proof of (13) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We conduct a similar procedure for the second  moment of X | S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Based on the Taylor expansion  1  x2 =  ∞  �  j=0  (j + 1)(−(x − µ))j  µj+2  = 1  µ2 − 2(x − µ)  µ3  + 3(x − µ)2  µ4  − 4(x − µ)3  µ5  + · · · ,  we have  E  � 1  X2 | S  �  =  ∞  �  j=0  j + 1  µj+2 E  �  (−(X − µ))j | S  �  =  ∞  �  j=0  2j + 1  µ2j+2 E  �  (X − µ)2j | S  �  =  k  �  j=0  2j + 1  µ2j+2 E  �  (X − µ)2j | S  �  + 1  µ2  ∞  �  j=k+1  (2j + 1)E  ��X − µ  µ  �2j  | S  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (28)  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  29  Due to (27),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' it follows that  ∞  �  j=k+1  (2j + 1)E  ��X − µ  µ  �2j  | S  �  ≤ E  ��X − µ  µ  �2k+2  | S  � ∞  �  j=0  (2k + 3 + 2j)  �µ − 1  µ  �2j  = E  ��X − µ  µ  �2k+2  | S  � �  �(2k + 3)  ∞  �  j=0  �µ − 1  µ  �2j  + 2  ∞  �  j=1  j  �µ − 1  µ  �2j  �  �  = E  ��X − µ  µ  �2k+2  | S  � �(2k + 3)µ2  2µ − 1  + 2µ2(µ − 1)2  (2µ − 1)2  �  = E  ��X − µ  µ  �2k+2  | S  �  O(µ2)  = O  � 1  µ2k  �  E[(X − µ)2k+2 | S],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (29)  where the term �∞  j=1 j  �  µ−1  µ  �2j  is a sum of an arithmetic–geometric sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1, (28) becomes  E  � 1  X2 | S  �  = 1  µ2  k  �  j=0  (2j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='σ2j  µ2j  + O  �σ2k+2  µ2k+2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (30)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1  Proof for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Under neighboring relation ∼ss, only one record changes  within one stratum and sample sizes remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Applying the Gaussian  mechanism to each stratum at the level of ρ gives ρ-zCDP guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' By post-  processing, the confidence interval is also ρ-zCDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Proof for Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The sensitivities of ˆp and �  Var(ˆp) are ∆p and ∆V , re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Applying the Gaussian mechanism, it follows that ˜p is ρ1-zCDP and  �V is ρ2-zCDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' By basic composition, the confidence interval ˜p ± z1− α  2  �  �V is  (ρ1 + ρ2)-zCDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Proof for Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' By the Gaussian mechanism and the basic composition  property of zCDP, we know that ˜ph is ρ-zCDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Under neighboring relation  ∼r, only one record changes within one stratum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Then, by post-processing, the  confidence interval is ρ-zCDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  30  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2  Before proving the theorem, we revisit the finite-population CLT first:  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2 (Theorem 1, [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Consider a finite population Π = {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=', XN}  of size N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let µ be the population mean and ¯Xn be the mean of a simple ran-  dom sample of size n from Π, and Var( ¯Xn) is the variance of ¯Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' The finite  population variance of Π is denoted by  v =  1  N − 1  N  �  i=1  (Xi − µ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  As N → ∞, if  1  min(n, N − n) · max1≤i≤N(Xi − µ)2  v  → 0,  (31)  we have  ¯Xn − µ  �  Var( ¯Xn)  d→ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (32)  The variance of ¯Xn is determined by the population variance v which is  unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Nevertheless, the sample variance �  Var( ¯Xn) can be used to estimate  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' To make sure the CLT still holds when substituting Var( ¯Xn) by �  Var( ¯Xn),  the consistency of �  Var( ¯Xn) is crucial, as stated in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let �  Var( ¯Xn) be the sample variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' �  Var( ¯Xn) is an unbiased  estimator for Var( ¯Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Moreover, under the condition in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2, as N →  ∞,  �  Var( ¯Xn)/Var( ¯Xn)  p→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Now we prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2:  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' It suffices to show  ˜ph−ph  √  �Vh  d→ N(0, 1) for all h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' By the  finite-population CLT in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2, we know  ˆph − ph  �  Var(ˆph)  d→ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Since ˜ph = ˆph + eh where eh ∼ N(0,  1  2ρn2  h ), we have  ˜ph − ph  �  Var(˜ph)  d→ N(0, 1)  (33)  where  Var(˜ph) = Var(ˆph) +  1  2ρn2  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  31  Let  �Vh =  �Nh − nh  Nh  � ˜ph(1 − ˜ph) +  1  2ρn2  h  nh − 1  +  1  2ρn2  h  = �  Var(ˆph) +  �Nh − nn  Nh  � eh − 2˜pheh − e2  h +  1  2ρn2  h  nh − 1  +  1  2ρn2  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (34)  If  1  ρnh → 0, we have eh  p→ 0 and then the second term of (34) is oP ( 1  nh ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Note that �  Var(ˆph) is of order  1  nh , and that by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='3, �  Var(ˆph)  p→ Var(ˆph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Therefore, �Vh  p→ Var(ˆph) +  1  2ρn2  h = Var(˜ph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Combining it with (33), we have  ˜ph − ph  �  �Vh  d→ N(0, 1)  (35)  by Slutsky’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Then, ˜p−p  √  �V  d→ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Therefore, the confidence interval  given by p ± z1−α/2  �  �V has asymptotic coverage level 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Note that, under the condition  1  ρnh → 0, it follows that  1  2ρn2  h = o(Var(ˆph))  and hence �Vh  p→ Var(ˆph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Therefore, �V  p→ Var(ˆp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='3  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Since  ˆp−p  √  Var(ˆp)  d→ N(0, 1) and ˜p = ˆp+N(0, ∆2  p/2ρ1) with ∆p = maxh  wh  nh ,  it follows that  ˜p − p  �  Var(˜p)  d→ N(0, 1),  and  Var(˜p) = Var(ˆp) + ∆2  p  2ρ1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  In Algorithm 2, we set  �V = �  Var(ˆp) + ∆2  p  2ρ1  + eV ,  (36)  where eV ∼ N(0, ∆2  V  2ρ2 ) with ∆V = maxh  �  Ch  nh  �  1 −  1  nh  ��  and Ch = w2  h  Nh−nh  Nh  1  nh−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  If  1  ρ2nh → 0 for all h, then eV = oP ( 1  nh ) for all h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Since �  Var(ˆp)  d→ Var(ˆp) by  finite-population CLT, we have �V  d→ Var(˜p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Therefore, by Slutsky’s Theorem,  ˜p − p  �  �V  d→ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (37)  Then, the confidence interval given by p ± z1−α/2  �  �V has the asymptotic cov-  erage level 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' In fact, �V  p→ Var(ˆp) if  1  ρ1nh → 0 for all h since  ∆2  p  2ρ1 =  o(Var(ˆp)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  32  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='4  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For ˜n ∼ N(n,  1  2ρ2 ), by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1, we derive the kth-order Taylor  series of the conditional expectation of ˜p given S = {1 ≤ ˜n ≤ 2n − 1}:  E (˜p | S) = p  k  �  j=0  (2j − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  n2j(2ρ2)j + O  �  1  n2k+1ρk+1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (38)  For example, when k = 2,  E (˜p | S) = p  �  1 +  1  2n2ρ2  +  3  4n4ρ2  2  �  + O  �  1  n5ρ3  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (39)  To obtain a Taylor expansion for the conditional variance, we plug  E  � 1  ˜n | S  �  = 1  n  k  �  j=0  (2j − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  n2j(2ρ2)j + O  �  1  n2k+2ρk+1  2  �  and  E  � 1  ˜n2 | S  �  = 1  n2  k  �  j=0  (2j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  n2j(2ρ2)j + O  �  1  n2k+2ρk+1  2  �  into  Var(˜p | S) = E(˜p2 | S) − (E(˜p | S))2 = E˜c2E  � 1  ˜n2 | S  �  − (E(˜p | S))2,  by which we derive a general expansion for the conditional variance:  Var(˜p | S) = Var(ˆp)  k  �  j=0  (2j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  n2j(2ρ2)j + p2  �  �  �  k  �  j=0  (2j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  n2j(2ρ2)j −  �  �  k  �  j=0  (2j − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  n2j(2ρ2)j  �  �  2�  �  �  +  1  2ρ1  k  �  j=0  (2j + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  n2j+2(2ρ2)j + O  �  1  n2kρk+1  2  �  + O  �  1  n2k+2ρ1ρk+1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (40)  When k = 2,  Var(˜p | S) = Var(ˆp)  �  1 +  3  2n2ρ2  +  15  4n4ρ2  2  �  + p2  �  1  2n2ρ2  +  2  n4ρ2  2  −  6  8n6ρ3  2  −  9  16n8ρ4  2  �  +  1  2ρ1  � 1  n2 +  3  2n4ρ2  +  15  4n6ρ2  2  �  + O  �  1  n4ρ3  2  �  + O  �  1  n6ρ1ρ3  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (41)  Based on Taylor expansion with k = 2 for both conditional mean and variance  given in (39) and (41), under the condition  1  ρ1n = o(1) and  1  ρ2n = o(1), we have  E(˜p | S) = p + o  � 1  n  �  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  33  and  Var(˜p | S) = Var(ˆp) + o  � 1  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Then, ˜p | S is asymptotically unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Note that Var(ˆp) is of order 1  n and thus  ˜p | S has a vanishing variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Therefore, ˜p | S converges to p in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Note also that Pr(S) → 1 as n → ∞, then for any ϵ > 0,  Pr(|˜p − p| > ϵ) = Pr(|˜p − p| > ϵ | S) + Pr(|˜p − p| > ϵ | Sc) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  That is, ˜p is a consistent estimator for p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5  To prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5, we need the following theorem and lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='4 (Theorem 1, [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let X be a normal random variable with  positive mean µx, variance σ2  x and coefficient of variation δx = σx/µx such that  0 < δx < λ ≤ 1, where λ is a known constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For every ϵ > 0, there exists  γ(ϵ) ∈ (0,  �  λ2 − δ2x) and also a normal random variable Y independent of X,  with positive mean µy, variance σ2  y and coefficient of variation δy = σy/µy that  satisfy the conditions,  0 < δy ≤ γ(ϵ) ≤  �  λ2 − δ2x < λ  (42)  for which the following result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Any z that belongs to the interval  I =  �  β − σz  λ , β + σz  λ  �  ,  where β = µx/µy, σz = β  �  δ2x + δ2y, satisfies that  |G(z) − FZ(z)| < ϵ,  where G(z) is the cumulative distribution function of N(β, σ2  z), and FZ is that  of Z = X/Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Note that once a given Y fulfills the closeness between the cor-  responding G to FZ , any other random variables with a smaller coefficient of  variation will satisfy this result too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' For a population of size N, let p be the true proportion in the  population with the attribute of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Consider simple random sampling with  sample size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let Z∗ ∼ N(p, V ) where V =  �  N−n  N−1  �  p(1−p)  n  +  1  2ρ1n2 +  p2  2ρ2n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' If  ρ1 = ω(1/n2) and ρ2 = ω(1/n), as N − n and n both tend to infinity, then for  any z ∈ (0, 2p),  |F˜p(z) − FZ∗(z)| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (43)  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  34  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' By the CLT in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='2, we know that ˆp ∼ AN(p, Var(ˆp)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Recall  that ˜c = nˆp + N(n,  1  2ρ1 ), then ˜c ∼ AN(np, n2 Var(ˆp) +  1  2ρ1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Let ˜X ∼ AN(np, n2 Var(ˆp)+ 1  2ρ1 ) and X ∼ N(np, n2 Var(ˆp)+ 1  2ρ1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Therefore,  for any ϵ > 0, there exists some n0 = n0(ϵ) such that for any x and n > n0,  |F ˜  X(x) − FX(x)| < ϵ,  (44)  where F denotes the cumulative density function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let Y ∼ N(n,  1  2ρ2 ), ˜Z = ˜X/Y  and Z = X/Y , then  F ˜  Z(z) = Pr  � ˜X  Y < z  �  = Pr( ˜X < Y z) =  � ∞  −∞  F ˜  X(yz)fy(y)dy,  where fy(y) is the density function of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' From (44), |F ˜  X(yx) − FX(yx)| < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' It  follows that,  � ∞  −∞  (FX(yx) − ϵ)fy(y)dy <  � ∞  −∞  F ˜  X(yx)fy(y)dy <  � ∞  −∞  (FX(yx) + ϵ)fy(y)dy,  which is equivalent to  ����  � ∞  −∞  F ˜  X(yx)fy(y)dy −  � ∞  −∞  FX(yx)fy(y)dy  ���� < ϵ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=',  |F ˜  Z(z) − FZ(z)| < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (45)  Let δx and δy be the coefficients of variation of X and Y , respectively, then  δ2  x = (Var(ˆp) +  1  2ρ1n2 )/p2 and δ2  y =  1  2ρ2n2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Under the condition  1  ρ1n = o(1),  we have δ2  x = O( 1  n) since Var(ˆp) = O( 1  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Under the condition  1  ρ2n = o(1),  we know δ2  y = o( 1  n) and then δy = o(δx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' When n is sufficiently large, δy is  sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let λ =  �  δ2x + 2δ2y and FZ∗(z) be the distribution function of  Z∗ ∼ N(p, Var(ˆp)+  1  2ρ1n2 +  p2  2ρ2n2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='4, for a normal random variable  Y independent of X, with small enough δy, the condition (42) is satisfied and  we have  |FZ(z) − FZ∗(z)| < ϵ,  (46)  for any z ∈ I =  �  p − σz∗  λ , p + σz∗  λ  �  where σz∗ = p  �  δ2x + δ2y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Hence, for z ∈ I,  |F ˜  Z(z) − FZ∗(z)| < |F ˜  Z(z) − FZ(z)| + |FZ(z) − FZ∗(z)| < 2ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (47)  Note also that as n → ∞, σz∗  λ  → p, and the limit of I is (0, 2p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  So far, we have shown that as n goes to infinity, under the conditions  1  ρ1n =  o(1) and  1  ρ2n = o(1), for z ∈ Ih,  |F ˜  Z(z) − FZ∗(z)| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (48)  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  35  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let Z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=', ZH and Z∗  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=', Z∗  H be independent continuous random  variables which depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Let F denote the distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' As n → ∞,  if  |FZh(z) − FZ∗  h(z)| → 0  holds for any h = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=', H and z in an interval (ah, bh) and Pr(Zh ∈ (ah, bh)) →  1, Pr(Z∗  h ∈ (ah, bh)) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Then,  ���F�H  h=1 chZh(z) − F�H  h=1 chZ∗  h(z)  ��� → 0  for any z ∈  ��H  h=1 chah, �H  h=1 chbh  �  , where ch’s are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' It suffices to show that, for any z ∈ (a1c1 + a2c2, b1c1 + b2c2),  ��Fc1Z1+c2Z2(z) − Fc1Z∗  1 +c2Z∗  2 (z)  �� → 0  as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' We have  Fc1Z1+c2Z2(z)  = Pr(c1Z1 + c2Z2 < z)  = Pr  �  Z1 < z − c2Z2  c1  �  =  �  R  FZ1  �z − c2x  c1  �  fZ2(x)dx  =  �  R  �  FZ1  �z − c2x  c1  �  − FZ∗  1  �z − c2x  c1  ��  fZ2(x)dx +  �  R  FZ∗  1  �z − c2x  c1  �  fZ2(x)dx  =  �  R  �  FZ1  �z − c2x  c1  �  − FZ∗  1  �z − c2x  c1  ��  fZ2(x)dx + Fc1Z∗  1 +c2Z2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (49)  When a1 < (z − c2x)/c1 < b1, we know  ����FZ1  �z − c2x  c1  �  − FZ∗  1  �z − c2x  c1  ����� → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (50)  Since FZ1(b1) − FZ1(a1) → 1 and FZ∗  1 (b1) − FZ∗  1 (a1) → 1, for any a < a1,  it holds that FZ1(a) → 0 and FZ∗  1 (a) → 0, and for any b > b1, FZ1(b) → 1  and FZ∗  1 (b) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Thus, (50) also holds when (z − c2x)/c1 is outside (a1, b1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Therefore, the first term of the right-hand side of (49) converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Then  ��Fc1Z1+c2Z2(z) − Fc1Z∗  1 +c2Z2(z)  �� → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Similarly, we have  ��Fc1Z∗  1 +c2Z2(z) − Fc1Z∗  1 +c2Z∗  2 (z)  �� → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  By the triangle inequality,  ��Fc1Z1+c2Z2(z) − Fc1Z∗  1 +c2Z∗  2 (z)  �� → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  36  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='5, for each stratum, under the conditions  ρ1 = ω(1/nh) and ρ2 = ω(1/nh), the distribution function of ˜ph converges to  that of N(ph, Vh) in the interval (0, 2ph) where  Vh =  �Nh − nh  Nh − 1  � ph(1 − ph)  nh  +  1  2ρ1n2  h  +  p2  h  2ρ2n2  h  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (51)  Let p∗ ∼ N(p, V ) where V = �H  h=1 w2  hVh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='6, in the interval (0, 2p),  we have  |F˜p(z) − Fp∗(z)| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (52)  where F˜p denotes the distribution function of ˜p designed in Algorithm 3 and  Fp∗ is the distribution function of p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Let L = p − z1−α/2  √  V , U = p + z1−α/2  √  V , ˜L = p − z1−α/2  �  �V and ˜U =  p + z1−α/2  �  �V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Note that L and U are constants whereas ˜L and ˜U are random  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Provided that nh’s are sufficiently large, U and L lie in the interval  where the following hold due to (52),  |F˜p(U) − Fp∗(U)| → 0  (53)  and  |F˜p(L) − Fp∗(L)| → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (54)  On the other hand, by Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='4, we know that ˜ph  p→ ph and  1  ˜nh  p→  1  nh under the conditions ρ1 = ω(1/nh) and ρ2 = ω(1/nh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' By the continuous  mapping theorem, �Vh  p→ Vh as nh → ∞, and, hence, �V  p→ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Therefore, ˜U  p→ U  and ˜L  p→ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Since F˜p is continuous, we have  |F˜p( ˜U) − F˜p(U)|  p→ 0  (55)  and  |F˜p(˜L) − F˜p(L)|  p→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  (56)  Therefore,  Pr  �  p ∈  �  ˜p − z1−α/2  �  �V , ˜p + z1−α/2  �  �V  ��  = Pr  �  p − z1−α/2  �  �V < ˜p < p + z1−α/2  �  �V  �  =  �  F˜p( ˜U) − F˜p(U)  �  + (F˜p(U) − Fp∗(U))  −  �  F˜p(˜L) − F˜p(L)  �  − (F˜p(L) − Fp∗(L)) + (Fp∗(U) − Fp∗(L)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Putting together (53) through (56) and Fp∗(U) − Fp∗(L) = 1 − α, we have  lim  n→∞ Pr  �  p ∈  �  ˜p − z1−α/2  �  �V , ˜p + z1−α/2  �  �V  ��  → 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  Since  1  ˜nh  p→  1  nh , under the conditions ρ1 = ω(1/nh) and ρ2 = ω(1/nh), it  holds that  1  2ρ1˜n2  h = oP ( 1  nh ) and  ˜p2  h  2ρ2˜n2  h = oP ( 1  nh ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Then �Vh  p→ Var(ˆph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Therefore,  �V  p→ Var(ˆp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='/Differentially Private Confidence Intervals  37  Acknowledgments  We are grateful for helpful conversations with and comments from (in no partic-  ular order) Rolando Rodriguez, Brian Finley, Jörg Drechsler, Gary Benedetto,  Michael Freiman, and Justin Doty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' This project was funded by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+page_content=' Confidentiality  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content='  [41] Wasserman, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' and Zhou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' A Statistical Framework for Dif-  ferential Privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
+page_content=' Journal of the American Statistical Association 105 375-  389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/btE_T4oBgHgl3EQfzhyy/content/2301.08324v1.pdf'}
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+Optical valley separation in two-dimensional semimetals with tilted Dirac cones
+Andrew Wild,∗ Eros Mariani,† and M. E. Portnoi‡
+Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom
+(Dated: January 12, 2023)
+Two-dimensional semimetals with tilted Dirac cones in the electronic band structure are shown
+to exhibit spatial separation of carriers belonging to different valleys under illumination. In stark
+contrast to gapped Dirac materials this optovalleytronic phenomenon occurs in systems with intact
+inversion and time-reversal symmetry that host massless Dirac cones in the band structure, thereby
+retaining the exceptional graphene-like transport properties. As a result we demonstrate that optical
+valley separation is possible at arbitrarily low photon frequencies including the deep infrared and
+terahertz regimes with full gate tunability via Pauli blocking. As a specific example of our theory, we
+demonstrate tunable valley separation in the proposed two-dimensional tilted Dirac cone semimetal
+8-Pmmn borophene for incident infrared photons at room temperature.
+I.
+INTRODUCTION
+Electrons in Dirac materials behave as massless
+fermions, existing in one of two inequivalent Dirac cones
+(known as valleys) with a low energy linear electronic
+dispersion. As the valleys in Dirac materials are widely
+separated in momentum space, carriers rarely scatter be-
+tween them in the absence of atomic-scale disorder. This
+makes Dirac materials ideal candidates for valleytronic
+applications where the valley index encodes quantum in-
+formation. For the realization of valleytronic devices it
+is vital to achieve independent control over carriers with
+different valley indices. The best known mechanism of
+optovalleytronics is in materials with broken inversion
+and preserved time-reversal symmetry that host gapped
+Dirac cones in their band structure.
+In such systems
+individual valleys can be addressed with different circu-
+larly polarized photons and under an external, in-plane
+electric field carriers from different valleys are steered in
+opposite directions yielding a finite photocurrent [1–4].
+The aforementioned mechanism does not offer optoval-
+leytronic applications for gapless two-dimensional (2D)
+Dirac fermions. Protecting the gapless nature of Dirac
+particles preserves their superior transport properties in
+the form of high mobility due to the suppression of back-
+scattering associated with Klein tunneling. These mer-
+its come at a cost – it becomes difficult to control the
+propagation of charge carriers.
+One solution to this
+problem utilizes the optical momentum alignment phe-
+nomenon in which photocarriers in Dirac materials such
+as graphene excited by linearly-polarized light propagate
+perpendicular to the polarization plane [5]. Momentum
+alignment could be exploited for valleytronic applica-
+tions in materials that exhibit a certain degree of val-
+ley anisotropy in the band structure.
+An example of
+such an anisotropy is the trigonal warping of the elec-
+tronic dispersion of graphene which becomes noticeable
+∗ A.Wild@exeter.ac.uk
+† E.Mariani@exeter.ac.uk
+‡ M.E.Portnoi@exeter.ac.uk
+from about 1eV above the apex of the Dirac cone [5, 6].
+However, this mechanism is limited to high excitation
+frequency preventing any control of valley separation by
+means of a gate voltage - the main asset of 2D materials
+for optoelectronic applications.
+In this work we propose a tunable mechanism of opti-
+cal valley separation in high-mobility 2D semimetals over
+a broad range of excitation frequencies including the elu-
+sive terahertz regime. This opportunity is offered by ma-
+terials hosting tilted Dirac cones in the electronic band
+structure where the two valleys are skewed in opposite
+directions (see inset of Fig. 1).
+Combining this intrin-
+sic valley anisotropy with optical momentum alignment
+and Pauli blocking effects it becomes possible to spatially
+separate photoexcited carriers with different valley index
+away from the light spot (see Fig. 1). The degree of valley
+polarization can be controlled via Pauli blocking which
+in 2D semimetals is readily tuned with a back gate. The
+spatial separation of valley carriers results in unequal val-
+ley populations at opposite sides of the light spot. This
+effect can be detected by measuring the degree of cir-
+cular polarization of the edge luminescence in a nearby
+gapped material [1, 2, 7], which ideally could be the same
+material with locally broken inversion symmetry.
+Tilted Dirac cones appear in three varieties:
+sub-
+critically tilted (type-I) with closed elliptical isoenergy
+contours, critically tilted (type-III) with open parabolic
+isoenergy contours and super-critically tilted (type-
+II) with open hyperbolic isoenergy contours.
+Two-
+dimensional materials hosting tilted Dirac cones are an
+ever growing family with candidate materials including
+8-Pmmn borophene [8–10], an organic salt α-(BEDT-
+TTF)2I3 [11] and many more [12–22].
+As a case study
+of our work we demonstrate tunable valley separation in
+8-Pmmn borophene upon illumination of infrared pho-
+tons at room temperature. We further demonstrate that
+type-II Dirac cone materials always possess perfect op-
+tical valley separation due to their super-critically tilted
+band dispersion. As an extension to our theory we show
+that type-III Dirac cones will display enhanced momen-
+tum alignment and emission of highly polarized terahertz
+photons via hot luminescence aided by the inclusion of
+carrier scattering.
+arXiv:2301.04564v1  [cond-mat.mes-hall]  11 Jan 2023
+
+2
+FIG. 1.
+Schematic of the suggested experimental setup
+for optically generating valley carrier separation in 2D tilted
+Dirac cone materials.
+A back-gate configuration with gate
+voltage VG can be used to the change the Fermi level EF.
+Linearly polarized photons are described by an electric field
+which propagates along the ˆz direction and is polarized at
+angle θ to the crystallographic ˆx axis. The inset shows the
+band structure of two tilted Dirac cones with valley index
+ξ = ± (sketched in green and orange). The incident photons
+induce interband transitions - in the shaded regions optical
+transitions are Pauli blocked. The resulting group velocity of
+photoexcited carriers depends on their valley index.
+II.
+MODEL
+We consider a 2D Dirac semimetal with tilted Dirac
+cones in the electronic band structure described by the
+Bloch Hamiltonian
+Hξ(q) = ℏvF
+�
+ξγηqx1 + ξηqxσx + qyσy
+�
+,
+(1)
+where σx and σy are Pauli matrices, 1 is the 2 × 2 iden-
+tity matrix and vF is the Fermi velocity along qy where
+q = (qx, qy) is the wavevector measured from the Dirac
+point in the Brillouin zone corresponding to the inequiv-
+alent valleys ξ = ±. The Dirac Hamiltonian has a tilt
+parameter γ which defines sub-critically tilted (|γ| < 1,
+type-I), critically tilted (|γ| = 1, type-III) and super-
+critically tilted (|γ| > 1, type-II) Dirac cones.
+The
+anisotropy parameter η > 0 scales the Dirac cone along
+the tilt axis.
+The valley-dependent eigenenergies and
+eigenvectors of the Hamiltonian are defined as
+Eξ
+±(q) = ℏvFq
+�
+ξγη cos(ϕq) ±
+�
+η2 cos2(ϕq) + sin2(ϕq)
+�
+,
+(2)
+and
+|Ψξ
+±(q)⟩ =
+1
+√
+2
+�
+±
+ξη cos(ϕq)−i sin(ϕq)
+√
+η2 cos2(ϕq)+sin2(ϕq)
+1
+�
+(3)
+respectively, for the conduction (+) and valence (−)
+bands.
+Here we have defined the wavevector in polar
+coordinates as qx = q cos(ϕq) and qy = q sin(ϕq) with
+q the radial wavevector and ϕq the wavevector angle.
+The semimetal has a Fermi level EF that can be tuned
+by means of a metallic back gate as shown in Fig. 1.
+The sample is incident upon by linearly polarized pho-
+tons with polarization ˆeθ = cos(θ)ˆx + sin(θ)ˆy and en-
+ergy hν.
+We treat the corresponding electric field as
+a time-dependent perturbation to the otherwise time-
+independent system using Fermi’s golden rule inducing
+vertical, interband transitions. In this work we do not
+consider intraband absorption as it requires knowledge
+of material-dependent scattering mechanisms and in the
+case of type-II Dirac cone materials, a detailed under-
+standing of the Fermi surface beyond the Dirac cone ap-
+proximation. We also note that our mechanism works
+for photons incident normally on the sample and does
+not rely on in-plane momentum transfer to electrons via
+phenomena such as photon-drag [23].
+There are three factors that govern the optical absorp-
+tion of photons. i) Initial and final states with wavevec-
+tor q must be separated by an energy of ∆E(q) =
+Eξ
++(q)−Eξ
+−(q) = hν. For a fixed frequency ν this condi-
+tion gives a set of wavevectors available for the transition
+given by
+∆E(q) = 2ℏvFq
+�
+η2 cos2(ϕq) + sin2(ϕq).
+(4)
+It can be seen that the states contributing to absorp-
+tion fall on the perimeter of an ellipse in wavevector
+space with semi-major and semi-minor axes (πν/vF and
+πν/ηvF) proportional to the frequency of the incident
+photon. For the case of the anisotropy parameter equal-
+ing unity (η = 1), this ellipse becomes a circle with ra-
+dius πν/vF. The geometry of this ellipse is independent
+of both the valley index (ξ) and tilt parameter (γ). ii)
+The transition rate describes the likelihood of an absorp-
+tion event occurring at a given wavevector. For linearly
+polarized photons the transition rate is proportional to
+the absolute value squared of the expectation value of the
+velocity operator projected along the axis of polarization
+[vcv(q) = ⟨Ψξ
+±(q)| ˆeθ · v |Ψξ
+∓(q)⟩] between the initial and
+final states[5, 24]. The velocity operator within the gra-
+dient approximation v = (1/ℏ)∇qHξ(q), leads to the
+squared velocity matrix element
+|vcv(ϕq)|2 =
+η2v2
+F
+η2 cos2(ϕq) + sin2(ϕq) sin2(ϕq − θ). (5)
+The formula above demonstrates momentum alignment
+in Dirac materials in which, upon absorption of photons
+polarized along θ, carriers are generated with wavevec-
+tor angle ϕq predominantly perpendicular to the polar-
+ization vector.
+The velocity matrix element is equiva-
+lent to that of non-tilted cones and hence is indepen-
+dent of both valley (ξ) and tilt parameter (γ). iii) For
+an absorption event to occur we must ensure that the
+initial state is occupied and the final state is empty to
+avoid Pauli blocking.
+We define these conditions with
+
+1=3
++=3
+Z
+E(z)
+<y
+0
+x
+DiracSemimetal
+VG
+Flectrode
+BackGate3
+FIG. 2. Distribution of photoexcited carriers (F ξ) in a type-I
+Dirac cone material with two valleys ξ = + and − sketched in
+green and orange respectively. In this plot the Dirac cone tilt
+is γ = 0.5, anisotropy is η = 1 and α ≈ 1/137 is the fine struc-
+ture constant. The Fermi level (EF) sits above the Dirac point
+creating regions of Pauli blocked transitions (dark gray). The
+group velocities of photoexcited carriers have been projected
+in to wavevector space and sketched as arrows. Due to Pauli
+blocking and the conical band structure, photoexcited carri-
+ers in the two valleys have differing group velocities. Optimal
+spatial separation of valley carriers is achieved when photon
+polarization is aligned with the crystallographic ˆy axis [panels
+(a) and (b)] compared to the ˆx axis [panels (c) and (d)].
+the Fermi-Dirac distributions fe(E) = 1 − fh(E) =
+�
+1+exp
+�
+(E−µ)/kBT
+��−1 for electrons (e) and holes (h)
+with Boltzmann constant kB, chemical potential µ and
+temperature T. Crucially, the regions of Pauli blocking
+are valley (ξ) and tilt parameter (γ) dependent leading
+to valley-dependent distributions of photoexcited carri-
+ers for certain values of the Fermi energy. Combining all
+of these factors we can write the angular distribution of
+excited carriers [5] at the instant of photocreation as
+Fξ(ϕq) = α gsℏ
+2πν |vcv(ϕq)|2 ×
+� ∞
+0
+δ
+�
+∆E(q) − hν
+�
+fe
+�
+Eξ
+−(q)
+�
+fh
+�
+Eξ
++(q)
+�
+qdq,
+(6)
+where gs = 2 accounts for the spin degeneracy, δ[...]
+is the Dirac delta function and α = e2/ℏc ≈ 1/137
+is the fine-structure constant in CGS units.
+We note
+that the normalization factor is chosen such that inte-
+grating over the wavevector angle yields the ratio of ab-
+sorbed photons. Accordingly, the absorption is defined
+as A = gv
+� 2π
+0
+Fξ(ϕq)dϕq, where gv = 2 is the valley de-
+generacy [25]. Note that for the case of graphene (γ = 0
+and η = 1) this expression simplifies to the well-known
+universal sheet absorption of A = πα ≈ 2.3%. For the
+full analytic expression of the distribution of photoex-
+cited carriers Fξ(ϕq) see Appendix A.
+III.
+RESULTS
+A.
+Type-I Dirac cones
+Initially, we consider a type-I (|γ| < 1) Dirac cone ma-
+terial with Fermi level sitting above the Dirac point (see
+Fig. 2). For incident photons polarized along the crystal-
+lographic ˆy axis (θ = π/2), the transition rate dictates
+that photoexcited electrons are created close to the tilt
+(qx) axis of the Dirac cone. However, any state inside the
+Pauli blocked regions does not undergo a transition. The
+resulting distribution of photoexcited carriers shows that
+if the tilt parameter takes on a positive value (γ > 0)
+the majority of carriers in valley ξ = + (−) are created
+on the right (left) side of the Dirac cone [see Figs. 2(a)
+and (b)]. The group velocities resulting from the coni-
+cal band structure dictate that at photocreation, carriers
+with valley number ξ = + will propagate to the right (ˆx
+direction) whilst carriers with valley number ξ = − will
+propagate to the left (−ˆx direction) towards the differ-
+ent sides of the illuminated light spot. We note that in
+general, the tilt parameter could take a negative value
+(γ < 0), in this case carriers from valley ξ will propagate
+in the −ξˆx direction.
+Although we have highlighted spatial separation of val-
+ley carriers for a specific polarization (θ = π/2), this phe-
+nomenon occurs, to a lesser extent, for all polarizations.
+As the polarization plane is rotated towards the crystal-
+lographic ˆx axis (θ = 0) an increased amount of carriers
+move along the ˆy axis, nevertheless, there is still a signif-
+icant amount of valley separation [see Figs. 2(c) and (d)].
+As it is possible to determine the orientation of the crys-
+tallographic axes of a Dirac semimetal with an optical
+procedure [25], aligning the incident photon polarization
+close to ˆy will yield optimal results.
+By tuning a gate-voltage we can modify the Fermi level
+which via Pauli blocking changes the distribution of pho-
+toexcited carriers.
+If the Fermi level sits close to the
+Dirac point then little to no transitions are Pauli blocked
+and carriers from both valleys propagate in all direc-
+tions, minimizing spatial separation of valley carriers [see
+Fig. 3(a)]. As the gate-voltage is increased, the regions
+of Pauli blocked transitions grow and we obtain consid-
+erable spatial separation of valley carriers [see Fig. 3(b)].
+If we were to increase the gate-voltage further, our ab-
+sorption would decrease [see Fig. 3(c)] before stopping
+altogether.
+
++=$
+EF
+é=y
+F-
+α
+Q
+2
+2
+(b)
+a
+é=x
+α
+Q
+2
+2
+1y
+(d)
+(c)
+qc
+qc4
+FIG. 3. (a)-(c) Distribution of photoexcited carriers (F +) in a single valley (ξ = +) of a type-I Dirac cone material for photons
+polarized along the crystallographic ˆy axis for a range of Fermi energies. Regions of Pauli blocked transitions are shaded in
+gray. In this plot the tilt parameter is γ = 0.5, the anisotropy parameter is η = 1 and α ≈ 1/137 is the fine structure constant.
+The group velocities of photoexcited carriers have been projected in to wavevector space and sketched as arrows. In panels
+(b) and (c), carriers with valley index ξ = + (−) will propagate along the x (−ˆx) direction - displaying spatial separation of
+valley carriers. On the contrary, in panel (a) the carriers move in each direction irrespective of their valley index. (d) The
+total absorption A (red) and the absolute value of the valley polarization degree
+��SR/L
+�� (purple) are plotted as functions of the
+Fermi level (EF) normalized by photon energy (hν). In panels (e) and (f), the degree of valley polarization and the absorption
+are plotted as functions of EF/hν and |γ|. The contours EF = hν(1 − |γ|2)/2 (dotted) and EF = hν(1 + |γ|)/2 (dot-dashed)
+correspond to the lines in panel (d).
+To quantify the valley separation we define the param-
+eter N ξ
+R(L) to be the percentage of photoexcited carriers
+in valley ξ that propagate to the right (left) side of the
+light spot along the crystallographic ˆx axis
+N ξ
+R =
+�
+Σ Fξ(ϕq)dϕq
+�
+Fξ(ϕq)dϕq
+.
+(7)
+The domain of integration Σ is defined as the set of angles
+ϕq corresponding to a positive ˆx component of the group
+velocity vξ
+x = (1/ℏ)∂qxEξ
++ where Fξ is defined by Eq. (6).
+The parameter N ξ
+L can be deduced from the normaliza-
+tion condition N ξ
+R + N ξ
+L = 1. Using these quantities, we
+can define the degree of valley polarization at the right-
+hand side of the light spot as
+SR = N +
+R − N −
+R
+N +
+R + N −
+R
+.
+(8)
+We provide an analytic expression for the valley polar-
+ization degree at either side of the light spot (SR/L) in
+Appendix B. If all photoexcited carriers at the right-hand
+side of the light spot are from valley ξ then the valley
+polarization takes on the value SR = ξ, in contrast, if
+there is an equal number of carriers from either valley
+then SR = 0. The valley polarization at the left-hand
+side of the light spot is the opposite of the right-hand
+side SL = −SR. The degree of valley polarization can
+be detected when photoexcited carriers propagate into a
+nearby gapped material, where they can recombine emit-
+ting circularly polarized photons with handedness deter-
+mined by their valley index ξ. The degree of valley po-
+larization maps on to the degree of circular polarization
+of the emitted light.
+We can now quantify the degree of valley polarization
+for a specific tilted Dirac cone geometry with different
+Fermi energies [see Fig. 3(d)]. In type-I Dirac cones the
+degree of valley polarization is maximal (
+��SR/L
+�� = 1)
+for gate-voltages greater than or equal to hν(1 − |γ|2)/2
+[see Fig. 3(e)].
+However, we note that for there to be
+any absorption, the Fermi energy must also be less than
+hν(1 + |γ|)/2 [see Fig. 3(f)].
+Therefore, it is theoreti-
+cally possible to achieve perfect valley carrier separation
+(
+��SL/R
+�� = 1 and A > 0) in type-I Dirac cones for gate
+voltages hν(1 − |γ|2)/2 ≤ EF < hν(1 + |γ|)/2 as demon-
+strated for a specific value of tilt in Figs. 3(b) and (c).
+This bound clearly vanishes in the limit of γ = 0 which
+emphasizes that this mechanism of spatial separation of
+valley carriers is not possible in non-tilted Dirac cone
+materials such as graphene.
+B.
+Special case: 8 − Pmmn borophene
+As a specific case study of our theory we demonstrate
+the spatial separation of valley carriers in the predicted
+tilted type-I Dirac cone material 8 − Pmmn borophene
+under illumination of infrared photons.
+In this mate-
+rial the Dirac cones have a Hamiltonian of the form
+given in Eq. (1) with Fermi velocity vF = 8.6 × 105ms−1,
+tilt parameter γ = 0.46 and anisotropy parameter η =
+0.80 [10]. It can be seen that with 8 − Pmmn borophene
+at room temperature (T = 300K) it will be possible to
+achieve valley separation by adding carriers to the sys-
+
+1.5
+F
+Type-II
+α-2
+SR/L
+qy
+Type-I
+ 1-2
+EF
+0.8
+hv
+(a)
+(b)
+(c)
+0
+0.6
+qα
+qα
+qc
+=0.5
+0.5
+TQ
+1
+0.4
+: 1-2
+11+1l
+[SR/L
+EF
+1+l
+A
+2
+hv
+0.2
+(d)
+[④]
+0
+0
+0
+0.25
+0.5
+0.75
+0
+0
+0.5
+0
+0.5
+1
+1
+EF/hv
+EF/hv
+EF/hv5
+FIG. 4. Distribution of photoexcited carriers (F ξ) in the can-
+didate type-I Dirac cone material 8 − Pmmn borophene. In
+this plot the Dirac cone tilt is γ = 0.46, the anisotropy param-
+eter is η = 0.80 and α ≈ 1/137 is the fine structure constant.
+The monolayer is incident upon by infrared photons polarized
+along the crystallographic ˆy axis with wavelength λ = 3µm at
+ambient temperature T = 300K. The finite temperature blurs
+the regions of Pauli blocked transitions, in conjunction with
+previous figures, the stronger the Pauli blocking the more
+opaque the gray shading. By utilizing a back-gate configu-
+ration, carriers can be added to the monolayer moving the
+Fermi level. The carrier density ∆n is defined as the density
+of carriers added from charge neutrality (∆n = 0). The group
+velocities of photoexcited carriers have been projected in to
+wavevector space and sketched as arrows.
+tem via a back-gate configuration [see Figs. 4(a) and (b)].
+By removing carriers, bringing the Fermi level back to
+charge neutrality, valley separation can be turned off [see
+Figs. 4(c) and (d)].
+C.
+Type-II Dirac cones
+Unlike their type-I counterparts, type-II (|γ| > 1)
+Dirac cones are super-critically tilted.
+The group ve-
+locity of these Dirac cones dictates that all photoexcited
+carriers will be spatially separated according to their val-
+ley index (see Fig. 5). In other words, as long as there
+is absorption [which requires EF < hν(1 + |γ|)/2] there
+will always be full spatial separation of valley carriers
+(
+��SR/L
+�� = 1) for any polarization of light [see Figs. 3(e)
+and (f)].
+As we always have full spatial separation of
+valley carriers in type-II Dirac cones, for demonstrative
+FIG. 5. Distribution of photoexcited carriers (F ξ) in a type-
+II Dirac cone material with valley indices ξ = + (−) sketched
+in green (orange). In this plot the Dirac cone tilt is γ = 1.1,
+the anisotropy parameter is η = 1 and α ≈ 1/137 is the fine
+structure constant. Regions of Pauli blocked transitions are
+shaded in gray. The group velocities of photoexcited carri-
+ers have been projected in to wavevector space and sketched
+as arrows. Due to the super-critically tilted band structure,
+all photoexcited carriers are spatially separated according to
+their valley index.
+purposes, we pick the polarization of light that maximizes
+the absorption and number of carriers which corresponds
+to θ = 0.
+D.
+Carrier relaxation enhanced momentum
+alignment in type-III Dirac cones
+Up until this point, critically tilted type-III Dirac cones
+have merely marked the boundary between type-I and II
+Dirac cones. However, when including the effects of car-
+rier relaxation, type-III Dirac cones offer an interesting
+mechanism of momentum alignment not possible in any
+other tilted Dirac cones.
+Critically tilted type-III (|γ| = 1) Dirac cones have
+a peculiar band structure in which the extrema of the
+upper and lower bands are one-dimensional lines in the
+wavevector space. First, we pump the material with ar-
+bitrarily polarized photons with energy hνp > EF (see
+Fig. 6). The resulting electrons and holes relax via a com-
+bination of carrier-carrier and carrier-phonon scattering
+processes to the most energetically favorable state. The
+holes aim to increase their energy, floating to the one-
+dimensional band maxima. The holes become stranded
+in these intermediate states which are perfectly aligned
+in momenta. Any holes that relaxed to a small wavevec-
+tor | qx | will be able to recombine with electrons in the
+upper band emitting photons of energy hνe < EF. Due
+to the momentum alignment of these holes, the emitted
+
+8-Pmmn borophene
+入= 3μm
+T = 300K
+△n = 5 × 1012 cm-2
++=3
+2n
+(b)
+a
+Q
+2m
+2m
+d
+C
+0+=3
+F
+α
+α
+12
+2
+qy
+0
+0
+qc
+qc6
+FIG. 6.
+Schematic of enhanced momentum alignment in a
+type-III Dirac cone with valley index ξ = + and tilt parameter
+γ = 1. Black arrows indicate interband absorption/emission
+and white arrows indicate relaxation via carrier-carrier and
+carrier-phonon scattering processes. After interband absorp-
+tion hν > EF holes float towards the Fermi level becoming
+trapped in an intermediate state with wavevector qy = 0.
+Upon recombination photons will be emitted with polar-
+ization aligned with the crystallographic ˆy axis and energy
+hν < EF.
+photons will have polarization aligned with the crystal-
+lographic ˆy axis.
+This mechanism of emission via an
+intermediate state is known as hot luminescence [26]. By
+modifying the Fermi level with a back-gate voltage, the
+emission energy of these photons can be tuned to the
+terahertz regime yielding a highly-polarized tunable ter-
+ahertz emitter.
+IV.
+CONCLUSION
+The realization of the valley-polarized currents via the
+valley Hall effect provided the elementary building block
+for valleytronic devices in gapped Dirac cone materials
+(see review articles [27–32] and references therein). This
+discovery sparked a desire for valleytronic components
+that in conjunction with the valley Hall effect could lead
+to valley-sensitive logic gates for classical and quantum
+computing applications [33]. In our work we demonstrate
+the spatial separation of valley carriers away from the
+light spot in gapless Dirac materials with tilted Dirac
+cones.
+Our discovery paves the way to the realization
+of novel valleytronic devices benefiting from the superior
+transport properties of massless Dirac fermions.
+With the recent burst of interest in massless tilted
+Dirac cone materials there have been several theoreti-
+cal works investigating the valley-dependent transport of
+carriers traversing gated junctions, waveguides and exter-
+nal fields [34–38]. Combining these transport techniques
+with the optical spatial separation of valley carriers pro-
+posed in our work could enable the design of valleytronic
+components such as valley filters and switches in gapless
+materials. It may also be possible to further direct the
+propagation of valley carriers across graphene-based in-
+terconnects based on electrostatic waveguides [39, 40],
+quantum wire leads [41] or gated junctions in externally
+applied fields [42]. Furthermore, the spatial separation of
+valley carriers in gapless tilted Dirac cone materials could
+be combined with valley-sensitive components of gapped
+Dirac cone materials such as valley transistors [43] or de-
+coding the valley index via emission of circularly polar-
+ized light [1, 2, 7]. This would require a detailed under-
+standing of the transport phenomena occurring at the
+interface between gapless and gapped Dirac cone materi-
+als. It is well-known that placing graphene on a hexago-
+nal boron nitride substrate induces a superlattice struc-
+ture inducing local regions with pseudo-gaps [44–46] - a
+similar technique for tilted Dirac materials should enable
+the seamless transport of valley carriers between gap-
+less and gapped regions in the spectrum allowing valley
+index measurement. Lastly, the theoretical and compu-
+tational predictions of two-dimensional materials hosting
+massless tilted Dirac cones are rapidly growing in num-
+ber [8–22]. The experimental efforts aiming at realizing
+these materials are catching up [47–49]. We hope that
+the prospect of optovalleytronics put forward in our work
+will stimulate further research into massless tilted Dirac
+cone materials.
+ACKNOWLEDGMENTS
+This work was supported by the EU H2020-MSCA-
+RISE projects TERASSE (Project No.
+823878) and
+DiSeTCom
+(Project
+No.
+823728)
+as
+well
+as
+by
+the NATO Science for Peace and Security project
+NATO.SPS.MYP.G5860.
+A.W. is supported by a UK
+EPSRC PhD studentship (Ref. 2239575). E.M. acknowl-
+edges financial support from the Royal Society (Grant
+No. IEC/R2/192166).
+Appendix A: Analytic expression for the distribution
+of photoexcited carriers in tilted Dirac cones
+In this Appendix, we present the expression for the
+distribution of photoexcited carriers Fξ(ϕq) for carriers
+with valley index ξ as a function of wavevector angle ϕq.
+Combining Eqs.(2) and (4)-(6) and solving the resultant
+integral yields
+
+hVe < EF
+hVp > EF7
+Fξ(ϕq) =α
+2
+η2 sin2(ϕq − θ)
+�
+η2 cos2(ϕq) + sin2(ϕq)
+�2
+�
+1 −
+�
+1 + exp
+�
+hνξηγ cos(ϕq)
+2kBT
+�
+η2 cos2(ϕq) + sin2(ϕq)
++
+hν
+2kBT −
+µ
+kBT
+��−1�
+×
+�
+1 + exp
+�
+hνξηγ cos(ϕq)
+2kBT
+�
+η2 cos2(ϕq) + sin2(ϕq)
+−
+hν
+2kBT −
+µ
+kBT
+��−1
+.
+(A1)
+Appendix B: Analytic expression for the
+polarization of valley carriers
+In this Appendix, we provide an analytic expression
+for the valley polarization of photoexcited carriers that
+propagate to the right-hand side of the light spot. As
+the Dirac cones in either valley are tilted in opposite di-
+rections, the percentage of carriers propagating to the
+right in valley ξ is equal to the percentage of carriers
+propagating to the left in valley −ξ yielding the iden-
+tity N ξ
+R = N −ξ
+L .
+Utilizing this expression, and the
+identity N ξ
+R + N ξ
+L = 1, Eq. (8) can be simplified to
+SR = ξ(N ξ
+R − N ξ
+L) which can be defined through the
+distribution of photoexcited carriers as
+SR = ξ
+� 2π
+0
+Fξ(ϕq)sign
+�
+vξ
+x(ϕq)
+�
+dϕq
+� 2π
+0
+Fξ(ϕq)dϕq
+,
+(B1)
+where sign(...) is the sign function and vξ
+x(ϕq) is the ˆx
+component of the group velocity.
+The valley polariza-
+tion at the left-hand side of the light spot is related to
+the right-hand side by the expression SL = −SR. Solv-
+ing Eq. (B1) for type-I (|γ| < 1) Dirac cone materials
+incident upon by photons polarized along the crystallo-
+graphic ˆy axis yields the degree of valley polarization
+SR = sign(γ)S where
+S =
+�
+�
+�
+�
+�
+S0,
+EF
+hν ≤ (1 − |γ|)/2,
+S1,
+(1 − |γ|)/2 < EF
+hν < (1 − |γ|2)/2,
+1,
+(1 − |γ|2)/2 ≤ EF
+hν < (1 + |γ|)/2,
+(B2)
+with
+S0 = 2
+π
+�
+arccos(− |γ|) − |γ|
+�
+1 − |γ|2�
+− 1,
+(B3)
+and
+S1 = 2
+�
+arccos(− |γ|) − |γ|
+�
+1 − |γ|2
+arccos(ψ) + ψ
+�
+1 − ψ2
+�
+− 1,
+(B4)
+where ψ = (1/ |γ|)(2EF/hν − 1). In this expression we
+have assumed the low temperature limit where kBT is
+small compared to the photon and Fermi energies.
+For type-II (|γ| > 1) Dirac cones, assuming there is
+absorption which requires EF < hν(1+|γ|)/2, the degree
+of valley polarization is equal to unity (
+��SR/L
+�� = 1).
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diff --git a/itE3T4oBgHgl3EQfgwov/content/tmp_files/load_file.txt b/itE3T4oBgHgl3EQfgwov/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf,len=686
+page_content='Optical valley separation in two-dimensional semimetals with tilted Dirac cones  Andrew Wild,∗ Eros Mariani,† and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Portnoi‡  Physics and Astronomy, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom  (Dated: January 12, 2023)  Two-dimensional semimetals with tilted Dirac cones in the electronic band structure are shown  to exhibit spatial separation of carriers belonging to different valleys under illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' In stark  contrast to gapped Dirac materials this optovalleytronic phenomenon occurs in systems with intact  inversion and time-reversal symmetry that host massless Dirac cones in the band structure, thereby  retaining the exceptional graphene-like transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' As a result we demonstrate that optical  valley separation is possible at arbitrarily low photon frequencies including the deep infrared and  terahertz regimes with full gate tunability via Pauli blocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' As a specific example of our theory, we  demonstrate tunable valley separation in the proposed two-dimensional tilted Dirac cone semimetal  8-Pmmn borophene for incident infrared photons at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  INTRODUCTION  Electrons in Dirac materials behave as massless  fermions, existing in one of two inequivalent Dirac cones  (known as valleys) with a low energy linear electronic  dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' As the valleys in Dirac materials are widely  separated in momentum space, carriers rarely scatter be-  tween them in the absence of atomic-scale disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' This  makes Dirac materials ideal candidates for valleytronic  applications where the valley index encodes quantum in-  formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' For the realization of valleytronic devices it  is vital to achieve independent control over carriers with  different valley indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The best known mechanism of  optovalleytronics is in materials with broken inversion  and preserved time-reversal symmetry that host gapped  Dirac cones in their band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  In such systems  individual valleys can be addressed with different circu-  larly polarized photons and under an external, in-plane  electric field carriers from different valleys are steered in  opposite directions yielding a finite photocurrent [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  The aforementioned mechanism does not offer optoval-  leytronic applications for gapless two-dimensional (2D)  Dirac fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Protecting the gapless nature of Dirac  particles preserves their superior transport properties in  the form of high mobility due to the suppression of back-  scattering associated with Klein tunneling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' These mer-  its come at a cost – it becomes difficult to control the  propagation of charge carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  One solution to this  problem utilizes the optical momentum alignment phe-  nomenon in which photocarriers in Dirac materials such  as graphene excited by linearly-polarized light propagate  perpendicular to the polarization plane [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Momentum  alignment could be exploited for valleytronic applica-  tions in materials that exhibit a certain degree of val-  ley anisotropy in the band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  An example of  such an anisotropy is the trigonal warping of the elec-  tronic dispersion of graphene which becomes noticeable  ∗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='Wild@exeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='uk  † E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='Mariani@exeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='uk  ‡ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='Portnoi@exeter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='uk  from about 1eV above the apex of the Dirac cone [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  However, this mechanism is limited to high excitation  frequency preventing any control of valley separation by  means of a gate voltage - the main asset of 2D materials  for optoelectronic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  In this work we propose a tunable mechanism of opti-  cal valley separation in high-mobility 2D semimetals over  a broad range of excitation frequencies including the elu-  sive terahertz regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' This opportunity is offered by ma-  terials hosting tilted Dirac cones in the electronic band  structure where the two valleys are skewed in opposite  directions (see inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Combining this intrin-  sic valley anisotropy with optical momentum alignment  and Pauli blocking effects it becomes possible to spatially  separate photoexcited carriers with different valley index  away from the light spot (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The degree of valley  polarization can be controlled via Pauli blocking which  in 2D semimetals is readily tuned with a back gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The  spatial separation of valley carriers results in unequal val-  ley populations at opposite sides of the light spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' This  effect can be detected by measuring the degree of cir-  cular polarization of the edge luminescence in a nearby  gapped material [1, 2, 7], which ideally could be the same  material with locally broken inversion symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Tilted Dirac cones appear in three varieties:  sub-  critically tilted (type-I) with closed elliptical isoenergy  contours, critically tilted (type-III) with open parabolic  isoenergy contours and super-critically tilted (type-  II) with open hyperbolic isoenergy contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Two-  dimensional materials hosting tilted Dirac cones are an  ever growing family with candidate materials including  8-Pmmn borophene [8–10], an organic salt α-(BEDT-  TTF)2I3 [11] and many more [12–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  As a case study  of our work we demonstrate tunable valley separation in  8-Pmmn borophene upon illumination of infrared pho-  tons at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' We further demonstrate that  type-II Dirac cone materials always possess perfect op-  tical valley separation due to their super-critically tilted  band dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' As an extension to our theory we show  that type-III Dirac cones will display enhanced momen-  tum alignment and emission of highly polarized terahertz  photons via hot luminescence aided by the inclusion of  carrier scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='04564v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='mes-hall]  11 Jan 2023  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Schematic of the suggested experimental setup  for optically generating valley carrier separation in 2D tilted  Dirac cone materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  A back-gate configuration with gate  voltage VG can be used to the change the Fermi level EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Linearly polarized photons are described by an electric field  which propagates along the ˆz direction and is polarized at  angle θ to the crystallographic ˆx axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The inset shows the  band structure of two tilted Dirac cones with valley index  ξ = ± (sketched in green and orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The incident photons  induce interband transitions - in the shaded regions optical  transitions are Pauli blocked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The resulting group velocity of  photoexcited carriers depends on their valley index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  MODEL  We consider a 2D Dirac semimetal with tilted Dirac  cones in the electronic band structure described by the  Bloch Hamiltonian  Hξ(q) = ℏvF  �  ξγηqx1 + ξηqxσx + qyσy  �  ,  (1)  where σx and σy are Pauli matrices, 1 is the 2 × 2 iden-  tity matrix and vF is the Fermi velocity along qy where  q = (qx, qy) is the wavevector measured from the Dirac  point in the Brillouin zone corresponding to the inequiv-  alent valleys ξ = ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The Dirac Hamiltonian has a tilt  parameter γ which defines sub-critically tilted (|γ| < 1,  type-I), critically tilted (|γ| = 1, type-III) and super-  critically tilted (|γ| > 1, type-II) Dirac cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  The  anisotropy parameter η > 0 scales the Dirac cone along  the tilt axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  The valley-dependent eigenenergies and  eigenvectors of the Hamiltonian are defined as  Eξ  ±(q) = ℏvFq  �  ξγη cos(ϕq) ±  �  η2 cos2(ϕq) + sin2(ϕq)  �  ,  (2)  and  |Ψξ  ±(q)⟩ =  1  √  2  �  ±  ξη cos(ϕq)−i sin(ϕq)  √  η2 cos2(ϕq)+sin2(ϕq)  1  �  (3)  respectively, for the conduction (+) and valence (−)  bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Here we have defined the wavevector in polar  coordinates as qx = q cos(ϕq) and qy = q sin(ϕq) with  q the radial wavevector and ϕq the wavevector angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  The semimetal has a Fermi level EF that can be tuned  by means of a metallic back gate as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  The sample is incident upon by linearly polarized pho-  tons with polarization ˆeθ = cos(θ)ˆx + sin(θ)ˆy and en-  ergy hν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  We treat the corresponding electric field as  a time-dependent perturbation to the otherwise time-  independent system using Fermi’s golden rule inducing  vertical, interband transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' In this work we do not  consider intraband absorption as it requires knowledge  of material-dependent scattering mechanisms and in the  case of type-II Dirac cone materials, a detailed under-  standing of the Fermi surface beyond the Dirac cone ap-  proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' We also note that our mechanism works  for photons incident normally on the sample and does  not rely on in-plane momentum transfer to electrons via  phenomena such as photon-drag [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  There are three factors that govern the optical absorp-  tion of photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' i) Initial and final states with wavevec-  tor q must be separated by an energy of ∆E(q) =  Eξ  +(q)−Eξ  −(q) = hν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' For a fixed frequency ν this condi-  tion gives a set of wavevectors available for the transition  given by  ∆E(q) = 2ℏvFq  �  η2 cos2(ϕq) + sin2(ϕq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  (4)  It can be seen that the states contributing to absorp-  tion fall on the perimeter of an ellipse in wavevector  space with semi-major and semi-minor axes (πν/vF and  πν/ηvF) proportional to the frequency of the incident  photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' For the case of the anisotropy parameter equal-  ing unity (η = 1), this ellipse becomes a circle with ra-  dius πν/vF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The geometry of this ellipse is independent  of both the valley index (ξ) and tilt parameter (γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' ii)  The transition rate describes the likelihood of an absorp-  tion event occurring at a given wavevector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' For linearly  polarized photons the transition rate is proportional to  the absolute value squared of the expectation value of the  velocity operator projected along the axis of polarization  [vcv(q) = ⟨Ψξ  ±(q)| ˆeθ · v |Ψξ  ∓(q)⟩] between the initial and  final states[5, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The velocity operator within the gra-  dient approximation v = (1/ℏ)∇qHξ(q), leads to the  squared velocity matrix element  |vcv(ϕq)|2 =  η2v2  F  η2 cos2(ϕq) + sin2(ϕq) sin2(ϕq − θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' (5)  The formula above demonstrates momentum alignment  in Dirac materials in which, upon absorption of photons  polarized along θ, carriers are generated with wavevec-  tor angle ϕq predominantly perpendicular to the polar-  ization vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  The velocity matrix element is equiva-  lent to that of non-tilted cones and hence is indepen-  dent of both valley (ξ) and tilt parameter (γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' iii) For  an absorption event to occur we must ensure that the  initial state is occupied and the final state is empty to  avoid Pauli blocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  We define these conditions with  1=3  +=3  Z  E(z)  <y  0  x  DiracSemimetal  VG  Flectrode  BackGate3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Distribution of photoexcited carriers (F ξ) in a type-I  Dirac cone material with two valleys ξ = + and − sketched in  green and orange respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' In this plot the Dirac cone tilt  is γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='5, anisotropy is η = 1 and α ≈ 1/137 is the fine struc-  ture constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The Fermi level (EF) sits above the Dirac point  creating regions of Pauli blocked transitions (dark gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The  group velocities of photoexcited carriers have been projected  in to wavevector space and sketched as arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Due to Pauli  blocking and the conical band structure, photoexcited carri-  ers in the two valleys have differing group velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Optimal  spatial separation of valley carriers is achieved when photon  polarization is aligned with the crystallographic ˆy axis [panels  (a) and (b)] compared to the ˆx axis [panels (c) and (d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  the Fermi-Dirac distributions fe(E) = 1 − fh(E) =  �  1+exp  �  (E−µ)/kBT  ��−1 for electrons (e) and holes (h)  with Boltzmann constant kB, chemical potential µ and  temperature T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Crucially, the regions of Pauli blocking  are valley (ξ) and tilt parameter (γ) dependent leading  to valley-dependent distributions of photoexcited carri-  ers for certain values of the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Combining all  of these factors we can write the angular distribution of  excited carriers [5] at the instant of photocreation as  Fξ(ϕq) = α gsℏ  2πν |vcv(ϕq)|2 ×  � ∞  0  δ  �  ∆E(q) − hν  �  fe  �  Eξ  −(q)  �  fh  �  Eξ  +(q)  �  qdq,  (6)  where gs = 2 accounts for the spin degeneracy, δ[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=']  is the Dirac delta function and α = e2/ℏc ≈ 1/137  is the fine-structure constant in CGS units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  We note  that the normalization factor is chosen such that inte-  grating over the wavevector angle yields the ratio of ab-  sorbed photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Accordingly, the absorption is defined  as A = gv  � 2π  0  Fξ(ϕq)dϕq, where gv = 2 is the valley de-  generacy [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Note that for the case of graphene (γ = 0  and η = 1) this expression simplifies to the well-known  universal sheet absorption of A = πα ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' For the  full analytic expression of the distribution of photoex-  cited carriers Fξ(ϕq) see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Type-I Dirac cones  Initially, we consider a type-I (|γ| < 1) Dirac cone ma-  terial with Fermi level sitting above the Dirac point (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' For incident photons polarized along the crystal-  lographic ˆy axis (θ = π/2), the transition rate dictates  that photoexcited electrons are created close to the tilt  (qx) axis of the Dirac cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' However, any state inside the  Pauli blocked regions does not undergo a transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The  resulting distribution of photoexcited carriers shows that  if the tilt parameter takes on a positive value (γ > 0)  the majority of carriers in valley ξ = + (−) are created  on the right (left) side of the Dirac cone [see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 2(a)  and (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The group velocities resulting from the coni-  cal band structure dictate that at photocreation, carriers  with valley number ξ = + will propagate to the right (ˆx  direction) whilst carriers with valley number ξ = − will  propagate to the left (−ˆx direction) towards the differ-  ent sides of the illuminated light spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' We note that in  general, the tilt parameter could take a negative value  (γ < 0), in this case carriers from valley ξ will propagate  in the −ξˆx direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Although we have highlighted spatial separation of val-  ley carriers for a specific polarization (θ = π/2), this phe-  nomenon occurs, to a lesser extent, for all polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  As the polarization plane is rotated towards the crystal-  lographic ˆx axis (θ = 0) an increased amount of carriers  move along the ˆy axis, nevertheless, there is still a signif-  icant amount of valley separation [see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 2(c) and (d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  As it is possible to determine the orientation of the crys-  tallographic axes of a Dirac semimetal with an optical  procedure [25], aligning the incident photon polarization  close to ˆy will yield optimal results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  By tuning a gate-voltage we can modify the Fermi level  which via Pauli blocking changes the distribution of pho-  toexcited carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  If the Fermi level sits close to the  Dirac point then little to no transitions are Pauli blocked  and carriers from both valleys propagate in all direc-  tions, minimizing spatial separation of valley carriers [see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 3(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' As the gate-voltage is increased, the regions  of Pauli blocked transitions grow and we obtain consid-  erable spatial separation of valley carriers [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 3(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  If we were to increase the gate-voltage further, our ab-  sorption would decrease [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 3(c)] before stopping  altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  +=$  EF  é=y  F-  α  Q  2  2  (b)  a  é=x  α  Q  2  2  1y  (d)  (c)  qc  qc4  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' (a)-(c) Distribution of photoexcited carriers (F +) in a single valley (ξ = +) of a type-I Dirac cone material for photons  polarized along the crystallographic ˆy axis for a range of Fermi energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Regions of Pauli blocked transitions are shaded in  gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' In this plot the tilt parameter is γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='5, the anisotropy parameter is η = 1 and α ≈ 1/137 is the fine structure constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  The group velocities of photoexcited carriers have been projected in to wavevector space and sketched as arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' In panels  (b) and (c), carriers with valley index ξ = + (−) will propagate along the x (−ˆx) direction - displaying spatial separation of  valley carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' On the contrary, in panel (a) the carriers move in each direction irrespective of their valley index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' (d) The  total absorption A (red) and the absolute value of the valley polarization degree  ��SR/L  �� (purple) are plotted as functions of the  Fermi level (EF) normalized by photon energy (hν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' In panels (e) and (f), the degree of valley polarization and the absorption  are plotted as functions of EF/hν and |γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The contours EF = hν(1 − |γ|2)/2 (dotted) and EF = hν(1 + |γ|)/2 (dot-dashed)  correspond to the lines in panel (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  To quantify the valley separation we define the param-  eter N ξ  R(L) to be the percentage of photoexcited carriers  in valley ξ that propagate to the right (left) side of the  light spot along the crystallographic ˆx axis  N ξ  R =  �  Σ Fξ(ϕq)dϕq  �  Fξ(ϕq)dϕq  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  (7)  The domain of integration Σ is defined as the set of angles  ϕq corresponding to a positive ˆx component of the group  velocity vξ  x = (1/ℏ)∂qxEξ  + where Fξ is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  The parameter N ξ  L can be deduced from the normaliza-  tion condition N ξ  R + N ξ  L = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Using these quantities, we  can define the degree of valley polarization at the right-  hand side of the light spot as  SR = N +  R − N −  R  N +  R + N −  R  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  (8)  We provide an analytic expression for the valley polar-  ization degree at either side of the light spot (SR/L) in  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' If all photoexcited carriers at the right-hand  side of the light spot are from valley ξ then the valley  polarization takes on the value SR = ξ, in contrast, if  there is an equal number of carriers from either valley  then SR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The valley polarization at the left-hand  side of the light spot is the opposite of the right-hand  side SL = −SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The degree of valley polarization can  be detected when photoexcited carriers propagate into a  nearby gapped material, where they can recombine emit-  ting circularly polarized photons with handedness deter-  mined by their valley index ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The degree of valley po-  larization maps on to the degree of circular polarization  of the emitted light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  We can now quantify the degree of valley polarization  for a specific tilted Dirac cone geometry with different  Fermi energies [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 3(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' In type-I Dirac cones the  degree of valley polarization is maximal (  ��SR/L  �� = 1)  for gate-voltages greater than or equal to hν(1 − |γ|2)/2  [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 3(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  However, we note that for there to be  any absorption, the Fermi energy must also be less than  hν(1 + |γ|)/2 [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 3(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Therefore, it is theoreti-  cally possible to achieve perfect valley carrier separation  (  ��SL/R  �� = 1 and A > 0) in type-I Dirac cones for gate  voltages hν(1 − |γ|2)/2 ≤ EF < hν(1 + |γ|)/2 as demon-  strated for a specific value of tilt in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 3(b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  This bound clearly vanishes in the limit of γ = 0 which  emphasizes that this mechanism of spatial separation of  valley carriers is not possible in non-tilted Dirac cone  materials such as graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Special case: 8 − Pmmn borophene  As a specific case study of our theory we demonstrate  the spatial separation of valley carriers in the predicted  tilted type-I Dirac cone material 8 − Pmmn borophene  under illumination of infrared photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  In this mate-  rial the Dirac cones have a Hamiltonian of the form  given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' (1) with Fermi velocity vF = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='6 × 105ms−1,  tilt parameter γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='46 and anisotropy parameter η =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='80 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' It can be seen that with 8 − Pmmn borophene  at room temperature (T = 300K) it will be possible to  achieve valley separation by adding carriers to the sys-  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='5  F  Type-II  α-2  SR/L  qy  Type-I  1-2  EF  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='8  hv  (a)  (b)  (c)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='6  qα  qα  qc  =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='5  TQ  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='4  : 1-2  11+1l  [SR/L  EF  1+l  A  2  hv  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='2  (d)  [④]  0  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='75  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='5  1  1  EF/hv  EF/hv  EF/hv5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Distribution of photoexcited carriers (F ξ) in the can-  didate type-I Dirac cone material 8 − Pmmn borophene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' In  this plot the Dirac cone tilt is γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='46, the anisotropy param-  eter is η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='80 and α ≈ 1/137 is the fine structure constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  The monolayer is incident upon by infrared photons polarized  along the crystallographic ˆy axis with wavelength λ = 3µm at  ambient temperature T = 300K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The finite temperature blurs  the regions of Pauli blocked transitions, in conjunction with  previous figures, the stronger the Pauli blocking the more  opaque the gray shading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' By utilizing a back-gate configu-  ration, carriers can be added to the monolayer moving the  Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The carrier density ∆n is defined as the density  of carriers added from charge neutrality (∆n = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The group  velocities of photoexcited carriers have been projected in to  wavevector space and sketched as arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  tem via a back-gate configuration [see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 4(a) and (b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  By removing carriers, bringing the Fermi level back to  charge neutrality, valley separation can be turned off [see  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 4(c) and (d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Type-II Dirac cones  Unlike their type-I counterparts, type-II (|γ| > 1)  Dirac cones are super-critically tilted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  The group ve-  locity of these Dirac cones dictates that all photoexcited  carriers will be spatially separated according to their val-  ley index (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' In other words, as long as there  is absorption [which requires EF < hν(1 + |γ|)/2] there  will always be full spatial separation of valley carriers  (  ��SR/L  �� = 1) for any polarization of light [see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 3(e)  and (f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  As we always have full spatial separation of  valley carriers in type-II Dirac cones, for demonstrative  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Distribution of photoexcited carriers (F ξ) in a type-  II Dirac cone material with valley indices ξ = + (−) sketched  in green (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' In this plot the Dirac cone tilt is γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='1,  the anisotropy parameter is η = 1 and α ≈ 1/137 is the fine  structure constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Regions of Pauli blocked transitions are  shaded in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The group velocities of photoexcited carri-  ers have been projected in to wavevector space and sketched  as arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Due to the super-critically tilted band structure,  all photoexcited carriers are spatially separated according to  their valley index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  purposes, we pick the polarization of light that maximizes  the absorption and number of carriers which corresponds  to θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Carrier relaxation enhanced momentum  alignment in type-III Dirac cones  Up until this point, critically tilted type-III Dirac cones  have merely marked the boundary between type-I and II  Dirac cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' However, when including the effects of car-  rier relaxation, type-III Dirac cones offer an interesting  mechanism of momentum alignment not possible in any  other tilted Dirac cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Critically tilted type-III (|γ| = 1) Dirac cones have  a peculiar band structure in which the extrema of the  upper and lower bands are one-dimensional lines in the  wavevector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' First, we pump the material with ar-  bitrarily polarized photons with energy hνp > EF (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The resulting electrons and holes relax via a com-  bination of carrier-carrier and carrier-phonon scattering  processes to the most energetically favorable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The  holes aim to increase their energy, floating to the one-  dimensional band maxima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The holes become stranded  in these intermediate states which are perfectly aligned  in momenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Any holes that relaxed to a small wavevec-  tor | qx | will be able to recombine with electrons in the  upper band emitting photons of energy hνe < EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Due  to the momentum alignment of these holes, the emitted  8-Pmmn borophene  入= 3μm  T = 300K  △n = 5 × 1012 cm-2  +=3  2n  (b)  a  Q  2m  2m  d  C  0+=3  F  α  α  12  2  qy  0  0  qc  qc6  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Schematic of enhanced momentum alignment in a  type-III Dirac cone with valley index ξ = + and tilt parameter  γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Black arrows indicate interband absorption/emission  and white arrows indicate relaxation via carrier-carrier and  carrier-phonon scattering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' After interband absorp-  tion hν > EF holes float towards the Fermi level becoming  trapped in an intermediate state with wavevector qy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Upon recombination photons will be emitted with polar-  ization aligned with the crystallographic ˆy axis and energy  hν < EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  photons will have polarization aligned with the crystal-  lographic ˆy axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  This mechanism of emission via an  intermediate state is known as hot luminescence [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' By  modifying the Fermi level with a back-gate voltage, the  emission energy of these photons can be tuned to the  terahertz regime yielding a highly-polarized tunable ter-  ahertz emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  CONCLUSION  The realization of the valley-polarized currents via the  valley Hall effect provided the elementary building block  for valleytronic devices in gapped Dirac cone materials  (see review articles [27–32] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' This  discovery sparked a desire for valleytronic components  that in conjunction with the valley Hall effect could lead  to valley-sensitive logic gates for classical and quantum  computing applications [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' In our work we demonstrate  the spatial separation of valley carriers away from the  light spot in gapless Dirac materials with tilted Dirac  cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Our discovery paves the way to the realization  of novel valleytronic devices benefiting from the superior  transport properties of massless Dirac fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  With the recent burst of interest in massless tilted  Dirac cone materials there have been several theoreti-  cal works investigating the valley-dependent transport of  carriers traversing gated junctions, waveguides and exter-  nal fields [34–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Combining these transport techniques  with the optical spatial separation of valley carriers pro-  posed in our work could enable the design of valleytronic  components such as valley filters and switches in gapless  materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' It may also be possible to further direct the  propagation of valley carriers across graphene-based in-  terconnects based on electrostatic waveguides [39, 40],  quantum wire leads [41] or gated junctions in externally  applied fields [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Furthermore, the spatial separation of  valley carriers in gapless tilted Dirac cone materials could  be combined with valley-sensitive components of gapped  Dirac cone materials such as valley transistors [43] or de-  coding the valley index via emission of circularly polar-  ized light [1, 2, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' This would require a detailed under-  standing of the transport phenomena occurring at the  interface between gapless and gapped Dirac cone materi-  als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' It is well-known that placing graphene on a hexago-  nal boron nitride substrate induces a superlattice struc-  ture inducing local regions with pseudo-gaps [44–46] - a  similar technique for tilted Dirac materials should enable  the seamless transport of valley carriers between gap-  less and gapped regions in the spectrum allowing valley  index measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Lastly, the theoretical and compu-  tational predictions of two-dimensional materials hosting  massless tilted Dirac cones are rapidly growing in num-  ber [8–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' The experimental efforts aiming at realizing  these materials are catching up [47–49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' We hope that  the prospect of optovalleytronics put forward in our work  will stimulate further research into massless tilted Dirac  cone materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  This work was supported by the EU H2020-MSCA-  RISE projects TERASSE (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  823878) and  DiSeTCom  (Project  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  823728)  as  well  as  by  the NATO Science for Peace and Security project  NATO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='SPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='MYP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='G5860.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' is supported by a UK  EPSRC PhD studentship (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 2239575).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' acknowl-  edges financial support from the Royal Society (Grant  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' IEC/R2/192166).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Appendix A: Analytic expression for the distribution  of photoexcited carriers in tilted Dirac cones  In this Appendix, we present the expression for the  distribution of photoexcited carriers Fξ(ϕq) for carriers  with valley index ξ as a function of wavevector angle ϕq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Combining Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' (2) and (4)-(6) and solving the resultant  integral yields  hVe < EF  hVp > EF7  Fξ(ϕq) =α  2  η2 sin2(ϕq − θ)  �  η2 cos2(ϕq) + sin2(ϕq)  �2  �  1 −  �  1 + exp  �  hνξηγ cos(ϕq)  2kBT  �  η2 cos2(ϕq) + sin2(ϕq)  +  hν  2kBT −  µ  kBT  ��−1�  ×  �  1 + exp  �  hνξηγ cos(ϕq)  2kBT  �  η2 cos2(ϕq) + sin2(ϕq)  −  hν  2kBT −  µ  kBT  ��−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  (A1)  Appendix B: Analytic expression for the  polarization of valley carriers  In this Appendix, we provide an analytic expression  for the valley polarization of photoexcited carriers that  propagate to the right-hand side of the light spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' As  the Dirac cones in either valley are tilted in opposite di-  rections, the percentage of carriers propagating to the  right in valley ξ is equal to the percentage of carriers  propagating to the left in valley −ξ yielding the iden-  tity N ξ  R = N −ξ  L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  Utilizing this expression, and the  identity N ξ  R + N ξ  L = 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' (8) can be simplified to  SR = ξ(N ξ  R − N ξ  L) which can be defined through the  distribution of photoexcited carriers as  SR = ξ  � 2π  0  Fξ(ϕq)sign  �  vξ  x(ϕq)  �  dϕq  � 2π  0  Fξ(ϕq)dϕq  ,  (B1)  where sign(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=') is the sign function and vξ  x(ϕq) is the ˆx  component of the group velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  The valley polariza-  tion at the left-hand side of the light spot is related to  the right-hand side by the expression SL = −SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Solv-  ing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' (B1) for type-I (|γ| < 1) Dirac cone materials  incident upon by photons polarized along the crystallo-  graphic ˆy axis yields the degree of valley polarization  SR = sign(γ)S where  S =  �  �  �  �  �  S0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  EF  hν ≤ (1 − |γ|)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  (1 − |γ|)/2 < EF  hν < (1 − |γ|2)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  (1 − |γ|2)/2 ≤ EF  hν < (1 + |γ|)/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  (B2)  with  S0 = 2  π  �  arccos(− |γ|) − |γ|  �  1 − |γ|2�  − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  (B3)  and  S1 = 2  �  arccos(− |γ|) − |γ|  �  1 − |γ|2  arccos(ψ) + ψ  �  1 − ψ2  �  − 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  (B4)  where ψ = (1/ |γ|)(2EF/hν − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' In this expression we  have assumed the low temperature limit where kBT is  small compared to the photon and Fermi energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  For type-II (|γ| > 1) Dirac cones, assuming there is  absorption which requires EF < hν(1+|γ|)/2, the degree  of valley polarization is equal to unity (  ��SR/L  �� = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Xiao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Yao, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Niu, Valley-contrasting physics  in graphene: Magnetic moment and topological trans-  port, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' 99, 236809 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content='  [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Yao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
+page_content=' Xiao, and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/itE3T4oBgHgl3EQfgwov/content/2301.04564v1.pdf'}
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+1
+RAPTOR: Advanced Persistent Threat Detection in
+Industrial IoT via Attack Stage Correlation
+Ayush Kumar and Vrizlynn L.L. Thing
+Cyber Security Strategic Technology Centre
+ST Engineering
+Email: ayush.kumar@u.nus.edu, vriz@ieee.org
+Abstract—IIoT (Industrial Internet-of-Things) systems are get-
+ting more prone to attacks by APT (Advanced Persistent Threat)
+adversaries. Past APT attacks on IIoT systems such as the 2016
+Ukrainian power grid attack which cut off the capital Kyiv
+off power for an hour and the 2017 Saudi petrochemical plant
+attack which almost shut down the plant’s safety controllers have
+shown that APT campaigns can disrupt industrial processes,
+shut down critical systems and endanger human lives. In this
+work, we propose RAPTOR, a system to detect APT campaigns
+in IIoT environments. RAPTOR detects and correlates various
+APT attack stages (adapted to IIoT) using multiple data sources.
+Subsequently, it constructs a high-level APT campaign graph
+which can be used by cybersecurity analysts towards attack
+analysis and mitigation. A performance evaluation of RAPTOR’s
+APT stage detection stages shows high precision and low false
+positive/negative rates. We also show that RAPTOR is able to
+construct the APT campaign graph for APT attacks (modelled
+after real-world attacks on ICS/OT infrastructure) executed on
+our IIoT testbed.
+Index Terms—Industrial Internet of Things, IoT, IIoT, Ad-
+vanced Persistent Threat, APT, APT Detection
+I. INTRODUCTION
+The Internet of Things (IoT) is a network of sensing
+devices with low-power and limited processing capability,
+which exchange data with each other and/or systems (e.g.,
+gateways, cloud servers), normally using wired and wireless
+technologies. Industrial IoT (IIoT) refers to the extension of
+IoT in industrial sectors and applications. With a strong focus
+on machine-to-machine (M2M) communication, big data, and
+machine learning, the IIoT enables industries and enterprises
+to have better efficiency and reliability in their operations.
+What makes IIoT distinct from IoT is the intersection of
+information technology (IT) and operational technology (OT).
+However, this convergence has widened the attack surface and
+increased the potential risks of cyberattacks being launched
+against such critical systems. A more significant concern
+relates to legacy OT systems (i.e., brownfield IIoTs) which are
+usually isolated but are becoming more connected with new
+IT technologies. Sophisticated attackers can easily gain access
+to such brownfield IIoT systems and damage their operation
+for lengthy periods of time.
+IIoT systems are more prone to attacks by APT adversaries
+than traditional ICS (Industrial Control Systems)/OT (Oper-
+ational Technology) networks [1] mainly due to addition of
+Corresponding author: A. Kumar (email: ayush.kumar@u.nus.edu).
+connectivity to IT networks (enabling lateral movement for
+attackers) and introduction of M2M communications which
+connect various new and intelligent devices. Furthermore, ICS
+assets themselves are prime targets for APT campaigns. This is
+because ICS devices often run on legacy, proprietary software
+which were not designed with security in mind and are not
+patched/updated regularly due to concerns over downtime in
+critical systems. APT attacks can be used to gather ICS-related
+intelligence, disrupt industrial processes, shut down critical
+systems and endanger human lives. Such APT attacks by well-
+resourced groups have happened a number of times in the
+past, e.g., the 2010 Stuxnet attack which damaged centrifuges
+in Iranian uranium enrichment plants and caused a significant
+setback to Iranian nuclear program [2], the 2014 attack on a
+German steel mill [3] which prevented the blast furnace from
+shutting down causing massive damage, the 2015 attack on
+Ukrainian power companies which disconnected substations
+from the power grid leaving 225,000+ customers without
+power for more than 6 hours [4], the 2016 attack on Ukranian
+transmission level substation which cut a fifth of the capital
+Kyiv off power for an hour [5] and the 2017 attack on a
+Saudi petrochemical plant which almost shut down the plant’s
+safety controllers which could have caused an explosion [6].
+Industroyer/Crashoverride, the malware behind 2016 Ukraine
+power grid attack and TRITON, the malware behind 2017
+Saudi petrochemical plant attack are still active, targeting
+electrical substations and energy utilities respectively [7], [8].
+New IIoT malware such as Incontroller/Pipedream [9], which
+was revealed as recently as 2022, contains modules that target
+specific ICS devices such as OPC servers, Schneider Electric
+PLCs using Modbus and Codesys protocols, and Omron PLCs
+and servo drives.
+Security solutions which are commonly deployed in IT net-
+works such as firewalls, NIDS (Network Intrusion Detection
+System), SIEM (Security Information and Event Management)
+products are not common in ICS/OT networks. Thus, there
+is an impending need to design systems which can detect
+ongoing APT attack campaigns in IIoT environments early
+before they cause substantial damage. Such systems should
+satisfy the following requirements:
+• Not dependent on deployment of proprietary, third-party
+security solutions whose features can vary across vendors.
+Instead, the system should use existing open-source and
+readily available information sources for its operation.
+arXiv:2301.11524v1  [cs.CR]  27 Jan 2023
+
+2
+• Detect the APT attack tactics with high accuracy and low
+number of false positives/negatives
+• Able to reconstruct the APT attack campaign with suffi-
+cient details and present them in a compact, high-level
+form which cybersecurity analysts can use for further
+analysis and attack mitigation.
+An APT campaign typically consists of various stages which
+occur one after another (this will be explained later in the
+paper). However, not all stages defined in existing attack
+frameworks are found together in real-world APT campaigns.
+Each stage in an APT campaign is linked to the previously
+executed stages, e.g., in terms of chronology of execution,
+target hosts affected. Thus, if we are able to detect some of the
+individual attack stages, the above fact can be exploited to cor-
+relate the detected stages and reconstruct the APT campaign
+with some acceptable margin of error. In this work, we present
+RAPTOR, a system for detecting ongoing APT campaigns
+in IIoT environments. RAPTOR’s main component is an
+APT attack-stage detection and correlation engine. It takes
+as input a variety of readily available, non-proprietary data
+sources (such as host logs, network traffic traces) and detects
+attack tactics (including those specific to IIoT environments)
+which are part of an ongoing APT campaign. The attack
+tactics are then stitched together based on their attributes to
+produce the APT attack campaign graph, which is a high-level
+representation of APT activity across the target IIoT network.
+The main contributions of our work are as follows:
+• We present, RAPTOR, a system for detecting ongoing
+APT campaigns in IIoT environments.
+• We employ a novel attack-stage detection and correlation
+approach which uses open source, readily available data
+sources for its operation.
+• RAPTOR constructs a compact, high level APT cam-
+paign graph which can be useful for cybersecurity ana-
+lysts.
+• We evaluate RAPTOR’s performance using a new dataset
+which includes attack TTPs close to real-world APT
+attacks on ICS/OT environments.
+II. RELATED WORK
+A. APT detection in Enterprise networks
+APT detection in enterprise network settings has received
+significant attention in computer security literature in recent
+years. Milajerdi et al. [10] have presented HOLMES, a system
+for APT campaign detection with high confidence. It maps
+activities in host audit logs and enterprise security alerts to
+the cyber kill-chain, correlates the alerts generated by APT
+steps based on information flow between low-level entities
+(files, processes, etc.) and builds a high-level scenario graph
+encapsulating the attack TTPs and information flows between
+entities involved in the TTPs. CONAN [11] is an APT
+detection system which makes two major modifications to
+HOLMES’s approach: one, it utilizes a state-based detection
+framework where all processes and files are represented as
+data structures similar to finite-state automata and two, it
+focuses on three constant attack phases of APTs- 1) deploy and
+execute the attacker’s code, 2) collect sensitive information or
+cause damage, and 3) communicate with the C&C server or
+exfiltrate sensitive data. Wilkens et al. [12] have proposed a
+method to construct APT attack graphs from IDS (Intrusion
+Detection System) alerts to assist human analysts. IDS alerts
+are clustered into meta-alerts and single alerts, assigned poten-
+tial attack stages and finally used to synthesize APT scenario
+graphs based on a kill chain state machine.
+Irshad et al. [13] have proposed TRACE, a provenance
+tracking system for enterprise-wide APT detection. TRACE
+offers host-level provenance tracking at the granularity of
+program executions units and integrates provenance collected
+from individual hosts to construct distributed enterprise-wide
+causal graphs. Hopper [14] instead focuses on the detection of
+a single APT attack stage: lateral movement, within enterprise
+networks. It does so by building a graph of login activity
+among machines in a network and identifying suspicious login
+sequences. A path inference algorithm is then deployed to
+identify the broader paths that each login belongs to, "caused"
+by the same user. Finally, an anomaly detection algorithm is
+applied to conservatively infer the set of login paths most
+likely to reflect lateral movement.
+However, all the above works suffer from certain limitations:
+•
+[10]–[14] focus on enterprise networks only.
+• While [10], [11], [13] have been designed using audit
+logs-based provenance and [14] has been designed using
+enterprise logs, [12] has been designed using network
+IDS alerts. Since each of these works is based on a single
+data source, they are unable to leverage the information
+provided by other data sources. Further, these works are
+only able to detect a subset of APT attack stages.
+• Exclusively graph-based approaches [10], [13] tend to
+suffer from dependency explosion problem which means
+that with time, each graph node gives rise to many
+edges which in turn give rise to new nodes and so on.
+A backward trace through the graph to infer a path
+exponentially increases the number of probable paths.
+• [12], [14] work offline and not in real-time which means
+that they assume that all the data required for APT
+detection is available at once. Though [10], [11], [13]
+claim that they work in real-time, they don’t provide the
+time delays encountered in APT detection.
+• Except for [12], no other work mentioned above detects
+an APT campaign from a whole-network perspective.
+Unfortunately, there has been no research work yet on
+detecting APT campaigns in IIoT settings which is the focus
+of this paper. Our work differs from [10]–[14] in the following
+aspects:
+• The focus is on APT detection in IIoT settings which
+combines both IT and OT environments. The APT attack
+stages used to design the detection system in our paper
+have been adapted to IIoT settings.
+• We leverage data from various sources such as network
+packet traces, host logs (including audit logs) as well
+as NIDS/HIDS alerts instead of limiting ourselves to
+a specific or proprietary data source. The optimal data
+source(s) for detection of each APT attack stage is(are)
+identified and used.
+
+3
+• We perform APT campaign detection from a whole-
+network perspective instead of detecting APT artifacts on
+individual hosts only.
+B. ICS/IIoT datasets
+In the past, researchers have built datasets (consisting of
+network traffic, system logs, etc.) to design IDSes for ICS
+(Industrial Control System). Morris et al. [15] have provided
+a labeled dataset which includes network traffic captured
+from a laboratory scale gas pipeline system during normal
+operation and during 35 cyber-attacks that affect the Modbus
+protocol such as reconnaissance, response injection, command
+injection and DoS. Similarly, Pan et al. [16] have produced
+a dataset consisting of synchrophasor measurement data and
+audit logs from a power system testbed, amounting to a total of
+10,000 simulated instances. The dataset includes 25 scenarios
+consisting of power system single-line-to-ground (SLG) faults,
+normal operation and cyber-attacks (relay trip command injec-
+tion, relay function disabling and SLG fault replay). Another
+dataset has been collected from the Secure Water Treatment
+(SWaT) testbed [17] and includes sensor/actuator readings
+both during normal operation and under 36 attacks such as
+bias attack, replay, single and multiple point attacks. The final
+dataset includes 946,772 samples consisting of 51 attributes.
+More recently, Myers et al. [18] have generated two datasets
+consisting of ICS device logs and network traffic captures
+from two separate industrial process setups respectively, during
+normal operation and during attacks on the testbed PLCs
+(command injection, flooding, etc.). Marcio et al. [19] have
+built a dataset consisting of six pre-decided ML-based flow-
+level features extracted from the network traffic captured
+during normal as well as under attack operation of a SCADA
+system testbed consisting of a water storage tank’s control
+system. Finally, Muna et al. [20] have presented the X-IIoTID
+intrusion dataset which targets modern IIoT systems. The
+dataset covers various attack scenarios and attacks related to
+newer IIoT connectivity protocols generated according to a
+pre-defined attack taxonomy. The data collected includes end-
+to-end network traffic (from physical field devices to the edge
+gateway and from edge gateway to the cloud and enterprise
+devices), host logs and computing resources, and IDS alert
+logs (OSSEC, Zeek) collected at the edge gateway for both
+normal and under attack operation of the testbed. The final
+dataset consists of 820,834 instances and 59 features.
+However, the above datasets suffer from certain limitations:
+• Many of the existing ICS datasets [15]–[19] highly de-
+pend on features related to sensor measurements, actua-
+tors’ statuses, and specific parameters of industrial pro-
+tocol packets, which limit their use for diverse industrial
+systems.
+• Most of those datasets lack a clear attack taxonomy and
+have not incorporated multi-stage attacks related to newer
+IIoT connectivity protocols and services (e.g., CoAP,
+MQTT).
+• The X-IIoTID dataset addresses the above shortcomings
+of previous datasets. However, all the above datasets
+including X-IIoTID provide pre-decided ML features
+extracted from network flows only and do not include
+raw traffic traces from which additional features can be
+extracted. Further, the X-IIoTID dataset provides host
+logs and IDS alert logs collected at the edge gateway only
+since according to the dataset authors, edge gateway is
+the highest priority target for attackers. The dataset does
+not include APT campaigns.
+To address the limitations of X-IIoTID dataset for APT
+detection, we build a new dataset using an IIoT testbed based
+on Brown-IIoTbed [21] architecture. Our dataset includes raw
+traffic traces from testbed hosts, host logs, audit-based system
+provenance at hosts as well as host/network IDS alerts col-
+lected during normal operation of the testbed, individual APT
+attack stages and APT campaigns based on real-world attacks.
+Our dataset also includes IIoT-specific attack stages/tactics.
+III. BACKGROUND
+A. Existing Attack Frameworks for IIoT
+The authors in [20] have proposed a generic IIoT attack
+life-cycle framework consisting of the following stages: re-
+connaissance, weaponization, exploitation, lateral movement,
+command & control, exfiltration and tampering. MITRE has
+also released an Adversarial Tactics, Techniques and Com-
+mon Knowledge (ATT&CK) framework for ICS [22] consist-
+ing of the following tactics/stages: initial access, execution,
+persistence, privilege escalation, evasion, discovery, lateral
+movement, collection, command and control, inhibit response
+function, impair process control, and impact.
+B. IIoT APT Invariant State Machine
+It should be noted that real-world APT campaigns in IIoT
+environments do not follow all the stages in attack frameworks
+mentioned above. Further, each tactic in an attack framework
+consists of many techniques, with new ones being added
+regularly. It is almost impossible to model all the possible
+attack techniques and then use them towards detection during
+ongoing APT campaigns. Therefore, using the IIoT attack life-
+cycle framework [20] and the MITRE ATT&CK framework
+for ICS [22], we have identified certain attack stages/tac-
+tics which are ‘invariant’: Command-and-control, Discovery,
+Lateral movement, Fieldbus scanning and CE communication
+spoofing. These attack tactics consist of only a few techniques
+and those techniques have not changed significantly across
+APT campaigns over the years (and are not expected to change
+significantly in future). It is easier to model these invariant
+attack tactics and use them towards APT detection. We propose
+an IIoT APT Invariant State Machine (IASM) as shown in
+Fig. 1 which models a typical APT campaign in an IIoT
+environment as a finite-state machine. The states in IASM
+represent the states of the APT campaign while the state
+transitions are brought about by the deployment of invariant
+APT tactics identified earlier. Most real-world APT campaigns
+in IIoT environments such as those described in Section I
+follow our proposed IASM.
+IASM Description: Once the APT attackers have acquired
+all the resources and information required for attack campaign
+
+4
+(Ready for attack state), they move by compromising one or
+more of the public network-facing hosts to gain entry into the
+target IIoT network (Infected entry host state) and establish
+communication with a Command-and-control or C&C server
+(Establish foothold state). Next, the attackers scan for other
+hosts connected to the compromised machine and attempt to
+gain control of one of the discovered hosts (Infected new host
+state) either by using CVE vulnerabilities or by stealing the
+remote access credentials for the discovered host and then
+logging in to it. The attackers may attempt to move across the
+IIoT network by gaining control of more hosts, thus remaining
+in the same Infected new host state and using the same tactics
+as outlined earlier. Once the attackers reach the edge gateway
+(Infected edge gateway stage), they scan for control elements
+(PLC/RTU/SIS) and their slave devices connected via fieldbus
+protocols. Fieldbus refers to Modbus and other similar open-
+source or proprietary vendor protocols (e.g., Profibus/Profinet,
+CAN) which are used to communicate with respective vendor
+control elements. Subsequently, they spoof communications
+with the discovered control elements and their slaves to
+gain more information about them (Collect ICS intelligence
+state) and execute commands on control elements remotely
+(Execute CE commands state). Depending on the target(s) set
+by the APT adversaries (just collection of ICS intelligence or
+execution of desired commands on CE), they might end the
+campaign wilfully or forcibly due to detection by cybersecurity
+analysts (Goals achieved/APT detected state).
+C. Threat Model
+We assume that only a few machines in the enterprise
+tier of the target IIoT network are connected to the Internet
+(through firewall/IDS). All other machines in the enterprise tier
+and other tiers (platform, edge) are isolated from the Internet
+though some of them can still communicate with the Internet-
+connected machines in enterprise tier. The APT attackers
+can enter the target IIoT network through the internet-facing
+enterprise tier machines (remote access) or other machines to
+which plant operators/engineers have access (insider attack).
+Once inside the enterprise tier network, the attackers can move
+laterally across machines till they reach the edge tier consist-
+ing of edge gateway, control elements and sensors/actuators.
+The attackers are assumed to be well-resourced in terms of
+computational resources, financial backing, hacking skills and
+time which is true of most APT groups.
+IV. SYSTEM OVERVIEW
+A. RAPTOR Architecture
+As shown in Fig. 2, RAPTOR consumes data from different
+sources such as network traffic traces, audit logs, HIDS/NIDS
+alerts and host logs. The data is processed prior to extraction of
+features and then the features are processed before being sent
+to the APT attack-stage detection & correlation engine which
+detects the invariant APT attack stages in IASM using the
+optimal data source(s) identified in the following sub-section.
+The detection methods employed for detection of the attacks-
+stages are explained in sub-section IV-C. The attack stages are
+detected and correlated using their attributes to re-construct the
+APT campaign which is presented in the form of a graph as
+described in sub-sections IV-E and IV-F. The APT Campaign
+Graph (ACG) thus constructed can be utilized by cybersecurity
+analysts to come up with appropriate actions to mitigate the
+attack campaign or for forensic analysis.
+B. Selection of optimal data sources
+As outlined in Section II-A, we utilize data from various
+sources towards APT detection instead of limiting ourselves
+to a specific or proprietary data source. However, not all of
+those data sources are required for detection of each APT
+stage. Some of the data sources might be redundant or have
+limitations compared to other data sources. Therefore, we
+analyse our data sources in the context of each invariant APT
+stage and select the optimal ones.
+• Command-and-Control: Most rootkits and malware regu-
+larly exchange keep-alive packets with a C&C server to
+maintain connection throughout the duration of infection
+which can be captured by network traffic traces. By
+default, network IDS rules are not configured to detect
+C&C server message exchange. Audit-based provenance
+which captures network socket file read/write operations
+can also be used to detect the establishment of a con-
+nection between the compromised host and public C&C
+server. However, due to the issues of pruning spurious
+dependencies and noise in provenance graphs due to
+benign activities, we feel that using audit-based prove-
+nance to extract C&C communication might not justify
+the computational cost incurred and therefore, network
+traffic traces alone can be sufficient for this purpose.
+• Discovery: This stage can be detected through network
+traffic traces since scanning for other targets results in
+TCP/UDP packets being sent from the compromised
+machine to other machines in the target network. This
+stage can also be detected from alerts generated by an
+open source network IDS such as Snort or Suricata.
+However, not all organizations can be expected to deploy
+network IDSes. Further, slow stealth scans might be able
+to evade IDSes altogether. In case network IDS alerts are
+available, we can use them to increase the accuracy of
+detection.
+• Lateral Movement: This stage can be detected using login
+logs on target machines since during lateral movement,
+the attacker logs in to a target machine which is connected
+to the same or different subnet as the compromised
+machine. IDS alerts would not be helpful in detecting
+lateral movement since by default, IDS rules are not
+configured to do so and lateral movements are not that
+frequent compared to the events that IDSes are expected
+to detect such as DoS attacks, port scans, SMB probes,
+etc. Network traffic traces can contain evidence of lateral
+movement in terms of TCP/UDP packets exchanged
+between compromised source and target machines and
+hence, they can be used to increase the accuracy of lateral
+movement detection.
+• Fieldbus scanning: This stage can be detected through
+network traffic traces since scanning for target control
+
+5
+Fig. 1: IIoT APT Invariant State Machine
+Fig. 2: RAPTOR Architecture
+elements over fieldbus protocols results in TCP packets
+being sent from the edge gateway to connected control
+elements at specific port numbers. This stage can also be
+detected from alerts generated by an open source network
+IDS such as Snort or Suricata configured with customized
+modules. In case network IDS alerts are available, we can
+use them to increase the accuracy of detection.
+• CE communication spoofing: Since the attacker needs to
+send appropriately crafted packets over the link between
+edge gateway and control elements, network traffic traces
+can be used to detect the attacker’s spoofing of commu-
+nications with control elements.
+Our findings regarding optimal data sources for detecting
+various APT attack stages for IIoT are summarized in Table
+I.
+C. APT attack-stage detection
+Detection of Command-and-control stage: In this stage,
+a compromised machine establishes connection with a C&C
+server, which is a public server, and communicates with it at
+regular time intervals. Therefore, multiple packet exchanges of
+a suspected host with public server IP address indicates com-
+munication with C&C server. We filter the TCP [PSH,ACK],
+[ACK]/UDP packets sent/received by public server IP ad-
+dresses (except VPN server) to/from IIoT testbed host IP
+addresses from network traffic traces (in pcap format). If
+the filter yields multiple such packets, we extract the packet
+arrival timings. Further, since many malware enforce periodic
+communication between the compromised machine and C&C
+server, we test for periodicity in packet arrival timings obtained
+earlier using the algorithm proposed in [23] which is based on
+discrete-time signal encoding and autocorrelation function. If
+the packet arrivals are found to be periodic, it can be inferred
+that the Command-and-control stage has been detected. It
+should be noted that there may be machines in the enterprise
+tier which talk to public servers, e.g., to retrieve a web
+page over HTTPS from a web server. The packets exchanged
+with such legitimate public servers contribute as noise to our
+extracted timings for periodicity detection. The ACF (auto-
+correlation function) can detect periodicity reliably in presence
+of such noise.
+Detection of Discovery stage: The network traffic traces
+
+[CommandandControl)
+Readyfor
+Infectedentry
+Established
+attack
+host
+foothold
+{Discovery,Lateral
+movement)
+Infected new
+host
+Discovery,Lateral
+movement)
+Fieldbusscanning,C
+(CEcomm.spoofing)
+comm.spoofing)
+Infectededge
+CollectICS
+Execute CE
+intelligence
+commands
+gateway
+7.Goals
+achieved/APT
+detectedDatasource#1
+Data source#2
+Data
+Feature
+Feature
+preparation
+extraction
+processing
+Data source #n-1
+Data source#n
+APTcampaign
+APTAttack-stageDetection
+graph
+&Correlationengine6
+TABLE I: Optimal data sources for detecting APT attack stages in IIoT. (P) indicates primary data source.
+Attack Stage
+Audit provenance
+Network traffic traces
+IDS alerts
+Host logs
+Command-and-control
+
+✓
+
+
+Discovery
+
+✓(P)
+✓
+
+Lateral Movement
+
+✓
+
+✓(P)
+Fieldbus scanning
+
+✓(P)
+✓
+
+CE communication spoofing
+
+✓
+
+
+collected at a host are split into smaller traces of fixed time
+duration. These traces are assumed to belong to either of two
+classes: normal or scanning. A normal trace is one which
+does not consist of network scanning packets while a scanning
+trace is one that does. Each trace is classified using an ML
+algorithm (e.g., Decision Trees, SVM, etc.). If a trace is
+classified as belonging to scanning class, it can be inferred
+that the Discovery attack stage has been detected. For each
+trace, the features for ML classification are extracted from
+TCP/UDP headers only and not the payloads of the respective
+packets since the network traffic might be encrypted. The ML
+features selected for detection of scanning traffic are shown in
+Table II.
+Motivation behind feature selection: The intuition behind
+selecting the first two features is that port scanning tools
+used by attackers send TCP/UDP requests to multiple IP
+addresses to find out open ports and the services running
+on them. The third set of features was selected because
+during port scan targeting an IP address, many of the ports
+to which TCP connection requests are sent are not open
+and no response/acknowledgement is sent back, so the TCP
+connections formed remain half-open. The fourth feature seeks
+to exploit the fact that once TCP connection is formed with
+an IP address at a certain port number, port scanning tools
+exchange data only for a short time and the connection is
+subsequently reset. Compared to packet length in normal TCP
+connections, port scanning packet lengths are generally shorter
+making the fifth set of features useful for classification. Finally,
+the sixth feature set targets the short time intervals between the
+transmission of port scanning packets as compared to normal
+packets.
+Detection of Lateral movement stage: According to our
+approach, the authentication logs on network hosts are used
+to detect this stage. We look for logins that satisfy the
+following suspicious property: the machine from which a user
+is initiating the login to another machine is a part of other
+detected APT attack stages that usually precede lateral move-
+ment, e.g., command-and-control and discovery. For example,
+if user1 logs in to machine-B from machine-A, machine-A
+is in a different subnet as machine-B and machine-A has
+been identified as part of Discovery stage detected earlier, we
+can conclude that lateral movement has been detected from
+machine-A to machine-B.
+Detection of Fieldbus scanning: Similar to the detection
+of Discovery stage, the network traffic traces collected at edge
+gateway network interfaces connected to control elements are
+split into smaller traces of fixed time duration. These traces
+are assumed to belong to either of two classes: normal or
+fieldbus scanning. A normal trace is one which does not
+consist of Fieldbus scanning packets while a fieldbus scanning
+trace is one that does. Each trace is classified using an ML
+algorithm. If a trace is classified as belonging to fieldbus
+scanning class, it can be inferred that the Fieldbus scanning
+attack stage has been detected. For each trace, the features for
+ML classification are extracted from TCP headers only and not
+the payloads of the respective packets since the network traffic
+might be encrypted. The ML features selected for detection of
+Fieldbus scanning traffic are shown in Table II.
+Motivation behind feature selection: The intuition behind
+selecting the first three features is that typically during fielbus
+scanning, attackers attempt to set up a TCP connection with a
+fieldbus device (PLC/RTU) which is unsuccessful for the first
+few times. Once the connection is set up, the fieldbus scanner
+requests device enumeration data and finally, the connection
+is closed. If the scanner is also trying to enumerate fieldbus
+slaves, it iterates through the list of slave IDs sequentially.
+Since slaves are not present at all SIDs, the TCP connection
+may be reset only for the fieldbus scanner to set up a new
+connection. Again, compared to packet length in normal TCP
+connections, fieldbus scanning packet lengths are generally
+shorter making the fourth set of features useful for classi-
+fication. Finally, the fifth feature set targets the short time
+intervals between the transmission of fieldbus scanning packets
+as compared to normal packets.
+Detection of CE communication spoofing: If an attacker is
+trying to spoof communications with a control element using
+one of the standard industrial automation (IA) protocols (e.g.,
+IEC 61850, IEC 61131-3), there would either be more than
+one TCP connections from the edge gateway to the control
+element at the destination port specific to that IA protocol,
+or the original TCP connection would be terminated by the
+attacker leaving only the attacker’s TCP connection active.
+Therefore, if there are more than one TCP connections from
+the edge gateway to the control element at the destination port
+specific to that IA protocol or the original TCP connection has
+been terminated, it can be inferred that the CE communication
+spoofing stage has been detected.
+D. Detection using multiple data sources
+As explained in Section IV-B, there can be more than
+one data sources which can be used to detect each APT
+attack stage. For an attack stage, a ∈ {Command-and-control,
+Discovery, Lateral movement, Fieldbus scanning, CE commu-
+nication spoofing}, we define the aggregate detection score
+as:
+da = ∑
+i
+wia1ia,
+(1)
+where wia is the weight assigned to the ith data source for
+detection of attack stage a, and the indicator function, 1ia is
+
+7
+Attack stage
+ML features
+Discovery
+Number of unique TCP SYN/UDP destina-
+tion IP addresses, Number of unique TCP
+SYN/UDP destination ports per destination IP
+address (maximum, minimum, mean), Number
+of half-open TCP connections, Number of TCP
+RESET packets, Packet length in bytes (max-
+imum, minimum, mean), Packet inter-arrival
+time in seconds (maximum, minimum, mean)
+Fieldbus scanning
+Number of TCP 3-way handshakes with a
+destination IP address (maximum, minimum,
+mean), Number of TCP RESET packets, Num-
+ber of TCP FIN packets, Packet length in bytes
+(maximum, minimum, mean), Packet inter-
+arrival time in seconds (maximum, minimum,
+mean)
+TABLE II: ML features selected for detection for APT attack stages
+defined as:
+1ia =
+�
+�
+�
+�
+�
+1,
+if ith data source is optimal for detection
+of attack stage a
+0,
+otherwise
+(2)
+The weight assigned to the primary optimal data source
+identified for an attack stage should be greater than the weight
+assigned to secondary data sources. If the aggregate detection
+score, da is greater than a pre-defined threshold, τ, then the
+attack stage a is considered as detected, otherwise not.
+E. Correlation of APT attack stages
+There are three conditions which need to be satisfied for an
+attack stage A to be followed by an attack stage B:
+• The source IP address for stage A should match with the
+source IP address for stage B.
+• When stage A involves movement of the attacker from
+one machine to another (e.g., Lateral movement), then
+the destination IP address for stage A should match with
+the source IP address for stage B.
+• The time stamp for stage A should fall earlier than the
+time stamp for stage B.
+The APT attack stage detection & correlation (ASDC) en-
+gine first checks if the initial stage of Command-and-control)
+in the proposed IASM can be detected at any of the Internet-
+facing hosts in enterprise tier. If the detection is successful,
+ASDC engine checks for the (Discovery) stage at the same
+host where Command-and-control) stage was detected. If the
+Discovery stage is detected, ASDC engine looks for signs of
+the Lateral movement stage at the same host. If the Lateral
+movement stage is detected, ASDC engine proceeds to check
+for the Discovery stage again followed by the Lateral move-
+ment stage as outlined above at the host accessed after lateral
+movement. If the host accessed after lateral movement is the
+edge gateway, ASDC engine starts looking for the Fieldbus
+scanning stage at the edge gateway and if it is detected, ASDC
+engine checks if CE communication spoofing stage can also be
+detected. The complete attack stage detection and correlation
+algorithm proposed above is shown in Algorithm 1.
+Handling false positives/negatives in ML classification:
+RAPTOR is designed to handle false positives/negatives in
+Algorithm 1 Detect_Correlate_Attack_Stages (list_host_IP_add)
+1: INPUT: list_host_IP_add (List of IP addresses of Internet-
+facing hosts)
+2: det_status = APT_DET_START
+3: for host_ip ∈ list_host_IP_add do
+4:
+if Check_C&C_stage (host_ip)[1] = TRUE then
+5:
+src_host_ip = host_ip
+6:
+end if
+7: end for
+8: if Check_Discovery_stage (src_host_ip)[1] = TRUE
+&&
+Check_Discovery_stage
+(src_host_ip)[2]
+>
+Check_C&C_stage (src_host_ip)[2] then
+9:
+for
+tgt_host_ip
+∈
+Check_Discovery_stage
+(src_host_ip)[4] do
+10:
+if
+Check_Lateral_Movement_stage
+(src_host_ip,
+tgt_host_ip)[1]
+=
+TRUE
+&&
+Check_Lateral_Movement_stage
+(src_host_ip,
+tgt_host_ip)[2]
+>
+Check_Discovery_stage
+(src_host_ip)[2] then
+11:
+if tgt_host_ip = edge_gw_IP then
+12:
+if
+Check_Fieldbus_scan_stage
+()[1]
+=
+TRUE
+&&
+Check_Fieldbus_scan_stage
+()[2]
+>
+Check_Lateral_Movement_stage
+(src_host_ip,
+tgt_host_ip)[2] then
+13:
+if
+Check_CE_comm_stage
+()[1]
+=
+TRUE
+&&
+Check_CE_comm_stage
+()[2]
+>
+Check_Fieldbus_scan_stage ()[2] then
+14:
+det_status = APT_DET_STOP
+15:
+return (det_status)
+16:
+end if
+17:
+end if
+18:
+end if
+19:
+end if
+20:
+end for
+21: end if
+22: Function
+Check_C&C_stage
+(host_ip),
+Returns
+{bool_val, time_det, C&C_server_IP}
+23: Function Check_Discovery_stage (host_ip), Returns
+{bool_val, time_det, scan_type, list_target_host_IPs}
+24: Function
+Check_Lateral_Movement_stage
+(src_host_ip, dst_host_ip), Returns {bool_val, time_det}
+25: Function
+Check_Fieldbus_scan_stage
+(),
+Returns
+{bool_val, time_det}
+26: Function
+Check_CE_comm_stage
+(),
+Returns
+{bool_val, time_det}
+ML-based detection. For example, let us assume that there is
+a false positive, i.e., a packet trace is classified as scanning
+though it is normal and the Discovery stage is marked as
+detected. The subsequent attack stages in IASM would not
+be detected by the ASDC engine and therefore, RAPTOR
+would know that there was a false positive in detection of
+an earlier attack stage. It is also possible that there is a false
+negative, i.e., a packet trace is classified as normal though it is
+scanning, and therefore, the ASDC engine would not invoke
+detection of subsequent attack stages in IASM. We propose to
+
+8
+handle both false positives and negatives by taking the mode
+of classification results for a packet trace for a sufficiently
+large number of iterations.
+F. APT campaign graph
+As the ASDC engine proceeds to detect various stages of an
+APT campaign, it uses the detected stages and their attributes
+to construct the APT campaign graph. It is a directed graph,
+G(V,E) where each node, vi ∈ V ∀i ∈ {1,2,...,Nv}, where Nv
+is the total number of nodes in the graph, corresponds to a
+machine (denoted by its IP address) which is a part of one of
+the detected attack stages. An edge, ej ∈ E ∀ j ∈ {1,2,...,Ne},
+where Ne is the total number of edges in the graph, is extended
+from node vi to another node vk in the graph if there is
+a connection from the machine corresponding to node vi to
+the machine corresponding to node vk during one or more of
+the APT attack stages. Each edge has an attribute {s1,s2,...}
+where sl ∀l ∈ {1,2,...} is an attack stage which enables the
+connection between the two machines corresponding to the
+nodes at either end of the edge.
+V. EXPERIMENTS
+A. IIoT Testbed
+To generate a realistic IIoT APT dataset which can be
+used to evaluate RAPTOR’s performance, we built an IIoT
+testbed modelled after Brown-IIoTbed [21] whose architecture
+is reproduced in Fig. 3. The implementation of IIoT testbeds
+is still in its early stages, with most existing implementations
+[4] being special projects and publicly unavailable. Brown-
+IIoTbed is designed based on the IIC (Industrial Internet
+Consortium)’s IIRA (Industrial Internet Reference Architec-
+ture) model and consists of three tiers- edge, platform and
+enterprise. It supports a number of real-world IIoT function-
+alities such as e-mail notifications to plant workers regarding
+important OT events, web-based SCADA interface (viewing
+real-time sensor values and trends, actuator status change
+notifications, tuning of PLC parameters), remote maintenance
+of edge gateway, query to edge data historian, etc. The testbed
+also supports a number of real-world IIoT protocols such as
+CoAP, MQTT, and Modbus.
+B. Experimental Methodology and Results
+We need to collect data from our IIoT testbed under normal
+operation as well during APT attack stages. We collect data
+from our testbed in the form of network traffic traces from
+hosts, audit-based provenance at each x86-based host, host
+logs (login records, authentication logs, syslog) and alerts from
+Snort network IDS. However, we use only the optimal data
+sources identified in Section IV-B towards the final attack
+detection.
+Command-and-control stage: To emulate this stage, we use
+open-source tools such as dnscat2 to create a communication
+channel between a C&C server and a compromised machine
+using tunneling over DNS protocol which is one of the most
+common C&C communication protocols used by attackers
+since most firewalls do not block it. We run a public dnscat2
+C&C server and the client on a Windows 10 Pro and a Ubuntu
+20.04 machine in our testbed. The packet traces generated
+on those client machines are collected using tcpdump in sets
+of 1 minute duration. A total of 1000 packet traces were
+collected for each type of host and fed as input to the detection
+phase of Algorithm 2 proposed in [23] for detection of C&C
+communication.
+The performance is evaluated in terms of detection rate
+(DR) and missed-detection rate (MDR). Detection Rate is
+the fraction of the total number of packet traces which have
+been correctly detected as containing C&C traffic, and Missed-
+detection Rate is the fraction of the total number of packet
+traces which have been incorrectly detected as not containing
+C&C traffic. Using the parameter values (given in Table III),
+the detection performance for both the scenarios specified
+above is shown in Table IV. It can be seen that the algorithm
+gives a DR of 1.0 and a MDR of 0.0.
+Traffic sampling frequency
+0.1
+Min. autocorrelation peak height
+0.7×(Max. peak height)
+Inter-peak gap variance threshold
+0.01
+TABLE III: Parameter Values for C&C Communication Periodicity Detection
+Discovery stage: To emulate this stage, we use open-source
+network scanning tools such as nmap which is either used
+directly or is the inspiration for customized port scanners used
+by most APT groups. We run the default nmap SYN scans
+on a Windows 10 Pro and a Ubuntu 20.04 machine in our
+testbed to enumerate connected hosts, their OS versions and
+the services running on them. Nmap is run in both normal
+mode as well as sneaky or as well call it, slow mode, with the
+latter mode targeted at evading IDSes [24]. The packet traces
+thus generated on those machines are collected using tcpdump
+in sets of 1 minute duration. In real-world APT campaigns,
+attackers may slow down network scanning to evade detection
+by IDS and therefore, we may need to increase the duration
+of packet captures. A total of 1000 packet traces are collected
+from both the Windows and Ubuntu machines under normal
+operation and further 1000 packet traces are collected during
+the network scanning operation. The packet traces are used
+to extract features mentioned in Section IV-C and appropriate
+class labels (’normal’ or ’scanning’) are assigned to them. The
+extracted features vectors are further processed (handling of
+missing values, scaling) and randomly divided into training
+and test datasets using an 80:20 split. Using χ2 test statistic,
+we select the best features (test statistic value above a pre-
+selected threshold) out of the existing ones. The final feature
+vectors thus obtained are used to train Support Vector Machine
+(SVM) and Random Forest (RForest) models. The trained
+ML models are then used to predict class labels for the test
+dataset and finally, the detection performance of the models is
+evaluated. We use a 10-fold cross validation approach to tune
+the hyper-parameters of the ML classifiers for achieving the
+DATASET
+METHOD
+DR
+MDR
+IIoT Testbed Ubuntu host
+Algorithm 2 [23]
+1.0
+0.0
+IIoT Testbed Windows host
+Algorithm 2 [23]
+1.0
+0.0
+TABLE IV: C&C Stage Detection Performance
+
+9
+Fig. 3: IIoT Testbed Architecture [21]
+DATASET
+MODEL
+PR
+RC
+IIoT testbed Ubuntu host
+(Discovery-normal)
+Rforest
+0.996
+1.0
+SVM
+1.0
+1.0
+IIoT testbed Ubuntu host
+(Discovery- slow)
+Rforest
+0.991
+1.0
+SVM
+0.978
+0.974
+IIoT testbed Windows host
+(Discovery-normal)
+Rforest
+0.996
+1.0
+SVM
+0.979
+1.0
+IIoT testbed Windows host
+(Discovery- slow)
+Rforest
+1.0
+1.0
+SVM
+0.912
+0.978
+IIoT testbed
+(Fieldbus (Modbus) Scanning- agg.)
+Rforest
+1.0
+0.992
+SVM
+1.0
+0.996
+IIoT testbed
+(Fieldbus (Modbus) Scanning- non-agg.)
+Rforest
+0.996
+1.0
+SVM
+0.996
+0.983
+IIoT testbed
+(Fieldbus (Profibus) Scanning)
+Rforest
+0.992
+1.0
+SVM
+0.988
+0.996
+TABLE V: Raptor’s ML performance for detection of Discovery and Fieldbus scanning
+stages
+highest possible CV scores. The cross validation is based on
+training data only without using any information from the test
+dataset.
+Fieldbus scanning stage: To emulate this stage, we run
+nmap with modbus-discover script on the edge-gateway for
+enumerating Modbus slave IDs and collecting details about the
+slave devices. We run the modbus-discover script in both
+’aggressive’ and ’non-aggressive’ modes, where the former
+mode refers to finding all slave IDs and the latter mode
+refers to finding just the first slave ID. Though Modbus is
+one of the common protocols used for communication with
+PLCs/RTUs, there are other protocols as well which are used
+in the industry, e.g., DNP3 (Distributed Network Protocol),
+Profibus/Profinet, CAN (Controller Area Network). Therefore,
+in a separate experiment, we connect a Seimens S7-1200 PLC
+to the edge gateway network and run nmap with s7-info script
+on the gateway for enumerating Seimens S7 PLC devices
+and collecting their device information. The steps for packet
+trace collection under normal and fieldbus scanning operations,
+feature vector extraction, processing and selection, ML model
+training and performance evaluation remain similar to the ones
+outlined for Discovery stage above.
+The performance of ML classifiers is typically evaluated in
+terms of precision (PR) and recall (RC) scores. Precision is
+the ratio
+TP
+TP+FP, where TP is the number of true positives
+and FP is the number of false positives. It represents the
+ability of a classifier to avoid labeling samples that are negative
+as positive. Recall is the ratio
+TP
+TP+FN , where TP is the
+number of true positives and FN is the number of false
+negatives. It represents the ability of a classifier to avoid
+labeling samples that are positive as negative. Using the tuned
+hyper-parameters’ values, the average classification precision
+(PR) and recall (RC) scores obtained for the final classifiers
+over 10 runs are shown in Table V. It can be observed that
+for the detection of Discovery stage on Ubuntu as well as
+Windows hosts using normal and slow scan speeds, Random
+Forest performs slightly better than SVM. In general, both
+the ML classifiers perform better with normal scanning speed
+compared to slow scanning speed which is expected since
+within the trace duration (1 min), more number of network
+scanning packets would be captured during normal versus slow
+speed scanning. For the detection of Fieldbus scanning stage
+using Modbus protocol in ’aggressive’ and ’non-aggressive’
+modes, Random Forest performs almost equally as SVM in
+terms of precision but SVM performs quite poorly compared
+to Random Forest in terms of recall for ’non-aggressive’
+mode. For the detection of Fieldbus scanning stage using
+Profibus protocol, Random Forest performs slightly better than
+SVM in terms of both precision and recall. Based on the
+performance results obtained above, it would be preferable to
+select Random Forest classifier for detection of Discovery and
+Fieldbus scanning stages in RAPTOR’s implementation since
+SVM’s performance degrades significantly at slow network
+scanning speed and for ’non-aggressive’ fieldbus scanning
+mode.
+Finally, to emulate an APT attack on our testbed for
+construction of APT campaign graphs, we develop three
+attack storylines from an APT group’s perspective, i.e., their
+background, motivation for attack, steps taken for attack and
+
+Edge Tier
+Platform Tier
+Enterprise Tier
+Local SCADA &
+Management devices
+Cloud Applications
+API
+End users
+Sensor & Actuator
+PLC
+Cloud Broker
+Dashboard
+End users
+Sensor & Actuator
+Edge gateway
+Cloud Storage
+0
+Connected Worker
+Remote
+Maintenance
+On-site Supervisor
+Local Servers
+Phvsical assets/field devices zone
+Edge gateway zone
+Cloud zone
+Edge mobile service zone
+Enterprise service zone
+- LANs, and router/firewall zone10
+final attack objective. The TTPs (Techniques, Tactics and
+Procedures) used in our storylines are close to the ones
+used in real-world APT attacks on IIoT environments such
+as those mentioned in Section I. For reasons of space, we
+present RAPTOR’s evaluation with only one of the APT
+attack storylines here. The rest of the storylines are presented
+in Appendix Sections A and B. Steps 1-2 (Initial Access),
+step 4 (Command-and-control), step 6 (Discovery) and step 10
+(Credential Access, Lateral Movement) of the attack storyline
+are based on the 2014 German steel mill and 2015/2016
+Ukraine power grid attacks. Step 16 (Fieldbus scanning, CE
+communication spoofing) of the attack storyline is based on
+the 2016 Ukraine power grid attack and the 2017 Saudi
+petrochemical plant attack. Steps 18-19 (Impact) of the attack
+storyline are based on the 2015 Ukraine power grid attack.
+We run the attack storyline on our IIoT testbed over the
+course of a few hours and collect the data generated from
+testbed hosts. Since real-world APT campaigns can stretch
+over months and it is not possible to emulate them on our
+testbed, we assume that our APT storylines are executed
+in an accelerated timeframe and therefore our performance
+evaluation of RAPTOR holds. The complete APT attack
+storyline used for evaluating RAPTOR is as follows.
+APT attack storyline 1
+Background: Attackers belong to a nation-state (or APT)
+group which has been tasked with targeting a prominent state-
+owned steel manufacturing plant. The APT group plans to
+steal ICS related data which can be used to understand the
+ICS design and components which can further be used to plan
+for later attacks.
+Goals: To steal sensitive OT data (e.g., blast furnace temper-
+ature sensor measurements, PLC configuration, credentials).
+OT data such as blast furnace temperature readings are sensi-
+tive because they can be used to learn the normal temperature
+range and temporal trends. Attackers can use this information
+to modify the settings of the furnace temperature controller
+to damage the furnace. The temperature readings can also be
+used to infer the furnace design.
+Steps:
+1) The attacker sends a spear phishing email (including
+a malicious VPN portal web link) to one of the steel
+plant employees posing as legitimate company email and
+obtains their VPN login credentials.
+2) It uses the employee’s company email address and
+phished credentials to remotely login to the maintenance
+machine connected to enterprise network (password re-
+use) through the VPN service.
+3) The attacker changes the employee’s VPN account pass-
+word for persistence.
+4) The compromised machine connects to an external C&C
+server through DNS tunnelling and forwards a shell to
+the attacker.
+5) Attacker installs a malware on the compromised machine
+which exploits software vulnerabilities to gain root ac-
+cess.
+6) The attacker controlling the compromised machine scans
+its local network and finds other hosts (firewall, MQTT
+server, external API machine) as well as the services
+running on them.
+7) Attacker tries to find CVE vulnerabilities corresponding
+to the services running on other hosts but can not exploit
+them successfully.
+8) It goes through the shell command history on com-
+promised machine and finds previous SSH connection
+attempts to the edge gateway containing username and
+hostname details.
+9) It tries to determine the SSH login password for the edge
+gateway as follows:
+a) Accesses the shadow password file on compromised
+machine (using root access obtained earlier) which
+stores password hashes and corresponding hashing
+algorithms used.
+b) Tries to crack the password hashes to obtain corre-
+sponding plaintext passwords.
+10) The attacker attempts to log in to the edge gateway by
+using one plaintext password at a time and is successful.
+It explores the files, folders (hidden and non-hidden) and
+the processes running on edge gateway.
+11) It finds a web server, a CoAP server and Node-red
+application running on edge gateway.
+12) The attacker tries to exploit CoAP related vulnerabilities
+but is unsuccessful. It remotely executes a script from the
+compromised maintenance machine to dump the CoAP
+resources.
+13) It executes a fake CoAP client code on the compromised
+maintenance machine to receive measurements from sen-
+sors directly connected to edge gateway.
+14) It scans devices connected to the edge gateway’s Wi-Fi
+hotspot network and finds a host running DNP service
+(PLC master).
+15) Attacker downloads a script from C&C server and copies
+it remotely to edge gateway.
+16) It extracts PLC configuration data (e.g., hardware,
+firmware, manufacturer, serial number, slave IDs) by
+running the script on edge gateway.
+17) The attacker compresses and encrypts all the targeted
+data collected in previous steps (e.g., PLC configuration,
+sensor measurements, login credentials) and exfiltrates it
+through the C&C channel.
+We assigned a weight, wia = 0.5 to the primary data sources
+for detecting an attack stage and a weight of of wia/2 to
+the secondary data sources. The threshold for detection is
+selected as τ = 0.5. The APT campaign graph generated by
+RAPTOR for Storyline 1 is shown in Fig. IV-F. The graph
+captures broad details of the APT campaign including the
+IP addresses of the machines affected and the tactics used
+during the campaign which is quite useful for cybersecurity
+analysts. This shows that our proposed attack stage detection
+and correlation algorithm in Section IV-E works as intended.
+However, the campaign graph does not capture all the tactics
+employed by the APT attackers since our focus is on detecting
+invariant APT tactics/stages only as explained in Section III-B.
+Further, the campaign graph does not contain any details on
+the specific techniques employed by the APT attackers since
+
+11
+our focus is not on detecting the individual techniques used
+for each tactic. The APT campaign graph can serve as a
+starting point for cybersecurity analysts to fill in the missing
+tactics based on the APT attack frameworks for ICS, further
+investigation and mitigation. The APT campaign graphs for
+Storyline 2 and Storyline 3 are shown in Fig. 4 and Fig. 6
+respectively in the Appendix.
+Fig. 4: APT campaign graph generated for Storyline 1
+VI. DISCUSSION
+A. Comparison with State-of-the-art
+We are unable to conduct a performance comparison of
+RAPTOR with [12] as they do not provide any source code for
+their proposed multi-stage attack detection algorithm. Further,
+the dataset used for performance evaluation in [12] which con-
+sist of a synthetic APT campaign injected into the CSE-IDC-
+2018 intrusion detection dataset [25] has not been publicly
+released. HOLMES [10] and CONAN [11] do not provide
+source codes for their proposed APT detection system as well
+though both use the DARPA Transparent Computing ((TC)
+Engagement dataset [26] for performance evaluation which has
+been released publicly. The DARPA TC dataset contains data
+from a red team deploying APT-style TTPs on a target system
+consisting of multiple interconnected hosts running different
+OSes and having exploitable CVE vulnerabilities. However,
+the DARPA TC dataset suffers from following limitations
+which reduce its applicability for RAPTOR’s performance
+evaluation:
+• The TC network setup is simple and does not emulate
+real-world enterprise/IIoT networks.
+• The dataset provides only json files but no raw pcap
+files for us to extract network traffic-based features for
+a meaningful comparison with RAPTOR’s performance
+on our IIoT testbed dataset1.
+• None of the TTPs used in TC dataset are IIoT-specific.
+B. Limitations
+The design of RAPTOR introduces a few limitations which
+are discussed in this section. RAPTOR uses supervised ML
+1There is an active unresolved issue with the DARPA TC dataset. While
+loading data from the compressed *.bin.1.gz files, the code get stuck at
+streaming records.
+algorithms for detection of certain APT attack stages which
+means that it can detect only known malicious traffic pat-
+terns produced by those attack stages. However, it should be
+noted that supervised ML algorithms are used much more
+commonly compared to unsupervised algorithms in real-world
+deployments of HIDS/NIDS. Advanced APT malware may
+attempt to evade detection by RAPTOR by slowing down the
+scanning activity (e.g., during Discovery, Fieldbus scanning
+stages) or changing the time period between scanning attempts
+to confuse the trained ML algorithms. This evasion technique
+can be countered by increasing the packet trace duration to
+capture enough attack packets though it may lead to longer
+classification delays. The ML classifiers used for detection
+of certain APT attack stages may have to be re-trained, for
+example, when the classification probability falls below a pre-
+defined threshold or the host OS is updated, and this may
+cause a delay in detection.
+Evasive APT malware may also use some other C&C
+server messaging mechanism than the TCP ([PSH,ACK],
+[ACK])/UDP one to escape filtering and/or force the C&C
+communication to be non-periodic (by adding noise traffic for
+example). If the attacker does change the C&C messaging
+mechanism, the detection method can be changed accordingly.
+Moreover, the C&C communication periodicity detection algo-
+rithm uses an ACF (Autocorrelation Function)-based approach
+which can detect periodicity in the presence of noise as well
+if the noise is uncorrelated with the desired signal (discrete-
+time sequence extracted from C&C server traffic). An attacker
+may evade the lateral movement detection logic by exploiting
+an RCE vulnerability instead of performing manual logins or
+using stealthy malware which blends the logins in its attack
+path with a previous legitimate user login. Finally, it is also
+possible that a few hosts may already be infected before
+RAPTOR is deployed in an IIoT environment.
+VII. CONCLUSION
+We have proposed RAPTOR, an APT detection system
+targeted at IIoT environments. It detects and correlates attack
+stages derived from an APT Attack Invariant State Machine us-
+ing optimal data sources selected for each stage. The correlated
+attack stages are utilized to generate a compact, high-level
+APT Campaign Graph which can be used by cybersecurity
+analysts to track the progress of the APT campaign and deploy
+appropriate mitigation measures. A performance evaluation of
+RAPTOR shows that it can detect APT campaigns modelled
+after real-world attacks with high precision and low false
+positive/negative rates.
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+APPENDIX A
+APT ATTACK STORYLINE 2
+Background: Attackers belong to a nation-state (or APT)
+group which has been tasked with targeting a prominent state-
+owned steel manufacturing plant. The APT group plans to
+disrupt the steel production and thereby affect other industries
+dependent on steel and exports.
+Goals: To shut the blast furnace down by controlling the
+furnace relays (LEDs in our testbed). This may damage the
+plant operations temporarily or permanently.
+Steps:
+1) An insider recruited by the APT group installs malware
+on the maintenance machine through a USB stick. The
+malware exploits software vulnerabilities on the machine
+to gain root access.
+2) The compromised machine connects to an external C&C
+server through DNS tunnelling and forwards a remote
+display to the attacker.
+3) Attacker installs a malware on the compromised machine
+which exploits software vulnerabilities to gain root ac-
+cess.
+4) The attacker controlling the compromised machine scans
+its local network and finds other hosts (firewall, MQTT
+server, external API machine) as well as the services
+running on them.
+5) Attacker tries to find CVE vulnerabilities corresponding
+to the services running on other hosts but can not exploit
+them successfully.
+6) It accesses the shadow password file on compromised
+machine (using root access obtained earlier) which stores
+password hashes and corresponding hashing algorithms
+used.
+7) Attacker successfully opens an RDP (Remote Desktop
+Protocol) session to the external API machine using one
+of the stolen password hashes.
+8) It accesses the SCADA/HMI web interface on the ex-
+ternal API machine and turns off the LEDs directly
+connected to edge gateway.
+APPENDIX B
+APT ATTACK STORYLINE 3
+Background: Attackers belong to a nation-state (or APT)
+group which has been tasked with targeting a prominent state-
+owned steel manufacturing plant. The APT group plans to
+
+13
+Fig. 5: APT campaign graph generated for Storyline 2
+disrupt the steel production and thereby affect other industries
+dependent on steel and exports.
+Goals: To damage the plant equipment by tampering with
+the operation of safety controllers which prevent the blast
+furnace from entering an unsafe state. The safety controllers
+may be reprogrammed to allow the blast furnace to enter
+a dangerous state without any corrective action leading to
+physical damage to the plant and even loss of human lives.
+Steps:
+1) The attacker sends a spear phishing email (including
+a malicious VPN portal web link) to one of the steel
+plant employees posing as legitimate company email and
+obtains their VPN login credentials.
+2) It uses the employee’s company email address and
+phished credentials to remotely login to the maintenance
+machine connected to enterprise network (password re-
+use) through the VPN service.
+3) The attacker changes the employee’s VPN account pass-
+word for persistence.
+4) The compromised machine connects to an external C&C
+server through DNS tunnelling and forwards a shell to
+the attacker.
+5) Attacker installs a malware on the compromised machine
+which exploits software vulnerabilities to gain root ac-
+cess.
+6) The attacker controlling the compromised machine scans
+its local network and finds other hosts (firewall, MQTT
+server, external API machine) as well as the services
+running on them.
+7) Attacker tries to find CVE vulnerabilities corresponding
+to the services running on other hosts but can not exploit
+them successfully.
+8) It goes through the shell command history on com-
+promised machine and finds previous SSH connection
+attempts to the edge gateway containing username and
+hostname details.
+9) It hijacks any future SSH session between the com-
+promised machine (started by an employee performing
+remote maintenance) and the edge gateway.
+10) The attacker explores the files, folders (hidden and non-
+hidden) and the processes running on edge gateway.
+11) It finds a web server, a CoAP server and Node-red
+application running on edge gateway.
+12) The attacker scans devices connected to the edge gate-
+way’s Wi-Fi hotspot network and finds a host running
+DNP service (PLC device).
+13) It downloads a payload from C&C server, copies it
+remotely to edge gateway and executes it.
+14) The attacker terminates the existing process which is
+communicating with the PLC device.
+15) It collects more information about the PLC device and
+enumerates all the slave IDs using the payload com-
+mands.
+16) The attacker uses the payload to send a command to the
+targeted slave to read its current state.
+17) It remotely uploads a new program to the PLC device
+while it continues to operate.
+Fig. 6: APT campaign graph generated for Storyline 3
+
+fCommandandControl
+192.168.10.2
+192.168.102.1
+[Discovery,Lateral
+Movement)
+192.168.10.4fCommandandControl)
+192.168.10.2
+192.168.102.1
+[Discovery,Lateral
+Movement)
+192.168.15.1
+[Discovery,Fieldbusscanning,
+CEcomm.spoofing)
+192.168.15.8
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+page_content='1  RAPTOR: Advanced Persistent Threat Detection in  Industrial IoT via Attack Stage Correlation  Ayush Kumar and Vrizlynn L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Thing  Cyber Security Strategic Technology Centre  ST Engineering  Email: ayush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='kumar@u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='edu, vriz@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='org  Abstract—IIoT (Industrial Internet-of-Things) systems are get-  ting more prone to attacks by APT (Advanced Persistent Threat)  adversaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Past APT attacks on IIoT systems such as the 2016  Ukrainian power grid attack which cut off the capital Kyiv  off power for an hour and the 2017 Saudi petrochemical plant  attack which almost shut down the plant’s safety controllers have  shown that APT campaigns can disrupt industrial processes,  shut down critical systems and endanger human lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' In this  work, we propose RAPTOR, a system to detect APT campaigns  in IIoT environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' RAPTOR detects and correlates various  APT attack stages (adapted to IIoT) using multiple data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Subsequently, it constructs a high-level APT campaign graph  which can be used by cybersecurity analysts towards attack  analysis and mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' A performance evaluation of RAPTOR’s  APT stage detection stages shows high precision and low false  positive/negative rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' We also show that RAPTOR is able to  construct the APT campaign graph for APT attacks (modelled  after real-world attacks on ICS/OT infrastructure) executed on  our IIoT testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Index Terms—Industrial Internet of Things, IoT, IIoT, Ad-  vanced Persistent Threat, APT, APT Detection  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' INTRODUCTION  The Internet of Things (IoT) is a network of sensing  devices with low-power and limited processing capability,  which exchange data with each other and/or systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=',  gateways, cloud servers), normally using wired and wireless  technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Industrial IoT (IIoT) refers to the extension of  IoT in industrial sectors and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' With a strong focus  on machine-to-machine (M2M) communication, big data, and  machine learning, the IIoT enables industries and enterprises  to have better efficiency and reliability in their operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  What makes IIoT distinct from IoT is the intersection of  information technology (IT) and operational technology (OT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  However, this convergence has widened the attack surface and  increased the potential risks of cyberattacks being launched  against such critical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' A more significant concern  relates to legacy OT systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=', brownfield IIoTs) which are  usually isolated but are becoming more connected with new  IT technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Sophisticated attackers can easily gain access  to such brownfield IIoT systems and damage their operation  for lengthy periods of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  IIoT systems are more prone to attacks by APT adversaries  than traditional ICS (Industrial Control Systems)/OT (Oper-  ational Technology) networks [1] mainly due to addition of  Corresponding author: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Kumar (email: ayush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='kumar@u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='nus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  connectivity to IT networks (enabling lateral movement for  attackers) and introduction of M2M communications which  connect various new and intelligent devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Furthermore, ICS  assets themselves are prime targets for APT campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' This is  because ICS devices often run on legacy, proprietary software  which were not designed with security in mind and are not  patched/updated regularly due to concerns over downtime in  critical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' APT attacks can be used to gather ICS-related  intelligence, disrupt industrial processes, shut down critical  systems and endanger human lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Such APT attacks by well-  resourced groups have happened a number of times in the  past, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' the 2010 Stuxnet attack which damaged centrifuges  in Iranian uranium enrichment plants and caused a significant  setback to Iranian nuclear program [2],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' the 2014 attack on a  German steel mill [3] which prevented the blast furnace from  shutting down causing massive damage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' the 2015 attack on  Ukrainian power companies which disconnected substations  from the power grid leaving 225,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='000+ customers without  power for more than 6 hours [4],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' the 2016 attack on Ukranian  transmission level substation which cut a fifth of the capital  Kyiv off power for an hour [5] and the 2017 attack on a  Saudi petrochemical plant which almost shut down the plant’s  safety controllers which could have caused an explosion [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Industroyer/Crashoverride, the malware behind 2016 Ukraine  power grid attack and TRITON, the malware behind 2017  Saudi petrochemical plant attack are still active, targeting  electrical substations and energy utilities respectively [7], [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  New IIoT malware such as Incontroller/Pipedream [9], which  was revealed as recently as 2022, contains modules that target  specific ICS devices such as OPC servers, Schneider Electric  PLCs using Modbus and Codesys protocols, and Omron PLCs  and servo drives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Security solutions which are commonly deployed in IT net-  works such as firewalls, NIDS (Network Intrusion Detection  System), SIEM (Security Information and Event Management)  products are not common in ICS/OT networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Thus, there  is an impending need to design systems which can detect  ongoing APT attack campaigns in IIoT environments early  before they cause substantial damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Such systems should  satisfy the following requirements:  Not dependent on deployment of proprietary, third-party  security solutions whose features can vary across vendors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Instead, the system should use existing open-source and  readily available information sources for its operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='11524v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='CR]  27 Jan 2023  2  Detect the APT attack tactics with high accuracy and low  number of false positives/negatives  Able to reconstruct the APT attack campaign with suffi-  cient details and present them in a compact, high-level  form which cybersecurity analysts can use for further  analysis and attack mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  An APT campaign typically consists of various stages which  occur one after another (this will be explained later in the  paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' However, not all stages defined in existing attack  frameworks are found together in real-world APT campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Each stage in an APT campaign is linked to the previously  executed stages, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=', in terms of chronology of execution,  target hosts affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Thus, if we are able to detect some of the  individual attack stages, the above fact can be exploited to cor-  relate the detected stages and reconstruct the APT campaign  with some acceptable margin of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' In this work, we present  RAPTOR, a system for detecting ongoing APT campaigns  in IIoT environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' RAPTOR’s main component is an  APT attack-stage detection and correlation engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' It takes  as input a variety of readily available, non-proprietary data  sources (such as host logs, network traffic traces) and detects  attack tactics (including those specific to IIoT environments)  which are part of an ongoing APT campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The attack  tactics are then stitched together based on their attributes to  produce the APT attack campaign graph, which is a high-level  representation of APT activity across the target IIoT network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  The main contributions of our work are as follows:  We present, RAPTOR, a system for detecting ongoing  APT campaigns in IIoT environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  We employ a novel attack-stage detection and correlation  approach which uses open source, readily available data  sources for its operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  RAPTOR constructs a compact, high level APT cam-  paign graph which can be useful for cybersecurity ana-  lysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  We evaluate RAPTOR’s performance using a new dataset  which includes attack TTPs close to real-world APT  attacks on ICS/OT environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' RELATED WORK  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' APT detection in Enterprise networks  APT detection in enterprise network settings has received  significant attention in computer security literature in recent  years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Milajerdi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' [10] have presented HOLMES, a system  for APT campaign detection with high confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' It maps  activities in host audit logs and enterprise security alerts to  the cyber kill-chain, correlates the alerts generated by APT  steps based on information flow between low-level entities  (files, processes, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=') and builds a high-level scenario graph  encapsulating the attack TTPs and information flows between  entities involved in the TTPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' CONAN [11] is an APT  detection system which makes two major modifications to  HOLMES’s approach: one, it utilizes a state-based detection  framework where all processes and files are represented as  data structures similar to finite-state automata and two, it  focuses on three constant attack phases of APTs- 1) deploy and  execute the attacker’s code, 2) collect sensitive information or  cause damage, and 3) communicate with the C&C server or  exfiltrate sensitive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Wilkens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' [12] have proposed a  method to construct APT attack graphs from IDS (Intrusion  Detection System) alerts to assist human analysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' IDS alerts  are clustered into meta-alerts and single alerts, assigned poten-  tial attack stages and finally used to synthesize APT scenario  graphs based on a kill chain state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Irshad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' [13] have proposed TRACE, a provenance  tracking system for enterprise-wide APT detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' TRACE  offers host-level provenance tracking at the granularity of  program executions units and integrates provenance collected  from individual hosts to construct distributed enterprise-wide  causal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Hopper [14] instead focuses on the detection of  a single APT attack stage: lateral movement, within enterprise  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' It does so by building a graph of login activity  among machines in a network and identifying suspicious login  sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' A path inference algorithm is then deployed to  identify the broader paths that each login belongs to, "caused"  by the same user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Finally, an anomaly detection algorithm is  applied to conservatively infer the set of login paths most  likely to reflect lateral movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  However, all the above works suffer from certain limitations: [10]–[14] focus on enterprise networks only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  While [10], [11], [13] have been designed using audit  logs-based provenance and [14] has been designed using  enterprise logs, [12] has been designed using network  IDS alerts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Since each of these works is based on a single  data source, they are unable to leverage the information  provided by other data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Further, these works are  only able to detect a subset of APT attack stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Exclusively graph-based approaches [10], [13] tend to  suffer from dependency explosion problem which means  that with time, each graph node gives rise to many  edges which in turn give rise to new nodes and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  A backward trace through the graph to infer a path  exponentially increases the number of probable paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  [12], [14] work offline and not in real-time which means  that they assume that all the data required for APT  detection is available at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Though [10], [11], [13]  claim that they work in real-time, they don’t provide the  time delays encountered in APT detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Except for [12], no other work mentioned above detects  an APT campaign from a whole-network perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Unfortunately, there has been no research work yet on  detecting APT campaigns in IIoT settings which is the focus  of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Our work differs from [10]–[14] in the following  aspects:  The focus is on APT detection in IIoT settings which  combines both IT and OT environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The APT attack  stages used to design the detection system in our paper  have been adapted to IIoT settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  We leverage data from various sources such as network  packet traces, host logs (including audit logs) as well  as NIDS/HIDS alerts instead of limiting ourselves to  a specific or proprietary data source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The optimal data  source(s) for detection of each APT attack stage is(are)  identified and used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  3  We perform APT campaign detection from a whole-  network perspective instead of detecting APT artifacts on  individual hosts only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' ICS/IIoT datasets  In the past, researchers have built datasets (consisting of  network traffic, system logs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=') to design IDSes for ICS  (Industrial Control System).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Morris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' [15] have provided  a labeled dataset which includes network traffic captured  from a laboratory scale gas pipeline system during normal  operation and during 35 cyber-attacks that affect the Modbus  protocol such as reconnaissance, response injection, command  injection and DoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Similarly, Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' [16] have produced  a dataset consisting of synchrophasor measurement data and  audit logs from a power system testbed, amounting to a total of  10,000 simulated instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The dataset includes 25 scenarios  consisting of power system single-line-to-ground (SLG) faults,  normal operation and cyber-attacks (relay trip command injec-  tion, relay function disabling and SLG fault replay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Another  dataset has been collected from the Secure Water Treatment  (SWaT) testbed [17] and includes sensor/actuator readings  both during normal operation and under 36 attacks such as  bias attack, replay, single and multiple point attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The final  dataset includes 946,772 samples consisting of 51 attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  More recently, Myers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' [18] have generated two datasets  consisting of ICS device logs and network traffic captures  from two separate industrial process setups respectively, during  normal operation and during attacks on the testbed PLCs  (command injection, flooding, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Marcio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' [19] have  built a dataset consisting of six pre-decided ML-based flow-  level features extracted from the network traffic captured  during normal as well as under attack operation of a SCADA  system testbed consisting of a water storage tank’s control  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Finally, Muna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' [20] have presented the X-IIoTID  intrusion dataset which targets modern IIoT systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The  dataset covers various attack scenarios and attacks related to  newer IIoT connectivity protocols generated according to a  pre-defined attack taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The data collected includes end-  to-end network traffic (from physical field devices to the edge  gateway and from edge gateway to the cloud and enterprise  devices), host logs and computing resources, and IDS alert  logs (OSSEC, Zeek) collected at the edge gateway for both  normal and under attack operation of the testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The final  dataset consists of 820,834 instances and 59 features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  However, the above datasets suffer from certain limitations:  Many of the existing ICS datasets [15]–[19] highly de-  pend on features related to sensor measurements, actua-  tors’ statuses, and specific parameters of industrial pro-  tocol packets, which limit their use for diverse industrial  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Most of those datasets lack a clear attack taxonomy and  have not incorporated multi-stage attacks related to newer  IIoT connectivity protocols and services (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=', CoAP,  MQTT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  The X-IIoTID dataset addresses the above shortcomings  of previous datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' However, all the above datasets  including X-IIoTID provide pre-decided ML features  extracted from network flows only and do not include  raw traffic traces from which additional features can be  extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Further, the X-IIoTID dataset provides host  logs and IDS alert logs collected at the edge gateway only  since according to the dataset authors, edge gateway is  the highest priority target for attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The dataset does  not include APT campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  To address the limitations of X-IIoTID dataset for APT  detection, we build a new dataset using an IIoT testbed based  on Brown-IIoTbed [21] architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Our dataset includes raw  traffic traces from testbed hosts, host logs, audit-based system  provenance at hosts as well as host/network IDS alerts col-  lected during normal operation of the testbed, individual APT  attack stages and APT campaigns based on real-world attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Our dataset also includes IIoT-specific attack stages/tactics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' BACKGROUND  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Existing Attack Frameworks for IIoT  The authors in [20] have proposed a generic IIoT attack  life-cycle framework consisting of the following stages: re-  connaissance, weaponization, exploitation, lateral movement,  command & control, exfiltration and tampering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' MITRE has  also released an Adversarial Tactics, Techniques and Com-  mon Knowledge (ATT&CK) framework for ICS [22] consist-  ing of the following tactics/stages: initial access, execution,  persistence, privilege escalation, evasion, discovery, lateral  movement, collection, command and control, inhibit response  function, impair process control, and impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' IIoT APT Invariant State Machine  It should be noted that real-world APT campaigns in IIoT  environments do not follow all the stages in attack frameworks  mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Further, each tactic in an attack framework  consists of many techniques, with new ones being added  regularly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' It is almost impossible to model all the possible  attack techniques and then use them towards detection during  ongoing APT campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Therefore, using the IIoT attack life-  cycle framework [20] and the MITRE ATT&CK framework  for ICS [22], we have identified certain attack stages/tac-  tics which are ‘invariant’: Command-and-control, Discovery,  Lateral movement, Fieldbus scanning and CE communication  spoofing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' These attack tactics consist of only a few techniques  and those techniques have not changed significantly across  APT campaigns over the years (and are not expected to change  significantly in future).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' It is easier to model these invariant  attack tactics and use them towards APT detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' We propose  an IIoT APT Invariant State Machine (IASM) as shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' 1 which models a typical APT campaign in an IIoT  environment as a finite-state machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The states in IASM  represent the states of the APT campaign while the state  transitions are brought about by the deployment of invariant  APT tactics identified earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Most real-world APT campaigns  in IIoT environments such as those described in Section I  follow our proposed IASM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  IASM Description: Once the APT attackers have acquired  all the resources and information required for attack campaign  4  (Ready for attack state), they move by compromising one or  more of the public network-facing hosts to gain entry into the  target IIoT network (Infected entry host state) and establish  communication with a Command-and-control or C&C server  (Establish foothold state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Next, the attackers scan for other  hosts connected to the compromised machine and attempt to  gain control of one of the discovered hosts (Infected new host  state) either by using CVE vulnerabilities or by stealing the  remote access credentials for the discovered host and then  logging in to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The attackers may attempt to move across the  IIoT network by gaining control of more hosts, thus remaining  in the same Infected new host state and using the same tactics  as outlined earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Once the attackers reach the edge gateway  (Infected edge gateway stage), they scan for control elements  (PLC/RTU/SIS) and their slave devices connected via fieldbus  protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Fieldbus refers to Modbus and other similar open-  source or proprietary vendor protocols (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=', Profibus/Profinet,  CAN) which are used to communicate with respective vendor  control elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Subsequently, they spoof communications  with the discovered control elements and their slaves to  gain more information about them (Collect ICS intelligence  state) and execute commands on control elements remotely  (Execute CE commands state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Depending on the target(s) set  by the APT adversaries (just collection of ICS intelligence or  execution of desired commands on CE), they might end the  campaign wilfully or forcibly due to detection by cybersecurity  analysts (Goals achieved/APT detected state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Threat Model  We assume that only a few machines in the enterprise  tier of the target IIoT network are connected to the Internet  (through firewall/IDS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' All other machines in the enterprise tier  and other tiers (platform, edge) are isolated from the Internet  though some of them can still communicate with the Internet-  connected machines in enterprise tier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The APT attackers  can enter the target IIoT network through the internet-facing  enterprise tier machines (remote access) or other machines to  which plant operators/engineers have access (insider attack).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Once inside the enterprise tier network, the attackers can move  laterally across machines till they reach the edge tier consist-  ing of edge gateway, control elements and sensors/actuators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  The attackers are assumed to be well-resourced in terms of  computational resources, financial backing, hacking skills and  time which is true of most APT groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' SYSTEM OVERVIEW  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' RAPTOR Architecture  As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' 2, RAPTOR consumes data from different  sources such as network traffic traces, audit logs, HIDS/NIDS  alerts and host logs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The data is processed prior to extraction of  features and then the features are processed before being sent  to the APT attack-stage detection & correlation engine which  detects the invariant APT attack stages in IASM using the  optimal data source(s) identified in the following sub-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  The detection methods employed for detection of the attacks-  stages are explained in sub-section IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The attack stages are  detected and correlated using their attributes to re-construct the  APT campaign which is presented in the form of a graph as  described in sub-sections IV-E and IV-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The APT Campaign  Graph (ACG) thus constructed can be utilized by cybersecurity  analysts to come up with appropriate actions to mitigate the  attack campaign or for forensic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Selection of optimal data sources  As outlined in Section II-A, we utilize data from various  sources towards APT detection instead of limiting ourselves  to a specific or proprietary data source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' However, not all of  those data sources are required for detection of each APT  stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Some of the data sources might be redundant or have  limitations compared to other data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Therefore, we  analyse our data sources in the context of each invariant APT  stage and select the optimal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Command-and-Control: Most rootkits and malware regu-  larly exchange keep-alive packets with a C&C server to  maintain connection throughout the duration of infection  which can be captured by network traffic traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' By  default, network IDS rules are not configured to detect  C&C server message exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Audit-based provenance  which captures network socket file read/write operations  can also be used to detect the establishment of a con-  nection between the compromised host and public C&C  server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' However, due to the issues of pruning spurious  dependencies and noise in provenance graphs due to  benign activities, we feel that using audit-based prove-  nance to extract C&C communication might not justify  the computational cost incurred and therefore, network  traffic traces alone can be sufficient for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Discovery: This stage can be detected through network  traffic traces since scanning for other targets results in  TCP/UDP packets being sent from the compromised  machine to other machines in the target network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' This  stage can also be detected from alerts generated by an  open source network IDS such as Snort or Suricata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  However, not all organizations can be expected to deploy  network IDSes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Further, slow stealth scans might be able  to evade IDSes altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' In case network IDS alerts are  available, we can use them to increase the accuracy of  detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Lateral Movement: This stage can be detected using login  logs on target machines since during lateral movement,  the attacker logs in to a target machine which is connected  to the same or different subnet as the compromised  machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' IDS alerts would not be helpful in detecting  lateral movement since by default, IDS rules are not  configured to do so and lateral movements are not that  frequent compared to the events that IDSes are expected  to detect such as DoS attacks, port scans, SMB probes,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Network traffic traces can contain evidence of lateral  movement in terms of TCP/UDP packets exchanged  between compromised source and target machines and  hence, they can be used to increase the accuracy of lateral  movement detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Fieldbus scanning: This stage can be detected through  network traffic traces since scanning for target control  5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' 1: IIoT APT Invariant State Machine  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' 2: RAPTOR Architecture  elements over fieldbus protocols results in TCP packets  being sent from the edge gateway to connected control  elements at specific port numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' This stage can also be  detected from alerts generated by an open source network  IDS such as Snort or Suricata configured with customized  modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' In case network IDS alerts are available, we can  use them to increase the accuracy of detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  CE communication spoofing: Since the attacker needs to  send appropriately crafted packets over the link between  edge gateway and control elements, network traffic traces  can be used to detect the attacker’s spoofing of commu-  nications with control elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Our findings regarding optimal data sources for detecting  various APT attack stages for IIoT are summarized in Table  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' APT attack-stage detection  Detection of Command-and-control stage: In this stage,  a compromised machine establishes connection with a C&C  server, which is a public server, and communicates with it at  regular time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Therefore, multiple packet exchanges of  a suspected host with public server IP address indicates com-  munication with C&C server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' We filter the TCP [PSH,ACK],  [ACK]/UDP packets sent/received by public server IP ad-  dresses (except VPN server) to/from IIoT testbed host IP  addresses from network traffic traces (in pcap format).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' If  the filter yields multiple such packets, we extract the packet  arrival timings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Further, since many malware enforce periodic  communication between the compromised machine and C&C  server, we test for periodicity in packet arrival timings obtained  earlier using the algorithm proposed in [23] which is based on  discrete-time signal encoding and autocorrelation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' If  the packet arrivals are found to be periodic, it can be inferred  that the Command-and-control stage has been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' It  should be noted that there may be machines in the enterprise  tier which talk to public servers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=', to retrieve a web  page over HTTPS from a web server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The packets exchanged  with such legitimate public servers contribute as noise to our  extracted timings for periodicity detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The ACF (auto-  correlation function) can detect periodicity reliably in presence  of such noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Detection of Discovery stage: The network traffic traces  [CommandandControl)  Readyfor  Infectedentry  Established  attack  host  foothold  {Discovery,Lateral  movement)  Infected new  host  Discovery,Lateral  movement)  Fieldbusscanning,C  (CEcomm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='spoofing)  comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='spoofing)  Infectededge  CollectICS  Execute CE  intelligence  commands  gateway  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='Goals  achieved/APT  detectedDatasource#1  Data source#2  Data  Feature  Feature  preparation  extraction  processing  Data source #n-1  Data source#n  APTcampaign  APTAttack-stageDetection  graph  &Correlationengine6  TABLE I: Optimal data sources for detecting APT attack stages in IIoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' (P) indicates primary data source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Attack Stage  Audit provenance  Network traffic traces  IDS alerts  Host logs  Command-and-control  \x17  ✓  \x17  \x17  Discovery  \x17  ✓(P)  ✓  \x17  Lateral Movement  \x17  ✓  \x17  ✓(P)  Fieldbus scanning  \x17  ✓(P)  ✓  \x17  CE communication spoofing  \x17  ✓  \x17  \x17  collected at a host are split into smaller traces of fixed time  duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' These traces are assumed to belong to either of two  classes: normal or scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' A normal trace is one which  does not consist of network scanning packets while a scanning  trace is one that does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Each trace is classified using an ML  algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=', Decision Trees, SVM, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' If a trace is  classified as belonging to scanning class, it can be inferred  that the Discovery attack stage has been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' For each  trace, the features for ML classification are extracted from  TCP/UDP headers only and not the payloads of the respective  packets since the network traffic might be encrypted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The ML  features selected for detection of scanning traffic are shown in  Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Motivation behind feature selection: The intuition behind  selecting the first two features is that port scanning tools  used by attackers send TCP/UDP requests to multiple IP  addresses to find out open ports and the services running  on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The third set of features was selected because  during port scan targeting an IP address, many of the ports  to which TCP connection requests are sent are not open  and no response/acknowledgement is sent back, so the TCP  connections formed remain half-open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The fourth feature seeks  to exploit the fact that once TCP connection is formed with  an IP address at a certain port number, port scanning tools  exchange data only for a short time and the connection is  subsequently reset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Compared to packet length in normal TCP  connections, port scanning packet lengths are generally shorter  making the fifth set of features useful for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Finally,  the sixth feature set targets the short time intervals between the  transmission of port scanning packets as compared to normal  packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Detection of Lateral movement stage: According to our  approach, the authentication logs on network hosts are used  to detect this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' We look for logins that satisfy the  following suspicious property: the machine from which a user  is initiating the login to another machine is a part of other  detected APT attack stages that usually precede lateral move-  ment, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=', command-and-control and discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' For example,  if user1 logs in to machine-B from machine-A, machine-A  is in a different subnet as machine-B and machine-A has  been identified as part of Discovery stage detected earlier, we  can conclude that lateral movement has been detected from  machine-A to machine-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Detection of Fieldbus scanning: Similar to the detection  of Discovery stage, the network traffic traces collected at edge  gateway network interfaces connected to control elements are  split into smaller traces of fixed time duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' These traces  are assumed to belong to either of two classes: normal or  fieldbus scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' A normal trace is one which does not  consist of Fieldbus scanning packets while a fieldbus scanning  trace is one that does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Each trace is classified using an ML  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' If a trace is classified as belonging to fieldbus  scanning class, it can be inferred that the Fieldbus scanning  attack stage has been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' For each trace, the features for  ML classification are extracted from TCP headers only and not  the payloads of the respective packets since the network traffic  might be encrypted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The ML features selected for detection of  Fieldbus scanning traffic are shown in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Motivation behind feature selection: The intuition behind  selecting the first three features is that typically during fielbus  scanning, attackers attempt to set up a TCP connection with a  fieldbus device (PLC/RTU) which is unsuccessful for the first  few times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Once the connection is set up, the fieldbus scanner  requests device enumeration data and finally, the connection  is closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' If the scanner is also trying to enumerate fieldbus  slaves, it iterates through the list of slave IDs sequentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Since slaves are not present at all SIDs, the TCP connection  may be reset only for the fieldbus scanner to set up a new  connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Again, compared to packet length in normal TCP  connections, fieldbus scanning packet lengths are generally  shorter making the fourth set of features useful for classi-  fication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Finally, the fifth feature set targets the short time  intervals between the transmission of fieldbus scanning packets  as compared to normal packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Detection of CE communication spoofing: If an attacker is  trying to spoof communications with a control element using  one of the standard industrial automation (IA) protocols (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=',  IEC 61850, IEC 61131-3), there would either be more than  one TCP connections from the edge gateway to the control  element at the destination port specific to that IA protocol,  or the original TCP connection would be terminated by the  attacker leaving only the attacker’s TCP connection active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Therefore, if there are more than one TCP connections from  the edge gateway to the control element at the destination port  specific to that IA protocol or the original TCP connection has  been terminated, it can be inferred that the CE communication  spoofing stage has been detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Detection using multiple data sources  As explained in Section IV-B, there can be more than  one data sources which can be used to detect each APT  attack stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' For an attack stage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' a ∈ {Command-and-control,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Discovery,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Lateral movement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Fieldbus scanning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' CE commu-  nication spoofing},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' we define the aggregate detection score  as:  da = ∑  i  wia1ia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  (1)  where wia is the weight assigned to the ith data source for  detection of attack stage a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' and the indicator function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' 1ia is  7  Attack stage  ML features  Discovery  Number of unique TCP SYN/UDP destina-  tion IP addresses,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Number of unique TCP  SYN/UDP destination ports per destination IP  address (maximum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' minimum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' mean),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Number  of half-open TCP connections,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Number of TCP  RESET packets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Packet length in bytes (max-  imum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' minimum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' mean),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Packet inter-arrival  time in seconds (maximum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' minimum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' mean)  Fieldbus scanning  Number of TCP 3-way handshakes with a  destination IP address (maximum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' minimum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  mean),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Number of TCP RESET packets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Num-  ber of TCP FIN packets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Packet length in bytes  (maximum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' minimum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' mean),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Packet inter-  arrival time in seconds (maximum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' minimum,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  mean)  TABLE II: ML features selected for detection for APT attack stages  defined as:  1ia =  �  �  �  �  �  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  if ith data source is optimal for detection  of attack stage a  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  otherwise  (2)  The weight assigned to the primary optimal data source  identified for an attack stage should be greater than the weight  assigned to secondary data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' If the aggregate detection  score, da is greater than a pre-defined threshold, τ, then the  attack stage a is considered as detected, otherwise not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Correlation of APT attack stages  There are three conditions which need to be satisfied for an  attack stage A to be followed by an attack stage B:  The source IP address for stage A should match with the  source IP address for stage B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  When stage A involves movement of the attacker from  one machine to another (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=', Lateral movement), then  the destination IP address for stage A should match with  the source IP address for stage B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  The time stamp for stage A should fall earlier than the  time stamp for stage B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  The APT attack stage detection & correlation (ASDC) en-  gine first checks if the initial stage of Command-and-control)  in the proposed IASM can be detected at any of the Internet-  facing hosts in enterprise tier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' If the detection is successful,  ASDC engine checks for the (Discovery) stage at the same  host where Command-and-control) stage was detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' If the  Discovery stage is detected, ASDC engine looks for signs of  the Lateral movement stage at the same host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' If the Lateral  movement stage is detected, ASDC engine proceeds to check  for the Discovery stage again followed by the Lateral move-  ment stage as outlined above at the host accessed after lateral  movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' If the host accessed after lateral movement is the  edge gateway, ASDC engine starts looking for the Fieldbus  scanning stage at the edge gateway and if it is detected, ASDC  engine checks if CE communication spoofing stage can also be  detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The complete attack stage detection and correlation  algorithm proposed above is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='Handling false positives/negatives in ML classification:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='RAPTOR is designed to handle false positives/negatives in  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='Algorithm 1 Detect_Correlate_Attack_Stages (list_host_IP_add)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='1: INPUT: list_host_IP_add (List of IP addresses of Internet-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='facing hosts)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='2: det_status = APT_DET_START  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='3: for host_ip ∈ list_host_IP_add do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='4:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='if Check_C&C_stage (host_ip)[1] = TRUE then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='5:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='src_host_ip = host_ip  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='6:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='end if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='7: end for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='8: if Check_Discovery_stage (src_host_ip)[1] = TRUE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='&&  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='Check_Discovery_stage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='(src_host_ip)[2]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='>  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='Check_C&C_stage (src_host_ip)[2] then  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='9:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='tgt_host_ip  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='∈  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='Check_Discovery_stage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='(src_host_ip)[4] do  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='10:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='if  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='Check_Lateral_Movement_stage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='(src_host_ip,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  tgt_host_ip)[1]  =  TRUE  &&  Check_Lateral_Movement_stage  (src_host_ip,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  tgt_host_ip)[2]  >  Check_Discovery_stage  (src_host_ip)[2] then  11:  if tgt_host_ip = edge_gw_IP then  12:  if  Check_Fieldbus_scan_stage  ()[1]  =  TRUE  &&  Check_Fieldbus_scan_stage  ()[2]  >  Check_Lateral_Movement_stage  (src_host_ip,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  tgt_host_ip)[2] then  13:  if  Check_CE_comm_stage  ()[1]  =  TRUE  &&  Check_CE_comm_stage  ()[2]  >  Check_Fieldbus_scan_stage ()[2] then  14:  det_status = APT_DET_STOP  15:  return (det_status)  16:  end if  17:  end if  18:  end if  19:  end if  20:  end for  21: end if  22: Function  Check_C&C_stage  (host_ip),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Returns  {bool_val,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' time_det,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' C&C_server_IP}  23: Function Check_Discovery_stage (host_ip),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Returns  {bool_val,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' time_det,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' scan_type,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' list_target_host_IPs}  24: Function  Check_Lateral_Movement_stage  (src_host_ip,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' dst_host_ip),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Returns {bool_val,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' time_det}  25: Function  Check_Fieldbus_scan_stage  (),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Returns  {bool_val,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' time_det}  26: Function  Check_CE_comm_stage  (),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Returns  {bool_val,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' time_det}  ML-based detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' For example, let us assume that there is  a false positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=', a packet trace is classified as scanning  though it is normal and the Discovery stage is marked as  detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The subsequent attack stages in IASM would not  be detected by the ASDC engine and therefore, RAPTOR  would know that there was a false positive in detection of  an earlier attack stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' It is also possible that there is a false  negative, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=', a packet trace is classified as normal though it is  scanning, and therefore, the ASDC engine would not invoke  detection of subsequent attack stages in IASM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' We propose to  8  handle both false positives and negatives by taking the mode  of classification results for a packet trace for a sufficiently  large number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' APT campaign graph  As the ASDC engine proceeds to detect various stages of an  APT campaign, it uses the detected stages and their attributes  to construct the APT campaign graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' It is a directed graph,  G(V,E) where each node, vi ∈ V ∀i ∈ {1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=',Nv}, where Nv  is the total number of nodes in the graph, corresponds to a  machine (denoted by its IP address) which is a part of one of  the detected attack stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' An edge, ej ∈ E ∀ j ∈ {1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=',Ne},  where Ne is the total number of edges in the graph, is extended  from node vi to another node vk in the graph if there is  a connection from the machine corresponding to node vi to  the machine corresponding to node vk during one or more of  the APT attack stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Each edge has an attribute {s1,s2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='}  where sl ∀l ∈ {1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='} is an attack stage which enables the  connection between the two machines corresponding to the  nodes at either end of the edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' EXPERIMENTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' IIoT Testbed  To generate a realistic IIoT APT dataset which can be  used to evaluate RAPTOR’s performance, we built an IIoT  testbed modelled after Brown-IIoTbed [21] whose architecture  is reproduced in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The implementation of IIoT testbeds  is still in its early stages, with most existing implementations  [4] being special projects and publicly unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Brown-  IIoTbed is designed based on the IIC (Industrial Internet  Consortium)’s IIRA (Industrial Internet Reference Architec-  ture) model and consists of three tiers- edge, platform and  enterprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' It supports a number of real-world IIoT function-  alities such as e-mail notifications to plant workers regarding  important OT events, web-based SCADA interface (viewing  real-time sensor values and trends, actuator status change  notifications, tuning of PLC parameters), remote maintenance  of edge gateway, query to edge data historian, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The testbed  also supports a number of real-world IIoT protocols such as  CoAP, MQTT, and Modbus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Experimental Methodology and Results  We need to collect data from our IIoT testbed under normal  operation as well during APT attack stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' We collect data  from our testbed in the form of network traffic traces from  hosts, audit-based provenance at each x86-based host, host  logs (login records, authentication logs, syslog) and alerts from  Snort network IDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' However, we use only the optimal data  sources identified in Section IV-B towards the final attack  detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Command-and-control stage: To emulate this stage, we use  open-source tools such as dnscat2 to create a communication  channel between a C&C server and a compromised machine  using tunneling over DNS protocol which is one of the most  common C&C communication protocols used by attackers  since most firewalls do not block it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' We run a public dnscat2  C&C server and the client on a Windows 10 Pro and a Ubuntu  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='04 machine in our testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The packet traces generated  on those client machines are collected using tcpdump in sets  of 1 minute duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' A total of 1000 packet traces were  collected for each type of host and fed as input to the detection  phase of Algorithm 2 proposed in [23] for detection of C&C  communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  The performance is evaluated in terms of detection rate  (DR) and missed-detection rate (MDR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Detection Rate is  the fraction of the total number of packet traces which have  been correctly detected as containing C&C traffic, and Missed-  detection Rate is the fraction of the total number of packet  traces which have been incorrectly detected as not containing  C&C traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Using the parameter values (given in Table III),  the detection performance for both the scenarios specified  above is shown in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' It can be seen that the algorithm  gives a DR of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0 and a MDR of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Traffic sampling frequency  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='1  Min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' autocorrelation peak height  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='7×(Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' peak height)  Inter-peak gap variance threshold  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='01  TABLE III: Parameter Values for C&C Communication Periodicity Detection  Discovery stage: To emulate this stage, we use open-source  network scanning tools such as nmap which is either used  directly or is the inspiration for customized port scanners used  by most APT groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' We run the default nmap SYN scans  on a Windows 10 Pro and a Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='04 machine in our  testbed to enumerate connected hosts, their OS versions and  the services running on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Nmap is run in both normal  mode as well as sneaky or as well call it, slow mode, with the  latter mode targeted at evading IDSes [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The packet traces  thus generated on those machines are collected using tcpdump  in sets of 1 minute duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' In real-world APT campaigns,  attackers may slow down network scanning to evade detection  by IDS and therefore, we may need to increase the duration  of packet captures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' A total of 1000 packet traces are collected  from both the Windows and Ubuntu machines under normal  operation and further 1000 packet traces are collected during  the network scanning operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The packet traces are used  to extract features mentioned in Section IV-C and appropriate  class labels (’normal’ or ’scanning’) are assigned to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The  extracted features vectors are further processed (handling of  missing values, scaling) and randomly divided into training  and test datasets using an 80:20 split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Using χ2 test statistic,  we select the best features (test statistic value above a pre-  selected threshold) out of the existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The final feature  vectors thus obtained are used to train Support Vector Machine  (SVM) and Random Forest (RForest) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The trained  ML models are then used to predict class labels for the test  dataset and finally, the detection performance of the models is  evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' We use a 10-fold cross validation approach to tune  the hyper-parameters of the ML classifiers for achieving the  DATASET  METHOD  DR  MDR  IIoT Testbed Ubuntu host  Algorithm 2 [23]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  IIoT Testbed Windows host  Algorithm 2 [23]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  TABLE IV: C&C Stage Detection Performance  9  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' 3: IIoT Testbed Architecture [21]  DATASET  MODEL  PR  RC  IIoT testbed Ubuntu host  (Discovery-normal)  Rforest  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='996  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  SVM  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  IIoT testbed Ubuntu host  (Discovery- slow)  Rforest  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='991  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  SVM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='978  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='974  IIoT testbed Windows host  (Discovery-normal)  Rforest  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='996  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  SVM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='979  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  IIoT testbed Windows host  (Discovery- slow)  Rforest  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  SVM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='912  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='978  IIoT testbed  (Fieldbus (Modbus) Scanning- agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=')  Rforest  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='992  SVM  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='996  IIoT testbed  (Fieldbus (Modbus) Scanning- non-agg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=')  Rforest  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='996  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  SVM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='996  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='983  IIoT testbed  (Fieldbus (Profibus) Scanning)  Rforest  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='992  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='0  SVM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='988  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='996  TABLE V: Raptor’s ML performance for detection of Discovery and Fieldbus scanning  stages  highest possible CV scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The cross validation is based on  training data only without using any information from the test  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Fieldbus scanning stage: To emulate this stage, we run  nmap with modbus-discover script on the edge-gateway for  enumerating Modbus slave IDs and collecting details about the  slave devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' We run the modbus-discover script in both  ’aggressive’ and ’non-aggressive’ modes, where the former  mode refers to finding all slave IDs and the latter mode  refers to finding just the first slave ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Though Modbus is  one of the common protocols used for communication with  PLCs/RTUs, there are other protocols as well which are used  in the industry, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=', DNP3 (Distributed Network Protocol),  Profibus/Profinet, CAN (Controller Area Network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Therefore,  in a separate experiment, we connect a Seimens S7-1200 PLC  to the edge gateway network and run nmap with s7-info script  on the gateway for enumerating Seimens S7 PLC devices  and collecting their device information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The steps for packet  trace collection under normal and fieldbus scanning operations,  feature vector extraction, processing and selection, ML model  training and performance evaluation remain similar to the ones  outlined for Discovery stage above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  The performance of ML classifiers is typically evaluated in  terms of precision (PR) and recall (RC) scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Precision is  the ratio  TP  TP+FP, where TP is the number of true positives  and FP is the number of false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' It represents the  ability of a classifier to avoid labeling samples that are negative  as positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Recall is the ratio  TP  TP+FN , where TP is the  number of true positives and FN is the number of false  negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' It represents the ability of a classifier to avoid  labeling samples that are positive as negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Using the tuned  hyper-parameters’ values, the average classification precision  (PR) and recall (RC) scores obtained for the final classifiers  over 10 runs are shown in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' It can be observed that  for the detection of Discovery stage on Ubuntu as well as  Windows hosts using normal and slow scan speeds, Random  Forest performs slightly better than SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' In general, both  the ML classifiers perform better with normal scanning speed  compared to slow scanning speed which is expected since  within the trace duration (1 min), more number of network  scanning packets would be captured during normal versus slow  speed scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' For the detection of Fieldbus scanning stage  using Modbus protocol in ’aggressive’ and ’non-aggressive’  modes, Random Forest performs almost equally as SVM in  terms of precision but SVM performs quite poorly compared  to Random Forest in terms of recall for ’non-aggressive’  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' For the detection of Fieldbus scanning stage using  Profibus protocol, Random Forest performs slightly better than  SVM in terms of both precision and recall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Based on the  performance results obtained above, it would be preferable to  select Random Forest classifier for detection of Discovery and  Fieldbus scanning stages in RAPTOR’s implementation since  SVM’s performance degrades significantly at slow network  scanning speed and for ’non-aggressive’ fieldbus scanning  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Finally, to emulate an APT attack on our testbed for  construction of APT campaign graphs, we develop three  attack storylines from an APT group’s perspective, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' their  background,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' motivation for attack,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' steps taken for attack and  Edge Tier  Platform Tier  Enterprise Tier  Local SCADA &  Management devices  Cloud Applications  API  End users  Sensor & Actuator  PLC  Cloud Broker  Dashboard  End users  Sensor & Actuator  Edge gateway  Cloud Storage  0  Connected Worker  Remote  Maintenance  On-site Supervisor  Local Servers  Phvsical assets/field devices zone  Edge gateway zone  Cloud zone  Edge mobile service zone  Enterprise service zone  LANs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' and router/firewall zone10  final attack objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The TTPs (Techniques, Tactics and  Procedures) used in our storylines are close to the ones  used in real-world APT attacks on IIoT environments such  as those mentioned in Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' For reasons of space, we  present RAPTOR’s evaluation with only one of the APT  attack storylines here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The rest of the storylines are presented  in Appendix Sections A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Steps 1-2 (Initial Access),  step 4 (Command-and-control), step 6 (Discovery) and step 10  (Credential Access, Lateral Movement) of the attack storyline  are based on the 2014 German steel mill and 2015/2016  Ukraine power grid attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Step 16 (Fieldbus scanning, CE  communication spoofing) of the attack storyline is based on  the 2016 Ukraine power grid attack and the 2017 Saudi  petrochemical plant attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Steps 18-19 (Impact) of the attack  storyline are based on the 2015 Ukraine power grid attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  We run the attack storyline on our IIoT testbed over the  course of a few hours and collect the data generated from  testbed hosts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Since real-world APT campaigns can stretch  over months and it is not possible to emulate them on our  testbed, we assume that our APT storylines are executed  in an accelerated timeframe and therefore our performance  evaluation of RAPTOR holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The complete APT attack  storyline used for evaluating RAPTOR is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  APT attack storyline 1  Background: Attackers belong to a nation-state (or APT)  group which has been tasked with targeting a prominent state-  owned steel manufacturing plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The APT group plans to  steal ICS related data which can be used to understand the  ICS design and components which can further be used to plan  for later attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Goals: To steal sensitive OT data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=', blast furnace temper-  ature sensor measurements, PLC configuration, credentials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  OT data such as blast furnace temperature readings are sensi-  tive because they can be used to learn the normal temperature  range and temporal trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Attackers can use this information  to modify the settings of the furnace temperature controller  to damage the furnace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The temperature readings can also be  used to infer the furnace design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Steps:  1) The attacker sends a spear phishing email (including  a malicious VPN portal web link) to one of the steel  plant employees posing as legitimate company email and  obtains their VPN login credentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  2) It uses the employee’s company email address and  phished credentials to remotely login to the maintenance  machine connected to enterprise network (password re-  use) through the VPN service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  3) The attacker changes the employee’s VPN account pass-  word for persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  4) The compromised machine connects to an external C&C  server through DNS tunnelling and forwards a shell to  the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  5) Attacker installs a malware on the compromised machine  which exploits software vulnerabilities to gain root ac-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  6) The attacker controlling the compromised machine scans  its local network and finds other hosts (firewall, MQTT  server, external API machine) as well as the services  running on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  7) Attacker tries to find CVE vulnerabilities corresponding  to the services running on other hosts but can not exploit  them successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  8) It goes through the shell command history on com-  promised machine and finds previous SSH connection  attempts to the edge gateway containing username and  hostname details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  9) It tries to determine the SSH login password for the edge  gateway as follows:  a) Accesses the shadow password file on compromised  machine (using root access obtained earlier) which  stores password hashes and corresponding hashing  algorithms used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  b) Tries to crack the password hashes to obtain corre-  sponding plaintext passwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  10) The attacker attempts to log in to the edge gateway by  using one plaintext password at a time and is successful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  It explores the files, folders (hidden and non-hidden) and  the processes running on edge gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  11) It finds a web server, a CoAP server and Node-red  application running on edge gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  12) The attacker tries to exploit CoAP related vulnerabilities  but is unsuccessful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' It remotely executes a script from the  compromised maintenance machine to dump the CoAP  resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  13) It executes a fake CoAP client code on the compromised  maintenance machine to receive measurements from sen-  sors directly connected to edge gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  14) It scans devices connected to the edge gateway’s Wi-Fi  hotspot network and finds a host running DNP service  (PLC master).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  15) Attacker downloads a script from C&C server and copies  it remotely to edge gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  16) It extracts PLC configuration data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=', hardware,  firmware, manufacturer, serial number, slave IDs) by  running the script on edge gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  17) The attacker compresses and encrypts all the targeted  data collected in previous steps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=', PLC configuration,  sensor measurements, login credentials) and exfiltrates it  through the C&C channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  We assigned a weight, wia = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='5 to the primary data sources  for detecting an attack stage and a weight of of wia/2 to  the secondary data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The threshold for detection is  selected as τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The APT campaign graph generated by  RAPTOR for Storyline 1 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' IV-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The graph  captures broad details of the APT campaign including the  IP addresses of the machines affected and the tactics used  during the campaign which is quite useful for cybersecurity  analysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' This shows that our proposed attack stage detection  and correlation algorithm in Section IV-E works as intended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  However, the campaign graph does not capture all the tactics  employed by the APT attackers since our focus is on detecting  invariant APT tactics/stages only as explained in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Further, the campaign graph does not contain any details on  the specific techniques employed by the APT attackers since  11  our focus is not on detecting the individual techniques used  for each tactic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The APT campaign graph can serve as a  starting point for cybersecurity analysts to fill in the missing  tactics based on the APT attack frameworks for ICS, further  investigation and mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The APT campaign graphs for  Storyline 2 and Storyline 3 are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' 6  respectively in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' 4: APT campaign graph generated for Storyline 1  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' DISCUSSION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Comparison with State-of-the-art  We are unable to conduct a performance comparison of  RAPTOR with [12] as they do not provide any source code for  their proposed multi-stage attack detection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Further,  the dataset used for performance evaluation in [12] which con-  sist of a synthetic APT campaign injected into the CSE-IDC-  2018 intrusion detection dataset [25] has not been publicly  released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' HOLMES [10] and CONAN [11] do not provide  source codes for their proposed APT detection system as well  though both use the DARPA Transparent Computing ((TC)  Engagement dataset [26] for performance evaluation which has  been released publicly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The DARPA TC dataset contains data  from a red team deploying APT-style TTPs on a target system  consisting of multiple interconnected hosts running different  OSes and having exploitable CVE vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' However,  the DARPA TC dataset suffers from following limitations  which reduce its applicability for RAPTOR’s performance  evaluation:  The TC network setup is simple and does not emulate  real-world enterprise/IIoT networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  The dataset provides only json files but no raw pcap  files for us to extract network traffic-based features for  a meaningful comparison with RAPTOR’s performance  on our IIoT testbed dataset1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  None of the TTPs used in TC dataset are IIoT-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Limitations  The design of RAPTOR introduces a few limitations which  are discussed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' RAPTOR uses supervised ML  1There is an active unresolved issue with the DARPA TC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' While  loading data from the compressed *.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='gz files, the code get stuck at  streaming records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  algorithms for detection of certain APT attack stages which  means that it can detect only known malicious traffic pat-  terns produced by those attack stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' However, it should be  noted that supervised ML algorithms are used much more  commonly compared to unsupervised algorithms in real-world  deployments of HIDS/NIDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Advanced APT malware may  attempt to evade detection by RAPTOR by slowing down the  scanning activity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=', during Discovery, Fieldbus scanning  stages) or changing the time period between scanning attempts  to confuse the trained ML algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' This evasion technique  can be countered by increasing the packet trace duration to  capture enough attack packets though it may lead to longer  classification delays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The ML classifiers used for detection  of certain APT attack stages may have to be re-trained, for  example, when the classification probability falls below a pre-  defined threshold or the host OS is updated, and this may  cause a delay in detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Evasive APT malware may also use some other C&C  server messaging mechanism than the TCP ([PSH,ACK],  [ACK])/UDP one to escape filtering and/or force the C&C  communication to be non-periodic (by adding noise traffic for  example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' If the attacker does change the C&C messaging  mechanism, the detection method can be changed accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Moreover, the C&C communication periodicity detection algo-  rithm uses an ACF (Autocorrelation Function)-based approach  which can detect periodicity in the presence of noise as well  if the noise is uncorrelated with the desired signal (discrete-  time sequence extracted from C&C server traffic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' An attacker  may evade the lateral movement detection logic by exploiting  an RCE vulnerability instead of performing manual logins or  using stealthy malware which blends the logins in its attack  path with a previous legitimate user login.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Finally, it is also  possible that a few hosts may already be infected before  RAPTOR is deployed in an IIoT environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' CONCLUSION  We have proposed RAPTOR, an APT detection system  targeted at IIoT environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' It detects and correlates attack  stages derived from an APT Attack Invariant State Machine us-  ing optimal data sources selected for each stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The correlated  attack stages are utilized to generate a compact, high-level  APT Campaign Graph which can be used by cybersecurity  analysts to track the progress of the APT campaign and deploy  appropriate mitigation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' A performance evaluation of  RAPTOR shows that it can detect APT campaigns modelled  after real-world attacks with high precision and low false  positive/negative rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
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+page_content='  [25] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Sharafaldin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Lashkari, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Ghorbani, “Toward  Generating a New Intrusion Detection Dataset and Intrusion Traffic  Characterization,” in Proceedings of the 4th International Conference  on Information Systems Security and Privacy, ICISSP 2018, Funchal,  Madeira - Portugal, January 22-24, 2018, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Mori, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Furnell, and  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Camp, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  SciTePress, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' 108–116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Available:  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='5220/0006639801080116  [26] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' Torrey, “Transparent Computing Engagement 5 Data Release,”  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='com/darpa-i2o/Transparent-Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  APPENDIX A  APT ATTACK STORYLINE 2  Background: Attackers belong to a nation-state (or APT)  group which has been tasked with targeting a prominent state-  owned steel manufacturing plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The APT group plans to  disrupt the steel production and thereby affect other industries  dependent on steel and exports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Goals: To shut the blast furnace down by controlling the  furnace relays (LEDs in our testbed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' This may damage the  plant operations temporarily or permanently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Steps:  1) An insider recruited by the APT group installs malware  on the maintenance machine through a USB stick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The  malware exploits software vulnerabilities on the machine  to gain root access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  2) The compromised machine connects to an external C&C  server through DNS tunnelling and forwards a remote  display to the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  3) Attacker installs a malware on the compromised machine  which exploits software vulnerabilities to gain root ac-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  4) The attacker controlling the compromised machine scans  its local network and finds other hosts (firewall, MQTT  server, external API machine) as well as the services  running on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  5) Attacker tries to find CVE vulnerabilities corresponding  to the services running on other hosts but can not exploit  them successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  6) It accesses the shadow password file on compromised  machine (using root access obtained earlier) which stores  password hashes and corresponding hashing algorithms  used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  7) Attacker successfully opens an RDP (Remote Desktop  Protocol) session to the external API machine using one  of the stolen password hashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  8) It accesses the SCADA/HMI web interface on the ex-  ternal API machine and turns off the LEDs directly  connected to edge gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  APPENDIX B  APT ATTACK STORYLINE 3  Background: Attackers belong to a nation-state (or APT)  group which has been tasked with targeting a prominent state-  owned steel manufacturing plant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The APT group plans to  13  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' 5: APT campaign graph generated for Storyline 2  disrupt the steel production and thereby affect other industries  dependent on steel and exports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Goals: To damage the plant equipment by tampering with  the operation of safety controllers which prevent the blast  furnace from entering an unsafe state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' The safety controllers  may be reprogrammed to allow the blast furnace to enter  a dangerous state without any corrective action leading to  physical damage to the plant and even loss of human lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Steps:  1) The attacker sends a spear phishing email (including  a malicious VPN portal web link) to one of the steel  plant employees posing as legitimate company email and  obtains their VPN login credentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  2) It uses the employee’s company email address and  phished credentials to remotely login to the maintenance  machine connected to enterprise network (password re-  use) through the VPN service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  3) The attacker changes the employee’s VPN account pass-  word for persistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  4) The compromised machine connects to an external C&C  server through DNS tunnelling and forwards a shell to  the attacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  5) Attacker installs a malware on the compromised machine  which exploits software vulnerabilities to gain root ac-  cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  6) The attacker controlling the compromised machine scans  its local network and finds other hosts (firewall, MQTT  server, external API machine) as well as the services  running on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  7) Attacker tries to find CVE vulnerabilities corresponding  to the services running on other hosts but can not exploit  them successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  8) It goes through the shell command history on com-  promised machine and finds previous SSH connection  attempts to the edge gateway containing username and  hostname details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  9) It hijacks any future SSH session between the com-  promised machine (started by an employee performing  remote maintenance) and the edge gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  10) The attacker explores the files, folders (hidden and non-  hidden) and the processes running on edge gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  11) It finds a web server, a CoAP server and Node-red  application running on edge gateway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  12) The attacker scans devices connected to the edge gate-  way’s Wi-Fi hotspot network and finds a host running  DNP service (PLC device).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  13) It downloads a payload from C&C server, copies it  remotely to edge gateway and executes it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  14) The attacker terminates the existing process which is  communicating with the PLC device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  15) It collects more information about the PLC device and  enumerates all the slave IDs using the payload com-  mands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  16) The attacker uses the payload to send a command to the  targeted slave to read its current state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  17) It remotely uploads a new program to the PLC device  while it continues to operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
+page_content=' 6: APT campaign graph generated for Storyline 3  fCommandandControl  192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
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+page_content='4fCommandandControl)  192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/j9FJT4oBgHgl3EQfYCxF/content/2301.11524v1.pdf'}
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+ 
+Xinchen Kang (22f3d431-dcc0-42a5-8e2b-c26464e0654d) 
+Beijing Union University, China 
+Dengfeng Yao (ae88317c-d091-4f41-97f1-b1e6be00ca68) 
+Beijing Union University, China 
+Minghu Jiang (ea1cc43b-eee9-4185-8d97-edeac9186268) 
+Tsinghua University, China 
+Yunlong Huang (cc1ddbf2-64b9-4c51-b553-0ebe52a8d645) 
+Tsinghua University, China 
+Fanshu Li (64642396-6e95-456f-a4f2-47ff81a23d6e) 
+Beijing Union University, China 
+ABSTRACT 
+In this study, the authors propose a computational cognitive model for sign language (SL) perception and 
+comprehension with detailed algorithmic descriptions based on cognitive functionalities in human 
+language processing. The semantic network model (SNM) that represents semantic relations between 
+concepts is used as a form of knowledge representation. The proposed model is applied in the 
+comprehension of sign language for classifier predicates. The spreading activation search method is 
+initiated by labeling a set of source nodes (e.g. concepts in the semantic network) with weights or 
+"activation," and then iteratively propagating or "spreading" that activation out to other nodes linked to 
+the source nodes. The results demonstrate that the proposed search method improves the performance of 
+sign language comprehension in the SNM. 
+Keywords: Attention, Cognitive Processing, Comprehension, Decision-Tree, Game Theory, Linguistics, 
+Perception, Semantic Network, Sign Language 
+INTRODUCTION 
+Sign language (SL) comprehension is a fundamental task for computational linguists. Two types of 
+algorithms have been proposed: (1) rule-based methods (Supalla, 1982), and (2) statistical methods 
+(Bauer & Heinz, 2000; Huenerfauth, 2005). Rule-based methods lack the capability of planning the 
+elements in the entire scene (Liddell, 2003). The method of modeling infinite natural language input 
+through finite rules, especially minor rules, barely meets all requirements of SL processing (Yao et al., 
+2017). Therefore, statistical methods are the preferred type of algorithm for SL comprehension. Statistical 
+models can be applied to spoken languages. Given the abundant data resources of spoken languages in the 
+digitalized Internet age, statistical models can be applied readily. However, the raw and annotated corpora 
+of SLs are insufficient because collecting and annotating SL videos are tedious and difficult. Data sparsity 
+consequently remains as the most serious problem when applying statistical models onto SLs. For 
+
+example, the real-time factor (RTF) of the SL video corpus is 100; that is, an hour corpus requires at least 
+100 hours of annotation (Dreuw et al., 2008b). 
+ 
+Simulating SL comprehension using traditional statistical models and machine-learning methods 
+isdifficult. Thus, reliable methods for establishing a signer’s 3-D model (which is the process of 
+developing a mathematical representation of any three-dimensional surface of moving trajectories of 
+signers in the space for SL via specialized software) for SL corpus building and technologies for 
+annotating a large-scale SL video corpus automatically must be developed. Unlike the spoken language 
+that is “a set of values that change with the passage of time” (Huenerfauth, 2005), SL does not have a 
+writing system and thus cannot be saved in any form of written texts. 
+  
+The natural language-processing system relies on texts to process spoken languages. This system records 
+only the written text that corresponds to speech flows and relies only on the literacy of the user. On the 
+other hand, the SL system comprises information from multiple modalities. Examples of such information 
+are the hand shape, hand location, hand movement, hand orientation, head tilting, shoulder tilting, eye 
+gazing, body gestures, and facial expressions. The considerable information from multiple channels in SL 
+conveys linguistic meaning. This multi-modality nature of SL poses difficulties for the coding of SLs into 
+a linear single-channeled character string. In addition, SLs have writing systems, such as the Sign Writing 
+system (Sutton, 2010), ASL-phabet (Supalla et al., 2008), and HamNoSys (Prillwitz et al., 1989). 
+However, these systems have a limited number of users (Johnston, 2004). 
+ 
+Many linguistic details are lost because of the multi-modality nature of SL during the translation of SL 
+into its corresponding writing system. SLs may be understood by directly matching the visual–spatial 
+characteristics of SL with the semantic units in the brain rather than applying written texts as an 
+interpreting medium. Here, semantic units are generally used for processing natural languages; these units 
+or nodes contain some information, which are used as knowledge representations form semantic units 
+(Geva et al., 2000). Such direct matching also represents the most natural way of comprehending SLs in 
+the brain. From this perspective, the authors present a computational cognitive model for SL 
+comprehension that is based on the cognitive functionalities of the human brain combined with a 
+knowledge representation theory of artificial intelligence (Shuklin, 2001). 
+  
+Visual–spatial mechanisms are exploited to express the grammatical structures and functions in SL. 
+Visual–spatial perception, memory, and mental transformations are prerequisites to grammatical 
+processing in SL (Emmorey & Corina, 1990) and are central to visual mental imagery (Farah, 1988). A 
+series of experiments have been conducted to investigate visual attention (Neville et al., 1998). Movement 
+recognition in peripheral vision is important in sign perception because the signers mainly look at the face 
+instead of tracking the hands when they communicate through SL (Siple, 1978). Therefore, identification 
+of lexical-level information depends on the peripheral vision system when signs are produced. The 
+recognition of movement directions is the selective function of peripheral vision (Bonnet, 1977). 
+ 
+Whether deaf people only have a strong peripheral vision or efficiently allocate attention to peripheral 
+vision remains unclear. Stivalet et al. (1998) showed that visual attention processing can be changed by 
+auditory deprivation. They determined that deaf people do not shift their attention when processing the 
+information (i.e., alphabet set) presented in the central vision field, whereas hearing subjects must shift 
+their attention to search for the alphabet set continuously. Smith et al. (1998) also found that lack of 
+auditory input causes weak and selective (or highly distributed) visual attention among deaf children. 
+Stivalet et al. (1998) proposed that effective visual processing is caused by intermodal sensory 
+compensation; that is, the strong allocation of visual attention can be attributed to neuron reorganization 
+caused by auditory deprivation from birth. Recent magnetic resonance imaging evidence supports this 
+hypothesis (Bavelier et al., 2000). 
+  
+
+These findings are selective attention cases, in which attention selectively processes certain stimuli but 
+ignores other stimuli. The cases refer to the selective orientation and concentration of the senses (i.e., 
+visual, auditory, taste, and tactile senses) and consciousness (i.e., awareness) of people on certain targets 
+(towards other factors). Studies on attention have failed to describe human attention at the biological level 
+in detail, as a person cannot focus continuously because the brain automatically suppresses activity when 
+attention reaches its limits. 
+  
+Emmorey and Reilly (2013) determined that when locations in a signing space (SL expressions streaks 
+the space) function topographically, spatial changes tend to be noticed easily. Thus, location information 
+indicating the spatial position of associated referents can be encoded and stored semantically in memory. 
+However, spatial locations with a primary distinguishing function of referents are encoded in a different 
+way and tend to be discarded from memory once the referential function is no longer required by context 
+(Emmorey & Reilly, 2013). Bavelier et al. (2001) claimed that only the posterior middle temporal gyrus 
+and the medial superior temporal cortex of deaf signers are highly active while perceiving movements in 
+peripheral vision. This phenomenon is unobservable in hearing signers who have skillfully grasped signs, 
+indicating that auditory deprivation results in a shift to stronger movement attention in the visual 
+periphery. Deaf people can easily reply to the attention and visual monitoring of their peri-personal space 
+(Bavelier et al., 2001). 
+ 
+Neville et al. (1998) determined that the classic language area in the left hemisphere, particularly the left 
+perisylvian, of both deaf and hearing subjects is activated when reading English sentences. The right 
+hemisphere, including the right perisylvian, of deaf people is also activated. They argued that spatial 
+processing is of great importance to sign grammar. Thus, the SL comprehension process of deaf people 
+employs neurons at both high and low levels in the neural network, which are connected with each other 
+by edges, and generates high-level features via feature combination processes that are realized by 
+combining the weight on the edges. For example, low-level visual edge features are assembled, processed, 
+and sent to the high level to form the angle, shape, and other higher features (Bertasius et al., 2015). High-
+level neurons form features that gradually approximate the semantics in turn, such as simple shapes, 
+simple targets, and real objects. The activation of high-level neurons during the reconstruction process 
+also reacts with the low-level neurons and adjusts and corrects deviations and losses (Bertasius et al., 
+2015); a temporal pattern appears in the horizontal structure connection. The neurons can make 
+predictions of the state at the next point of every time point through a horizontal connection based on the 
+information of their current status (Hawkins et al., 2009). 
+SEMANTIC NETWORK MODEL (SNM) 
+Model of semantic networks (SNs) are generally used for processing natural languages (Shuklin, 2001). 
+SNs, as knowledge representations, are extensible and have been used to model mental disturbances 
+(Geva et al., 2000). The semantic network (which is a network that represents semantic relations between 
+concepts, is used as a form of knowledge representation, here it is based on SL information processing of 
+human brain cortex. The edges connect different nodes in the network and represent the strength or 
+weakness of the correlation. After being set up, the semantic network is stored in long-term memory for 
+future retrieval and extraction to be encoded as semantic memory. Outside stimuli at a certain time can be 
+the demand of a person on specific knowledge and information to activate the demand on the extraction 
+of useful information of long-term memory (Sedikides & Skowronski, 1991). The activation process of 
+the stored network works in a form of spreading in the memory (Collins & Quillian, 1972). 
+  
+The semantic model, which is based on SL information processing of human brain cortex, is developed 
+accordingly. Different areas of the brain cortex are involved in the processing and are connected in a 
+hierarchical manner. Low-level information from sense organs is first processed in the primary 
+information-processing regions of the brain cortex and is then transferred to high-level regions for further 
+
+processing, such as abstracting, integrating, and interpreting. The detailed description and illustration of 
+this hierarchical structure are summarized in Figure 1. 
+Figure 1. Hierarchical structure. Low-level areas in the hierarchy generate specific information that 
+increases speed and contain further details, whereas high-level areas form stable spatial invariance, 
+change slowly, and show high-level semantic object expression (Adapted from Yao et al., 2015) 
+ 
+ 
+In SL communication, both substantial and semantic information (substantial information includes hand 
+shape, hand location, hand movement, hand orientation, head tilting, shoulder tilting, eye gazing, body 
+gestures, and facial expressions, and semantic information is represented into semantic concepts by these 
+substantial SL information) almost exclusively relies on signs. However, accurate SL information 
+analysis and prediction remain as challenging tasks in the field of natural SL processing. Three main tasks 
+are, namely, capturing, decoding, and extracting the physical characteristics and relationship of signs 
+(perception stage), matching the decoded cognitive representations with the stored semantic information 
+(memory stage), and completing the machine translation process of SL information (judgment stage). 
+This process of cognitive processing and understanding during SL communication is based on the PMJ 
+principle of  “from the definition and extraction/annotation of cognitive representation (Stage P) to the 
+feature storage in line with the cognitive economy principles (Stage M), and then to the output of the 
+classification and judgment (Stage J).” 
+  
+The P→M→J (PMJ) principle exhibits a complete fine processing frame, the detailed illustration, and 
+description of SL comprehension frame based on the PMJ principle is summarized in Figure 2. 
+Figure 2. SL comprehension frame based on the PMJ principle. Perception refers to acquiring sign 
+information through selective attention. The information is limitedly processed by the brain if prominence 
+is given to useful and important information. Other information may be filtered out or suppressed when 
+sources for information processing are limited. Memory refers to the spreading activation process, in 
+which input information is coded, and one intends to store the information for a short period. Judgment 
+
+targeted
+wer
+targeted sen antics
+sign semantic concept
+semantics
+rea
+representation
+Area H
+hign deve semantid
+high-level semartic feature
+feature detectors
+Area A
+low-evel senantic
+low-level semantic feature
+low-level s emanticf eature
+feature detectors
+cogritive
+representation
+Higher leve visual
+handshape
+location
+oriention
+moveme rt
+No-manual feature
+AreaV4
+abstractions
+Edge
+outine
+Simple shape
+Area V1
+detecton
+ edge feat ure
+ed ge feature
+Retina
+oves
+pixel
+pixel
+pixel
+pixel
+physical
+characteristics
+一refers to the process in which the perceived information or the information stored in memory is 
+compared, matched, or classified, and a decision or prediction is made. After the spreading activation, 
+the network records the attention features of users and activates their future preferences (Yao et al., 
+2015) 
+ 
+ 
+Concepts are in the form of network storage. The different concepts are stored in different functional 
+areas in both hemispheres of the human brain. The same or similar concepts are stored in same or 
+adjacent regions of brain. Specific information of entities in the outside world, such as humans, animals, 
+or tools, is represented by the concept network in the human brain. This concept network (A concept is an 
+abstract idea representing the fundamental characteristics of what it represents. Concept network consists 
+of these abstract concepts) is, in turn, connected with the lexical network from mental lexicon in the left 
+temporal lobe. Such specific information from mental lexicon will be employed to facilitate SL 
+production during which the SL users generate classifier hand shapes under the guidance of the 
+knowledge and rules of SL classifier predicates (Valli & Lucas, 2000). Here, classifier predicates are 
+made by combining small meaningful unites to create bigger units, the main units being the hand shape 
+and the movement. This condition implies that findings from brain research can provide knowledge and 
+guidance for the cognitive computational modeling of classifier predicate comprehension. In order to 
+obtain a deep understanding of sign lexical semantics, a cognitive processing model, which is based on 
+the cognitive mechanism of human brain, is established. The cognitive processing model would activate 
+the concept network of the associated classifier hand shapes in the brain. Here, classifier predicates differ 
+from traditional linguistic units. Traditional methods, such as the syntactic tree, cannot satisfy the 
+generation of the classifier predicates (Huenerfauth et al., 2006).  
+DECISION-TREE BASED ALGORITHMIC METHODS 
+The authors use SNs as the knowledge representation and organization mode of SL. The relationship in 
+semantic networks represents a type of information among nodes. Nodes with a complicated relationship 
+with other nodes contain additional information. Such nodes require further effort to be understood. 
+Consequently, the authors simulate selective attention (i.e., the processing of visual or auditory input 
+based on whether it is relevant or important). They selected particular representations to enter perceptual 
+awareness and therefore guide behavior. Through this process, less relevant information is suppressed by 
+humans using the proposed algorithmic methods to accentuate the nodes selectively and suppress the 
+unessential nodes (Chelazzi et al., 2013). 
+ 
+The emergence of 3-D-based sensors, such as Kinect by Microsoft and Leap Motion (Yao et al., 2014), 
+has improved studies on sign recognition from video-based to 3-D-based sign recognition. However, this 
+transformation makes traditional video-based SL recognition methods inapplicable to 3-D-based SL 
+
+Perception
+Memon
+Judgment
+Attentional enha ncement and suppress ion
+0
+Spread activation
+Interactive activationrecognition technologies. Large training data are required for valid recognition in 3-D-based SL 
+recognition technologies because of the low operation efficiency of the rotatable joint-based sorter and the 
+matching techniques for sign signal recognition. Yao et al. (2014) proposed a decision tree-based 
+algorithm. The algorithm aims to achieve a high-precision and real-time performance of SL automatic 
+perception according to the features of Leap Motion. The authors adopted this method as the first step of 
+SL comprehension. 
+Attention Function 
+The authors propose the following attention function: 
+ 
+                                                                              ������������������������ =
+∑
+������������������������������������
+������������������������������������ ������������
+∑ ������������������������������������
+������������������������  
+(1) 
+ 
+where ∑
+������������������������������������
+������������������������������������ ������������
+ denotes the sum of the semantic relation weights around the semantic node x, ∑ ������������������������������������ 
+denotes the sum of all semantic relation weights, and ������������������������ represents the activation value on the semantic 
+node x after the spreading activation process. 
+Semantic Matching 
+Cognitive units in the memory network compete with one another based on certain rules to obtain more of 
+the limited attention resources and more energy for a more active state. SL comprehension supports 
+interactive activation models (Gutierrez et al., 2012). Therefore, judgment is the outcome of the attention 
+competition game in the spreading activation, which is a search algorithm. The search algorithm is 
+initiated by labeling a set of source nodes (e.g. concepts in a semantic network) with weights 
+or ”activation,” and then iteratively propagating or “spreading” that activation out to other nodes linked to 
+the source nodes processes of the human brain (Crestani, 1997; Preece, 1981). 
+ 
+A semantic matching algorithm based on activation spreading modes is proposed to determine the most 
+appropriate semantic information. Activation starts to spread from the corresponding nodes of the signs 
+presented by the signer. The activation value of the stimulus node (i.e., signs to be perceived before the 
+start of spreading) must be calculated first. In particular, the increment in the interest value of object 
+concept must be calculated, and this concept node must be used as an initial node for the spread study. 
+Activation spreads to the neighboring nodes, which usually have a lower activation value than the source 
+value. Therefore, introducing an activation attenuation factor for decreasing activation over the path 
+length in the closed interval [0…1] is mandatory. That is, for every propagation through an edge a loss of 
+activation is considered (Neumann et al., 1993; Rocha et al., 2004). The activation spreading process can 
+be expressed as follows:  
+ 
+ ������������������������(������������ + 1) = ������������������������(������������)������������������������������������(1 − ������������)
+(2) 
+ 
+where  ������������������������(������������ + 1) represents that the value is spread from node x to y at time, t+1, ������������������������(������������) represents the 
+activation value that was spread at node x at time, t, ������������������������������������ signifies the link between nodes x and y, and δ is 
+an attenuation factor used to describe the energy loss caused in the activation spreading process (Jiang & 
+Tan, 2006). 
+  
+Spreading activation theory states that the activation of human memory “chunks” (the content of any 
+buffer is limited to a single declarative unit of knowledge, called a chunk) is determined by two factors 
+(Anderson et. al., 2004; Anderson, 2013), namely, the use history of the memory chunk and the 
+correlation between the memory chunk and the current retrieval information. These two factors calculate 
+the weights and determine whether the chunk is activated and selected. This assumption has been verified 
+
+by experimental cognitive psychology, and the calculation model has been established (Roelofs, 1992). 
+The authors must use moments to express the distance in each activation time with the current time. Time 
+units may be per hour as a unit and may also be the day. With the day as a unit, we can count the 
+historical value in the previous day as the activation value of the first day. The algorithm based on the 
+theory of memory activation can improve SL understanding, which is sometimes highly sensitive to time. 
+ 
+At node y, the largest number of neighbor nodes is (n-1); thus, the maximum of  ������������������������(������������ + 1) can be 
+expressed as: 
+  
+ ������������������������(������������) = [������������1, ������������2, … , ������������������������]������������ 
+ 
+i.e., the initial value of the semantic network. Where I1, I2, …, In are these activation value of neighbor 
+nodes.  
+ 
+If activation spreads from a node in many directions, then its adjacent nodes obtain a low activation value. 
+The adjacent nodes give a feedback value of their resonance energy (i.e., contributing structure with the 
+lowest potential energy) to the co-adjacent nodes after they absorb the activation value. The following 
+equation is therefore used: 
+ 
+Iz(t + 1) = Oz(t) + ∑
+Ox(t)Λxz(1-δ)
+all actived x
+                                       (3) 
+ 
+where Oz(t) denotes the activation value of node z at time t.  
+ 
+Given that the quantity of activated information is limited, the nodes that obtain less resonance 
+information are equivalently inhibited and are less likely to be activated. The activation value distribution 
+in the resonance process conforms to the human attention model. 
+Attention Game Process 
+Cognitive units in memory network compete with one another by certain rules to increase the possibility 
+of obtaining more human attention resources and more energy that will improve activity. This 
+phenomenon is called a game process. The authors use game theory (Myerson, 1997), which is the study 
+of mathematical models of conflict and cooperation between intelligent rational decision-makers and 
+attempts to achieve the largest cognitive gains with the least energy possible, as a reference to simulate 
+the attention enhancement and suppression processes that are selective attention processes. In other 
+words, when visually searching for a non-spatial feature or a perceptual feature, selectively enhancing the 
+sensitivity to that specific feature plays a role in directing attention. When people are told to look for 
+motion, then motion will capture their attention, but attention is not captured by motion if they are told to 
+look for color (Reynolds & Chelazzi, 2004). Activated results consistent with cognitive features can then 
+be obtained. The authors assume that the game contains n nodes. ������������������������
+′ and ������������������������
+" are the two selectable 
+strategies for node i, and they represent the acceptance and non-acceptance of the change in the attention 
+function (i.e., ������������������������
+′, ������������������������
+" ∈ ������������������������ ). The corresponding gain can be represented by ������������������������
+′ and ������������������������
+" , and ������������������������
+′, ������������������������
+" ∈ ������������������������. N 
+nodes are assumed to reach an agreement before participating in the game to introduce the Nash 
+equilibrium (i.e., each node only selects a specific strategy). The authors let ������������∗ = (������������1
+∗, … , ������������������������∗) represent the 
+agreement, where ������������������������
+∗ is the strategy of the node i specified in the agreement. Nodes comply with this 
+agreement only when the benefit from complying with the agreement is larger than that from not 
+complying. This agreement constitutes Nash equilibrium if any node abides by this agreement. Thus, the 
+Nash equilibrium is written as follows: 
+ 
+ ������������������������(������������������������
+∗, ������������−������������
+∗ ) ≥ ������������������������(������������������������, ������������−������������
+∗ ),               ∀si ∈ Si
+(4) 
+
+ 
+where the combination of strategy  ������������∗ = (������������1
+∗, … , ������������������������∗) is a Nash equilibrium. Given that other nodes select  
+������������−������������
+∗ = (������������1
+∗, … , ������������������������−1
+∗
+, ������������������������+1
+∗
+, … , ������������������������∗), ������������������������
+∗ is the optimal strategy of each node i (Myerson, 1997).  
+ 
+The attention game process determines whether the nodes need adjustment or need to be changed on the 
+basis of the attention function. The activation energy distribution will reach a state consistent with the 
+human attentive distribution after adjusting the activated value distribution. Nodes of the spread SNs have 
+their own activation energy threshold values. The source node in the attention game process that 
+represents a presented sign has the maximum activation value O in the present SNs. All equidistant nodes 
+will participate in the game based on the attention function. The nodes with low activation energy 
+(defined as the minimum energy required to start a chemical reaction) of a reaction is denoted by Ea and 
+given in units of kilojoules per mole (kJ/mol) or kilocalories per mole (kcal/mol)), threshold must be 
+removed through a screening process to prevent them from participating in the enhancement and 
+suppression processes of activating the most likely node. In the proposed screening, the authors ignore the 
+nodes with a significantly low activation value to be activated in the enhancement process instead of 
+lowering the possibility for other nodes to be activated. 
+ 
+The difference between the attentive readjustment in the present attention game process and the previous 
+attentive allocation causes the instability in the overall cognitive structure of users to decrease knowledge 
+credibility. Thus, a new cognitive structure must be determined at a cost as follows: 
+ 
+Cost(t, i, si, ui, SN) = �n−1 ∑
+�Ii(t + 1) − Oi(t)�
+2
+n
+i=1
+                                       (5) 
+ 
+where ������������������������(������������ + 1) denotes the activation value that is conveyed from one node at time t+1 to node i, and 0i 
+(t) denotes the activation value of node i at time t. Therefore, the total cost is attributed to the change in 
+the activation energy of all nodes in the SN. The goal of judgment is to achieve the overall optimal gain 
+with a minimal computing cost. The gain function in the attention game process must then be determined. 
+As the optimal strategy for node i, ������������������������
+∗ must minimize the distribution change that refers to the distribution 
+change in the activation values of the overall network changed by the decision. The amount of spreading 
+activation energy is fixed in the total process of activation spread in the SN; thus, the semantic node 
+energy enhancement must be accompanied by reduced node energy. The attention parameters are affected 
+by the overall distribution change in activation energy. The activation energy enhancement increases the 
+impossibility of activating this node. Such activation is the ultimate purpose of each node (i.e., the node 
+obtains the gain). Accordingly, the gain function is presented as follows:  
+ 
+Gain(t, i, si, ui, SN) =
+�∑
+Ix∈{neighbor node}(t+1)
+num(all x)
+j=1
+−∑
+Ox∈{neighbor node}(t)
+num(all x)
+j=1
+�(1−δ)
+num(all x)
+ 
+(6) 
+ 
+where SN represents the current semantic network, num(all x) represents the number of neighbor nodes x 
+of node i, ∑
+Ox∈{neighbor node}(t)
+num(all x)
+j=1
+ denotes the sum of the activation value that was spread of all 
+node i neighboring nodes at time t, the gain function is expressed as the attention gain of  neighbor nodes 
+x of node i after the enhancement and suppression processes, it represents the benefit a node gets by 
+unilaterally changing their strategy. 
+ 
+The utility function of the attention game process can be determined as follows: 
+ 
+Max�������������������������(������������, ������������������������
+∗, ������������−������������
+∗ )� = Gain(������������, ������������, ������������������������, ������������������������, ������������������������������������) − Cost(������������, ������������, ������������������������, ������������������������, ������������������������������������)
+(7) 
+
+ 
+where ������������������������
+∗ is the optimal strategy of each node i, ������������−������������
+∗ is the strategies set of other nodes except node i. only 
+when   ������������������������(������������, ������������������������
+∗, ������������−������������
+∗ ) reaches the maximum, ������������������������
+∗ is a Nash equilibrium of node i. The utility of the other 
+nodes will be affected by the decision of all other nodes because of the fact that the total quantity of 
+activation energy is fixed (i.e., attention is limited) in the attention game process. When each node selects 
+a decision for itself, it also considers the possible decision of other nodes and selects the “Nash 
+equilibrium point” with maximum utility. This scenario is consistent with classical game theory. The 
+authors select a Nash equilibrium decision for each node through the utility function of the attention game 
+that is defined by Equation (7). 
+METHODS 
+Data Sets and Experimental Settings 
+All data from the authors’ experiments are obtained from the Tsinghua University–Chinese SL Corpus 
+(TH–SLC). The data mainly comprise SL expressions of idiom stories and life fragments of deaf students. 
+No automatic annotation software based on videos is currently available because the annotation process 
+for SL videos is time consuming and requires expert knowledge in dual language (i.e., Chinese language 
+and Chinese SL). Video annotation is also time consuming. Specifically, it takes about 30 hours for the 
+annotation RTF (real-time factor) of a parliamentary speech (i.e., One hour of speech requires 30 hours of 
+annotation). However, the annotation RTF (real-time factor) for a full annotation of all manual and non-
+manual components of an SL video can reach up to 100 hours (Dreuw & Ney, 2008a). Therefore, such a 
+corpus is significantly small. For example, the Aachen Boston database contains American SL and has 
+annotated 201 English sentences (Dreuw & Ney, 2008a). The authors spent a year collecting more than 
+2000 sentences, but only 416 sentences containing 2496 signs were marked. 
+ 
+The authors asked 20 deaf students to select 300 sign pairs from 2469 annotated signs in TH–SLC and to 
+judge the relevance of the sign pairs. The correlation values range from 0.0 to 1.0. For convenience, a 
+five-point scale is used to assess the correlation. The sign pairs were obtained using a marked correlation. 
+The authors establish an SN based on the word similarity computing method of HowNet (Liu & Li, 2002) 
+to determine the connection weight of the network to validate the effects of the proposed model. The 
+authors introduce the continuous bag-of-words (CBOW that predicts the current word from a window of 
+surrounding context words. The order of context words does not influence the prediction (CBOW 
+assumption) model (Mikolov et al., 2013), and the HowNet (Liu & Li, 2002) method as the baseline 
+methods using the same recommended parameters. The efficiency of the utility function of the attention 
+game process is evaluated in terms of word correlation computation, and the model complexity is 
+analyzed. 
+Word Relatedness Computation 
+Each model in this task needs to compute the semantic correlation of the given sign pair. The correlation 
+between the experimental results of the model and human judgment reflects upon the model’s 
+performance. The authors selected 290 signs for the closed set and 10 signs for the open set. 
+ 
+Spearman’s correlation between model correlation score and human judgment correlation score was 
+calculated for comparison. Spearman correlation coefficient is defined as the Pearson correlation 
+coefficient among the ranked variables (Myers & Well, 2003). For a sample of size N, original data  ������������������������, ������������������������ 
+are converted into grade data������������������������, ������������������������, the correlation coefficient ρ is defined as follows: 
+ 
+ρ = 1 −
+6 ∑ di
+2
+n(n2−1)
+(8) 
+
+ 
+where the difference between the observations of the two variable levels is set as  ������������������������ = ������������������������ − ������������������������. If there is 
+no duplicate value in the data, and two variables are completely monotonic correlation, the Spearman 
+correlation coefficient is +1 or -1. 
+RESULTS 
+For CBOW, the correlation scores of the two words are calculated using the cosine similarity of word 
+embedding (Mikolov et al., 2013). The evaluative results of the baseline methods and the proposed SNM 
+method in the closed test and in all test sets are shown in Table 1. 
+Table 1. Evaluative results 
+Data Set 
+Closed Test 
+All Test Sets  
+(Including Open Test) 
+Spearman’s Rank 
+Correlation Coefficient 
+Method 
+290 pairs 
+300 pairs 
+ 
+CBOW (baseline method) 
+0.4843 
+0.4869 
+0.4136 
+Word similarity computing 
+based on HowNet 
+0.6157 
+0.6174 
+0.6052 
+Proposed SNM method 
+0.6951 
+0.7063 
+0.6437 
+ 
+The evaluation results show that the proposed SNM method is better than the baseline method in 290 and 
+300 word pairs. This finding indicates that the cognitive mechanism of sign comprehension is essential to 
+understanding the meaning of signs. The internal structure, such as location, orientation, hand shape, and 
+movement, contains rich semantic information. However, deep learning methods, such as CBOW, 
+consider the external context, but ignore the internal structure.  
+ 
+Using the computing method of word similarity based on HowNet results in only a rough semantic 
+computation. For example, adding 10 new sign pairs negligibly changes the performance of these 
+methods. In other words, these methods can still handle new signs with improved performance. The 
+semantic correlation of these new sign pairs calculated by the proposed method is close to human 
+judgment. Figure 3 shows the quantitative analysis of the attention game process for two signs. Each hand 
+shape of the two signs has at least 20 related semantic lexicons. The stimulus information and 
+permutation of each node are shown in the first and second columns from high to low according to the 
+activated value after the activation spreading process. Only 10 semantic lexicons that are maximally 
+activated are shown. The permutation of each node is shown in columns three to seven from high to low 
+according to the activation value after the end of the first to fifth attention games. The top 10 lexicons are 
+also shown. The semantic lexicons in the blue background rank high after the games, those in the green 
+background rank low after the games, and those in the white background are unchanged. 
+Figure 3. Examples of attention games. The semantic lexicons in the blue background rank high after the 
+games, those in the green background rank low after the games, and those in the white background are 
+unchanged. This trend shows that the ranking of other semantic lexicons below slightly changes after the 
+semantic lexicon that ranks highest becomes unchanged. This condition is due to the source that 
+corresponds to the attention model being determined after several game processes.  
+
+ 
+ 
+Figure 3 also shows that significant changes occur during the ranking of the semantic lexicons in the first 
+and second instances after the first several games, whereas only a few changes occur in the following 
+stimulus games. This trend shows that the ranking of lower semantic lexicons slightly change after the 
+semantic lexicon that ranks highest becomes unchanged. This condition is due to the source that 
+corresponds to the attention model being determined after several game processes. Attention is also 
+assigned to other nodes in accordance with the attention game process. Humans reach a steady state after 
+thinking about problems constantly, and the result negligibly changes if they rethink. Nearly no change is 
+observed in the result after several rounds. Several semantic lexicons related to the signs are contained in 
+the text set; thus, a few possible changes occur. The result of the attention game model conforms to 
+human cognitive rules to a certain degree. 
+  
+Attention is also assigned to other nodes in accordance with the attention game process (here, efforts have 
+been made in modeling according to the mechanism of human attention). The result of the SNM conforms 
+to human cognitive rules to a certain degree (Gutierrez et al., 2012). For example, the authors assume that 
+deaf people understand the signs shown in Figure 3. Deaf people usually search for many familiar and 
+specific nouns or signs in a spreading activation mode to comprehend classifier predicates. After all 
+activated values are calculated; the activated nodes are graded and sorted. A high-activated value of the 
+node indicates the importance of the interested object or concept represented by the node. This shows that 
+deaf people are familiar with the concept node. Similar to the attention game process shown in Figure 3, 
+
+activation
+activation
+activation
+activation
+activation
+activation
+value
+value
+value
+value
+value
+value
+sorting
+sorting after
+sorting after
+sorting after
+sorting after
+sorting after
+after
+the first
+the second
+the 3rd
+the 4th
+the 5th
+spread
+attention
+attention
+attention
+attention
+attention
+input
+activation
+game
+game
+game
+game
+game
+A
+handshape
+good
+General
+General
+General
+General
+poog
+Reliable
+Beheaded
+Beheaded
+Beheaded
+Beheaded
+Defend
+Advanced
+Support
+Support
+Support
+Support
+Maintain
+Strange
+Keep
+Keep
+Keep
+Keep
+Protect
+General
+General
+General
+General
+General
+Beheaded
+Teacher
+Teacher
+Teacher
+Teacher
+Teacher
+Support
+Marshal
+Reliable
+Reliable
+Reliable
+Reliable
+General
+Madam
+Advanced
+Advanced
+Advanced
+Advanced
+Teacher
+Ancestors
+Protect
+Protect
+Protect
+Protect
+Reliable
+Defend
+Maintain
+Maintain
+Maintain
+Maintain
+Y
+handshape
+human
+Animal
+cat
+cat
+cat
+stand
+animal
+Burial
+dog
+Sop
+dog
+run
+burial
+Frustration
+horse
+horse
+horse
+lie
+Frustration
+Future
+human
+human
+human
+Resistance
+future
+Voltage
+Ambassador
+Ambassador
+Ambassador
+Protest
+Voltage
+Ambassador
+coach
+coach
+coach
+Control
+Ambassador
+Coach
+stand
+stand
+stand
+Incite
+coach
+Guide
+run
+run
+run
+Exploitation
+Guide
+Blind
+lie
+lie
+lie
+Recalcitrant
+Blind
+Opponent
+Resistance
+Resistance
+Resistancethe high-ranked semantic lexicon is a cat or dog after several rounds. This result shows that the most 
+common subjects for deaf people are typical subjects that represent classifier predicates. 
+DISCUSSION 
+Compared with that of existing models, the complexity of the proposed model is reflected mainly on the 
+computational cost of the memory stage and the judgment stage (i.e., the computational cost of spreading 
+activation and the attention game at time (t + 1)). The cost is a dynamic value and related to two factors, 
+namely, the activation state of the current sign and the current cycle as the first activation of the sign. 
+Therefore, the value changes regardless of the choice of the user. This outcome is consistent with the 
+strong dynamics of sign information, which can reflect the influence of information in different periods. 
+In the memory stage, the time complexity of computing ������������������������(������������) is unity; thus, the time complexity is 
+related to the total amount N of activation energy and cycle times. The time complexity of each activation 
+in each cycle is n × 1 = n. Space complexity is the storage space of each node and the semantic relation 
+weight according to semantic similarity (semantic similarity can be estimated by defining a topological 
+similarity, by using ontologies to define the distance between terms/concepts). Therefore, unlike the 
+general model such as cobweb theorem model and vector space model, where the SNM increases the 
+overhead in time complexity and space complexity. The model also increases the matching time of query 
+nodes and weights in the current activation. However, the overhead at this time can provide more 
+effective results than an invalid spreading and can be accepted by users. 
+ 
+In the judgment stage, when the node selects the game strategy to change its activated energy value, the 
+convergence speed of adjusting the cognitive benefits to its own utility maximum “Nash equilibrium” is 
+an important measure of evaluating the SNM (i.e., the cycle times of an attention game process). For the 
+attention game, the Nash decision of different semantic nodes must minimize the change cost of the 
+activation energy distribution of the entire network. The Nash equilibrium point decision for each node is 
+selected using the utility function defined in the SNM. This process is repeated until the overall network 
+activation energy distribution change is less than the specified threshold. The node needs to solve n-order 
+nonlinear equations in every cycle. Therefore, the performance of the convergence speed of the SNM is 
+indicated by the number of game cycles that the network requires to reach the Nash equilibrium point 
+(i.e., the computing times of calculating the corresponding equation by each node in a game process). The 
+square root of the sum of the variance of activation value ������������������������(������������ + 1) of each adjusted node is directly 
+reflected by the rate of convergence in the game process. 
+ 
+To verify its effectiveness, the attention game model is compared with the traditional model in terms of 
+load balancing. In the traditional method, the activation value of each node is certain (i.e., the value is not 
+enhanced or inhibited). The experimental results are shown in Figure 4. The results show that the load 
+balance performance of the attention game model is better than that of the traditional model because the 
+attention game model adjusts the activation strategy after the activation of each node. When the change 
+cost of the energy distribution of the entire network activation is larger than the specified threshold, the 
+human brain adjusts the strategy to inhibit the activation energy value in the next cycle. In doing so, the 
+free competition and distribution of attention for each node according to the attention game model can be 
+assured. The result is obtained through the overall competition. The load of attention of the network is 
+balanced. The traditional model assumes that the activation energy value of each node is certain because 
+the brain activation energy resource amount is constant in a period of time. The brain selects the node 
+with a low activation energy value and performs the allocation of attention. This allocation causes the 
+attention load of several nodes to be excessively large or unutilized. 
+Figure 4. Comparison of load balance. The load balance performance of the SNM is better than that of 
+the traditional model because the SNM adjusts the activation strategy after the activation of each node 
+
+ 
+ 
+The proposed SNM model used Nash equilibrium to simulate the energy activation process. In order to 
+quantitatively analyze the effects of Nash equilibrium, the authors compared the SNM with the cobweb 
+theorem model (Pashigian, 2008) in terms of different activation energy amounts. The cobweb theorem is 
+expressed as follows:  
+ 
+������������(������������ + 1) = ������������(������������) + ������������ ��������������������������(������������)� − �������������������������′(������������)��
+(9) 
+ 
+where r is the adjustment parameter of the activation value,  �������������������������(������������)� is the activation function of a node, 
+������������(������������) is the activation value at time t,  �������������������������′(������������)� is the attention allocation function,  ������������′(������������)  is the 
+expectation activation value at time t, and   �������������������������(������������)� − �������������������������′(������������)�is the excessive demand function that 
+represents the actual gaps between the activation value and activated allocated value. A large gap 
+indicates a high activation value of the Nash Equilibrium of node. The parameter (r) indicates the actual 
+speed and strength of adjusting the activation value according to the attention distribution condition in the 
+last moment. When r > 0 it indicates that the adjustment direction of the activation value is consistent 
+with the direction of the demand function. 
+ 
+The amount of activation energy Ea is assumed to be 100 kJ/mol. Figure 5 shows the result of comparing 
+the attention utilization between the game model and the cobweb model. The attention amount (attention 
+is the behavioral and cognitive process of selectively concentrating on a discrete aspect of information, 
+while ignoring other perceivable information. Attention amount refers to as the allocation size of limited 
+processing resources), is less than 100 kJ/mol. If the attention amount is insufficient, then attention 
+resources can only meet part of the node demand, and the resource utilization rate of the SNM will 
+become higher than that of the cobweb model. When attention supply exceeds the demand of a node, the 
+cobweb model achieves balance to meet the needs of several nodes after a repetitive cycle. The SNM 
+meets the needs of all nodes, and the utilization rate of attention resources is higher than that of the 
+cobweb model. 
+
+Comparisonofloadbalance
+9
+8
+nodes
+hhhl
+6
+Numberof
+L
+4
+3
+1
+0
+No.1
+No.2
+No.3
+No.4
+No.5
+No.6No.7
+No.8
+No.9
+No.10
+Numberofactivationenergy amount
+Iattentiongamemodel
+cobwebtheoremmodeFigure 5. Comparison of activation energy values. After the change in the initial value of the activation 
+energy, the number of iterations increases depending on the difference between the initial activation 
+energy value in the cobweb model and the balanced energy value. The iteration of the attention game 
+model can be adjusted according to the difference in the activation energy between supply and demand. A 
+sizeable adjustment is required to reach the Nash equilibrium state if a large difference exists between the 
+supply and demand 
+ 
+Figure 6 shows the cycle times of the SNM and the cobweb model that needs to achieve the Nash 
+equilibrium. As shown in the figure, the equilibrium activation energy value of the nodes is 20 kJ/mol in 
+the SNM, and the activation energy is 120 kJ/mol in total. If the initial value of the activation energy is 
+changed, then the initial activation energy value of the cobweb model is higher than the energy 
+equilibrium value and requires abundant cycle time. The SNM in each cycle can adjust the activation 
+energy according to the variance of the activation energy. The variance and adjustment range are large, 
+and the SNM eventually reaches the Nash equilibrium point. 
+Figure 6. Cycle times of the SNM and the cobweb model that are needed to achieve the Nash equilibrium. 
+When the supply falls short of demand, attention resources can only meet the demands of several nodes, 
+and the resource utilization rate of the SNM becomes higher than that of the cobweb model. If the supply 
+exceeds demand, then the cobweb model can reach equilibrium after repeated iterations and can meet 
+only part of the demands of nodes. However, the SNM can meet the demands of all nodes, and its 
+resource utilization rate is higher than that of the cobweb model 
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+40
+60
+80
+100
+120
+Attention Resource Utilization(percentage)
+Amount of activation energy(KJ/mol)
+Comparison of activation energy values
+attention game model
+cobweb theorem model
+
+ 
+CONCLUSION 
+The authors presented a new model for SL comprehension based on spatial information. This process uses 
+game theory to simulate the human attention suppression and enhancement process. This process also 
+joins the forgetting function of human memory traces to compute the initial state of the node. Memory is 
+encoded with specific (semantic) meaning, or refers to information that is encoded along a spatial and 
+temporal plane. Although the semantic network provides a functional view of how knowledge may be 
+organized in the brain, it does not provide a clear model of how semantic memory might be presented in 
+the brain (see Cacha et al., 2017). Spreading activation reveals that information can be stored in SNs for a 
+long time, in which a network node is a linguistic concept and the nodes are connected through the 
+correlation. An algorithmic method is proposed according to selective functions, and its effectiveness was 
+verified using an example. The results show that the proposed method improves the performance of SL 
+comprehension. 
+ACKNOWLEDGMENT 
+The authors would like to thank Chunda Liu from the National Center for Sign Language and Braille for 
+helping in stimulus preparation and data collection. This paper forms an expanded and revised version of 
+a conference paper at the 14th IEEE International Conference on Cognitive Informatics & Cognitive 
+Computing (ICCI* CC) at Tsinghua University, Beijing, July 6-8, 2015. The authors are grateful to Dr. 
+Raymond Chiong, and two anonymous referees for their helpful comments. 
+Conflict of Interest 
+The authors of this publication declare there is no conflict of interest. 
+Funding Agency 
+This research was supported by the Beijing Municipal Natural Science Foundation [4202028]; National 
+Social Science Foundation of China [21BYY106]; National Natural Science Foundation of China 
+[62036001, 61866035, 61966033]; Premium Funding Project for Academic Human Resources 
+
+Comparison of cycle times
+40
+Number of Iterations
+35
+30
+25
+20
+15
+10
+5
+0
+1
+5
+10
+15
+20
+25
+30
+Activationvalue
+attentiongamemodel
+cobwebtheoremmodelDevelopment in Beijing Union University [BPHR2019CZ05]; Jiangsu Province Key R&D Program 
+(Industry Prospects and Key Core Technologies) [BE2020047]; and the characteristic-disciplines oriented 
+research project in Beijing Union University [KYDE40201702].  
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+alphabetic system. In Poster presented at the First Sign Typ Conference, Storrs CT. 
+Supalla, T. R. (1982). Structure and acquisition of verbs of motion and location in American Sign 
+Language. University of California, San Diego. 
+Sutton. V. (2010). The SignWriting Alphabet, International SignWriting Alphabet 2010. ISWA 2010. 
+http://www.signwriting.org/archive/docs7/sw0636_SignWriting_Alphabet_Manual_2010.pdf 
+Yao, D., Jiang, M., Abulizi, A., & You, X. (2014, October) Decision-tree-based algorithm for 3D sign 
+classification. In 2014 12th International Conference on Signal Processing (ICSP) (pp. 1200-
+1204). IEEE. 
+
+Yao, D., Jiang, M., Abulizi, A., & Li, H. (2015, July). Cognitive computing on Chinese Sign Language 
+perception and comprehension. In 2015 IEEE 14th International Conference on Cognitive 
+Informatics & Cognitive Computing (ICCI* CC) (pp. 90-97). IEEE. 
+Yao, D., Jiang, M., Huang, Y., Abulizi, A., & Li, H. (2017). Study of sign segmentation in the text of 
+Chinese sign language. Universal Access in the Information Society, 16(3), 725-737.  
+Valli, C., & Lucas, C. (2000) Linguistics of American Sign Language: An introduction. Gallaudet 
+University Press. 
+ 
+ 
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf,len=1025
+page_content='Semantic Network Model for Sign  Language Comprehension  Xinchen Kang (22f3d431-dcc0-42a5-8e2b-c26464e0654d)  Beijing Union University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' China  Dengfeng Yao (ae88317c-d091-4f41-97f1-b1e6be00ca68)  Beijing Union University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' China  Minghu Jiang (ea1cc43b-eee9-4185-8d97-edeac9186268)  Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' China  Yunlong Huang (cc1ddbf2-64b9-4c51-b553-0ebe52a8d645)  Tsinghua University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' China  Fanshu Li (64642396-6e95-456f-a4f2-47ff81a23d6e)  Beijing Union University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' China  ABSTRACT  In this study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' the authors propose a computational cognitive model for sign language (SL) perception and  comprehension with detailed algorithmic descriptions based on cognitive functionalities in human  language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The semantic network model (SNM) that represents semantic relations between  concepts is used as a form of knowledge representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The proposed model is applied in the  comprehension of sign language for classifier predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The spreading activation search method is  initiated by labeling a set of source nodes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' concepts in the semantic network) with weights or  "activation," and then iteratively propagating or "spreading" that activation out to other nodes linked to  the source nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The results demonstrate that the proposed search method improves the performance of  sign language comprehension in the SNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Keywords: Attention, Cognitive Processing, Comprehension, Decision-Tree, Game Theory, Linguistics,  Perception, Semantic Network, Sign Language  INTRODUCTION  Sign language (SL) comprehension is a fundamental task for computational linguists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Two types of  algorithms have been proposed: (1) rule-based methods (Supalla, 1982), and (2) statistical methods  (Bauer & Heinz, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Huenerfauth, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Rule-based methods lack the capability of planning the  elements in the entire scene (Liddell, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The method of modeling infinite natural language input  through finite rules, especially minor rules, barely meets all requirements of SL processing (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=',  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Therefore, statistical methods are the preferred type of algorithm for SL comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Statistical  models can be applied to spoken languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Given the abundant data resources of spoken languages in the  digitalized Internet age, statistical models can be applied readily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' However, the raw and annotated corpora  of SLs are insufficient because collecting and annotating SL videos are tedious and difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Data sparsity  consequently remains as the most serious problem when applying statistical models onto SLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' For  example, the real-time factor (RTF) of the SL video corpus is 100;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' that is, an hour corpus requires at least  100 hours of annotation (Dreuw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2008b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Simulating SL comprehension using traditional statistical models and machine-learning methods  isdifficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Thus, reliable methods for establishing a signer’s 3-D model (which is the process of  developing a mathematical representation of any three-dimensional surface of moving trajectories of  signers in the space for SL via specialized software) for SL corpus building and technologies for  annotating a large-scale SL video corpus automatically must be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Unlike the spoken language  that is “a set of values that change with the passage of time” (Huenerfauth, 2005), SL does not have a  writing system and thus cannot be saved in any form of written texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  The natural language-processing system relies on texts to process spoken languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This system records  only the written text that corresponds to speech flows and relies only on the literacy of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' On the  other hand, the SL system comprises information from multiple modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Examples of such information  are the hand shape, hand location, hand movement, hand orientation, head tilting, shoulder tilting, eye  gazing, body gestures, and facial expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The considerable information from multiple channels in SL  conveys linguistic meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This multi-modality nature of SL poses difficulties for the coding of SLs into  a linear single-channeled character string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' In addition, SLs have writing systems, such as the Sign Writing  system (Sutton, 2010), ASL-phabet (Supalla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2008), and HamNoSys (Prillwitz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  However, these systems have a limited number of users (Johnston, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Many linguistic details are lost because of the multi-modality nature of SL during the translation of SL  into its corresponding writing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' SLs may be understood by directly matching the visual–spatial  characteristics of SL with the semantic units in the brain rather than applying written texts as an  interpreting medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Here, semantic units are generally used for processing natural languages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' these units  or nodes contain some information, which are used as knowledge representations form semantic units  (Geva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Such direct matching also represents the most natural way of comprehending SLs in  the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' From this perspective, the authors present a computational cognitive model for SL  comprehension that is based on the cognitive functionalities of the human brain combined with a  knowledge representation theory of artificial intelligence (Shuklin, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Visual–spatial mechanisms are exploited to express the grammatical structures and functions in SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Visual–spatial perception, memory, and mental transformations are prerequisites to grammatical  processing in SL (Emmorey & Corina, 1990) and are central to visual mental imagery (Farah, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' A  series of experiments have been conducted to investigate visual attention (Neville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Movement  recognition in peripheral vision is important in sign perception because the signers mainly look at the face  instead of tracking the hands when they communicate through SL (Siple, 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Therefore, identification  of lexical-level information depends on the peripheral vision system when signs are produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The  recognition of movement directions is the selective function of peripheral vision (Bonnet, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Whether deaf people only have a strong peripheral vision or efficiently allocate attention to peripheral  vision remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Stivalet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' (1998) showed that visual attention processing can be changed by  auditory deprivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' They determined that deaf people do not shift their attention when processing the  information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', alphabet set) presented in the central vision field, whereas hearing subjects must shift  their attention to search for the alphabet set continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' (1998) also found that lack of  auditory input causes weak and selective (or highly distributed) visual attention among deaf children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Stivalet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' (1998) proposed that effective visual processing is caused by intermodal sensory  compensation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' that is, the strong allocation of visual attention can be attributed to neuron reorganization  caused by auditory deprivation from birth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Recent magnetic resonance imaging evidence supports this  hypothesis (Bavelier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  These findings are selective attention cases, in which attention selectively processes certain stimuli but  ignores other stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The cases refer to the selective orientation and concentration of the senses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=',  visual, auditory, taste, and tactile senses) and consciousness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', awareness) of people on certain targets  (towards other factors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Studies on attention have failed to describe human attention at the biological level  in detail, as a person cannot focus continuously because the brain automatically suppresses activity when  attention reaches its limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Emmorey and Reilly (2013) determined that when locations in a signing space (SL expressions streaks  the space) function topographically, spatial changes tend to be noticed easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Thus, location information  indicating the spatial position of associated referents can be encoded and stored semantically in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  However, spatial locations with a primary distinguishing function of referents are encoded in a different  way and tend to be discarded from memory once the referential function is no longer required by context  (Emmorey & Reilly, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Bavelier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' (2001) claimed that only the posterior middle temporal gyrus  and the medial superior temporal cortex of deaf signers are highly active while perceiving movements in  peripheral vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This phenomenon is unobservable in hearing signers who have skillfully grasped signs,  indicating that auditory deprivation results in a shift to stronger movement attention in the visual  periphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Deaf people can easily reply to the attention and visual monitoring of their peri-personal space  (Bavelier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Neville et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' (1998) determined that the classic language area in the left hemisphere, particularly the left  perisylvian, of both deaf and hearing subjects is activated when reading English sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The right  hemisphere, including the right perisylvian, of deaf people is also activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' They argued that spatial  processing is of great importance to sign grammar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Thus, the SL comprehension process of deaf people  employs neurons at both high and low levels in the neural network, which are connected with each other  by edges, and generates high-level features via feature combination processes that are realized by  combining the weight on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' For example, low-level visual edge features are assembled, processed,  and sent to the high level to form the angle, shape, and other higher features (Bertasius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' High-  level neurons form features that gradually approximate the semantics in turn, such as simple shapes,  simple targets, and real objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The activation of high-level neurons during the reconstruction process  also reacts with the low-level neurons and adjusts and corrects deviations and losses (Bertasius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=',  2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' a temporal pattern appears in the horizontal structure connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The neurons can make  predictions of the state at the next point of every time point through a horizontal connection based on the  information of their current status (Hawkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  SEMANTIC NETWORK MODEL (SNM)  Model of semantic networks (SNs) are generally used for processing natural languages (Shuklin, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  SNs, as knowledge representations, are extensible and have been used to model mental disturbances  (Geva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The semantic network (which is a network that represents semantic relations between  concepts, is used as a form of knowledge representation, here it is based on SL information processing of  human brain cortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The edges connect different nodes in the network and represent the strength or  weakness of the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' After being set up, the semantic network is stored in long-term memory for  future retrieval and extraction to be encoded as semantic memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Outside stimuli at a certain time can be  the demand of a person on specific knowledge and information to activate the demand on the extraction  of useful information of long-term memory (Sedikides & Skowronski, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The activation process of  the stored network works in a form of spreading in the memory (Collins & Quillian, 1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  The semantic model, which is based on SL information processing of human brain cortex, is developed  accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Different areas of the brain cortex are involved in the processing and are connected in a  hierarchical manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Low-level information from sense organs is first processed in the primary  information-processing regions of the brain cortex and is then transferred to high-level regions for further  processing, such as abstracting, integrating, and interpreting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The detailed description and illustration of  this hierarchical structure are summarized in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Hierarchical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Low-level areas in the hierarchy generate specific information that  increases speed and contain further details, whereas high-level areas form stable spatial invariance,  change slowly, and show high-level semantic object expression (Adapted from Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2015)  In SL communication, both substantial and semantic information (substantial information includes hand  shape, hand location, hand movement, hand orientation, head tilting, shoulder tilting, eye gazing, body  gestures, and facial expressions, and semantic information is represented into semantic concepts by these  substantial SL information) almost exclusively relies on signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' However, accurate SL information  analysis and prediction remain as challenging tasks in the field of natural SL processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Three main tasks  are, namely, capturing, decoding, and extracting the physical characteristics and relationship of signs  (perception stage), matching the decoded cognitive representations with the stored semantic information  (memory stage), and completing the machine translation process of SL information (judgment stage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  This process of cognitive processing and understanding during SL communication is based on the PMJ  principle of  “from the definition and extraction/annotation of cognitive representation (Stage P) to the  feature storage in line with the cognitive economy principles (Stage M),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' and then to the output of the  classification and judgment (Stage J).”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  The P→M→J (PMJ) principle exhibits a complete fine processing frame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' the detailed illustration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' and  description of SL comprehension frame based on the PMJ principle is summarized in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' SL comprehension frame based on the PMJ principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Perception refers to acquiring sign  information through selective attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The information is limitedly processed by the brain if prominence  is given to useful and important information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Other information may be filtered out or suppressed when  sources for information processing are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Memory refers to the spreading activation process, in  which input information is coded, and one intends to store the information for a short period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Judgment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='targeted  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='wer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='targeted sen antics  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='sign semantic concept  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='semantics  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='rea  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='representation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Area H  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='hign deve semantid  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='high-level semartic feature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='feature detectors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Area A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='low-evel senantic  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='low-level semantic feature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='low-level s emanticf eature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='feature detectors  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='cogritive  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='representation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Higher leve visual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='handshape  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='location  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='oriention  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='moveme rt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='No-manual feature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='AreaV4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='abstractions  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Edge  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='outine  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Simple shape  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Area V1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='detecton  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='edge feat ure  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='ed ge feature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Retina  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='oves  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='pixel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='pixel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='pixel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='pixel  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='physical  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='characteristics  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='一refers to the process in which the perceived information or the information stored in memory is  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='compared,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' matched,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' or classified,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' and a decision or prediction is made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' After the spreading activation,  the network records the attention features of users and activates their future preferences (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=',  2015)  Concepts are in the form of network storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The different concepts are stored in different functional  areas in both hemispheres of the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The same or similar concepts are stored in same or  adjacent regions of brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Specific information of entities in the outside world, such as humans, animals,  or tools, is represented by the concept network in the human brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This concept network (A concept is an  abstract idea representing the fundamental characteristics of what it represents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Concept network consists  of these abstract concepts) is, in turn, connected with the lexical network from mental lexicon in the left  temporal lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Such specific information from mental lexicon will be employed to facilitate SL  production during which the SL users generate classifier hand shapes under the guidance of the  knowledge and rules of SL classifier predicates (Valli & Lucas, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Here, classifier predicates are  made by combining small meaningful unites to create bigger units, the main units being the hand shape  and the movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This condition implies that findings from brain research can provide knowledge and  guidance for the cognitive computational modeling of classifier predicate comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' In order to  obtain a deep understanding of sign lexical semantics, a cognitive processing model, which is based on  the cognitive mechanism of human brain, is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The cognitive processing model would activate  the concept network of the associated classifier hand shapes in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Here, classifier predicates differ  from traditional linguistic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Traditional methods, such as the syntactic tree, cannot satisfy the  generation of the classifier predicates (Huenerfauth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  DECISION-TREE BASED ALGORITHMIC METHODS  The authors use SNs as the knowledge representation and organization mode of SL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The relationship in  semantic networks represents a type of information among nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Nodes with a complicated relationship  with other nodes contain additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Such nodes require further effort to be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Consequently, the authors simulate selective attention (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', the processing of visual or auditory input  based on whether it is relevant or important).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' They selected particular representations to enter perceptual  awareness and therefore guide behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Through this process, less relevant information is suppressed by  humans using the proposed algorithmic methods to accentuate the nodes selectively and suppress the  unessential nodes (Chelazzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  The emergence of 3-D-based sensors, such as Kinect by Microsoft and Leap Motion (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2014),  has improved studies on sign recognition from video-based to 3-D-based sign recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' However, this  transformation makes traditional video-based SL recognition methods inapplicable to 3-D-based SL  Perception  Memon  Judgment  Attentional enha ncement and suppress ion  0  Spread activation  Interactive activationrecognition technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Large training data are required for valid recognition in 3-D-based SL  recognition technologies because of the low operation efficiency of the rotatable joint-based sorter and the  matching techniques for sign signal recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' (2014) proposed a decision tree-based  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The algorithm aims to achieve a high-precision and real-time performance of SL automatic  perception according to the features of Leap Motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The authors adopted this method as the first step of  SL comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Attention Function  The authors propose the following attention function:  ������������������������ =  ∑  ������������������������������������  ������������������������������������ ������������  ∑ ������������������������������������  ������������������������  (1)  where ∑  ������������������������������������  ������������������������������������ ������������  denotes the sum of the semantic relation weights around the semantic node x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ∑ ������������������������������������  denotes the sum of all semantic relation weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' and ������������������������ represents the activation value on the semantic  node x after the spreading activation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Semantic Matching  Cognitive units in the memory network compete with one another based on certain rules to obtain more of  the limited attention resources and more energy for a more active state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' SL comprehension supports  interactive activation models (Gutierrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Therefore, judgment is the outcome of the attention  competition game in the spreading activation, which is a search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The search algorithm is  initiated by labeling a set of source nodes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' concepts in a semantic network) with weights  or ”activation,” and then iteratively propagating or “spreading” that activation out to other nodes linked to  the source nodes processes of the human brain (Crestani, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Preece, 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  A semantic matching algorithm based on activation spreading modes is proposed to determine the most  appropriate semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Activation starts to spread from the corresponding nodes of the signs  presented by the signer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The activation value of the stimulus node (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', signs to be perceived before the  start of spreading) must be calculated first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' In particular, the increment in the interest value of object  concept must be calculated, and this concept node must be used as an initial node for the spread study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Activation spreads to the neighboring nodes, which usually have a lower activation value than the source  value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Therefore, introducing an activation attenuation factor for decreasing activation over the path  length in the closed interval [0…1] is mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' That is, for every propagation through an edge a loss of  activation is considered (Neumann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Rocha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The activation spreading process can  be expressed as follows:  ������������������������(������������ + 1) = ������������������������(������������)������������������������������������(1 − ������������)  (2)  where  ������������������������(������������ + 1) represents that the value is spread from node x to y at time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' t+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ������������������������(������������) represents the  activation value that was spread at node x at time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ������������������������������������ signifies the link between nodes x and y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' and δ is  an attenuation factor used to describe the energy loss caused in the activation spreading process (Jiang &  Tan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Spreading activation theory states that the activation of human memory “chunks” (the content of any  buffer is limited to a single declarative unit of knowledge, called a chunk) is determined by two factors  (Anderson et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Anderson, 2013), namely, the use history of the memory chunk and the  correlation between the memory chunk and the current retrieval information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' These two factors calculate  the weights and determine whether the chunk is activated and selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This assumption has been verified  by experimental cognitive psychology, and the calculation model has been established (Roelofs, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  The authors must use moments to express the distance in each activation time with the current time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Time  units may be per hour as a unit and may also be the day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' With the day as a unit, we can count the  historical value in the previous day as the activation value of the first day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The algorithm based on the  theory of memory activation can improve SL understanding, which is sometimes highly sensitive to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  At node y, the largest number of neighbor nodes is (n-1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' thus, the maximum of  ������������������������(������������ + 1) can be  expressed as:  ������������������������(������������) = [������������1, ������������2, … , ������������������������]������������  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', the initial value of the semantic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Where I1, I2, …, In are these activation value of neighbor  nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  If activation spreads from a node in many directions, then its adjacent nodes obtain a low activation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  The adjacent nodes give a feedback value of their resonance energy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', contributing structure with the  lowest potential energy) to the co-adjacent nodes after they absorb the activation value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The following  equation is therefore used:  Iz(t + 1) = Oz(t) + ∑  Ox(t)Λxz(1-δ)  all actived x  (3)  where Oz(t) denotes the activation value of node z at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Given that the quantity of activated information is limited, the nodes that obtain less resonance  information are equivalently inhibited and are less likely to be activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The activation value distribution  in the resonance process conforms to the human attention model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Attention Game Process  Cognitive units in memory network compete with one another by certain rules to increase the possibility  of obtaining more human attention resources and more energy that will improve activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This  phenomenon is called a game process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The authors use game theory (Myerson, 1997), which is the study  of mathematical models of conflict and cooperation between intelligent rational decision-makers and  attempts to achieve the largest cognitive gains with the least energy possible, as a reference to simulate  the attention enhancement and suppression processes that are selective attention processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' In other  words, when visually searching for a non-spatial feature or a perceptual feature, selectively enhancing the  sensitivity to that specific feature plays a role in directing attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' When people are told to look for  motion, then motion will capture their attention, but attention is not captured by motion if they are told to  look for color (Reynolds & Chelazzi, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Activated results consistent with cognitive features can then  be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The authors assume that the game contains n nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ������������������������  ′ and ������������������������  " are the two selectable  strategies for node i, and they represent the acceptance and non-acceptance of the change in the attention  function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', ������������������������  ′, ������������������������  " ∈ ������������������������ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The corresponding gain can be represented by ������������������������  ′ and ������������������������  " , and ������������������������  ′, ������������������������  " ∈ ������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' N  nodes are assumed to reach an agreement before participating in the game to introduce the Nash  equilibrium (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', each node only selects a specific strategy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The authors let ������������∗ = (������������1  ∗, … , ������������������������∗) represent the  agreement, where ������������������������  ∗ is the strategy of the node i specified in the agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Nodes comply with this  agreement only when the benefit from complying with the agreement is larger than that from not  complying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This agreement constitutes Nash equilibrium if any node abides by this agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Thus, the  Nash equilibrium is written as follows:  ������������������������(������������������������  ∗, ������������−������������  ∗ ) ≥ ������������������������(������������������������, ������������−������������  ∗ ),               ∀si ∈ Si  (4)  where the combination of strategy  ������������∗ = (������������1  ∗, … , ������������������������∗) is a Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Given that other nodes select  ������������−������������  ∗ = (������������1  ∗, … , ������������������������−1  ∗  , ������������������������+1  ∗  , … , ������������������������∗), ������������������������  ∗ is the optimal strategy of each node i (Myerson, 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  The attention game process determines whether the nodes need adjustment or need to be changed on the  basis of the attention function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The activation energy distribution will reach a state consistent with the  human attentive distribution after adjusting the activated value distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Nodes of the spread SNs have  their own activation energy threshold values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The source node in the attention game process that  represents a presented sign has the maximum activation value O in the present SNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' All equidistant nodes  will participate in the game based on the attention function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The nodes with low activation energy  (defined as the minimum energy required to start a chemical reaction) of a reaction is denoted by Ea and  given in units of kilojoules per mole (kJ/mol) or kilocalories per mole (kcal/mol)), threshold must be  removed through a screening process to prevent them from participating in the enhancement and  suppression processes of activating the most likely node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' In the proposed screening, the authors ignore the  nodes with a significantly low activation value to be activated in the enhancement process instead of  lowering the possibility for other nodes to be activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  The difference between the attentive readjustment in the present attention game process and the previous  attentive allocation causes the instability in the overall cognitive structure of users to decrease knowledge  credibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Thus, a new cognitive structure must be determined at a cost as follows:  Cost(t, i, si, ui, SN) = �n−1 ∑  �Ii(t + 1) − Oi(t)�  2  n  i=1  (5)  where ������������������������(������������ + 1) denotes the activation value that is conveyed from one node at time t+1 to node i, and 0i  (t) denotes the activation value of node i at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Therefore, the total cost is attributed to the change in  the activation energy of all nodes in the SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The goal of judgment is to achieve the overall optimal gain  with a minimal computing cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The gain function in the attention game process must then be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  As the optimal strategy for node i, ������������������������  ∗ must minimize the distribution change that refers to the distribution  change in the activation values of the overall network changed by the decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The amount of spreading  activation energy is fixed in the total process of activation spread in the SN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' thus, the semantic node  energy enhancement must be accompanied by reduced node energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The attention parameters are affected  by the overall distribution change in activation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The activation energy enhancement increases the  impossibility of activating this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Such activation is the ultimate purpose of each node (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', the node  obtains the gain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Accordingly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' the gain function is presented as follows:  Gain(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ui,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' SN) =  �∑  Ix∈{neighbor node}(t+1)  num(all x)  j=1  −∑  Ox∈{neighbor node}(t)  num(all x)  j=1  �(1−δ)  num(all x)  (6)  where SN represents the current semantic network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' num(all x) represents the number of neighbor nodes x  of node i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ∑  Ox∈{neighbor node}(t)  num(all x)  j=1  denotes the sum of the activation value that was spread of all  node i neighboring nodes at time t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' the gain function is expressed as the attention gain of  neighbor nodes  x of node i after the enhancement and suppression processes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' it represents the benefit a node gets by  unilaterally changing their strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  The utility function of the attention game process can be determined as follows:  Max�������������������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ������������������������  ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ������������−������������  ∗ )� = Gain(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ������������������������������������) − Cost(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ������������������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ������������������������������������)  (7)  where ������������������������  ∗ is the optimal strategy of each node i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' ������������−������������  ∗ is the strategies set of other nodes except node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' only  when   ������������������������(������������, ������������������������  ∗, ������������−������������  ∗ ) reaches the maximum, ������������������������  ∗ is a Nash equilibrium of node i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The utility of the other  nodes will be affected by the decision of all other nodes because of the fact that the total quantity of  activation energy is fixed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', attention is limited) in the attention game process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' When each node selects  a decision for itself, it also considers the possible decision of other nodes and selects the “Nash  equilibrium point” with maximum utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This scenario is consistent with classical game theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The  authors select a Nash equilibrium decision for each node through the utility function of the attention game  that is defined by Equation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  METHODS  Data Sets and Experimental Settings  All data from the authors’ experiments are obtained from the Tsinghua University–Chinese SL Corpus  (TH–SLC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The data mainly comprise SL expressions of idiom stories and life fragments of deaf students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  No automatic annotation software based on videos is currently available because the annotation process  for SL videos is time consuming and requires expert knowledge in dual language (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', Chinese language  and Chinese SL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Video annotation is also time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Specifically, it takes about 30 hours for the  annotation RTF (real-time factor) of a parliamentary speech (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', One hour of speech requires 30 hours of  annotation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' However, the annotation RTF (real-time factor) for a full annotation of all manual and non-  manual components of an SL video can reach up to 100 hours (Dreuw & Ney, 2008a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Therefore, such a  corpus is significantly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' For example, the Aachen Boston database contains American SL and has  annotated 201 English sentences (Dreuw & Ney, 2008a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The authors spent a year collecting more than  2000 sentences, but only 416 sentences containing 2496 signs were marked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  The authors asked 20 deaf students to select 300 sign pairs from 2469 annotated signs in TH–SLC and to  judge the relevance of the sign pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The correlation values range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' For convenience, a  five-point scale is used to assess the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The sign pairs were obtained using a marked correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  The authors establish an SN based on the word similarity computing method of HowNet (Liu & Li, 2002)  to determine the connection weight of the network to validate the effects of the proposed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The  authors introduce the continuous bag-of-words (CBOW that predicts the current word from a window of  surrounding context words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The order of context words does not influence the prediction (CBOW  assumption) model (Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2013), and the HowNet (Liu & Li, 2002) method as the baseline  methods using the same recommended parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The efficiency of the utility function of the attention  game process is evaluated in terms of word correlation computation, and the model complexity is  analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Word Relatedness Computation  Each model in this task needs to compute the semantic correlation of the given sign pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The correlation  between the experimental results of the model and human judgment reflects upon the model’s  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The authors selected 290 signs for the closed set and 10 signs for the open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Spearman’s correlation between model correlation score and human judgment correlation score was  calculated for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Spearman correlation coefficient is defined as the Pearson correlation  coefficient among the ranked variables (Myers & Well, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' For a sample of size N, original data  ������������������������, ������������������������  are converted into grade data������������������������, ������������������������, the correlation coefficient ρ is defined as follows:  ρ = 1 −  6 ∑ di  2  n(n2−1)  (8)  where the difference between the observations of the two variable levels is set as  ������������������������ = ������������������������ − ������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' If there is  no duplicate value in the data, and two variables are completely monotonic correlation, the Spearman  correlation coefficient is +1 or -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  RESULTS  For CBOW, the correlation scores of the two words are calculated using the cosine similarity of word  embedding (Mikolov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The evaluative results of the baseline methods and the proposed SNM  method in the closed test and in all test sets are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Evaluative results  Data Set  Closed Test  All Test Sets  (Including Open Test)  Spearman’s Rank  Correlation Coefficient  Method  290 pairs  300 pairs  CBOW (baseline method)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='4843  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='4869  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='4136  Word similarity computing  based on HowNet  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='6157  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='6174  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='6052  Proposed SNM method  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='6951  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='7063  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='6437  The evaluation results show that the proposed SNM method is better than the baseline method in 290 and  300 word pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This finding indicates that the cognitive mechanism of sign comprehension is essential to  understanding the meaning of signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The internal structure, such as location, orientation, hand shape, and  movement, contains rich semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' However, deep learning methods, such as CBOW,  consider the external context, but ignore the internal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Using the computing method of word similarity based on HowNet results in only a rough semantic  computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' For example, adding 10 new sign pairs negligibly changes the performance of these  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' In other words, these methods can still handle new signs with improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The  semantic correlation of these new sign pairs calculated by the proposed method is close to human  judgment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Figure 3 shows the quantitative analysis of the attention game process for two signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Each hand  shape of the two signs has at least 20 related semantic lexicons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The stimulus information and  permutation of each node are shown in the first and second columns from high to low according to the  activated value after the activation spreading process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Only 10 semantic lexicons that are maximally  activated are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The permutation of each node is shown in columns three to seven from high to low  according to the activation value after the end of the first to fifth attention games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The top 10 lexicons are  also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The semantic lexicons in the blue background rank high after the games, those in the green  background rank low after the games, and those in the white background are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Examples of attention games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The semantic lexicons in the blue background rank high after the  games, those in the green background rank low after the games, and those in the white background are  unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This trend shows that the ranking of other semantic lexicons below slightly changes after the  semantic lexicon that ranks highest becomes unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This condition is due to the source that  corresponds to the attention model being determined after several game processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Figure 3 also shows that significant changes occur during the ranking of the semantic lexicons in the first  and second instances after the first several games, whereas only a few changes occur in the following  stimulus games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This trend shows that the ranking of lower semantic lexicons slightly change after the  semantic lexicon that ranks highest becomes unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This condition is due to the source that  corresponds to the attention model being determined after several game processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Attention is also  assigned to other nodes in accordance with the attention game process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Humans reach a steady state after  thinking about problems constantly, and the result negligibly changes if they rethink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Nearly no change is  observed in the result after several rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Several semantic lexicons related to the signs are contained in  the text set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' thus, a few possible changes occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The result of the attention game model conforms to  human cognitive rules to a certain degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Attention is also assigned to other nodes in accordance with the attention game process (here, efforts have  been made in modeling according to the mechanism of human attention).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The result of the SNM conforms  to human cognitive rules to a certain degree (Gutierrez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' For example, the authors assume that  deaf people understand the signs shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Deaf people usually search for many familiar and  specific nouns or signs in a spreading activation mode to comprehend classifier predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' After all  activated values are calculated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' the activated nodes are graded and sorted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' A high-activated value of the  node indicates the importance of the interested object or concept represented by the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This shows that  deaf people are familiar with the concept node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Similar to the attention game process shown in Figure 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='activation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='activation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content='activation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content='Protect  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Protect  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content='Defend  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content='Maintain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content='Y  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='handshape  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='human  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Animal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='cat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='cat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='cat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content='Burial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='dog  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Sop  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='dog  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='run  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='burial  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Frustration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='horse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='horse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='horse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='lie  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Frustration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Future  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='human  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='human  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='human  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Resistance  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='future  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Voltage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Ambassador  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Ambassador  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Ambassador  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Protest  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Voltage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Ambassador  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='coach  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='coach  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='coach  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Control  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Ambassador  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Coach  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='stand  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='stand  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='stand  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Incite  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='coach  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Guide  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='run  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='run  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='run  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Exploitation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Guide  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Blind  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='lie  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='lie  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='lie  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Recalcitrant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Blind  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Opponent  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Resistance  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Resistance  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Resistancethe high-ranked semantic lexicon is a cat or dog after several rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This result shows that the most  common subjects for deaf people are typical subjects that represent classifier predicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  DISCUSSION  Compared with that of existing models, the complexity of the proposed model is reflected mainly on the  computational cost of the memory stage and the judgment stage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', the computational cost of spreading  activation and the attention game at time (t + 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The cost is a dynamic value and related to two factors,  namely, the activation state of the current sign and the current cycle as the first activation of the sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Therefore, the value changes regardless of the choice of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This outcome is consistent with the  strong dynamics of sign information, which can reflect the influence of information in different periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  In the memory stage, the time complexity of computing ������������������������(������������) is unity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' thus, the time complexity is  related to the total amount N of activation energy and cycle times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The time complexity of each activation  in each cycle is n × 1 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Space complexity is the storage space of each node and the semantic relation  weight according to semantic similarity (semantic similarity can be estimated by defining a topological  similarity, by using ontologies to define the distance between terms/concepts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Therefore, unlike the  general model such as cobweb theorem model and vector space model, where the SNM increases the  overhead in time complexity and space complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The model also increases the matching time of query  nodes and weights in the current activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' However, the overhead at this time can provide more  effective results than an invalid spreading and can be accepted by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  In the judgment stage, when the node selects the game strategy to change its activated energy value, the  convergence speed of adjusting the cognitive benefits to its own utility maximum “Nash equilibrium” is  an important measure of evaluating the SNM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', the cycle times of an attention game process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' For the  attention game, the Nash decision of different semantic nodes must minimize the change cost of the  activation energy distribution of the entire network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The Nash equilibrium point decision for each node is  selected using the utility function defined in the SNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This process is repeated until the overall network  activation energy distribution change is less than the specified threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The node needs to solve n-order  nonlinear equations in every cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Therefore, the performance of the convergence speed of the SNM is  indicated by the number of game cycles that the network requires to reach the Nash equilibrium point  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', the computing times of calculating the corresponding equation by each node in a game process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The  square root of the sum of the variance of activation value ������������������������(������������ + 1) of each adjusted node is directly  reflected by the rate of convergence in the game process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  To verify its effectiveness, the attention game model is compared with the traditional model in terms of  load balancing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' In the traditional method, the activation value of each node is certain (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', the value is not  enhanced or inhibited).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The experimental results are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The results show that the load  balance performance of the attention game model is better than that of the traditional model because the  attention game model adjusts the activation strategy after the activation of each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' When the change  cost of the energy distribution of the entire network activation is larger than the specified threshold, the  human brain adjusts the strategy to inhibit the activation energy value in the next cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' In doing so, the  free competition and distribution of attention for each node according to the attention game model can be  assured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The result is obtained through the overall competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The load of attention of the network is  balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The traditional model assumes that the activation energy value of each node is certain because  the brain activation energy resource amount is constant in a period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The brain selects the node  with a low activation energy value and performs the allocation of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This allocation causes the  attention load of several nodes to be excessively large or unutilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Comparison of load balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The load balance performance of the SNM is better than that of  the traditional model because the SNM adjusts the activation strategy after the activation of each node  The proposed SNM model used Nash equilibrium to simulate the energy activation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' In order to  quantitatively analyze the effects of Nash equilibrium, the authors compared the SNM with the cobweb  theorem model (Pashigian, 2008) in terms of different activation energy amounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The cobweb theorem is  expressed as follows:  ������������(������������ + 1) = ������������(������������) + ������������ ��������������������������(������������)� − �������������������������′(������������)��  (9)  where r is the adjustment parameter of the activation value,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  �������������������������(������������)� is the activation function of a node,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  ������������(������������) is the activation value at time t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  �������������������������′(������������)� is the attention allocation function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  ������������′(������������)  is the  expectation activation value at time t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' and   �������������������������(������������)� − �������������������������′(������������)�is the excessive demand function that  represents the actual gaps between the activation value and activated allocated value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' A large gap  indicates a high activation value of the Nash Equilibrium of node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The parameter (r) indicates the actual  speed and strength of adjusting the activation value according to the attention distribution condition in the  last moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' When r > 0 it indicates that the adjustment direction of the activation value is consistent  with the direction of the demand function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  The amount of activation energy Ea is assumed to be 100 kJ/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Figure 5 shows the result of comparing  the attention utilization between the game model and the cobweb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The attention amount (attention  is the behavioral and cognitive process of selectively concentrating on a discrete aspect of information,  while ignoring other perceivable information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Attention amount refers to as the allocation size of limited  processing resources), is less than 100 kJ/mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' If the attention amount is insufficient, then attention  resources can only meet part of the node demand, and the resource utilization rate of the SNM will  become higher than that of the cobweb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' When attention supply exceeds the demand of a node, the  cobweb model achieves balance to meet the needs of several nodes after a repetitive cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The SNM  meets the needs of all nodes, and the utilization rate of attention resources is higher than that of the  cobweb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Comparisonofloadbalance  9  8  nodes  hhhl  6  Numberof  L  4  3  1  0  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='1  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='2  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='3  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='4  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='5  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='6No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='7  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='8  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='9  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='10  Numberofactivationenergy amount  Iattentiongamemodel  cobwebtheoremmodeFigure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Comparison of activation energy values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' After the change in the initial value of the activation  energy, the number of iterations increases depending on the difference between the initial activation  energy value in the cobweb model and the balanced energy value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The iteration of the attention game  model can be adjusted according to the difference in the activation energy between supply and demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' A  sizeable adjustment is required to reach the Nash equilibrium state if a large difference exists between the  supply and demand  Figure 6 shows the cycle times of the SNM and the cobweb model that needs to achieve the Nash  equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' As shown in the figure, the equilibrium activation energy value of the nodes is 20 kJ/mol in  the SNM, and the activation energy is 120 kJ/mol in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' If the initial value of the activation energy is  changed, then the initial activation energy value of the cobweb model is higher than the energy  equilibrium value and requires abundant cycle time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The SNM in each cycle can adjust the activation  energy according to the variance of the activation energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The variance and adjustment range are large,  and the SNM eventually reaches the Nash equilibrium point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Cycle times of the SNM and the cobweb model that are needed to achieve the Nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  When the supply falls short of demand, attention resources can only meet the demands of several nodes,  and the resource utilization rate of the SNM becomes higher than that of the cobweb model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' If the supply  exceeds demand, then the cobweb model can reach equilibrium after repeated iterations and can meet  only part of the demands of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' However, the SNM can meet the demands of all nodes, and its  resource utilization rate is higher than that of the cobweb model  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='2  40  60  80  100  120  Attention Resource Utilization(percentage)  Amount of activation energy(KJ/mol)  Comparison of activation energy values  attention game model  cobweb theorem model  CONCLUSION  The authors presented a new model for SL comprehension based on spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This process uses  game theory to simulate the human attention suppression and enhancement process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This process also  joins the forgetting function of human memory traces to compute the initial state of the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Memory is  encoded with specific (semantic) meaning, or refers to information that is encoded along a spatial and  temporal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Although the semantic network provides a functional view of how knowledge may be  organized in the brain, it does not provide a clear model of how semantic memory might be presented in  the brain (see Cacha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Spreading activation reveals that information can be stored in SNs for a  long time, in which a network node is a linguistic concept and the nodes are connected through the  correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' An algorithmic method is proposed according to selective functions, and its effectiveness was  verified using an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The results show that the proposed method improves the performance of SL  comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  ACKNOWLEDGMENT  The authors would like to thank Chunda Liu from the National Center for Sign Language and Braille for  helping in stimulus preparation and data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' This paper forms an expanded and revised version of  a conference paper at the 14th IEEE International Conference on Cognitive Informatics & Cognitive  Computing (ICCI* CC) at Tsinghua University, Beijing, July 6-8, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' The authors are grateful to Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Raymond Chiong, and two anonymous referees for their helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Conflict of Interest  The authors of this publication declare there is no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Funding Agency  This research was supported by the Beijing Municipal Natural Science Foundation [4202028];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' National  Social Science Foundation of China [21BYY106];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' National Natural Science Foundation of China  [62036001, 61866035, 61966033];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Premium Funding Project for Academic Human Resources  Comparison of cycle times  40  Number of Iterations  35  30  25  20  15  10  5  0  1  5  10  15  20  25  30  Activationvalue  attentiongamemodel  cobwebtheoremmodelDevelopment in Beijing Union University [BPHR2019CZ05];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Jiangsu Province Key R&D Program  (Industry Prospects and Key Core Technologies) [BE2020047];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' and the characteristic-disciplines oriented  research project in Beijing Union University [KYDE40201702].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content='  Johnston, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' (2006) W(h)ither the Deaf community?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Population, genetics, and the future of Australian  Sign Language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Sign Language Studies, 6(2), 137-173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Lei, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content=' Computers in Biology and Medicine, 39(11), 978-988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Not cited in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Chelazzi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content=', Santandrea, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Rewards teach visual selective  attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Vision research, 85, 58-72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content=' Grammar, gesture, and meaning in American Sign Language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content=' Cambridge  University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content=' Computational  Linguistics and Chinese Language Processing, 7(2), 59-76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='Myers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content=' Harvard University Press, MA, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
+page_content='  Mikolov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content=', Chen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+page_content='  Pashigian, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9FKT4oBgHgl3EQfCy2S/content/2301.11709v1.pdf'}
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+The splashback radius of groups and clusters of galaxies
+at low redshifts
+F.G. Kopylova𝑎,∗ and A.I. Kopylov𝑎
+𝑎SAO RAS, Nizhny Arkhyz, Russia
+E-mail: flera@sao.ru, akop@sao.ru
+We present a study of the distribution of galaxies along the radius of 157 groups and clusters of
+galaxies (200 km s−1 < 𝜎 < 1100 km s−1) of the local Universe (0.01 < 𝑧 < 0.1). We introduced
+a new boundary of galaxy systems and identified it with the splashback radius 𝑅𝑠𝑝. We also
+identified the central region of galaxy systems with a radius of 𝑅𝑐. These radii are defined by the
+observed integrated distribution of the total number of galaxies depending on the squared distance
+from the center of the groups/clusters coinciding, as a rule, with the brightest galaxy. We show
+that the radius 𝑅𝑠𝑝 is proportional to the 𝑅200𝑐 (radius of the virialized region of a galaxy cluster)
+and to the radius of the central region 𝑅𝑐 with a slope close to 1. Among the obtained dependences
+of the radii on X-ray luminosity, the log 𝑅𝑠𝑝 - log 𝐿𝑋 relation has the lowest scatter. We measured
+< 𝑅𝑠𝑝 > = 1.67 ± 0.05 Mpc for the total sample, < 𝑅𝑠𝑝 > = 1.14 ± 0.14 Mpc for galaxy groups
+with 𝜎 ≤ 400 km s−1, < 𝑅𝑠𝑝 > = 2.00 ± 0.20 Mpc for galaxy clusters with 𝜎 > 400 km s−1. We
+found the average ratio of the radii 𝑅𝑠𝑝/𝑅200𝑐 = 1.40 ± 0.02 or 𝑅𝑠𝑝/𝑅200𝑚 = 0.88 ± 0.02.
+The Multifaceted Universe: Theory and Observations - 2022 (MUTO2022)
+23-27 May 2022
+SAO RAS, Nizhny Arkhyz, Russia
+∗Speaker
+© Copyright owned by the author(s) under the terms of the Creative Commons
+Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
+https://pos.sissa.it/
+arXiv:2301.05432v1  [astro-ph.CO]  13 Jan 2023
+
+The splashback radius of groups and clusters of galaxies at low redshifts
+F.G. Kopylova
+1.
+Introduction
+Clusters of galaxies are the largest gravitationally bound objects in the Universe. They are
+collapsing structures known as dark matter halos. Clusters of galaxies are continuously increasing
+in mass both as a result of the merging with individual galaxies and smaller groups of galaxies,
+and as a result of continuous infall of dark matter from the environment. Clusters of galaxies do
+not have clear boundaries, which are often determined by density contrast relative to the critical or
+average density of the universe. Their evolution is considered within the framework of the spherical
+collapse model in the expanding universe (e.g. [6, 7]).
+Using N-body simulations of the motion of particles of a dark matter halo (galaxies), it was
+found that a significant part of them (up to 50%) located outside the virialized regions of galaxy
+clusters (up to 2𝑅𝑣𝑖𝑟 or 2𝑅200) have already been inside [2, 5, 14].
+[8] presents the results of Millennium simulations for 75 galaxy clusters (𝑧 = 0.0), where it is
+shown that a significant part of the galaxies bounce up to 3𝑟 𝑝𝑟𝑜 𝑗/𝑟200 in the phase-space diagram.
+The radius of galaxy clusters (physical halo boundary), the splashback radius 𝑅𝑠𝑝, was intro-
+duced in [1] as the radius at which newly accreted dark matter particles are piled up within the
+apocenters of their orbits. The 𝑅𝑠𝑝 radius is clearly visible on the dark matter halo density profiles
+as a sharp density drop [1, 4]. In simulations performed in [15], it is shown that the localization of
+𝑅𝑠𝑝 depends on the rate of mass accretion into the cluster: in the halo with a rapid accretion rate
+𝑅𝑠𝑝 =∼ 0.8 - 1𝑅200𝑚, in the halo with a slow rate — ∼ 1.5𝑅200𝑚.
+In this work, we are looking for observational manifestations of splashback features in a sample
+of groups and clusters of galaxies (data from the SDSS catalog). In [9] we show the edge of the
+galaxy clusters, clearly identified by the integral distribution of the number of all galaxies in a
+cluster depending on the squared clustercentric distance, which we call the radius of the halo, 𝑅ℎ.
+This radius is usually larger than the radius 𝑅200𝑐 and is measured along the projected profile when
+a sharp increase in the number of galaxies in the center of clusters ends. We identified them later
+with the splashback radius 𝑅𝑠𝑝 and gave the results of its measurements for ∼ 100 groups/clusters
+of galaxies [9, 10, 12, 13]. In the works [12, 13] we have shown that the distribution of early-type
+galaxies in clusters allows for a more accurate estimate of the desired radius. We have measured
+for 40 galaxy systems the average radius < 𝑅𝑠𝑝 >= 1.54 ± 0.06 𝑅200𝑐 or 𝑅𝑠𝑝 = 0.96 ± 0.06 𝑅200𝑚
+(if we take into account 4𝑅200𝑐 ≈ 2.5𝑅200𝑚), which varies from 1.10 Mpc for the group NGC 5627
+with 𝜎 = 314 km s−1 to 4.17 Mpc for the cluster Coma (A 1656) with 𝜎 = 921 km s−1. Here 𝑅200𝑐
+(hereinafter 𝑅200) is the radius of a cluster inside which the density exceeds the critical density of
+the universe by a factor 200. This radius in our works is determined by the dispersion of radial
+velocities of galaxies in the systems. In model simulations, the 𝑅200𝑚 radius is often used, within
+which the density in the syatem exceeds the average density of the universe by a factor 200.
+In this study, we used a sample of 157 groups/clusters of galaxies from the regions of super-
+clusters of galaxies Leo (N=12), Hercules (N=27), Ursa Major (N=19), Corona Borealis (N=8),
+Bootes (N=13), from other smaller superclusters (N=11) and fields (N=20), groups of galaxies
+from the region of the A 1656/ A1367 supercluster (N=48). For these systems of galaxies we have
+determined radii 𝑅𝑠𝑝 (splachback radius) and 𝑅𝑐 (core radius) from the observed profile and found
+the dependences of the radii on other galaxy cluster characteristics. This study uses the data of the
+SDSS (Sloan Digital Sky Survey Data Releases 7, 8) and 2MASS XSC (Two-Micron ALL-Sky
+2
+
+The splashback radius of groups and clusters of galaxies at low redshifts
+F.G. Kopylova
+Survey Extended Source Catalog) catalogs and NED (NASA Extragalactic Database). Throughout
+this study we adopted the following values of cosmological parameters: Ω𝑚 = 0.3, ΩΛ = 0.7,
+𝐻0 = 70 km s−1 ípc−1.
+2.
+Method and data
+In the papers [9–11, 13] we presented the dynamical characteristics for a region with a radius
+of 𝑅200 for almost the entire sample of groups and clusters studied in this work. This radius can be
+estimated by the formula 𝑅200 =
+√
+3𝜎/(10𝐻(𝑧) Mpc [3]. Then, if a cluster can be considered to be
+virialized within this radius, its mass 𝑀200 can be computed by the formula 𝑀200 = 3𝐺−1𝑅200𝜎2,
+where 𝜎 is the dispersion of the line-of-sight velocities of the galaxies located within the 𝑅200
+radius, and G is the gravitational constant. Thus, the measured mass of the cluster is 𝑀200 ∝ 𝜎3. In
+our work, we first estimated the average line-of-sight velocity 𝑐𝑧 of the cluster and its dispersion 𝜎,
+and then use the inferred dispersion to determine the 𝑅200 radius. We then determine the number
+of galaxies within this radius and redetermine 𝑐𝑧, 𝜎, 𝑅200, etc. We move from the cluster and
+determine iteratively the dispersion of line-of-sight velocities of the galaxies and other parameters
+of the clusters within this radius. We consider galaxies with velocities deviating by more than 2.7𝜎
+from the mean velocity of the group (see, e.g., [14]) as field objects.
+To find the radius 𝑅𝑠𝑝, it is important to highlight the outskirts of galaxy systems. For this
+purpose, we present Figure 1 which describes in detail the structure and kinematics of galaxy cluster
+A 1318 (as an example). The panels of Figure 1 show: a) deviations of line-of-sight velocities of
+cluster members and field galaxies from the average radial velocity of the system plotted as a function
+of the squared clustercentric radius; b) the integrated distribution of the number of galaxies as a
+function of the squared clustercentric distance; c) location of galaxies in the sky plane in equatorial
+coordinates; d) the histogram of the line-of-sight velocities of all galaxies within the 𝑅200 radius.
+The solid line shows the Gaussian corresponding to the dispersion of line-of-sight velocities of
+galaxies. The vertical lines show the radii: 𝑅𝑠𝑝 (dashed-and-dotted), 𝑅200 (short dashed), 𝑅𝑐 (long
+dashed).
+We are especially interested in panel b), where the projected cluster profile is shown, the
+integrated distribution of the number of galaxies as a function of the squared clustercentric distance.
+This distribution makes it possible to visually reveal the dense core of the cluster, its more tenuous
+shell, and the external region where the distribution becomes linear (shown by the straight magenta
+lines in the figure) in the adopted coordinates, i.e., where the distrubution of surrounding galaxies
+becomes uniform on the average [9]. The figure shows the radius of the virialized region, 𝑅200,
+and the radius, 𝑅𝑠𝑝, beyond which the steep growth of the cluster members ends and linear growth
+begins. We also marked with a long dashed line the central part (core) of the cluster of radius 𝑅𝑐,
+where the main steep increase in the number of galaxies is observed. The lower curve in the same
+figure shows the distribution of early-type galaxies brighter tnan 𝑀𝐾 = −21 .m5, which we use to
+refine the radius in question. Such galaxies are located, as a rule, in the central virialized regions
+of galaxy systems. 𝑅𝑠𝑝, the splashback radius of the galaxy systems [1, 4] found by us, is the
+radius of the apocenters of the orbits of galaxies, originating in the central region of galaxy clusters.
+Thus, the radius 𝑅𝑠𝑝 separates galaxies which fall onto the cluster for the first time from collapsing
+galaxies that are already participating in establishing virial equilibrium. For our sample we measured
+3
+
+The splashback radius of groups and clusters of galaxies at low redshifts
+F.G. Kopylova
+Figure 1: Distribution of galaxies in the cluster A 1318. The panel a) shows the deviation of radial velocities
+of galaxies from the average cluster radial velocity determined from galaxies within the radius 𝑅200. The
+horizontal dashed red lines correspond to ±2.7𝜎 deviations, and the vertical blue short-dashed line marks the
+𝑅200 radius; the blue long-dashed line marks the 𝑅𝑐 radius, and the green dot-dashed line – the 𝑅𝑠𝑝 radius.
+The larger circles, plus signs, and crosses mark the galaxies brighter than 𝑀𝐾 = −24𝑚, background galaxies,
+and foreground galaxies, respectively. The panel b) presents the integrated distribution of the total number of
+galaxies (the upper curve) as a function of the squared projected distance from the cluster center. The lower
+curve shows the distribution of early-type galaxies brighter than 𝑀𝐾 < −21 .m5. The circles correspond to
+the galaxies denoted by the circles in the top panel a), the crosses show the distribution of field galaxies.
+The solid magenta lines show linear sections of the profile. The panel c) shows the sky distribution (in
+quatorial coordinates) of galaxies presented in the top panel a) (same designations are used). The colored
+circles highlight the regions with radii 𝑅𝑐, 𝑅200, and 𝑅𝑠𝑝. The studied region is bounded by a circle of
+radius 3.5𝑅200 (solid black line). The large cross indicates the center of the cluster. The panel d) presents
+the distribution of radial velocities of all galaxies within 𝑅200 (the solid line shows the Gaussian for cluster
+members). The solid vertical line indicates the average radial velocity of the cluster, and the dashed lines
+correspond to ±2.7𝜎 deviations.
+< 𝑅𝑠𝑝 >= 1.67±0.05 Mpc with a variation range of 0.75÷4.24, < 𝑅𝑐 >= 0.78±0.03 (0.30÷2.00)
+and < 𝑅𝑠𝑝/𝑅200𝑐 >= 1.40 ± 0.02, or < 𝑅𝑠𝑝/𝑅200𝑚 >= 0.88 ± 0.02 (given that 4𝑅200𝑐 ≈ 2.5𝑅200𝑚.
+This value is consistent with results of simulations (see, e.g., [15]).
+3.
+Results
+We stutied the dependences of the found radii 𝑅𝑠𝑝 and 𝑅𝑐 on the properties of galaxy clusters.
+Figure 2 shows the dependence of log 𝑅𝑠𝑝 on the X-ray luminosity, log 𝐿𝑋, and for comparison a
+4
+
+A1318
+3000
++'
++
+Dec
+***芊
++
++
+++
+t
+OD
+2000
++
++
++a
++
++
++ ++
++
+t
+55030'
++
+++
++
++
++
++
+S-1
+Xo
+1000
+6=
+0
++
+**
+km
+00
+Q
++
+0
+1
+'zo
+1000
+55000*
+-2000
+a)
+X
+-3000
+arcmin20
+695
+1390
+X
+54030*
++ ×
+80
+c)
+11h40m
+11h36m
+11h32m
+60
+RA
+15
+N
+40
+8
+10
+ooo
+20
+5
+b)
+d)
+0
+0.0
+3.0
+6.0
+14000
+16000
+18000
+r, Mpc2
+cz,km s-1The splashback radius of groups and clusters of galaxies at low redshifts
+F.G. Kopylova
+similar dependence for the radius log 𝑅200 is shown. The relations shown (straight lines) represent
+the average between forward and inverse regressions, when the independent variables are inter-
+changed. The dashed lines show 1𝜎 deviations from them. Note that rms deviation depends on the
+log 𝑅𝑠𝑝 radius less than on log 𝑅200. Red circles show merging clusters of galaxies with bimodal
+radial velocity distribution within the radius 𝑅200. We can be see that these syatems do not differ
+from normal clusters of galaxies by location on the log 𝑅𝑠𝑝–log 𝐿𝑋 relation. The Table 1 shows the
+parameters of the relations we obtained: slopes, normalizations, and scatters.
+We have noted in Section 1 the results of other authors (e.g., [15]), from which it follows that
+𝑅𝑠𝑝 depends on the rate of accretion of dark matter particles on the clusters: at a rapid rate of mass
+accretion, this radius is close to the virial radius, that is, in our case to 𝑅200.
+Figure 3 shows the dependence of the radii ratios 𝑅𝑠𝑝/𝑅200 and 𝑅200/𝑅𝑐 on the X-ray luminosity
+of systems of galaxies. Variations of the 𝑅𝑠𝑝/𝑅200 ratio have distinct boundaries for most systems
+of galaxies: 𝑅𝑠𝑝/𝑅200 = 1.2 ÷ 1.6 and log 𝐿𝑋 = 42.5 ÷ 44.5. We refer to galaxy systems with a
+radius ratio less than 1.15 (or 𝑅𝑠𝑝/𝑅200𝑚 < 0.71) as objects with a rapid rate of mass accretion
+(RA), and the systems with a radii ratio greater than 1.6 (or 𝑅𝑠𝑝/𝑅200𝑚 > 0.99) - as objects with
+a slow accretion rate (SA). There are 21 RA and 34 SA systems in our sample. RA systems are
+usually groups/clusters of galaxies with a non-Gaussian radial velocity distribution, with signs of
+merging with other groups and galaxies near the virial radius, for example: A 1270, A 1904, A 1991,
+NGC 2563. Among SA groups/clusters of galaxies there are rich galaxy systems such as A 1656,
+A 1795, A 2142, A 2029, poor galaxy systems such as NGC 7237, IC 2476, MCG-01-29, which
+collect matter (groups, galaxies, gas) from great distances from the center.
+Figure 2: The radii 𝑅200 (a) and 𝑅𝑠𝑝 (b) as functions of the X-ray luminosity. The solid lines correspond
+to the relations 𝑅200 ∝ 𝐿0.25
+𝑋
+and 𝑅𝑠𝑝 ∝ 𝐿0.24
+𝑋
+. Dashed lines correspond to 1𝜎 deviations from them. Red
+circles show groups/clusters of galaxies with bimodal distribution of radial velocities.
+We have obtained the following results:
+1. The boundary of the dark halo of groups/clusters of galaxies, the radius 𝑅𝑠𝑝 determined
+by galaxies, is proportional to the radius of the virialized region 𝑅200 and to the radius of the core
+region 𝑅𝑐 with a slope close to 1.
+5
+
+3.6
+3.4
+rms=0.097
+rms=0.092
+/o
+3.4
+kpc
+kpc
+QQ
+3.2
+R200
+C
+Rsp
+3.2
+log
+3.0
+log
+0
+0
+0
+3.0
+0
+2.8
+0
+6
+0
+a)
+0
+b)
+2.8
+I
+42.00
+43.00
+44.00
+45.00
+42.00
+43.00
+44.00
+45.00
+log Lx, erg s-1
+1ogLx.
+erg s-1The splashback radius of groups and clusters of galaxies at low redshifts
+F.G. Kopylova
+Figure 3: The ratios 𝑅𝑠𝑝/𝑅200 (a) and 𝑅200/𝑅𝑐 (b) as functions of X-ray luminosity. Red circles show
+groups/clusters of galaxies with bimodal distribution of radial velocities.
+2. All the radii we measured correlate with the X-ray luminosity of groups/clusters of galaxies
+and have similar slopes. Dependencies of the splashback radius on mass 𝑀200 and luminosity
+𝐿𝐾 ,200 have a lower scatter.
+Table 1: Best fit parameters
+Relation
+Slope
+Normalization
+Scatter
+log 𝑅𝑠𝑝 – log 𝐿𝑋
+0.24 ± 0.03
+−7.39 ± 0.33
+0.092
+log 𝑅200 – log 𝐿𝑋
+0.25 ± 0.04
+−7.60 ± 0.34
+0.097
+log 𝑅𝑐 – log 𝐿𝑋
+0.26 ± 0.04
+−8.45 ± 0.37
+0.110
+log 𝑅𝑠𝑝 – log 𝑀200/𝑀⊙
+0.32 ± 0.02
+−1.42 ± 0.13
+0.066
+log 𝑅𝑐 – log 𝑀200/𝑀⊙
+0.35 ± 0.03
+−2.08 ± 0.17
+0.086
+log 𝑅𝑠𝑝 – log 𝐿𝐾/𝐿 ⊙
+0.42 ± 0.03
+−2.00 ± 0.17
+0.074
+log 𝑅𝑐 – log 𝐿𝐾/𝐿 ⊙
+0.44 ± 0.03
+−2.66 ± 0.19
+0.088
+log 𝑅𝑠𝑝 – log 𝑅200
+1.00 ± 0.04
++0.17 ± 0.11
+0.064
+log 𝑅𝑠𝑝 – log 𝑅𝑐
+0.95 ± 0.05
++0.46 ± 0.14
+0.088
+Acknowledgments
+This research has made use of the NASA/IPAC Extragalactic Database (NED, http://
+nedwww.ipac.caltech.edu), which is operated by the Jet Propulsion Laboratory, California
+Institute of Technology, under contract with the National Aeronautics and Space Administra-
+tion, Sloan Digital Sky Survey (SDSS, http://www.sdss.org), which is supported by Alfred
+P. Sloan Foundation, the participant institutes of the SDSS collaboration, National Science Foun-
+dation, and the United States Department of Energy and Two Micron All Sky Survey (2MASS,
+http://www.ipac.caltech.edu/2mass/releases/allsky/).
+6
+
+2.0
+3.0
+0
+1.8
+0
+%
+2.5
+0
+0
+0
+(0)
+0
+0
+1.6
+0
+0
+0
+/ R200
+0
+00
+00
+0
+0
+0
+0
+0
+2.0
+1.4
+0
+0
+00
+0
+0
+o
+0
+000
+0
+0
+0
+0
+0
+8
+8。°
+0
+00
+8
+%
+.0
+0
+1.2
+0
+8
+00
+09
+8
+1.5
+0
+0
+0
+60
+0
+0
+00
+0
+0
+0
+o
+9
+8
+0
+00
+0
+0
+1.0
+0
+a)
+%0
+b)
+90
+n
+1.0
+42.00
+43.00
+44.00
+45.00
+42.00
+43.00
+44.00
+45.00
+logLx.
+erg s-1The splashback radius of groups and clusters of galaxies at low redshifts
+F.G. Kopylova
+References
+[1] Adhikari, S., Dalal, N., Chamberlain, R. T. 2014, JCAP, 11, 19
+[2] Balogh, M. L., Navarro, J. F., Morris, S. L. 2000, Astrophys. J. , 540, 113
+[3] Carlberg, R. G., Yee, H. K. C., Ellingson, E., et al. 1997, Astrophys. J. , 485, L13
+[4] Diemer, B. & Kravtsov, A. V. 2014, Astrophys. J. , 789, 1
+[5] Gill, S. P. D., Knebe, A., Gibson, B. K. 2005, Mon. Not. R. Astron. Soc. , 356, 1327
+[6] Gott, J. R,III 1973, Astrophys. J. , 186, 481
+[7] Gunn, J. E., & Gott, J. R,III 1972, Astrophys. J. , 176, 1
+[8] Haines, C. P., Pereira, M. J., Smith, G. P., et al. 2015, Astrophys. J. , 806, 101
+[9] Kopylov, A. I., Kopylova, F. G. 2015, Asrophysical Bulletin, 70, 243
+[10] Kopylova, F. G., Kopylov, A. I. 2016, Asrophysical Bulletin, 71, 129
+[11] Kopylova, F. G., Kopylov, A. I. 2017, Asrophysical Bulletin, 72, 363
+[12] Kopylova, F. G., Kopylov, A. I. 2018, Asrophysical Bulletin, 73, 267
+[13] Kopylova, F. G., Kopylov, A. I. 2019, Asrophysical Bulletin, 74, 365
+[14] Mamon, G. A., Sanchis, T., Salvador-Sole, E., Solanes, M. J. 2004, A&A, 414, 445
+[15] More, S., Diemer, B., Kravtsov, A. V. 2015, Astrophys. J. , 810, 36
+7
+
diff --git a/kNE5T4oBgHgl3EQfGg61/content/tmp_files/load_file.txt b/kNE5T4oBgHgl3EQfGg61/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf,len=376
+page_content='The splashback radius of groups and clusters of galaxies  at low redshifts  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Kopylova𝑎,∗ and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Kopylov𝑎  𝑎SAO RAS, Nizhny Arkhyz, Russia  E-mail: flera@sao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='ru, akop@sao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='ru  We present a study of the distribution of galaxies along the radius of 157 groups and clusters of  galaxies (200 km s−1 < 𝜎 < 1100 km s−1) of the local Universe (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='01 < 𝑧 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' We introduced  a new boundary of galaxy systems and identified it with the splashback radius 𝑅𝑠𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' We also  identified the central region of galaxy systems with a radius of 𝑅𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' These radii are defined by the  observed integrated distribution of the total number of galaxies depending on the squared distance  from the center of the groups/clusters coinciding, as a rule, with the brightest galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' We show  that the radius 𝑅𝑠𝑝 is proportional to the 𝑅200𝑐 (radius of the virialized region of a galaxy cluster)  and to the radius of the central region 𝑅𝑐 with a slope close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Among the obtained dependences  of the radii on X-ray luminosity, the log 𝑅𝑠𝑝 - log 𝐿𝑋 relation has the lowest scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' We measured  < 𝑅𝑠𝑝 > = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='05 Mpc for the total sample, < 𝑅𝑠𝑝 > = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='14 Mpc for galaxy groups  with 𝜎 ≤ 400 km s−1, < 𝑅𝑠𝑝 > = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='20 Mpc for galaxy clusters with 𝜎 > 400 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' We  found the average ratio of the radii 𝑅𝑠𝑝/𝑅200𝑐 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='02 or 𝑅𝑠𝑝/𝑅200𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  The Multifaceted Universe: Theory and Observations - 2022 (MUTO2022)  23-27 May 2022  SAO RAS, Nizhny Arkhyz, Russia  ∗Speaker  © Copyright owned by the author(s) under the terms of the Creative Commons  Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='it/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='05432v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='CO]  13 Jan 2023  The splashback radius of groups and clusters of galaxies at low redshifts  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Kopylova  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  Introduction  Clusters of galaxies are the largest gravitationally bound objects in the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' They are  collapsing structures known as dark matter halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Clusters of galaxies are continuously increasing  in mass both as a result of the merging with individual galaxies and smaller groups of galaxies,  and as a result of continuous infall of dark matter from the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Clusters of galaxies do  not have clear boundaries, which are often determined by density contrast relative to the critical or  average density of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Their evolution is considered within the framework of the spherical  collapse model in the expanding universe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' [6, 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  Using N-body simulations of the motion of particles of a dark matter halo (galaxies), it was  found that a significant part of them (up to 50%) located outside the virialized regions of galaxy  clusters (up to 2𝑅𝑣𝑖𝑟 or 2𝑅200) have already been inside [2, 5, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  [8] presents the results of Millennium simulations for 75 galaxy clusters (𝑧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='0), where it is  shown that a significant part of the galaxies bounce up to 3𝑟 𝑝𝑟𝑜 𝑗/𝑟200 in the phase-space diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  The radius of galaxy clusters (physical halo boundary), the splashback radius 𝑅𝑠𝑝, was intro-  duced in [1] as the radius at which newly accreted dark matter particles are piled up within the  apocenters of their orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The 𝑅𝑠𝑝 radius is clearly visible on the dark matter halo density profiles  as a sharp density drop [1, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' In simulations performed in [15], it is shown that the localization of  𝑅𝑠𝑝 depends on the rate of mass accretion into the cluster: in the halo with a rapid accretion rate  𝑅𝑠𝑝 =∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='8 - 1𝑅200𝑚, in the halo with a slow rate — ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='5𝑅200𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  In this work, we are looking for observational manifestations of splashback features in a sample  of groups and clusters of galaxies (data from the SDSS catalog).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' In [9] we show the edge of the  galaxy clusters, clearly identified by the integral distribution of the number of all galaxies in a  cluster depending on the squared clustercentric distance, which we call the radius of the halo, 𝑅ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  This radius is usually larger than the radius 𝑅200𝑐 and is measured along the projected profile when  a sharp increase in the number of galaxies in the center of clusters ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' We identified them later  with the splashback radius 𝑅𝑠𝑝 and gave the results of its measurements for ∼ 100 groups/clusters  of galaxies [9, 10, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' In the works [12, 13] we have shown that the distribution of early-type  galaxies in clusters allows for a more accurate estimate of the desired radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' We have measured  for 40 galaxy systems the average radius < 𝑅𝑠𝑝 >= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='06 𝑅200𝑐 or 𝑅𝑠𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='06 𝑅200𝑚  (if we take into account 4𝑅200𝑐 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='5𝑅200𝑚), which varies from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='10 Mpc for the group NGC 5627  with 𝜎 = 314 km s−1 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='17 Mpc for the cluster Coma (A 1656) with 𝜎 = 921 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Here 𝑅200𝑐  (hereinafter 𝑅200) is the radius of a cluster inside which the density exceeds the critical density of  the universe by a factor 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' This radius in our works is determined by the dispersion of radial  velocities of galaxies in the systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' In model simulations, the 𝑅200𝑚 radius is often used, within  which the density in the syatem exceeds the average density of the universe by a factor 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  In this study, we used a sample of 157 groups/clusters of galaxies from the regions of super-  clusters of galaxies Leo (N=12), Hercules (N=27), Ursa Major (N=19), Corona Borealis (N=8),  Bootes (N=13), from other smaller superclusters (N=11) and fields (N=20), groups of galaxies  from the region of the A 1656/ A1367 supercluster (N=48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' For these systems of galaxies we have  determined radii 𝑅𝑠𝑝 (splachback radius) and 𝑅𝑐 (core radius) from the observed profile and found  the dependences of the radii on other galaxy cluster characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' This study uses the data of the  SDSS (Sloan Digital Sky Survey Data Releases 7, 8) and 2MASS XSC (Two-Micron ALL-Sky  2  The splashback radius of groups and clusters of galaxies at low redshifts  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Kopylova  Survey Extended Source Catalog) catalogs and NED (NASA Extragalactic Database).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Throughout  this study we adopted the following values of cosmological parameters: Ω𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='3, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='7,  𝐻0 = 70 km s−1 ípc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  Method and data  In the papers [9–11, 13] we presented the dynamical characteristics for a region with a radius  of 𝑅200 for almost the entire sample of groups and clusters studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' This radius can be  estimated by the formula 𝑅200 =  √  3𝜎/(10𝐻(𝑧) Mpc [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Then, if a cluster can be considered to be  virialized within this radius, its mass 𝑀200 can be computed by the formula 𝑀200 = 3𝐺−1𝑅200𝜎2,  where 𝜎 is the dispersion of the line-of-sight velocities of the galaxies located within the 𝑅200  radius, and G is the gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Thus, the measured mass of the cluster is 𝑀200 ∝ 𝜎3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' In  our work, we first estimated the average line-of-sight velocity 𝑐𝑧 of the cluster and its dispersion 𝜎,  and then use the inferred dispersion to determine the 𝑅200 radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' We then determine the number  of galaxies within this radius and redetermine 𝑐𝑧, 𝜎, 𝑅200, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' We move from the cluster and  determine iteratively the dispersion of line-of-sight velocities of the galaxies and other parameters  of the clusters within this radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' We consider galaxies with velocities deviating by more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='7𝜎  from the mean velocity of the group (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=', [14]) as field objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  To find the radius 𝑅𝑠𝑝, it is important to highlight the outskirts of galaxy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' For this  purpose, we present Figure 1 which describes in detail the structure and kinematics of galaxy cluster  A 1318 (as an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The panels of Figure 1 show: a) deviations of line-of-sight velocities of  cluster members and field galaxies from the average radial velocity of the system plotted as a function  of the squared clustercentric radius;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' b) the integrated distribution of the number of galaxies as a  function of the squared clustercentric distance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' c) location of galaxies in the sky plane in equatorial  coordinates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' d) the histogram of the line-of-sight velocities of all galaxies within the 𝑅200 radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  The solid line shows the Gaussian corresponding to the dispersion of line-of-sight velocities of  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The vertical lines show the radii: 𝑅𝑠𝑝 (dashed-and-dotted), 𝑅200 (short dashed), 𝑅𝑐 (long  dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  We are especially interested in panel b), where the projected cluster profile is shown, the  integrated distribution of the number of galaxies as a function of the squared clustercentric distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  This distribution makes it possible to visually reveal the dense core of the cluster, its more tenuous  shell, and the external region where the distribution becomes linear (shown by the straight magenta  lines in the figure) in the adopted coordinates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=', where the distrubution of surrounding galaxies  becomes uniform on the average [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The figure shows the radius of the virialized region, 𝑅200,  and the radius, 𝑅𝑠𝑝, beyond which the steep growth of the cluster members ends and linear growth  begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' We also marked with a long dashed line the central part (core) of the cluster of radius 𝑅𝑐,  where the main steep increase in the number of galaxies is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The lower curve in the same  figure shows the distribution of early-type galaxies brighter tnan 𝑀𝐾 = −21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='m5, which we use to  refine the radius in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Such galaxies are located, as a rule, in the central virialized regions  of galaxy systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' 𝑅𝑠𝑝, the splashback radius of the galaxy systems [1, 4] found by us, is the  radius of the apocenters of the orbits of galaxies, originating in the central region of galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  Thus, the radius 𝑅𝑠𝑝 separates galaxies which fall onto the cluster for the first time from collapsing  galaxies that are already participating in establishing virial equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' For our sample we measured  3  The splashback radius of groups and clusters of galaxies at low redshifts  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Kopylova  Figure 1: Distribution of galaxies in the cluster A 1318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The panel a) shows the deviation of radial velocities  of galaxies from the average cluster radial velocity determined from galaxies within the radius 𝑅200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The  horizontal dashed red lines correspond to ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='7𝜎 deviations, and the vertical blue short-dashed line marks the  𝑅200 radius;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' the blue long-dashed line marks the 𝑅𝑐 radius, and the green dot-dashed line – the 𝑅𝑠𝑝 radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  The larger circles, plus signs, and crosses mark the galaxies brighter than 𝑀𝐾 = −24𝑚, background galaxies,  and foreground galaxies, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The panel b) presents the integrated distribution of the total number of  galaxies (the upper curve) as a function of the squared projected distance from the cluster center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The lower  curve shows the distribution of early-type galaxies brighter than 𝑀𝐾 < −21 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='m5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The circles correspond to  the galaxies denoted by the circles in the top panel a), the crosses show the distribution of field galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  The solid magenta lines show linear sections of the profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The panel c) shows the sky distribution (in  quatorial coordinates) of galaxies presented in the top panel a) (same designations are used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The colored  circles highlight the regions with radii 𝑅𝑐, 𝑅200, and 𝑅𝑠𝑝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The studied region is bounded by a circle of  radius 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='5𝑅200 (solid black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The large cross indicates the center of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The panel d) presents  the distribution of radial velocities of all galaxies within 𝑅200 (the solid line shows the Gaussian for cluster  members).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The solid vertical line indicates the average radial velocity of the cluster, and the dashed lines  correspond to ±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='7𝜎 deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  < 𝑅𝑠𝑝 >= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='67±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='05 Mpc with a variation range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='75÷4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='24, < 𝑅𝑐 >= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='78±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='03 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='30÷2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='00)  and < 𝑅𝑠𝑝/𝑅200𝑐 >= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='02, or < 𝑅𝑠𝑝/𝑅200𝑚 >= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='88 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='02 (given that 4𝑅200𝑐 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='5𝑅200𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  This value is consistent with results of simulations (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=', [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  Results  We stutied the dependences of the found radii 𝑅𝑠𝑝 and 𝑅𝑐 on the properties of galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content="  Figure 2 shows the dependence of log 𝑅𝑠𝑝 on the X-ray luminosity, log 𝐿𝑋, and for comparison a  4  A1318  3000  +'  +  Dec  ***芊  +  +  ++  t  OD  2000  +  +  +a  +  +  + ++  +  t  55030'  +  ++  +  +  +  +  S-1  Xo  1000  6=  0  +  **  km  00  Q  +  0  1  'zo  1000  55000*  2000  a)  X  3000  arcmin20  695  1390  X  54030*  + ×  80  c)  11h40m  11h36m  11h32m  60  RA  15  N  40  8  10  ooo  20  5  b)  d)  0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='0  14000  16000  18000  r, Mpc2  cz,km s-1The splashback radius of groups and clusters of galaxies at low redshifts  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Kopylova  similar dependence for the radius log 𝑅200 is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The relations shown (straight lines) represent  the average between forward and inverse regressions, when the independent variables are inter-  changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The dashed lines show 1𝜎 deviations from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Note that rms deviation depends on the  log 𝑅𝑠𝑝 radius less than on log 𝑅200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Red circles show merging clusters of galaxies with bimodal  radial velocity distribution within the radius 𝑅200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' We can be see that these syatems do not differ  from normal clusters of galaxies by location on the log 𝑅𝑠𝑝–log 𝐿𝑋 relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The Table 1 shows the  parameters of the relations we obtained: slopes, normalizations, and scatters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  We have noted in Section 1 the results of other authors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=', [15]), from which it follows that  𝑅𝑠𝑝 depends on the rate of accretion of dark matter particles on the clusters: at a rapid rate of mass  accretion, this radius is close to the virial radius, that is, in our case to 𝑅200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  Figure 3 shows the dependence of the radii ratios 𝑅𝑠𝑝/𝑅200 and 𝑅200/𝑅𝑐 on the X-ray luminosity  of systems of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Variations of the 𝑅𝑠𝑝/𝑅200 ratio have distinct boundaries for most systems  of galaxies: 𝑅𝑠𝑝/𝑅200 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='2 ÷ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='6 and log 𝐿𝑋 = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='5 ÷ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' We refer to galaxy systems with a  radius ratio less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='15 (or 𝑅𝑠𝑝/𝑅200𝑚 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='71) as objects with a rapid rate of mass accretion  (RA), and the systems with a radii ratio greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='6 (or 𝑅𝑠𝑝/𝑅200𝑚 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='99) - as objects with  a slow accretion rate (SA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' There are 21 RA and 34 SA systems in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' RA systems are  usually groups/clusters of galaxies with a non-Gaussian radial velocity distribution, with signs of  merging with other groups and galaxies near the virial radius, for example: A 1270, A 1904, A 1991,  NGC 2563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Among SA groups/clusters of galaxies there are rich galaxy systems such as A 1656,  A 1795, A 2142, A 2029, poor galaxy systems such as NGC 7237, IC 2476, MCG-01-29, which  collect matter (groups, galaxies, gas) from great distances from the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  Figure 2: The radii 𝑅200 (a) and 𝑅𝑠𝑝 (b) as functions of the X-ray luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The solid lines correspond  to the relations 𝑅200 ∝ 𝐿0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='25  𝑋  and 𝑅𝑠𝑝 ∝ 𝐿0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='24  𝑋  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Dashed lines correspond to 1𝜎 deviations from them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Red  circles show groups/clusters of galaxies with bimodal distribution of radial velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  We have obtained the following results:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' The boundary of the dark halo of groups/clusters of galaxies, the radius 𝑅𝑠𝑝 determined  by galaxies, is proportional to the radius of the virialized region 𝑅200 and to the radius of the core  region 𝑅𝑐 with a slope close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='4  rms=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='097  rms=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='092  /o  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='4  kpc  kpc  QQ  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='2  R200  C  Rsp  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='2  log  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='0  log  0  0  0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='0  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='8  0  6  0  a)  0  b)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='8  I  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='00  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='00  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='00  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='00  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='00  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='00  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='00  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='00  log Lx, erg s-1  1ogLx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  erg s-1The splashback radius of groups and clusters of galaxies at low redshifts  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Kopylova  Figure 3: The ratios 𝑅𝑠𝑝/𝑅200 (a) and 𝑅200/𝑅𝑐 (b) as functions of X-ray luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Red circles show  groups/clusters of galaxies with bimodal distribution of radial velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' All the radii we measured correlate with the X-ray luminosity of groups/clusters of galaxies  and have similar slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content=' Dependencies of the splashback radius on mass 𝑀200 and luminosity  𝐿𝐾 ,200 have a lower scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='  Table 1: Best fit parameters  Relation  Slope  Normalization  Scatter  log 𝑅𝑠𝑝 – log 𝐿𝑋  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='03  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='33  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='092  log 𝑅200 – log 𝐿𝑋  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='04  −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='34  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
+page_content='097  log 𝑅𝑐 – log 𝐿𝑋  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE5T4oBgHgl3EQfGg61/content/2301.05432v1.pdf'}
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+Knowledge Reasoning via Jointly Modeling Knowledge Graphs and
+Soft Rules
+Yinyu Lan, Shizhu He, Kang Liu, Jun Zhao
+Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China.
+Abstract
+Knowledge graphs (KGs) play a crucial role in many applications, such as question answering, but incom-
+pleteness is an urgent issue for their broad application. Much research in knowledge graph completion (KGC) has been
+performed to resolve this issue. The methods of KGC can be classified into two major categories: rule-based reasoning and
+embedding-based reasoning. The former has high accuracy and good interpretability, but a major challenge is to obtain
+effective rules on large-scale KGs. The latter has good efficiency and scalability, but it relies heavily on data richness and
+cannot fully use domain knowledge in the form of logical rules. We propose a novel method that injects rules and learns
+representations iteratively to take full advantage of rules and embeddings. Specifically, we model the conclusions of rule
+groundings as 0-1 variables and use a rule confidence regularizer to remove the uncertainty of the conclusions. The proposed
+approach has the following advantages: 1) It combines the benefits of both rules and knowledge graph embeddings (KGEs)
+and achieves a good balance between efficiency and scalability. 2) It uses an iterative method to continuously improve KGEs
+and remove incorrect rule conclusions. Evaluations on two public datasets show that our method outperforms the current
+state-of-the-art methods, improving performance by 2.7% and 4.3% in mean reciprocal rank (MRR).
+Keywords
+Distributed representation, Knowledge graph, Link prediction, Logical rule
+1
+Introduction
+A knowledge graph (KG) organizes knowledge as
+a set of interlinked triples, and a triple ((head entity,
+relation, tail entity), simply represented as (h, r, t)) in-
+dicates the fact that two entities have a certain relation.
+Rich structured and formalized knowledge has become
+a valuable resource to support downstream tasks, for
+example, question answering [1, 2] and recommender
+systems [3, 4].
+Although KGs such as DBpedia [5], Freebase [6] and
+NELL [7] contain large amounts of entities, relations,
+and triples, they are far from complete, which is an
+urgent issue for their broad application.
+To address
+this, the task of knowledge graph completion (KGC)
+has been proposed and has attracted growing atten-
+tion; it utilizes knowledge reasoning techniques to per-
+form automatic discovery of new facts based on existing
+facts in a KG [8].
+At present, the methods of KGC can be classified
+into two major categories: 1) One type of method uses
+explicit reasoning rules; it obtains the reasoning rules
+through inductive learning and then deductively infers
+new facts. 2) Another method is based on representa-
+tion learning instead of directly modeling rules, aiming
+to learn a distributed embedding for entities and rela-
+tions and perform generalization in numerical space.
+Rule-based reasoning is accurate and can provide
+interpretability for the inference results. Domain ex-
+perts can handcraft these rules [9] or can mine them
+from the KG with an induction algorithm such as
+AMIE [10].
+Traditional methods such as expert sys-
+tems [11, 12] use hard logical rules to make predictions.
+For example, as shown in Fig. 1, given the logical rule
+(x, born in, y)∧(y, city of, z) ⇒ (x, nationality, z) and
+the two facts that (Chicago, city of, USA) and (Mary,
+born in, Chicago), we can infer the fact (Mary, nation-
+ality, USA). A large number of new facts (conclusions)
+can be derived based on forward chaining inference.
+Regular Paper
+arXiv:2301.02781v1  [cs.AI]  7 Jan 2023
+
+2
+However, for large-scale KGs, sufficient and effective
+reasoning rules are difficult and expensive to obtain.
+Moreover, in many cases, the logical rules may be im-
+perfect or even self-contradictory. Therefore, it is es-
+sential to model the uncertainty of (soft) logical rules
+effectively.
+The methods of determining KGEs learn to embed
+entities and relations into a continuous low-dimensional
+space [13, 14]. These embeddings retain the semantic
+meaning of entities and relations, which can be used to
+predict missing triples. In addition, they can be effec-
+tively trained using stochastic gradient descent. How-
+ever, this kind of method cannot fully use logical rules,
+which compactly encode domain knowledge and are
+helpful in various applications. Good embedding relies
+heavily on data richness, so these methods have diffi-
+culty learning useful representations for sparse entities
+[15, 16].
+In fact, both rule-based methods and embedding-
+based methods have advantages and disadvantages in
+the KGC task.
+Logical rules are accurate and inter-
+pretable, and embedding is flexible and computation-
+ally efficient. To achieve more precise knowledge com-
+pletion, recently, there has also been research on com-
+bining the advantages of logical rules and KGEs. Mixed
+techniques can infer missing triples effectively by ex-
+ploiting and modeling uncertain logical rules.
+Some
+existing methods have aimed to iteratively learn KGEs
+and rules [16], and some other methods also utilize soft
+rules or groundings of rules to regularize the learning
+of KGEs [17, 18].
+Knowledge Graph
+Rules
+Weights
+∀𝑥, 𝑦 ∶ (𝑥, child_of, 𝑦) ⇒ (𝑦, parent_of, 𝑥)
+0.9
+∀𝑥, 𝑦, 𝑧 ∶ (𝑥, born_in, 𝑦) ⋀ (𝑦, city_of, 𝑧)
+⇒ (𝑥, nationality, 𝑧)
+0.8
+∀𝑥, 𝑦, 𝑧 ∶ (𝑥, child_of,𝑧) ⋀ (𝑦, child_of,𝑧) 
+⇒ (𝑥, sister_of,𝑧)
+0.5
+…
+…
+Facts
+Weights
+(Mary, parent_of, Lisa)
+0.9
+(Mary, parent_of, Mike)
+0.9
+(Mary, nationality, USA)
+0.8
+(Nacy, nationality, USA)
+0.8
+(Lisa, sister_of, Nancy)
+0.5
+(Mike, sister_of, Nancy)
+0.5
+…
+…
+Facts
+Weights
+(Mary, parent_of, Lisa)
+1
+(Mary, parent_of, Mike)
+1
+(Mary, nationality, USA)
+1
+(Nacy, nationality, USA)
+0
+(Lisa, sister_of, Nancy)
+1
+(Mike, sister_of, Nancy)
+0
+…
+Chicago
+Boston
+Mary
+Lisa
+Nancy
+Mike
+USA
+born_in
+nationality
+child_of
+city_of
+Previous Works
+This Work
+…
+…
+KGE
+Uncertain Conclusions
+Deterministic Conclusions
+Uncertain Rules
+induction
+deduction
+completion
+Fig. 1. We propose a novel iterative knowledge reasoning frame-
+work by fusing logical rules into a KGE. Previous methods asso-
+ciate each conclusion with a weight derived from the correspond-
+ing rule. In contrast, our method can infer which conclusion is
+true via jointly modeling the deterministic KG and uncertain
+soft rules.
+The integration of logical rules and knowledge graph
+embeddings can achieve more efficient and accurate
+knowledge completion. Current methods model uncer-
+tain rules and add soft labels to conclusions by t-norm-
+based fuzzy logic [19]; they further utilize the conclu-
+sions to perform forward reasoning [17] or to enhance
+the KGE [16].
+However, in most KGs, the facts are
+deterministic. Therefore, we believe that rules are un-
+certain but conclusions are deterministic in knowledge
+reasoning, as shown in Fig. 1; each fact is only abso-
+lutely true or false. Previous methods associate each
+conclusion with a weight derived from the correspond-
+ing rule.
+In contrast, we propose inferring that all
+conclusions are true (e.g., (Mary, nationality, USA))
+or not (e.g., (Mike, sister of, Nancy)) (the other fact,
+i.e., (Mike, gender, male), indicates that Mike is not
+Nancy’s sister) by jointly modeling the deterministic
+KG and soft rules.
+Specifically, we first mine soft rules from the knowl-
+edge graph and then infer conclusions as candidate
+facts. Second, the KG, conclusions, and weighted rules
+are also used as resources to learn embeddings. Third,
+through the definition of deterministic conclusion loss,
+the conclusion labels are modeled as 0-1 variables, and
+
+MODEL KNOWLEDGE GRAPH AND RULES FOR REASONING
+3
+the confidence loss of a rule is also used to constrain
+the conclusions. Finally, the embedding learning stage
+removes the noise in the candidate conclusions, and
+then the proper conclusions are added back to the orig-
+inal KG. The above steps are performed iteratively.
+We empirically evaluate the proposed method on pub-
+lic datasets from two real large-scale KGs: DBpedia
+and Freebase. The experimental results show that our
+method Iterlogic-E (Iterative using logic rule for rea-
+soning and learning Embedding) achieves state-of-the-
+art results on multiple evaluation metrics. Iterlogic-E
+also achieves improvements of 2.7%/4.3% in mean re-
+ciprocal rank (MRR) and 2.0%/4.0% in HITS@1 com-
+pared to the state-of-the-art model.
+In summary, our main contributions are as follows:
+• We propose a novel KGC method, Iterlogic-E,
+which jointly models logical rules and KGs in the
+framework of a KGE. Iterlogic-E combines the ad-
+vantages of both rules and embeddings in knowl-
+edge reasoning. Iterlogic-E models the conclusion
+labels as 0-1 variables and uses a confidence reg-
+ularizer to eliminate the uncertain conclusions.
+• We propose a novel iterative learning paradigm
+that achieves a good balance between efficiency
+and scalability.
+Iterlogic-E not only makes the
+KG denser but can also filter incorrect conclu-
+sions.
+• Compared with traditional reasoning methods,
+Iterlogic-E is more interpretable in determining
+conclusions. It not only knows why the conclu-
+sion holds but also knows which is true and which
+is false.
+• We empirically evaluate Iterlogic-E with the
+task of link prediction on multiple benchmark
+datasets. The experimental results indicate that
+Iterlogic-E can achieve state-of-the-art results on
+multiple evaluation metrics. The qualitative anal-
+ysis proves that Iterlogic-E is more robust for
+rules with different confidence levels.
+2
+Related Work
+Knowledge reasoning aims to infer certain entities
+over KGs as the answers to a given query. A query in
+KGC is a head entity h (or a tail entity t) and a rela-
+tion r. Given (h, r, ?) (or (?, r, t)), KGC aims to find
+the right tail entity t (or head entity h) in the KG that
+satisfies the triple (h, r, t). Next, we review the three
+most relevant classes of KGC methods.
+2.1
+Rule-Based Reasoning
+Logical rules can encode human knowledge com-
+pactly, and early knowledge reasoning was primarily
+based on first-order logical rules. Existing rule-based
+reasoning methods have primarily utilized search-based
+inductive logic programming (ILP) methods, usually
+searching and pruning rules. Based on the partial com-
+pleteness assumption, AMIE [10] introduces a revised
+confidence metric, which is well suited for modeling
+KGs.
+By query rewriting and pruning, AMIE+ [20]
+is optimized to expand to larger KGs.
+Additionally,
+AMIE+ improves the precision of the forecasts by using
+joint reasoning and type information. In this paper, we
+employ AMIE+1 to mine horn rules from a KG. Rule-
+based reasoning methods can be combined with multi-
+ple probability graph models. A Markov logic network
+(MLN) [21] is a typical model. Based on preprovided
+rules, it builds a probabilistic graph model and then
+learns the weights of rules. However, due to the com-
+plicated graph structure among triples, the reasoning in
+an MLN is time-consuming and difficult, and the incom-
+1https://github.com/lajus/amie
+
+4
+pleteness of KGs also impacts the inference results. In
+contrast, Iterlogic-E uses rules to enhance KGEs with
+more effective inference.
+2.2
+Embedding-Based Reasoning
+Recently, embedding-based methods have attracted
+much attention; they aim to learn distributed em-
+beddings for entities and relations in KGs.
+Gener-
+ally, current KGE methods can be divided into three
+classes:
+1) translation-based models that learn em-
+beddings by translating one entity into another en-
+tity through a specific relation [22, 23]; 2) composi-
+tional models that use simple mathematical operations
+to model facts, including linear mapping [24], bilinear
+mapping [25, 26, 27], and circular correlation [28]; 3)
+neural network-based models that utilize a multilayer
+neural structure to learn embeddings and estimate the
+plausibility of triples with nonlinear features, for exam-
+ple, R-GCN [29], ConvE [30] and and so on [31, 32, 33].
+The above methods learn representations based only on
+the triples existing in KGs, and the sparsity of data lim-
+its them. To solve this problem and learn semantic-rich
+representations, recent works further attempted to in-
+corporate information beyond triples, e.g., contextual
+information [34], entity type information [35, 36], on-
+tological information [37], taxonomic information [38],
+textual descriptions [39] and hierarchical information
+[49].
+In contrast, the proposed Iterlogic-E uses em-
+beddings to remove incorrect conclusions obtained by
+rules, which combines the advantages of rules and em-
+beddings.
+2.3
+Hybrid Reasoning
+Both rule-based and embedding-based methods
+have advantages and disadvantages. Recent works have
+integrated these two kinds of reasoning methods. Guo
+et al. [40] attempted to learn a KGE from rule ground-
+ings and triples together. Wang et al. [41] used asym-
+metric and transitive information to approximately or-
+der relations by maximizing the margin between neg-
+ative and positive logical rules. Zhang et al. [17] and
+Guo et al. [42] obtained KGEs with supervision from
+soft rules, proving the effectiveness of logical rules. Qu
+et al. [43] used an MLN to model logical rules and in-
+ferred new triples to enhance KGEs. Guo et al. [18]
+enhanced KGEs by injecting grounding rules. Niu et
+al.
+[50] enhanced KGEs by extracting commonsense
+from factual triples with entity concepts. In addition,
+some previous methods that enhance embeddings by it-
+erative learning were studied in early works. Zhang et
+al. [16] aimed to improve a sparse entity representation
+through iterative learning and update the confidence
+of rules through embeddings. In contrast, Iterlogic-E
+models the conclusion labels as 0-1 variables and uses
+confidence regularization loss to eliminate the uncertain
+conclusions. Such labels are easier to train on.
+3
+The Proposed Method
+This
+section
+introduces
+our
+proposed
+method
+Iterlogic-E. We first give an overview of our method,
+including the entire iterative learning process. Then,
+we detail the two parts of Iterlogic-E: rule mining and
+reasoning and embedding learning.
+Last, we discuss
+the space and time complexity of Iterlogic-E and dis-
+cuss connections to related works [16, 17].
+3.1
+Overview
+Given a KG G = {E, R, T }, T = {(h, r, t)}, r ∈ R
+is a relation and h, t ∈ E are entities. As discussed in
+Section 1, on the one hand, embedding learning meth-
+ods do not make full use of logical rules and suffer from
+data sparsity. On the other hand, precise rules are dif-
+ficult to obtain efficiently and cannot cover all facts in
+KGs. Our goal is to improve the embedding quality by
+
+MODEL KNOWLEDGE GRAPH AND RULES FOR REASONING
+5
+Chicago
+Boston
+Mary
+Lisa
+Nancy
+Mike
+USA
+born_in
+nationality
+child_of
+city_of
+Rules
+Weights
+∀𝑥, 𝑦 : (𝑥, child_of, 𝑦) ⇒ (𝑦, parent_of, 𝑥)
+0.9
+∀𝑥, 𝑦, 𝑧 : (𝑥, born_in, 𝑦) ⋀ (𝑦, city_of, 𝑧) ⇒ (𝑥, nationality, 𝑧)
+0.8
+∀𝑥, 𝑦, 𝑧 : (𝑥, child_of,𝑧) ⋀ (𝑦, child_of,𝑧) ⇒ (𝑥, sister_of,𝑧)
+0.5
+…
+…
+Conclusions
+Adjusted weights
+(Mary, parent_of, Lisa)
+1
+(Mary, parent_of, Mike)
+1
+(Mary, nationality, USA)
+1
+(Nacy, nationality, USA)
+0
+(Lisa, sister_of, Nancy)
+1
+(Mike, sister_of, Nancy)
+0
+…
+…
+Groundings of rules
+Rule weights
+(Mary, born_in,Chicago) ⋀ (Chicago, city_of, USA) ⇒
+(Mary, nationality, USA)
+0.9
+(Nacy, born_in,Bostan) ⋀ (Bostan, city_of, USA) ⇒
+(Nacy, nationality, USA)
+0.9
+(Lisa, child_of,Mary) ⇒ (Mary, parent_of, Lisa)
+0.8
+(Mike, child_of,Mary) ⇒ (Mary, parent_of, Mike)
+0.8
+(Lisa, child_of,Mary) ⋀ (Nancy, child_of, Mary)  ⇒
+(Lisa, sister_of, Nancy)
+0.5
+(Lisa, child_of,Mary) ⋀ (Mike, child_of, Mary)  ⇒
+(Mike, sister_of, Nancy)
+0.5
+…
+…
+…
+…
+joint embedding learning 
+(§3.2.2)
+triple 
+injecting (§3.1)
+rule mining 
+(§3.1.1)
+rule reasoning 
+(§3.1.2)
+conclusion filtering 
+(§3.1)
+Knowledge Graph
+Knowledge Graph Embedding
+Groundings of Rules
+Soft Rules
+Reasoning Conclusions
+Fig.2. The framework details Iterlogic-E with two iterative stages: (i) rule mining and reasoning and (ii) embedding learning. Stage
+(i) generates rules and grounding rules to obtain new conclusions. Stage (ii) jointly models the conclusions of grounding rules and the
+KG in learning embeddings. After embedding learning, the conclusions are injected into the KG, and then the rule reasoning module
+is executed to start the next round of iterative training
+explicitly modeling the reasoned conclusions of logical
+rules, removing incorrect conclusions, and improving
+the confidence of the rules at the same time. Figure 2
+shows the overview of Iterlogic-E given a toy knowledge
+graph. Iterlogic-E is a general framework that can fuse
+different KGE models.
+Iterlogic-E has two iterative steps: (i) rule mining
+and reasoning and (ii) embedding learning. In the rule
+mining and reasoning step, there are two modules: rule
+mining and rule reasoning. The mining configuration,
+such as the maximum length of rules and the confi-
+dence threshold of rules, and KG triples are input to
+the rule mining module.
+Then, it automatically ob-
+tains the soft rules from these inputs. The KG triples
+and the extracted soft rules are input into the rule rea-
+soning module to infer new triples. After that, the new
+triples are appended to the embedding learning step as
+candidate conclusions. In the embedding learning step,
+relations are modeled as a linear mapping operation,
+and triple plausibility is represented as the correlation
+between the head and tail entities after the operation.
+Finally, the incorrect conclusions are filtered out by la-
+beling the conclusions with their scores2. The right con-
+clusion will be added back to the original KG triples,
+and then the rule reasoning module is performed to
+start the next cycle of iterative training.
+3.2
+Rule Mining and Reasoning
+The first step is composed of the rule mining mod-
+ule and the rule reasoning module. We introduce these
+two modules in detail below.
+3.2.1
+Rule Mining
+We extract soft rules from the KG using the state-of-
+the-art rule mining method AMIE+ [20] in this module.
+2In the experiments, we choose the conclusion with a normalized score of more than 0.99 as the true conclusion.
+
+6
+AMIE+ applies principal component analysis (PCA)
+confidence to estimate the reliability of a rule since its
+partial completeness assumption is more suited to real-
+world KGs. Additionally, AMIE+ defines a variety of
+restriction types to help extract applicable rules, e.g.,
+the maximum length of the rule. After the rule mining
+module receives the KG triples and the mining config-
+uration, it executes the AMIE+ algorithm and outputs
+soft rules. Although rules can be re-mined in each it-
+eration, we only run the rule mining module once for
+efficiency reasons.
+3.2.2
+Rule Reasoning
+The logical rule set is denoted as F = {(f, c)}. f
+is in the form of ∀x, y, z : (x, rp1, y) ∧ (y, rp2, z)
+c⇒
+(x, rc, z).
+x, y and z represent variables of different
+entities, and rp1, rp2 and rrc represent different rela-
+tions.
+The left side of the symbol ⇒ is the premise
+of the rule, which is composed of several connected
+atoms. The right side is only a single atom, which is
+the rule’s conclusion. The horn rules are closed [25],
+where continuous relations share the intermediate en-
+tity and the first and last entities of the premise ap-
+pear as the head and tail entities of the conclusion.
+Such rules can provide interpretive insights. A rule’s
+length is equal to the number of atoms in the premise.
+For example, ∀x, y, z : (x, bornin, y) ∧ (y, city, z)
+0.8
+⇒
+(x, nationality, z) is a length-2 rule. This rule reflects
+the reality that, most likely, a person’s nationality is
+the country in which he or she was born. The rule f
+has a confidence level of 0.8. The higher the confidence
+of the rule, the more likely it is to hold.
+The reasoning procedure consists of instantiating
+the rule’s premise and obtaining a large number of fresh
+conclusion triples. One of the most common approaches
+is forward chaining, also known as the match-select-
+act cycle, which works in three-phase cycles. Forward
+chaining matches the currently existing facts in the KG
+with all known rule premises in one cycle to determine
+the rules that can be satisfied.
+Finally, the selected
+rule’s conclusions are derived, and if the conclusions
+are not already in the KG, they are added as new facts.
+This cycle should be repeated until no new conclusions
+emerge. However, if soft rules are used, forward chain
+reasoning will lead to incorrect conclusions. Therefore,
+we run one reasoning cycle in every iteration.
+3.3
+Embedding Learning
+In this section, we present a joint embedding learn-
+ing approach that allows the embedding model to learn
+from KG triples, conclusion triples, and soft rule con-
+fidence all at the same time.
+First, we will examine
+a basic KGE model, and then we will describe how to
+incorporate soft rule conclusions. Finally, we detail the
+overall training goal.
+3.3.1
+A Basic KGE Model
+Different KGE models have different score func-
+tions that aim to obtain a suitable function to map
+the triple score to a continuous true value in [0, 1], i.e.,
+φ : E × R × E → (0, 1), which indicates the probabil-
+ity that the triple holds. We follow [17, 18] and choose
+ComplEx [26] as a basic KGE model. It is important
+to note that our proposed framework can be combined
+with an arbitrary KGE model. Theoretically, using a
+better base model can continue improving performance.
+Therefore, We also experiment with RotatE as a base
+model. Below we take ComplEx as an example to in-
+troduce. ComplEx assumes that the entity and relation
+embeddings exist in a complex space, i.e., e ∈ Cd and
+r ∈ Cd, where d is the dimensionality of the complex
+space. Using plurals to represent entities and relations
+can better model antisymmetric and symmetric rela-
+tions (e.g., kinship and marriage) [26]. Through a mul-
+
+MODEL KNOWLEDGE GRAPH AND RULES FOR REASONING
+7
+tilinear dot product, ComplEx scores every triple:
+F(h, r, t) = Re(hTdiag(r)¯t) = Re(�
+i[h]i[r]i[t]i),
+(1)
+where the Re(·) function takes the real part of a
+complex value and the diag(·) function constructs a di-
+agonal matrix from r; ¯t is the conjugate of t; and [·]i is
+the i-th entry of a vector. To predict the probability,
+ComplEx further uses the sigmoid function for normal-
+ization:
+φ(h, r, t) = σ(F(h, r, t)) = σ(Re(hTdiag(r)¯t)),
+(2)
+where σ(·) is the sigmoid function. By minimizing the
+logistic loss function, ComplEx learns the relation and
+entity embeddings:
+�
+(h,r,t)∈T ∪T ′
+log(1 + exp(−yhrt · f(h, r, t))),
+(3)
+where T ′ is a set of sampled negative examples and yhrt
+is the label of a positive or negative triple.
+3.3.2
+Joint Modeling KG and Conclusions of Soft
+Rules
+To model the conclusion label as a 0-1 variable,
+based on the current KGE model’s scoring function,
+we follow ComplEx and use the function f(·) as the
+scoring function for conclusion triples:
+Si = σ(F(hi, ri, ti)), (hi, ri, ti) ∈ Cf,
+(4)
+where Cf is the set of conclusion triples derived from
+rule f and F(·) is the score function defined in Equa-
+tion (1). Aiming to regularize this scoring function so
+that it approaches 0 or 1, and to distinguish between
+true and false conclusions, we use a quadratic function
+with a symmetry axis of 0.5. Therefore, the conclusion
+score is the smallest when it is close to 0 or 1. There-
+fore, we define the deterministic conclusion loss Ldc as
+follows:
+Ldc = − 1
+|Cf|
+�
+(hi,ri,ti)∈Cf
+∥Si − 0.5∥2 .
+(5)
+According to the definition of rule confidence in [10],
+the confidence of a rule f in a KB G is the proportion of
+true conclusions among the true conclusions and false
+conclusions.
+Therefore, we can define the confidence
+loss of a rule as follows:
+Lrc =
+������
+1
+|Cf|
+�
+(hi,ri,ti)∈Cf
+Si − cf
+������
+2
+,
+(6)
+where cf is the confidence of rule f. Therefore, the loss
+of the conclusions of all the rules Lac can be defined as
+follows:
+Lac =
+1
+|F|
+�
+f∈F
+(Ldc + Lrc),
+(7)
+where F is the set of all rules.
+To learn the KGE
+and rule conclusions at the same time, we minimize the
+global loss over a soft rule set F and a labeled triple set
+L = (xl, yl) (including negative and positive examples).
+The overall training objective of Iterlogic-E is:
+min
+θ
+1
+|L|
+�
+(xl,yl)∈L
+L(−f(xl) · yl)
++ 1
+|F|
+�
+f∈F
+(− 1
+|Cf|
+�
+(hi,ri,ti)∈Cf
+∥Si − 0.5∥2
++
+������
+1
+|Cf|
+�
+(hi,ri,ti)∈Cf
+Si − cf
+������
+2
+).
+(8)
+where the f(·) function denotes the score function and
+L(x) = log(1 + exp(x)) is the soft-plus function. In Al-
+gorithm 1, we detail the embedding learning procedure
+of our method. To avoid overfitting, we further impose
+l2 regularization on embedding Θ. Following [44, 18],
+we also imposed nonnegative constraints (NNE) on the
+
+8
+entity embedding to learn more effective features.
+Algorithm 1: Iterative learning algorithm of
+Iterlogic-E
+Input: KG triples T = {(ei, rk, ej)}, logical
+rules F = {(fp, cp)}, the number of
+iterative learning steps M.
+Output: Relation and entity embeddings Θ
+1 Randomly initialize relation and entity
+embeddings Θ(0);
+2 for n ← 1 to N do
+3
+C ← ∅;
+4
+if n/[N/M] == 0 then
+5
+Generate a set of conclusions
+C′ = {(e′
+i, r′
+k, e′
+j)} by rule grounding
+from T , F;
+6
+C = C ∪ C′;
+7
+end
+8
+Sample a mini-batch T b, Cb from T , C;
+9
+Generate a set of negative triples T b
+neg;
+10
+Lb ← ∅;
+11
+for each xl ∈ T b
+neg ∪ T b ← 1 to N do
+12
+yl = +1/ − 1;
+13
+Lb ← Lb ∪ (xl, yl);
+14
+end
+15
+Θ(n) ← Θ(n−1) −
+η(
+1
+|Lb|
+�
+(xl,yl)∈Lb ▽ΘL(−f(xl) · yl) +
+1
+|F|
+�
+f∈F(−
+1
+|Cf |
+�
+(hi,ri,ti)∈Cf ▽Θ ∥Si − 0.5∥2+
+▽Θ
+���
+1
+|Cf |
+�
+(hi,ri,ti)∈Cf Si − cf
+���
+2
+) ; !
+cf.
+Eq.
+(8)
+16
+Ct ← ∅;
+17
+for each xm ∈ C do
+18
+if σ(f(xm)) >= 0.99 then
+19
+Ct ∪ (xm);
+20
+end
+21
+end
+22
+T = T ∪ Ct;
+23 end
+24 return Θ(N)
+3.4
+Discussion
+3.4.1
+Complexity
+In the embedding learning step, we represent re-
+lations and entities as complex value vectors, follow-
+ing ComplEx.
+As a result, the space complexity is
+O(ned + nrd), where d is the embedding space’s di-
+mensionality. The number of relations is nr, and the
+number of entities is ne. Each iteration of the learning
+process has a time complexity of O(nld + ncd), where
+nl/nc is the number of new conclusions or the number of
+labeled triples in a mini-batch, as shown in Algorithm
+1. Iterlogic-E is similar to ComplEx in that its space
+and time complexity increase linearly with d. The num-
+ber of new conclusions in a minibatch is usually con-
+siderably lower than the number of initial triples; i.e.,
+nc ≪ nl. As a result, Iterlogic-E’s time complexity is
+very close to that of ComplEx, which needs only O(nld)
+per iteration. Because of the rule mining module’s great
+efficiency and practical constraints, such as the PCA
+confidence threshold not being lower than 0.5 and the
+length of rules not exceeding two, the rule grounding
+stage’s space and time complexity is trivial compared
+to that of the embedding learning stage. Therefore, we
+may disregard it when considering the space and time
+complexity of Iterlogic-E.
+3.4.2
+Connection to Related Works
+IterE [16] also uses iterative learning, which defines
+several types of rules with OWL2, but IterE does not
+change the process of embedding learning and is limited
+by rules that will yield many noisy conclusions. IterE
+uses a pruning strategy that utilizes traversal and ran-
+dom selection to obtain rules. Moreover, they only im-
+prove the prediction effect of sparse entities but not well
+on standard datasets. By contrast, Iterlogic-E uses the
+SOTA rule mining system [20] to mine high-confidence
+rules, and the quality of the rules obtained in this way
+is higher because it uses the KG to fully evaluate the
+reliability of the rules.
+SoLE [17] enhances KGE by
+jointly modeling the groundings of rules and facts and
+directly utilizes uncertain rules for forward chain rea-
+soning without eliminating incorrect grounding. More-
+over, SoLE uses t-norm based fuzzy logic [19] to model
+grounding, which will greatly increase the time com-
+plexity. The method we propose avoids the above men-
+
+MODEL KNOWLEDGE GRAPH AND RULES FOR REASONING
+9
+tioned problems without increasing the number of pa-
+rameters. SLRE [18] uses rule-based regularization that
+merely enforces relation to satisfying constraints intro-
+duced by soft rules. However, it does not use rules for
+reasoning and can not benefit from the interpretability
+and accuracy advantages. Moreover, SLRE has strict
+requirements on the form of the rules, while our method
+can utilize various forms of rules more simply and flex-
+ibly via the rule reasoning module.
+4
+Experiments and Results
+4.1
+Datasets
+Iterlogic-E is tested on two common datasets:
+FB15K and DB100K. The first is based on Freebase,
+which was released by Bordes et al.
+[22].
+The sec-
+ond was taken from DBpedia by Ding et al.
+[44],
+and it includes 99,604 entities and 470 relations. For
+model training, hyperparameter tuning and evaluation,
+we utilize fixed training, validation, and test sets on
+both datasets.
+With each training dataset, we obtain soft rules and
+examine rules with a length of no more than 2 to allow
+efficient extraction.
+These rules, together with their
+confidence levels, are automatically retrieved from each
+dataset’s training set using AMIE+ [20], and only those
+with confidence levels greater than 0.8 are used. Shorter
+rules are thought to more directly represent logical con-
+nections among relations. Therefore, we remove longer
+rules when all of their relations also exist in shorter
+ones.
+Table 1 summarizes the datasets’ comprehen-
+sive statistics, and Table 2 also includes several rule
+instances. We can observe from the statistics that the
+number of rules on both datasets is extremely minimal
+when compared to the number of triples.
+Table 1.
+Statistics of the datasets, where the columns repre-
+sent the numbers of entities, relations, training/validation/test
+triples, and soft rules
+Dataset # Ent # Rel
+# Train/Valid/Test # Rule
+FB15K 14,951
+1,345 483,142/50,000/59,071
+441
+DB100K 99,604
+470 597,572/50,000/50,000
+25
+Table 2.
+Examples of rules, with confidences, that were ex-
+tracted from FB15K (top) and DB100K (bottom)
+/location/contains(y, x)
+0.84
+⇒ /location/containedby(x,y)
+/production company/films(y, x)
+0.89
+⇒ /location/
+containedby(x,y)/hud county place/place(x, y)∧
+hud county place/county(y, z)
+1.0
+⇒ /hud county place/county(x, z)
+sisterNewspaper(x, y)∧sisterNewspaper(z, y)
+0.82
+⇒
+sisterNewspaper(x, z)
+distributingCompany(x, y)
+0.91
+⇒ distributingLabel(x, y)
+nationality(x, y)
+0.99
+⇒ stateOfOrigin(x, y)
+4.2
+Link Prediction
+Our method was evaluated on link prediction. The
+goal of this task was to restore a missing triple (ei, rk, ?)
+with the tail entity ej or (?, rk, ej) with the head entity
+ei.
+4.2.1
+Evaluation Protocol
+The standard protocol established by [22] is used
+for evaluation. The head entity ei is replaced with each
+entity for every test triple (ei, rk, ej), and the corrupted
+triple’s score is calculated. We record the rank of the
+right entity ei by ranking these scores in decreasing
+order. The mean reciprocal rank (MRR) and the per-
+centage of ranks no greater than N (H@N, N = 1, 3,
+10) are used to evaluate the ranking quality of all test
+triples.
+4.2.2
+Comparison Settings
+We compare the performance of our method to that
+of a number of previous KGE models, as shown in Table
+3. The translation-based model (row 1), the composi-
+
+10
+Table 3. Link prediction results on the test sets of FB15K and DB100K
+Method
+FB15K
+DB100K
+MRR
+HITS@1
+HITS@3
+HITS@10
+MRR
+HITS@1
+HITS@3
+HITS@10
+1
+TransE [22]
+0.380
+0.231
+0.472
+0.641
+0.111
+0.016
+0.164
+0.270
+2
+DistMult [25]
+0.654
+0.546
+0.733
+0.824
+0.233
+0.115
+0.301
+0.448
+3
+HolE [28]
+0.524
+0.402
+0.613
+0.739
+0.260
+0.182
+0.309
+0.4118
+4
+ComplEx [26]
+0.627
+0.550
+0.671
+0.766
+0.272
+0.218
+0.303
+0.362
+5
+ANALOGY [27]
+0.725
+0.646
+0.785
+0.854
+0.252
+0.143
+0.323
+0.427
+6
+ComplEx-NNE [44]
+0.727
+0.659
+0.772
+0.845
+0.298
+0.229
+0.330
+0.426
+7
+ComplEx-CAS [51]
+-
+-
+-
+0.866
+-
+-
+-
+-
+8
+RotatE [24]
+0.664
+0.551
+0.751
+0.841
+0.327
+0.200
+0.417
+0.526
+9
+HAKE[49]
+0.408
+0.312
+0.463
+0.579
+-
+-
+-
+-
+10
+CAKE[50]
+0.741
+0.646
+0.825
+0.896
+-
+-
+-
+-
+11
+R-GCN+ [29]
+0.696
+0.601
+0.760
+0.842
+-
+-
+-
+-
+12
+ConvE [30]
+0.745
+0.670
+0.801
+0.873
+-
+-
+13
+DPMPN[45]
+0.764
+0.726
+0.784
+0.834
+-
+-
+-
+-
+14
+BLP [46]
+0.242
+0.151
+0.269
+0.424
+-
+-
+-
+-
+15
+MLN [21]
+0.321
+0.210
+0.370
+0.550
+-
+-
+-
+-
+16
+PTransE [47]
+0.679
+0.565
+0.768
+0.855
+0.195
+0.063
+0.278
+0.416
+17
+KALE [40]
+0.518
+0.382
+0.606
+0.756
+0.249
+0.100
+0.346
+0.497
+18
+ComplExR [48]
+-
+-
+-
+-
+0.253
+0.167
+0.294
+0.420
+19
+TARE [41]
+0.781
+0.617
+0.728
+0.842
+-
+-
+-
+-
+20
+RUGE [42]
+0.768
+0.703
+0.815
+0.865
+0.246
+0.129
+0.325
+0.433
+21
+ComplEx-NNE+AER [44]
+0.801
+0.757
+0.829
+0.873
+0.311
+0.249
+0.339
+0.426
+22
+IterE [16]
+0.576
+0.443
+0.665
+0.818
+0.274
+0.215
+0.299
+0.386
+23
+pLogicNet [43]
+0.776
+0.706
+0.817
+0.885
+-
+-
+-
+-
+24
+SoLE [17]
+0.801
+0.764
+0.821
+0.867
+0.306
+0.248
+0.328
+0.418
+25
+SLRE [18]
+0.810
+0.774
+0.829
+0.871
+0.340
+0.261
+0.372
+0.490
+Iterlogic-E(ComplEx)
+0.814
+0.778
+0.835
+0.873
+0.374
+0.301
+0.409
+0.509
+Iterlogic-E(RotatE)
+0.837
+0.794
+0.868
+0.904
+0.387
+0.287
+0.449
+0.559
+Table 4. Link prediction results on the test sets of FB15K-sparse
+Method
+FB15K-sparse
+MRR
+HITS@1
+HITS@3
+HITS@10
+TransE
+0.398
+0.258
+0.486
+0.645
+DistMult
+0.600
+0.618
+0.651
+0.759
+ComplEx
+0.616
+0.540
+0.657
+0.761
+IterE
+0.628
+0.551
+0.673
+0.771
+SoLE
+0.668
+0.604
+0.699
+0.794
+Iterlogic-E(ComplEx)
+0.674
+0.611
+0.701
+0.800
+tional models utilizing basic mapping operations (rows
+2-10) and the neural network-based models (rows 11-13)
+are among the first batch of baselines that rely only on
+triples seen in the KGs. The second batch of baselines
+are rule-based methods (rows 14-15). The last batch of
+baselines further incorporates logical rules (rows 16-25).
+4.2.3
+Implementation Details
+On FB15K and DB100K, we compared Iterlogic-
+E against all of the baselines.
+We immediately ob-
+tained the results of a set of baselines on FB15K and
+
+MODEL KNOWLEDGE GRAPH AND RULES FOR REASONING
+11
+DB100K from SoLR, SLRE and CAKE. We reimple-
+mented ComplEx on the PyTorch framework based
+on thebased on the code code3 supplied by [24] since
+our approach was dependent on it. Then, depending
+on our implementation, we provided the ComplEx re-
+sult. Furthermore, the IterE result was tested on the
+sparse version of FB15K (FB15K-sparse) released by
+[16], which included only sparse entities with 18,544
+and 22,013 triples in the validation and test sets. There-
+fore, we reimplemented IterE on FB15K and DB100K
+based on the code and hyperparameters4 released by
+the author. As a result, we compared our approach to
+IterE and SoLE on the FB15K-sparse dataset. Both
+approaches use a logistic loss and optimize in the same
+way (SGD with AdaGrad).
+The other results of the
+baselines were obtained directly from prior literature.
+We tuned the embedding dimensionality d within {100,
+150, 200, 250, 300}, the number of negatives per posi-
+tive triple η within {2, 4, 6, 8, 10}, the initial learning
+rate γ within {10−4, 10−3, 10−2, 5 × 10−2, 10−1}, and
+the L2 regularization coefficient µ within {10−5, 3 ×
+10−5, 10−4, 10−3, 3−3, 10−2}.
+We further tuned the
+margin within {0.1, 0.2, 0.5, 1, 2, 5, 12, 18, 24} for
+the approaches that utilize the margin-based ranking
+loss. The best hyperparameters were selected to maxi-
+mize the MRR on the validation set. The best settings
+for Iterlogic-E were d = 300, γ = 10−3, η = 10, and
+µ = 3×10−5 on FB15K and d = 300, γ = 10−4, η = 10,
+and µ = 10−4 on DB100K.
+4.2.4
+Main Results
+The results of all compared methods on the test
+sets of FB15K, DB100K, and FB15K-sparse are shown
+in Tables 3 and 4. For each test triple, the mean re-
+ciprocal rank or H@N value with N = 1, 3, 10 is uti-
+lized as paired data.
+The experimental results show
+that 1) Iterlogic-E outperforms numerous strong base-
+lines in the vast majority of cases.
+This shows that
+Iterlogic-E can achieve very good accuracy. 2) Iterlogic-
+E significantly outperforms the basic models that use
+triples alone, and the improvement comes from the abil-
+ity to learn the conclusions obtained by soft rules. 3)
+Iterlogic-E also beats many baselines that incorporate
+logical rules. Specifically, Iterlogic-E performs better
+than SoLE and IterE under most metrics. This demon-
+strates the superiority of Iterlogic-E in reducing the
+noise of candidate conclusions. 4) IterE can only en-
+hance sparse entities, so the experimental results are
+much lower than those of other baseline models. How-
+ever, Iterlogic-E is also effective on FB15K-sparse. 5)
+On DB100K, the improvements over SLRE and SoLE
+are more significant than those on FB15K. The reason
+for this is probably that the groundings of the rules
+on DB100K contain more incorrect conclusions. Sim-
+ple rules between a pair of relations are adequate to
+capture these simple patterns on the FB15K dataset.
+6) The performance of Iterlogic-E(ComplEx) is worse
+than some baselines in HIT@10, and we consider that
+this limitation is mainly due to the shortcomings of the
+base model ComplEx [26]. Sun et al. [24] point out that
+ComplEx can not model the composition relation. We
+have experimented with replacing the base model with
+the RotatE model(Iterlogic-E(RotatE)), which is capa-
+ble of modeling four relation patterns. The experimen-
+tal results have been further improved, and our method
+consistently achieves the best results in all evaluation
+metrics.
+3https://github.com/DeepGraphLearning/KnowledgeGraphEmbedding
+4https://github.com/wencolani/IterE
+
+12
+4.2.5
+Ablation Study
+Table 5. Ablation study
+DB100K
+MRR HITS@1 HITS@3 HITS@10
+Iterlogic-E
+0.374
+0.301
+0.409
+0.509
+1
+w/o l2 on Θ
+0.323
+0.278
+0.342
+0.409
+2
+w/o NNE
+0.371
+0.295
+0.409
+0.509
+3
+1+2
+0.324
+0.279
+0.342
+0.410
+4
+w/o IL
+0.372
+0.300
+0.407
+0.505
+5
+w/o IL+Ldc
+0.347
+0.258
+0.395
+0.510
+6
+w/o IL+Lrc
+0.351
+0.257
+0.406
+0.515
+7
+w/o IL+Ldc+Lrc
+0.328
+0.279
+0.348
+0.422
+8
+Iterlogic-E*
+0.369
+0.298
+0.404
+0.502
+9
+ComplEx+AC+IL 0.285
+0.223
+0.312
+0.407
+10 ComplEx+WC+IL 0.295
+0.231
+0.327
+0.422
+ComplEx
+0.272
+0.218
+0.303
+0.362
+To explore the influence of different constraints and
+iterative learning, we perform an ablation study of
+Iterlogic-E on DB100K with 9 configurations in Ta-
+ble 5. The first and second variants, compared to the
+completed model Iterlogic-E, remove the non-negativity
+and l2-norm constraints (l2).
+The third setting is
+the combination setting. The fourth removes iterative
+learning (IL), which uses a rule to reason only once.
+The fifth, sixth, and seventh variants remove the addi-
+tional loss item based on the sixth variant. The eighth
+setting is another variant of the Iterlogic-E model based
+on the ninth setting, which is that after the ComplEx
+fitting, according to the scores of the conclusions, the
+top n (where n is the rounded product of the rule and
+its confidence) conclusions of each rule are selected to
+be added to the KG to continue training. The ninth
+setting (AC) is to add all conclusions inferred from the
+rules as positive examples, and the tenth setting (WC)
+is to use the rule confidence as soft labels of conclusions.
+As seen in Table 5, we can conclude the following:
+1) When removing NNE constraints, the performance
+of Iterlogic-E decreases slightly. Without l2-norm con-
+straints on entities, the performance of Iterlogic-E de-
+grades by 2.3% in H@1 and by 5.1% in MRR. One ex-
+planation may be that l2-norm constraints are sufficient
+to constrain embedding norms on DB100K. However,
+the performance will suffer dramatically if there are
+no l2-norm constraints.
+2) Removing iterative learn-
+ing decreases performance slightly.
+One reason may
+be that the number of rules on DB100K is relatively
+small, so the number of conclusions added through it-
+erative learning is relatively small.
+3) Removing the
+additional loss item of the conclusions decreases per-
+formance slightly. This illustrates that Iterlogic-E can
+filter out incorrect conclusions and makes the KG dense.
+Surprisingly, even if we directly use ComplEx to filter
+and learn the conclusions that can achieve such high
+performance, this method is not as flexible as Iterlogic-
+E. 4) Compared with the basic model, all variants have
+different degrees of improvement.
+This demonstrates
+the critical importance of logical rules in link predic-
+tion tasks.
+4.3
+Influence of the Number of Iterations
+To demonstrate how Iterlogic-E can enhance the
+embedding effect during the training process, we show
+the link prediction results on DB100K with different
+iterations. Figure 3 shows that as the number of train-
+ing iterations increases, the prediction results, including
+Hit@1, Hit@3, Hit@10, and MRR, will improve. From
+Fig. 3, we can infer the following: 1) Iterative learn-
+ing enhances embedding learning since the quality of
+embeddings improved with time.
+2) In the first two
+iterations, the embedding learning module was quickly
+fitted to the conclusions of the rules and the triples of
+the training set, and the prediction accuracy rapidly
+improved.
+3) After two iterations, as the number of
+new conclusions decreased, the results of the inference
+tended to be stable, and the true conclusions and the
+
+MODEL KNOWLEDGE GRAPH AND RULES FOR REASONING
+13
+initial KG triples were well preserved in the embedding.
+0
+2
+4
+6
+8
+10
+Number of iterations
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+HIT@1
+HIT@3
+HIT@10
+MRR
+Fig.3. Link prediction results in different iterations
+4.4
+Influence of Confidence Levels
+Additionally, we evaluate the effect of the rules’ con-
+fidence thresholds on FB15K. Since there are different
+rules and ComplEx does not merge rules, we refer only
+to their fixed results on FB15K. We set all hyperparam-
+eters to their optimum values and change the confidence
+threshold within [0.5, 1] in 0.05-step increments. Both
+SoLE and Iterlogic-E use rules with confidence levels
+greater than this threshold. Figure 4 displays the MRR
+and H@1 values obtained by Iterlogic-E and other base-
+lines on the FB15K test set. We make the following
+observations: 1) Iterlogic-E beats both ComplEx and
+IterE at varying confidence levels. This demonstrates
+that Iterlogic-E is sufficiently robust to deal with un-
+certain soft rules.
+4.5
+Case Study
+In Table 6,
+we present a case study with 4
+conclusions (true or false as predicted by Iterlogic-
+E), which are inferred with 2 rules during train-
+ing.
+Table 6 shows some conclusions derived from
+rule inference and the score change (the average of
+the head entity prediction score and the tail en-
+tity prediction score) of the conclusions.
+Using
+the first conclusion as an example, the true conclu-
+sion
+is
+(Albany Devils,/hockey roster position/posit
+on,
+Centerman),
+which
+is
+obtained
+by
+the
+rule
+”/sports team roster/position(x, y)
+0.808
+⇒ /
+hockey roster position/position(x, y)”.
+The Albany
+Devils are a professional ice hockey team in the
+American Hockey League
+5, and the centerman is
+the center in ice hockey.
+Therefore,
+this is in-
+deed a true conclusion.
+Compared to ComplEx,
+Iterlogic-E
+increased
+the
+score
+of
+this
+fact
+from
+5.48 to 9.40.
+Also, (San Diego State Aztecs football,
+/hockey roster position/position,Linebacker) can be in-
+ferred from the fact (San Diego State Aztecs football,
+/sports team roster/ team, Linebacker) by the same
+rule. However, the San Diego State Aztecs are a foot-
+ball team, not a hockey team, and this is an incorrect
+conclusion.
+Compared to ComplEx, Iterlogic-E de-
+creased the score of this fact from -4.65 to -8.35. This
+illustrates that Iterlogic-E can distinguish whether the
+conclusion is true and can improve the prediction per-
+formance.
+Furthermore, Iterlogic-E has good inter-
+pretability, and we can understand why the conclusion
+is inferred by it.
+5
+Conclusion and Future Work
+This paper proposes a novel framework that itera-
+tively learns logical rules and embeddings, which mod-
+els the conclusion labels as 0-1 variables. The proposed
+Iterlogic-E uses the confidences of rules and the context
+of the KG to eliminate the uncertainty of the conclu-
+sion in the stage of learning embeddings. Specifically,
+our method is based on iterative learning, which not
+only supplements conclusions but also filters incorrect
+conclusions, resulting in a good balance between effi-
+5https://en.wikipedia.org/wiki/Albany_Devils
+
+14
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Confidence threshold
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+0.85
+MRR
+ComplEx
+IterE
+SoLE
+Iterlogic-E
+(a)
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Confidence threshold
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+HIT@1
+ComplEx
+IterE
+SoLE
+Iterlogic-E
+(b)
+Fig.4. Results of MRR (a) and HITS@1 (b) achieved by Iterlogic-E with different confidence thresholds on FB15K
+Table 6. A case study with 4 conclusions (true or false as predicted by Iterlogic-E), which are reasoned on by 2 rules
+True
+False
+Conclusions
+(Albany Devils,/hockey roster position/
+position,Centerman)
+(San Diego State Aztecs football,/hockey
+roster position/position,Linebacker)
+Predicted by rules
+/sports team roster/position(x, y)
+0.808
+⇒ /hockey roster position/position(x, y)
+Score change
+5.4888 → 9.4039
+-4.6518 → -8.3547
+Conclusions
+(Chris Nurse,/football roster position/
+team,Stevenage F.C.)
+(Brett Favre,/football roster position/
+team,Green Bay Packers)
+Predicted by rules
+/sports team roster/team(x, y)
+0.847
+⇒ /football roster position/team(x, y)
+Score change
+5.8123 → 9.5991
+-3.0131 → -11.3952
+ciency and scalability. The evaluation on benchmark
+KGs demonstrates that the method can learn correct
+conclusions and improve against a variety of strong
+baselines. In the future, we would like to explore how
+to use embeddings to learn better rules and rule con-
+fidences than AMIE+. Additionally, we will continu-
+ously explore more advanced models to integrate rules
+and KGEs for knowledge reasoning.
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diff --git a/ldE0T4oBgHgl3EQf7wLZ/content/tmp_files/load_file.txt b/ldE0T4oBgHgl3EQf7wLZ/content/tmp_files/load_file.txt
new file mode 100644
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+++ b/ldE0T4oBgHgl3EQf7wLZ/content/tmp_files/load_file.txt
@@ -0,0 +1,947 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf,len=946
+page_content='Knowledge Reasoning via Jointly Modeling Knowledge Graphs and  Soft Rules  Yinyu Lan, Shizhu He, Kang Liu, Jun Zhao  Institute of Automation, Chinese Academy of Sciences, Beijing 100190, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Abstract  Knowledge graphs (KGs) play a crucial role in many applications, such as question answering, but incom-  pleteness is an urgent issue for their broad application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Much research in knowledge graph completion (KGC) has been  performed to resolve this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The methods of KGC can be classified into two major categories: rule-based reasoning and  embedding-based reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The former has high accuracy and good interpretability, but a major challenge is to obtain  effective rules on large-scale KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The latter has good efficiency and scalability, but it relies heavily on data richness and  cannot fully use domain knowledge in the form of logical rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' We propose a novel method that injects rules and learns  representations iteratively to take full advantage of rules and embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Specifically, we model the conclusions of rule  groundings as 0-1 variables and use a rule confidence regularizer to remove the uncertainty of the conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The proposed  approach has the following advantages: 1) It combines the benefits of both rules and knowledge graph embeddings (KGEs)  and achieves a good balance between efficiency and scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 2) It uses an iterative method to continuously improve KGEs  and remove incorrect rule conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Evaluations on two public datasets show that our method outperforms the current  state-of-the-art methods, improving performance by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='7% and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='3% in mean reciprocal rank (MRR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Keywords  Distributed representation, Knowledge graph, Link prediction, Logical rule  1  Introduction  A knowledge graph (KG) organizes knowledge as  a set of interlinked triples, and a triple ((head entity,  relation, tail entity), simply represented as (h, r, t)) in-  dicates the fact that two entities have a certain relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Rich structured and formalized knowledge has become  a valuable resource to support downstream tasks, for  example, question answering [1, 2] and recommender  systems [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Although KGs such as DBpedia [5], Freebase [6] and  NELL [7] contain large amounts of entities, relations,  and triples, they are far from complete, which is an  urgent issue for their broad application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  To address  this, the task of knowledge graph completion (KGC)  has been proposed and has attracted growing atten-  tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' it utilizes knowledge reasoning techniques to per-  form automatic discovery of new facts based on existing  facts in a KG [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  At present, the methods of KGC can be classified  into two major categories: 1) One type of method uses  explicit reasoning rules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' it obtains the reasoning rules  through inductive learning and then deductively infers  new facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 2) Another method is based on representa-  tion learning instead of directly modeling rules, aiming  to learn a distributed embedding for entities and rela-  tions and perform generalization in numerical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Rule-based reasoning is accurate and can provide  interpretability for the inference results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Domain ex-  perts can handcraft these rules [9] or can mine them  from the KG with an induction algorithm such as  AMIE [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Traditional methods such as expert sys-  tems [11, 12] use hard logical rules to make predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  For example, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 1, given the logical rule  (x, born in, y)∧(y, city of, z) ⇒ (x, nationality, z) and  the two facts that (Chicago, city of, USA) and (Mary,  born in, Chicago), we can infer the fact (Mary, nation-  ality, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' A large number of new facts (conclusions)  can be derived based on forward chaining inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Regular Paper  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='02781v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='AI]  7 Jan 2023  2  However, for large-scale KGs, sufficient and effective  reasoning rules are difficult and expensive to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Moreover, in many cases, the logical rules may be im-  perfect or even self-contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Therefore, it is es-  sential to model the uncertainty of (soft) logical rules  effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  The methods of determining KGEs learn to embed  entities and relations into a continuous low-dimensional  space [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' These embeddings retain the semantic  meaning of entities and relations, which can be used to  predict missing triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In addition, they can be effec-  tively trained using stochastic gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' How-  ever, this kind of method cannot fully use logical rules,  which compactly encode domain knowledge and are  helpful in various applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Good embedding relies  heavily on data richness, so these methods have diffi-  culty learning useful representations for sparse entities  [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  In fact, both rule-based methods and embedding-  based methods have advantages and disadvantages in  the KGC task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Logical rules are accurate and inter-  pretable, and embedding is flexible and computation-  ally efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' To achieve more precise knowledge com-  pletion, recently, there has also been research on com-  bining the advantages of logical rules and KGEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Mixed  techniques can infer missing triples effectively by ex-  ploiting and modeling uncertain logical rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Some  existing methods have aimed to iteratively learn KGEs  and rules [16], and some other methods also utilize soft  rules or groundings of rules to regularize the learning  of KGEs [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Knowledge Graph  Rules  Weights  ∀𝑥, 𝑦 ∶ (𝑥, child_of, 𝑦) ⇒ (𝑦, parent_of, 𝑥)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='9  ∀𝑥, 𝑦, 𝑧 ∶ (𝑥, born_in, 𝑦) ⋀ (𝑦, city_of, 𝑧)  ⇒ (𝑥, nationality, 𝑧)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='8  ∀𝑥, 𝑦, 𝑧 ∶ (𝑥, child_of,𝑧) ⋀ (𝑦, child_of,𝑧)  ⇒ (𝑥, sister_of,𝑧)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5  …  …  Facts  Weights  (Mary, parent_of, Lisa)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='9  (Mary, parent_of, Mike)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='9  (Mary, nationality, USA)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='8  (Nacy, nationality, USA)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='8  (Lisa, sister_of, Nancy)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5  (Mike, sister_of, Nancy)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5  …  …  Facts  Weights  (Mary, parent_of, Lisa)  1  (Mary, parent_of, Mike)  1  (Mary, nationality, USA)  1  (Nacy, nationality, USA)  0  (Lisa, sister_of, Nancy)  1  (Mike, sister_of, Nancy)  0  …  Chicago  Boston  Mary  Lisa  Nancy  Mike  USA  born_in  nationality  child_of  city_of  Previous Works  This Work  …  …  KGE  Uncertain Conclusions  Deterministic Conclusions  Uncertain Rules  induction  deduction  completion  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' We propose a novel iterative knowledge reasoning frame-  work by fusing logical rules into a KGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Previous methods asso-  ciate each conclusion with a weight derived from the correspond-  ing rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In contrast, our method can infer which conclusion is  true via jointly modeling the deterministic KG and uncertain  soft rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  The integration of logical rules and knowledge graph  embeddings can achieve more efficient and accurate  knowledge completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Current methods model uncer-  tain rules and add soft labels to conclusions by t-norm-  based fuzzy logic [19];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' they further utilize the conclu-  sions to perform forward reasoning [17] or to enhance  the KGE [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  However, in most KGs, the facts are  deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Therefore, we believe that rules are un-  certain but conclusions are deterministic in knowledge  reasoning, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' each fact is only abso-  lutely true or false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Previous methods associate each  conclusion with a weight derived from the correspond-  ing rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  In contrast, we propose inferring that all  conclusions are true (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=', (Mary, nationality, USA))  or not (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=', (Mike, sister of, Nancy)) (the other fact,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=', (Mike, gender, male), indicates that Mike is not  Nancy’s sister) by jointly modeling the deterministic  KG and soft rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Specifically, we first mine soft rules from the knowl-  edge graph and then infer conclusions as candidate  facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Second, the KG, conclusions, and weighted rules  are also used as resources to learn embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Third,  through the definition of deterministic conclusion loss,  the conclusion labels are modeled as 0-1 variables, and  MODEL KNOWLEDGE GRAPH AND RULES FOR REASONING  3  the confidence loss of a rule is also used to constrain  the conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Finally, the embedding learning stage  removes the noise in the candidate conclusions, and  then the proper conclusions are added back to the orig-  inal KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The above steps are performed iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  We empirically evaluate the proposed method on pub-  lic datasets from two real large-scale KGs: DBpedia  and Freebase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The experimental results show that our  method Iterlogic-E (Iterative using logic rule for rea-  soning and learning Embedding) achieves state-of-the-  art results on multiple evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Iterlogic-E  also achieves improvements of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='7%/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='3% in mean re-  ciprocal rank (MRR) and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='0%/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='0% in HITS@1 com-  pared to the state-of-the-art model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  In summary, our main contributions are as follows:  We propose a novel KGC method, Iterlogic-E,  which jointly models logical rules and KGs in the  framework of a KGE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Iterlogic-E combines the ad-  vantages of both rules and embeddings in knowl-  edge reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Iterlogic-E models the conclusion  labels as 0-1 variables and uses a confidence reg-  ularizer to eliminate the uncertain conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  We propose a novel iterative learning paradigm  that achieves a good balance between efficiency  and scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Iterlogic-E not only makes the  KG denser but can also filter incorrect conclu-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Compared with traditional reasoning methods,  Iterlogic-E is more interpretable in determining  conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' It not only knows why the conclu-  sion holds but also knows which is true and which  is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  We empirically evaluate Iterlogic-E with the  task of link prediction on multiple benchmark  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The experimental results indicate that  Iterlogic-E can achieve state-of-the-art results on  multiple evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The qualitative anal-  ysis proves that Iterlogic-E is more robust for  rules with different confidence levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  2  Related Work  Knowledge reasoning aims to infer certain entities  over KGs as the answers to a given query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' A query in  KGC is a head entity h (or a tail entity t) and a rela-  tion r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Given (h, r, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=') (or (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=', r, t)), KGC aims to find  the right tail entity t (or head entity h) in the KG that  satisfies the triple (h, r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Next, we review the three  most relevant classes of KGC methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='1  Rule-Based Reasoning  Logical rules can encode human knowledge com-  pactly, and early knowledge reasoning was primarily  based on first-order logical rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Existing rule-based  reasoning methods have primarily utilized search-based  inductive logic programming (ILP) methods, usually  searching and pruning rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Based on the partial com-  pleteness assumption, AMIE [10] introduces a revised  confidence metric, which is well suited for modeling  KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  By query rewriting and pruning, AMIE+ [20]  is optimized to expand to larger KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Additionally,  AMIE+ improves the precision of the forecasts by using  joint reasoning and type information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In this paper, we  employ AMIE+1 to mine horn rules from a KG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Rule-  based reasoning methods can be combined with multi-  ple probability graph models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' A Markov logic network  (MLN) [21] is a typical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Based on preprovided  rules, it builds a probabilistic graph model and then  learns the weights of rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' However, due to the com-  plicated graph structure among triples, the reasoning in  an MLN is time-consuming and difficult, and the incom-  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='com/lajus/amie  4  pleteness of KGs also impacts the inference results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In  contrast, Iterlogic-E uses rules to enhance KGEs with  more effective inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2  Embedding-Based Reasoning  Recently, embedding-based methods have attracted  much attention;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' they aim to learn distributed em-  beddings for entities and relations in KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Gener-  ally, current KGE methods can be divided into three  classes:  1) translation-based models that learn em-  beddings by translating one entity into another en-  tity through a specific relation [22, 23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 2) composi-  tional models that use simple mathematical operations  to model facts, including linear mapping [24], bilinear  mapping [25, 26, 27], and circular correlation [28];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 3)  neural network-based models that utilize a multilayer  neural structure to learn embeddings and estimate the  plausibility of triples with nonlinear features, for exam-  ple, R-GCN [29], ConvE [30] and and so on [31, 32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  The above methods learn representations based only on  the triples existing in KGs, and the sparsity of data lim-  its them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' To solve this problem and learn semantic-rich  representations, recent works further attempted to in-  corporate information beyond triples, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=', contextual  information [34], entity type information [35, 36], on-  tological information [37], taxonomic information [38],  textual descriptions [39] and hierarchical information  [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  In contrast, the proposed Iterlogic-E uses em-  beddings to remove incorrect conclusions obtained by  rules, which combines the advantages of rules and em-  beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='3  Hybrid Reasoning  Both rule-based and embedding-based methods  have advantages and disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Recent works have  integrated these two kinds of reasoning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Guo  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' [40] attempted to learn a KGE from rule ground-  ings and triples together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' [41] used asym-  metric and transitive information to approximately or-  der relations by maximizing the margin between neg-  ative and positive logical rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' [17] and  Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' [42] obtained KGEs with supervision from  soft rules, proving the effectiveness of logical rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Qu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' [43] used an MLN to model logical rules and in-  ferred new triples to enhance KGEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' [18]  enhanced KGEs by injecting grounding rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Niu et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  [50] enhanced KGEs by extracting commonsense  from factual triples with entity concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In addition,  some previous methods that enhance embeddings by it-  erative learning were studied in early works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Zhang et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' [16] aimed to improve a sparse entity representation  through iterative learning and update the confidence  of rules through embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In contrast, Iterlogic-E  models the conclusion labels as 0-1 variables and uses  confidence regularization loss to eliminate the uncertain  conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Such labels are easier to train on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  3  The Proposed Method  This  section  introduces  our  proposed  method  Iterlogic-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' We first give an overview of our method,  including the entire iterative learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Then,  we detail the two parts of Iterlogic-E: rule mining and  reasoning and embedding learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Last, we discuss  the space and time complexity of Iterlogic-E and dis-  cuss connections to related works [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='1  Overview  Given a KG G = {E, R, T }, T = {(h, r, t)}, r ∈ R  is a relation and h, t ∈ E are entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' As discussed in  Section 1, on the one hand, embedding learning meth-  ods do not make full use of logical rules and suffer from  data sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' On the other hand, precise rules are dif-  ficult to obtain efficiently and cannot cover all facts in  KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Our goal is to improve the embedding quality by  MODEL KNOWLEDGE GRAPH AND RULES FOR REASONING  5  Chicago  Boston  Mary  Lisa  Nancy  Mike  USA  born_in  nationality  child_of  city_of  Rules  Weights  ∀𝑥, 𝑦 : (𝑥, child_of, 𝑦) ⇒ (𝑦, parent_of, 𝑥)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='9  ∀𝑥, 𝑦, 𝑧 : (𝑥, born_in, 𝑦) ⋀ (𝑦, city_of, 𝑧) ⇒ (𝑥, nationality, 𝑧)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='8  ∀𝑥, 𝑦, 𝑧 : (𝑥, child_of,𝑧) ⋀ (𝑦, child_of,𝑧) ⇒ (𝑥, sister_of,𝑧)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5  …  …  Conclusions  Adjusted weights  (Mary, parent_of, Lisa)  1  (Mary, parent_of, Mike)  1  (Mary, nationality, USA)  1  (Nacy, nationality, USA)  0  (Lisa, sister_of, Nancy)  1  (Mike, sister_of, Nancy)  0  …  …  Groundings of rules  Rule weights  (Mary, born_in,Chicago) ⋀ (Chicago, city_of, USA) ⇒  (Mary, nationality, USA)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='9  (Nacy, born_in,Bostan) ⋀ (Bostan, city_of, USA) ⇒  (Nacy, nationality, USA)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='9  (Lisa, child_of,Mary) ⇒ (Mary, parent_of, Lisa)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='8  (Mike, child_of,Mary) ⇒ (Mary, parent_of, Mike)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='8  (Lisa, child_of,Mary) ⋀ (Nancy, child_of, Mary)  ⇒  (Lisa, sister_of, Nancy)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5  (Lisa, child_of,Mary) ⋀ (Mike, child_of, Mary)  ⇒  (Mike, sister_of, Nancy)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5  …  …  …  …  joint embedding learning  (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2)  triple  injecting (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='1)  rule mining  (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='1)  rule reasoning  (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2)  conclusion filtering  (§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='1)  Knowledge Graph  Knowledge Graph Embedding  Groundings of Rules  Soft Rules  Reasoning Conclusions  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The framework details Iterlogic-E with two iterative stages: (i) rule mining and reasoning and (ii) embedding learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Stage  (i) generates rules and grounding rules to obtain new conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Stage (ii) jointly models the conclusions of grounding rules and the  KG in learning embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' After embedding learning, the conclusions are injected into the KG, and then the rule reasoning module  is executed to start the next round of iterative training  explicitly modeling the reasoned conclusions of logical  rules, removing incorrect conclusions, and improving  the confidence of the rules at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Figure 2  shows the overview of Iterlogic-E given a toy knowledge  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Iterlogic-E is a general framework that can fuse  different KGE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Iterlogic-E has two iterative steps: (i) rule mining  and reasoning and (ii) embedding learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In the rule  mining and reasoning step, there are two modules: rule  mining and rule reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The mining configuration,  such as the maximum length of rules and the confi-  dence threshold of rules, and KG triples are input to  the rule mining module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Then, it automatically ob-  tains the soft rules from these inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The KG triples  and the extracted soft rules are input into the rule rea-  soning module to infer new triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' After that, the new  triples are appended to the embedding learning step as  candidate conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In the embedding learning step,  relations are modeled as a linear mapping operation,  and triple plausibility is represented as the correlation  between the head and tail entities after the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Finally, the incorrect conclusions are filtered out by la-  beling the conclusions with their scores2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The right con-  clusion will be added back to the original KG triples,  and then the rule reasoning module is performed to  start the next cycle of iterative training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2  Rule Mining and Reasoning  The first step is composed of the rule mining mod-  ule and the rule reasoning module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' We introduce these  two modules in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='1  Rule Mining  We extract soft rules from the KG using the state-of-  the-art rule mining method AMIE+ [20] in this module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  2In the experiments, we choose the conclusion with a normalized score of more than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='99 as the true conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  6  AMIE+ applies principal component analysis (PCA)  confidence to estimate the reliability of a rule since its  partial completeness assumption is more suited to real-  world KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Additionally, AMIE+ defines a variety of  restriction types to help extract applicable rules, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=',  the maximum length of the rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' After the rule mining  module receives the KG triples and the mining config-  uration, it executes the AMIE+ algorithm and outputs  soft rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Although rules can be re-mined in each it-  eration, we only run the rule mining module once for  efficiency reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2  Rule Reasoning  The logical rule set is denoted as F = {(f, c)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' f  is in the form of ∀x, y, z : (x, rp1, y) ∧ (y, rp2, z)  c⇒  (x, rc, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  x, y and z represent variables of different  entities, and rp1, rp2 and rrc represent different rela-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  The left side of the symbol ⇒ is the premise  of the rule, which is composed of several connected  atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The right side is only a single atom, which is  the rule’s conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The horn rules are closed [25],  where continuous relations share the intermediate en-  tity and the first and last entities of the premise ap-  pear as the head and tail entities of the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Such rules can provide interpretive insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' A rule’s  length is equal to the number of atoms in the premise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  For example, ∀x, y, z : (x, bornin, y) ∧ (y, city, z)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='8  ⇒  (x, nationality, z) is a length-2 rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' This rule reflects  the reality that, most likely, a person’s nationality is  the country in which he or she was born.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The rule f  has a confidence level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The higher the confidence  of the rule, the more likely it is to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  The reasoning procedure consists of instantiating  the rule’s premise and obtaining a large number of fresh  conclusion triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' One of the most common approaches  is forward chaining, also known as the match-select-  act cycle, which works in three-phase cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Forward  chaining matches the currently existing facts in the KG  with all known rule premises in one cycle to determine  the rules that can be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Finally, the selected  rule’s conclusions are derived, and if the conclusions  are not already in the KG, they are added as new facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  This cycle should be repeated until no new conclusions  emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' However, if soft rules are used, forward chain  reasoning will lead to incorrect conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Therefore,  we run one reasoning cycle in every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='3  Embedding Learning  In this section, we present a joint embedding learn-  ing approach that allows the embedding model to learn  from KG triples, conclusion triples, and soft rule con-  fidence all at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  First, we will examine  a basic KGE model, and then we will describe how to  incorporate soft rule conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Finally, we detail the  overall training goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='1  A Basic KGE Model  Different KGE models have different score func-  tions that aim to obtain a suitable function to map  the triple score to a continuous true value in [0, 1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=',  φ : E × R × E → (0, 1), which indicates the probabil-  ity that the triple holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' We follow [17, 18] and choose  ComplEx [26] as a basic KGE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' It is important  to note that our proposed framework can be combined  with an arbitrary KGE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Theoretically, using a  better base model can continue improving performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Therefore, We also experiment with RotatE as a base  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Below we take ComplEx as an example to in-  troduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' ComplEx assumes that the entity and relation  embeddings exist in a complex space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=', e ∈ Cd and  r ∈ Cd, where d is the dimensionality of the complex  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Using plurals to represent entities and relations  can better model antisymmetric and symmetric rela-  tions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=', kinship and marriage) [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Through a mul-  MODEL KNOWLEDGE GRAPH AND RULES FOR REASONING  7  tilinear dot product, ComplEx scores every triple:  F(h, r, t) = Re(hTdiag(r)¯t) = Re(�  i[h]i[r]i[t]i),  (1)  where the Re(·) function takes the real part of a  complex value and the diag(·) function constructs a di-  agonal matrix from r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' ¯t is the conjugate of t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' and [·]i is  the i-th entry of a vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' To predict the probability,  ComplEx further uses the sigmoid function for normal-  ization:  φ(h, r, t) = σ(F(h, r, t)) = σ(Re(hTdiag(r)¯t)),  (2)  where σ(·) is the sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' By minimizing the  logistic loss function, ComplEx learns the relation and  entity embeddings:  �  (h,r,t)∈T ∪T ′  log(1 + exp(−yhrt · f(h, r, t))),  (3)  where T ′ is a set of sampled negative examples and yhrt  is the label of a positive or negative triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2  Joint Modeling KG and Conclusions of Soft  Rules  To model the conclusion label as a 0-1 variable,  based on the current KGE model’s scoring function,  we follow ComplEx and use the function f(·) as the  scoring function for conclusion triples:  Si = σ(F(hi, ri, ti)), (hi, ri, ti) ∈ Cf,  (4)  where Cf is the set of conclusion triples derived from  rule f and F(·) is the score function defined in Equa-  tion (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Aiming to regularize this scoring function so  that it approaches 0 or 1, and to distinguish between  true and false conclusions, we use a quadratic function  with a symmetry axis of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Therefore, the conclusion  score is the smallest when it is close to 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' There-  fore, we define the deterministic conclusion loss Ldc as  follows:  Ldc = − 1  |Cf|  �  (hi,ri,ti)∈Cf  ∥Si − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  (5)  According to the definition of rule confidence in [10],  the confidence of a rule f in a KB G is the proportion of  true conclusions among the true conclusions and false  conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Therefore, we can define the confidence  loss of a rule as follows:  Lrc =  ������  1  |Cf|  �  (hi,ri,ti)∈Cf  Si − cf  ������  2  ,  (6)  where cf is the confidence of rule f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Therefore, the loss  of the conclusions of all the rules Lac can be defined as  follows:  Lac =  1  |F|  �  f∈F  (Ldc + Lrc),  (7)  where F is the set of all rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  To learn the KGE  and rule conclusions at the same time, we minimize the  global loss over a soft rule set F and a labeled triple set  L = (xl, yl) (including negative and positive examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  The overall training objective of Iterlogic-E is:  min  θ  1  |L|  �  (xl,yl)∈L  L(−f(xl) · yl)  + 1  |F|  �  f∈F  (− 1  |Cf|  �  (hi,ri,ti)∈Cf  ∥Si − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5∥2  +  ������  1  |Cf|  �  (hi,ri,ti)∈Cf  Si − cf  ������  2  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  (8)  where the f(·) function denotes the score function and  L(x) = log(1 + exp(x)) is the soft-plus function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In Al-  gorithm 1, we detail the embedding learning procedure  of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' To avoid overfitting, we further impose  l2 regularization on embedding Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Following [44, 18],  we also imposed nonnegative constraints (NNE) on the  8  entity embedding to learn more effective features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Algorithm 1: Iterative learning algorithm of  Iterlogic-E  Input: KG triples T = {(ei, rk, ej)}, logical  rules F = {(fp, cp)}, the number of  iterative learning steps M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Output: Relation and entity embeddings Θ  1 Randomly initialize relation and entity  embeddings Θ(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  2 for n ← 1 to N do  3  C ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  4  if n/[N/M] == 0 then  5  Generate a set of conclusions  C′ = {(e′  i, r′  k, e′  j)} by rule grounding  from T , F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  6  C = C ∪ C′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  7  end  8  Sample a mini-batch T b, Cb from T , C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  9  Generate a set of negative triples T b  neg;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  10  Lb ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  11  for each xl ∈ T b  neg ∪ T b ← 1 to N do  12  yl = +1/ − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  13  Lb ← Lb ∪ (xl, yl);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  14  end  15  Θ(n) ← Θ(n−1) −  η(  1  |Lb|  �  (xl,yl)∈Lb ▽ΘL(−f(xl) · yl) +  1  |F|  �  f∈F(−  1  |Cf |  �  (hi,ri,ti)∈Cf ▽Θ ∥Si − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5∥2+  ▽Θ  ���  1  |Cf |  �  (hi,ri,ti)∈Cf Si − cf  ���  2  ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  (8)  16  Ct ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  17  for each xm ∈ C do  18  if σ(f(xm)) >= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='99 then  19  Ct ∪ (xm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  20  end  21  end  22  T = T ∪ Ct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  23 end  24 return Θ(N)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='4  Discussion  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='1  Complexity  In the embedding learning step, we represent re-  lations and entities as complex value vectors, follow-  ing ComplEx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  As a result, the space complexity is  O(ned + nrd), where d is the embedding space’s di-  mensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The number of relations is nr, and the  number of entities is ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Each iteration of the learning  process has a time complexity of O(nld + ncd), where  nl/nc is the number of new conclusions or the number of  labeled triples in a mini-batch, as shown in Algorithm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Iterlogic-E is similar to ComplEx in that its space  and time complexity increase linearly with d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The num-  ber of new conclusions in a minibatch is usually con-  siderably lower than the number of initial triples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=',  nc ≪ nl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' As a result, Iterlogic-E’s time complexity is  very close to that of ComplEx, which needs only O(nld)  per iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Because of the rule mining module’s great  efficiency and practical constraints, such as the PCA  confidence threshold not being lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5 and the  length of rules not exceeding two, the rule grounding  stage’s space and time complexity is trivial compared  to that of the embedding learning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Therefore, we  may disregard it when considering the space and time  complexity of Iterlogic-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2  Connection to Related Works  IterE [16] also uses iterative learning, which defines  several types of rules with OWL2, but IterE does not  change the process of embedding learning and is limited  by rules that will yield many noisy conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' IterE  uses a pruning strategy that utilizes traversal and ran-  dom selection to obtain rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Moreover, they only im-  prove the prediction effect of sparse entities but not well  on standard datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' By contrast, Iterlogic-E uses the  SOTA rule mining system [20] to mine high-confidence  rules, and the quality of the rules obtained in this way  is higher because it uses the KG to fully evaluate the  reliability of the rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  SoLE [17] enhances KGE by  jointly modeling the groundings of rules and facts and  directly utilizes uncertain rules for forward chain rea-  soning without eliminating incorrect grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' More-  over, SoLE uses t-norm based fuzzy logic [19] to model  grounding, which will greatly increase the time com-  plexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The method we propose avoids the above men-  MODEL KNOWLEDGE GRAPH AND RULES FOR REASONING  9  tioned problems without increasing the number of pa-  rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' SLRE [18] uses rule-based regularization that  merely enforces relation to satisfying constraints intro-  duced by soft rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' However, it does not use rules for  reasoning and can not benefit from the interpretability  and accuracy advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Moreover, SLRE has strict  requirements on the form of the rules, while our method  can utilize various forms of rules more simply and flex-  ibly via the rule reasoning module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  4  Experiments and Results  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='1  Datasets  Iterlogic-E is tested on two common datasets:  FB15K and DB100K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The first is based on Freebase,  which was released by Bordes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  The sec-  ond was taken from DBpedia by Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  [44],  and it includes 99,604 entities and 470 relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' For  model training, hyperparameter tuning and evaluation,  we utilize fixed training, validation, and test sets on  both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  With each training dataset, we obtain soft rules and  examine rules with a length of no more than 2 to allow  efficient extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  These rules, together with their  confidence levels, are automatically retrieved from each  dataset’s training set using AMIE+ [20], and only those  with confidence levels greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='8 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Shorter  rules are thought to more directly represent logical con-  nections among relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Therefore, we remove longer  rules when all of their relations also exist in shorter  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Table 1 summarizes the datasets’ comprehen-  sive statistics, and Table 2 also includes several rule  instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' We can observe from the statistics that the  number of rules on both datasets is extremely minimal  when compared to the number of triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Statistics of the datasets, where the columns repre-  sent the numbers of entities, relations, training/validation/test  triples, and soft rules  Dataset # Ent # Rel  # Train/Valid/Test # Rule  FB15K 14,951  1,345 483,142/50,000/59,071  441  DB100K 99,604  470 597,572/50,000/50,000  25  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Examples of rules, with confidences, that were ex-  tracted from FB15K (top) and DB100K (bottom)  /location/contains(y, x)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='84  ⇒ /location/containedby(x,y)  /production company/films(y, x)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='89  ⇒ /location/  containedby(x,y)/hud county place/place(x, y)∧  hud county place/county(y, z)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='0  ⇒ /hud county place/county(x, z)  sisterNewspaper(x, y)∧sisterNewspaper(z, y)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='82  ⇒  sisterNewspaper(x, z)  distributingCompany(x, y)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='91  ⇒ distributingLabel(x, y)  nationality(x, y)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='99  ⇒ stateOfOrigin(x, y)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2  Link Prediction  Our method was evaluated on link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The  goal of this task was to restore a missing triple (ei, rk, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=')  with the tail entity ej or (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=', rk, ej) with the head entity  ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='1  Evaluation Protocol  The standard protocol established by [22] is used  for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The head entity ei is replaced with each  entity for every test triple (ei, rk, ej), and the corrupted  triple’s score is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' We record the rank of the  right entity ei by ranking these scores in decreasing  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The mean reciprocal rank (MRR) and the per-  centage of ranks no greater than N (H@N, N = 1, 3,  10) are used to evaluate the ranking quality of all test  triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2  Comparison Settings  We compare the performance of our method to that  of a number of previous KGE models, as shown in Table  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The translation-based model (row 1), the composi-  10  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Link prediction results on the test sets of FB15K and DB100K  Method  FB15K  DB100K  MRR  HITS@1  HITS@3  HITS@10  MRR  HITS@1  HITS@3  HITS@10  1  TransE [22]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
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+page_content='559  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Link prediction results on the test sets of FB15K-sparse  Method  FB15K-sparse  MRR  HITS@1  HITS@3  HITS@10  TransE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='398  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='258  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='486  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='645  DistMult  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
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+page_content='761  IterE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='628  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
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+page_content='699  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='794  Iterlogic-E(ComplEx)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='674  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='611  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
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+page_content='800  tional models utilizing basic mapping operations (rows  2-10) and the neural network-based models (rows 11-13)  are among the first batch of baselines that rely only on  triples seen in the KGs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The second batch of baselines  are rule-based methods (rows 14-15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The last batch of  baselines further incorporates logical rules (rows 16-25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='3  Implementation Details  On FB15K and DB100K, we compared Iterlogic-  E against all of the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  We immediately ob-  tained the results of a set of baselines on FB15K and  MODEL KNOWLEDGE GRAPH AND RULES FOR REASONING  11  DB100K from SoLR, SLRE and CAKE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' We reimple-  mented ComplEx on the PyTorch framework based  on thebased on the code code3 supplied by [24] since  our approach was dependent on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Then, depending  on our implementation, we provided the ComplEx re-  sult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Furthermore, the IterE result was tested on the  sparse version of FB15K (FB15K-sparse) released by  [16], which included only sparse entities with 18,544  and 22,013 triples in the validation and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' There-  fore, we reimplemented IterE on FB15K and DB100K  based on the code and hyperparameters4 released by  the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' As a result, we compared our approach to  IterE and SoLE on the FB15K-sparse dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Both  approaches use a logistic loss and optimize in the same  way (SGD with AdaGrad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  The other results of the  baselines were obtained directly from prior literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  We tuned the embedding dimensionality d within {100,  150, 200, 250, 300}, the number of negatives per posi-  tive triple η within {2, 4, 6, 8, 10}, the initial learning  rate γ within {10−4, 10−3, 10−2, 5 × 10−2, 10−1}, and  the L2 regularization coefficient µ within {10−5, 3 ×  10−5, 10−4, 10−3, 3−3, 10−2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  We further tuned the  margin within {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5, 1, 2, 5, 12, 18, 24} for  the approaches that utilize the margin-based ranking  loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The best hyperparameters were selected to maxi-  mize the MRR on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The best settings  for Iterlogic-E were d = 300, γ = 10−3, η = 10, and  µ = 3×10−5 on FB15K and d = 300, γ = 10−4, η = 10,  and µ = 10−4 on DB100K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='4  Main Results  The results of all compared methods on the test  sets of FB15K, DB100K, and FB15K-sparse are shown  in Tables 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' For each test triple, the mean re-  ciprocal rank or H@N value with N = 1, 3, 10 is uti-  lized as paired data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  The experimental results show  that 1) Iterlogic-E outperforms numerous strong base-  lines in the vast majority of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  This shows that  Iterlogic-E can achieve very good accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 2) Iterlogic-  E significantly outperforms the basic models that use  triples alone, and the improvement comes from the abil-  ity to learn the conclusions obtained by soft rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 3)  Iterlogic-E also beats many baselines that incorporate  logical rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Specifically, Iterlogic-E performs better  than SoLE and IterE under most metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' This demon-  strates the superiority of Iterlogic-E in reducing the  noise of candidate conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 4) IterE can only en-  hance sparse entities, so the experimental results are  much lower than those of other baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' How-  ever, Iterlogic-E is also effective on FB15K-sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 5)  On DB100K, the improvements over SLRE and SoLE  are more significant than those on FB15K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The reason  for this is probably that the groundings of the rules  on DB100K contain more incorrect conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Sim-  ple rules between a pair of relations are adequate to  capture these simple patterns on the FB15K dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  6) The performance of Iterlogic-E(ComplEx) is worse  than some baselines in HIT@10, and we consider that  this limitation is mainly due to the shortcomings of the  base model ComplEx [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' [24] point out that  ComplEx can not model the composition relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' We  have experimented with replacing the base model with  the RotatE model(Iterlogic-E(RotatE)), which is capa-  ble of modeling four relation patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The experimen-  tal results have been further improved, and our method  consistently achieves the best results in all evaluation  metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='com/DeepGraphLearning/KnowledgeGraphEmbedding  4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='com/wencolani/IterE  12  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5  Ablation Study  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Ablation study  DB100K  MRR HITS@1 HITS@3 HITS@10  Iterlogic-E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
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+page_content='505  5  w/o IL+Ldc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='347  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='258  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='395  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='510  6  w/o IL+Lrc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='351  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='257  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='406  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='515  7  w/o IL+Ldc+Lrc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='328  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='279  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='348  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='422  8  Iterlogic-E*  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='369  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='298  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='404  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='502  9  ComplEx+AC+IL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='285  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='223  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='312  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='407  10 ComplEx+WC+IL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='295  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='231  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='327  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='422  ComplEx  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='272  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='218  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='303  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='362  To explore the influence of different constraints and  iterative learning, we perform an ablation study of  Iterlogic-E on DB100K with 9 configurations in Ta-  ble 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The first and second variants, compared to the  completed model Iterlogic-E, remove the non-negativity  and l2-norm constraints (l2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  The third setting is  the combination setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The fourth removes iterative  learning (IL), which uses a rule to reason only once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  The fifth, sixth, and seventh variants remove the addi-  tional loss item based on the sixth variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The eighth  setting is another variant of the Iterlogic-E model based  on the ninth setting, which is that after the ComplEx  fitting, according to the scores of the conclusions, the  top n (where n is the rounded product of the rule and  its confidence) conclusions of each rule are selected to  be added to the KG to continue training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The ninth  setting (AC) is to add all conclusions inferred from the  rules as positive examples, and the tenth setting (WC)  is to use the rule confidence as soft labels of conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  As seen in Table 5, we can conclude the following:  1) When removing NNE constraints, the performance  of Iterlogic-E decreases slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Without l2-norm con-  straints on entities, the performance of Iterlogic-E de-  grades by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='3% in H@1 and by 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='1% in MRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' One ex-  planation may be that l2-norm constraints are sufficient  to constrain embedding norms on DB100K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' However,  the performance will suffer dramatically if there are  no l2-norm constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  2) Removing iterative learn-  ing decreases performance slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  One reason may  be that the number of rules on DB100K is relatively  small, so the number of conclusions added through it-  erative learning is relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  3) Removing the  additional loss item of the conclusions decreases per-  formance slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' This illustrates that Iterlogic-E can  filter out incorrect conclusions and makes the KG dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Surprisingly, even if we directly use ComplEx to filter  and learn the conclusions that can achieve such high  performance, this method is not as flexible as Iterlogic-  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 4) Compared with the basic model, all variants have  different degrees of improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  This demonstrates  the critical importance of logical rules in link predic-  tion tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='3  Influence of the Number of Iterations  To demonstrate how Iterlogic-E can enhance the  embedding effect during the training process, we show  the link prediction results on DB100K with different  iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Figure 3 shows that as the number of train-  ing iterations increases, the prediction results, including  Hit@1, Hit@3, Hit@10, and MRR, will improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' From  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 3, we can infer the following: 1) Iterative learn-  ing enhances embedding learning since the quality of  embeddings improved with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  2) In the first two  iterations, the embedding learning module was quickly  fitted to the conclusions of the rules and the triples of  the training set, and the prediction accuracy rapidly  improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  3) After two iterations, as the number of  new conclusions decreased, the results of the inference  tended to be stable, and the true conclusions and the  MODEL KNOWLEDGE GRAPH AND RULES FOR REASONING  13  initial KG triples were well preserved in the embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  0  2  4  6  8  10  Number of iterations  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='6  HIT@1  HIT@3  HIT@10  MRR  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Link prediction results in different iterations  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='4  Influence of Confidence Levels  Additionally, we evaluate the effect of the rules’ con-  fidence thresholds on FB15K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Since there are different  rules and ComplEx does not merge rules, we refer only  to their fixed results on FB15K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' We set all hyperparam-  eters to their optimum values and change the confidence  threshold within [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5, 1] in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='05-step increments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Both  SoLE and Iterlogic-E use rules with confidence levels  greater than this threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Figure 4 displays the MRR  and H@1 values obtained by Iterlogic-E and other base-  lines on the FB15K test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' We make the following  observations: 1) Iterlogic-E beats both ComplEx and  IterE at varying confidence levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' This demonstrates  that Iterlogic-E is sufficiently robust to deal with un-  certain soft rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5  Case Study  In Table 6,  we present a case study with 4  conclusions (true or false as predicted by Iterlogic-  E), which are inferred with 2 rules during train-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Table 6 shows some conclusions derived from  rule inference and the score change (the average of  the head entity prediction score and the tail en-  tity prediction score) of the conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Using  the first conclusion as an example, the true conclu-  sion  is  (Albany Devils,/hockey roster position/posit  on,  Centerman),  which  is  obtained  by  the  rule  ”/sports team roster/position(x, y)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='808  ⇒ /  hockey roster position/position(x, y)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  The Albany  Devils are a professional ice hockey team in the  American Hockey League  5, and the centerman is  the center in ice hockey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Therefore,  this is in-  deed a true conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Compared to ComplEx,  Iterlogic-E  increased  the  score  of  this  fact  from  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='48 to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Also, (San Diego State Aztecs football,  /hockey roster position/position,Linebacker) can be in-  ferred from the fact (San Diego State Aztecs football,  /sports team roster/ team, Linebacker) by the same  rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' However, the San Diego State Aztecs are a foot-  ball team, not a hockey team, and this is an incorrect  conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Compared to ComplEx, Iterlogic-E de-  creased the score of this fact from -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='65 to -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' This  illustrates that Iterlogic-E can distinguish whether the  conclusion is true and can improve the prediction per-  formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  Furthermore, Iterlogic-E has good inter-  pretability, and we can understand why the conclusion  is inferred by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  5  Conclusion and Future Work  This paper proposes a novel framework that itera-  tively learns logical rules and embeddings, which mod-  els the conclusion labels as 0-1 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The proposed  Iterlogic-E uses the confidences of rules and the context  of the KG to eliminate the uncertainty of the conclu-  sion in the stage of learning embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Specifically,  our method is based on iterative learning, which not  only supplements conclusions but also filters incorrect  conclusions, resulting in a good balance between effi-  5https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='org/wiki/Albany_Devils  14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
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+page_content='80  HIT@1  ComplEx  IterE  SoLE  Iterlogic-E  (b)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Results of MRR (a) and HITS@1 (b) achieved by Iterlogic-E with different confidence thresholds on FB15K  Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' A case study with 4 conclusions (true or false as predicted by Iterlogic-E), which are reasoned on by 2 rules  True  False  Conclusions  (Albany Devils,/hockey roster position/  position,Centerman)  (San Diego State Aztecs football,/hockey  roster position/position,Linebacker)  Predicted by rules  /sports team roster/position(x, y)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='808  ⇒ /hockey roster position/position(x, y)  Score change  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='4888 → 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='4039  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='6518 → -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='3547  Conclusions  (Chris Nurse,/football roster position/  team,Stevenage F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=')  (Brett Favre,/football roster position/  team,Green Bay Packers)  Predicted by rules  /sports team roster/team(x, y)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='847  ⇒ /football roster position/team(x, y)  Score change  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='8123 → 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='5991  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='0131 → -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='3952  ciency and scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' The evaluation on benchmark  KGs demonstrates that the method can learn correct  conclusions and improve against a variety of strong  baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In the future, we would like to explore how  to use embeddings to learn better rules and rule con-  fidences than AMIE+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Additionally, we will continu-  ously explore more advanced models to integrate rules  and KGEs for knowledge reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  References  [1] Berant J, Chou A, Frostig R, Liang P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Semantic parsing on  freebase from question-answer pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In EMNLP, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  1533–1544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  [2] Huang X, Zhang J, Li D, Li P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Knowledge graph embedding  based question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In WSDM, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 105–113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  [3] Wang X, Wang D, Xu C, He X, Cao Y, Chua T S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Explain-  able reasoning over knowledge graphs for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  In AAAI, 01, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 5329–5336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  [4] Cao Y, Wang X, He X, Hu Z, Chua T S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Unifying knowl-  edge graph learning and recommendation: Towards a better  understanding of user preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In WWW, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
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+page_content=' In AAAI, number 1, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
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+page_content=' A survey on  knowledge graphs: Representation, acquisition and applica-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
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+page_content=' Modeling re-  lation paths for representation learning of knowledge bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  In EMNLP, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 705–714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  [48] Minervini P, Costabello L, Mu˜noz E, Nov´acek V, Vanden-  bussche P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Regularizing knowledge graph embeddings via  equivalence and inversion axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In ECML-PKDD, 2017,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 668–683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  [49] Zhang Z, Cai J, Zhang Y, Wang J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Learning hierarchy-aware  knowledge graph embeddings for link prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In AAAI,  2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 3065–3072.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  [50] Niu G, Li B, Zhang Y, Pu S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  CAKE: A Scalable  Commonsense-Aware Framework For Multi-View Knowl-  edge Graph Completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In ACL, 2022, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 2867–2877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content='  [51] Yang J, Ying X, Shi Y, Tong X, Wang R, Chen T, Xing  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' Knowledge Graph Embedding by Adaptive Limit Scor-  ing Loss Using Dynamic Weighting Strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' In ACL, 2022,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
+page_content=' 1153–1163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE0T4oBgHgl3EQf7wLZ/content/2301.02781v1.pdf'}
diff --git a/ldE2T4oBgHgl3EQfeAcm/content/tmp_files/2301.03911v1.pdf.txt b/ldE2T4oBgHgl3EQfeAcm/content/tmp_files/2301.03911v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..601e2457b22c2dbc3192be31b0a31e0d565188e5
--- /dev/null
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@@ -0,0 +1,1945 @@
+1 
+Title 
+Variable bandwidth, high efficiency microwave resonator 
+for control of spin-qubits in nitrogen-vacancy centers 
+Authors 
+Dr. Anton Savitsky, Dr. Jingfu Zhang, Prof. Dieter Suter 
+Faculty of Physics, Technical University Dortmund, Otto-Hahn-Str. 4a, 44227 Dortmund, 
+Germany 
+Keywords 
+Planar microresonator, NV center, spin qubit, microwave magnetic field 
+
+2 
+ 
+Abstract 
+Nitrogen-Vacancy (NV) centers in diamond are attractive tools for sensing and quantum 
+information. Realisation of this potential requires effective tools for controlling the spin degree 
+of freedom by microwave (mw) magnetic fields. In this work we present a planar microwave 
+resonator optimised for microwave-optical double resonance experiments on single nitrogen-
+vacancy (NV) centers in diamond. It consists of a piece of wide microstrip line which is 
+symmetrically connected to two 50 Ω microstrip feed lines. In the center of the resonator, an  Ω-
+shaped loop focuses the current and the mw magnetic field. It generates a relatively 
+homogeneous magnetic field over a volume of 0.07mm2×0.1mm. It can be operated at 2.9 GHz 
+in both transmission and reflection modes with bandwidths of 1000 MHz and 400 MHz, 
+respectively. The high power-to-magnetic field conversion efficiency allows to produce π-
+pulses with a duration of 50 ns with only about 200 mW and 50 mW microwave power in 
+transmission and reflection, respectively. The transmission mode also offers capability for 
+efficient radio frequency excitation. The resonance frequency can be tuned between 1.3 GHz 
+and 6 GHz by adjusting the length of the resonator. This will be useful for experiments on NV-
+centers at higher external magnetic fields and on different types of optically active spin centres. 
+1. Introduction 
+The NV center in diamond is used in various fields, such as quantum information, quantum 
+sensing, magnetometry, bioimaging, etc. 1-5. In all applications, efficient manipulation of the 
+electron spin is an essential prerequisite, both for continuous wave (cw) experiments and for 
+fast spin control in pulsed experiments. Therefore, maximising the coupling between the control 
+microwave (mw) magnetic field and the electron spins is a fundamental concern. Here, we 
+consider specifically applications where single NV centers are excited by a laser and the spin 
+state is read out by collecting photoluminescence (PL) through a microscope objective with 
+large numerical aperture (NA > 1). Measuring a cw ODMR spectrum in such requires a mw 
+magnetic field of ������������ = 150 A/m (������������ = ������������0������������ = 0.19 mT) to reach the maximum fluorescence 
+contrast 6. This field corresponds to a Rabi frequency of ������������������������ = 2 MHz if the oscillating magnetic 
+field is perpendicular to the NV axis. In pulsed experiments, typical π-pulse durations of ������������������������ =
+50 ns are used, which correspond to an excitation bandwidth of 1.2/������������������������ = 24 MHz (������������������������ =
+10 MHz). This is sufficient for complete electron spin flip of the selected triplet spin transition 
+6. This pulse length requires magnetic field amplitude of ������������ = 800 A/m.  
+Currently, most experimental setups rely on a wire positioned over the sample to achieve such 
+mw magnetic field amplitudes. If this wire is placed in a short gap of a transmission line with 
+
+3 
+ 
+������������������������ =50 ٠impedance, the electric current in the wire has an amplitude ������������ = �2������������������������������������/������������������������ = 
+0.2 A/√W ∙ �������������������������������������. This current produces the tangential magnetic field ������������������������������������������������������������ =
+������������
+2������������������������ outside the 
+wire of radius ������������0. For a typical wire diameter of 20 µm, the maximum magnetic field generated 
+at the wire surface is ������������������������������������������������������������(������������0) = 3200
+������������
+������������ at 1W of mw power. The wire is, however, optically 
+opaque. Therefore, only NV-centers at a distance > √2������������0 from the center of the wire center can 
+be optically probed. At this position, the magnetic field is reduced to 2600 
+������������
+������������ at 1W. Even at 
+realistic distances from the wire of 40 µm, which still reduces the fluorescence collection 
+efficiency due to obstruction by the wire, pulsed experiments can be performed with 1 W mw 
+power, reaching �������������������������������������������������������������40 µm� = 800 
+������������
+������������. Therefore, the microwire system is used in many 
+laboratories, including our laboratory7. A major disadvantage is that the 1/������������ dependence of the 
+magnetic field amplitude significantly restricts the volume and the surface area available for 
+probing single NV-centers and requires precise initial positioning of the microscope objective. 
+Moreover, readjustment of the mw power settings is necessary for any new probed NV-center 
+which is not only due to the distance dependence of the magnetic field amplitude but also to the 
+magnetic field direction dependence on distance. This problem can be overcome using a 
+microloop integrated into a transmission line instead of the microwire 8, 9. The magnetic field 
+amplitude in the center of the ideal loop is given by ������������������������������������������������������������ =
+������������
+������������ where ������������ is the loop diameter. 
+Thus, the field of 1000 
+������������
+������������ can be generated by a loop with 200 µm diameter at 1 W mw power, 
+which fulfills the requirements for ODMR. Advantages of the loop include a good homogeneity 
+and directivity of the magnetic field over a 100 ×100 µm2 area, much higher than the microwire. 
+Compared to a wire, the loop magnetic field is, however, less tolerant to the effect of the metal 
+case of the microscope objective, which must be positioned close to the sample. We discuss 
+means to avoid this issue below. 
+There are several additional significant handicaps of loop, as well as wire, systems in 
+conjunction with transmission line, (i) the construction does not allow quick replacement or 
+reposition of the diamond spaceman; (ii) significant heating effects (the resistance of 10 mm 
+copper wire with 20 µm diameter is 0.25 Ω at 3 GHz, which leads to power dissipation at the 
+mw power level required for ODMR, but the thermal conductivity is limited due to the small 
+conductor cross-section); (iii) substantial power return losses caused by the discontinuity in the 
+transmission line. The main handicap is, however, the high mw powers required to fulfill the 
+magnetic field requirements for ODMR experiments. 
+
+4 
+ 
+The power limitation can be overcome using planar resonators to generate the magnetic field. 
+In the past numerous reflection mode resonators were proposed based on different planar 
+structures, for instance double-split ring 10, 11, triple-split-ring 12, loop-gap 13, strip-line 14, and 
+several other types 15. All these resonators allow to store the mw energy for a time proportional 
+to the unloaded quality factor ������������0, which leads to increase of the current and, therefore, an 
+enhanced magnetic field amplitude proportional to �������������0  as compared to a non-resonant 
+structure with the same geometry. The resonator, however, introduces additional limitation. The 
+substantial magnetic field enhancement is achieved only within the resonator bandwidth. For 
+instance, for a matched (critically coupled) reflection resonator with ������������0=100 would require 100 
+times less mw power at the resonance frequency of ������������0 = 3 GHz to generate the same mw 
+magnetic field amplitude as compared to the non-resonant structure of same geometry. This 
+effect is, however, only obtained within Δ������������1/2 = 2 ∙ ������������0
+−1 ∙ ������������0= 60 MHz around the resonance 
+frequency. This would substantially limit its general applicability for ODMR on single NV-
+centers. The bandwidth of a matched resonator can only be increased by lowering the ������������0-value 
+13. This, however, leads to lower mw power to magnetic field conversion, i.e. it reduces the 
+usefulness of the resonator. 
+The aim of this work is to develop a device that avoids these issues and can be used, e.g., for 
+ODMR spectroscopy of single NV centers. It overcomes high power requirements and 
+compromises limitations of transmission line systems as well as previously reported resonant 
+structures. We design it to fulfill the following requirements: (i) easy and cheap fabrication with 
+reliable resonator parameters; (ii) compatibility with standard coaxial mw delivery system; (iii) 
+high mw power to magnetic field conversion efficiency; (iv) high magnetic field directivity and 
+spatial homogeneity at least in the area accessible by nanopositioners based on piezoelectric 
+actuator (100 ×100 µm2); (v) small mw magnetic field frequency dependence, i.e. large 
+bandwidth; (vi) possibility for microwave and radio frequency excitations; (vii) high 
+mechanical and thermal stability; (viii) easy and reliable replacement or reposition of the 
+diamond specimen; and (very important !) (ix) it must be compatible with high resolution 
+confocal objectives, i.e. tolerate the presence of dielectric and conductive parts in very close 
+vicinity (200 µm to 300 µm) of the magnetic fields of the resonator. 
+2. Results and discussion 
+2.1 Resonator design 
+Figure 1 depicts the design of the developed half-wave resonator. It consists of a piece of wide 
+microstrip line symmetrically connected to two 50 Ω microstrip feed lines terminated by SMA 
+
+5 
+ 
+connectors. In the center of the microstrip, the Ω-loop concentrates the current and the mw 
+magnetic field. The geometrical parameters of the resonator investigated in this study are 
+summarized in the figure caption. The diamond is placed on the loop, as shown in Fig 1b. Thus, 
+the lower diamond surface is exposed to the mw magnetic field near the maximum amplitude. 
+The resonator is fabricated using standard PCB lithography on low loss Rogers RO3003 
+laminate with 166 µm overall thickness. This design allows us to explore the diamond down to 
+least 100 µm above the surface. The resonator is designed for operation in transmission (the 
+output SMA is terminated by a 50 Ω load) or reflection (the output SMA is open to free space) 
+modes of operation. 
+ 
+Figure 1. (a) Full model of the microwave resonator. It is fabricated on 60×26 mm2 Rogers 
+RO3003 low loss laminate ( ����������� = 3.00 , tan ������������ = 0.001  at 10 GHz, dielectric 
+thickness 0.13 mm and 18 µm copper cladding) mounted on a 60×26×1mm3 copper 
+holder and two standard PCB SMA flat tab connectors . (b) Resonator design. The 
+geometrical parameters are: ������������������������ =  0.3 mm  - width of the microstrip feed line (������������������������ =
+50 Ω); ������������������������ = 3 mm - width of the resonator, ������������ =  17 mm - length of the resonator; 
+������������ = 0.1 mm - gap width, ������������������������ = 0.4 mm - inner diameter of the loop, ������������������������ = 0.1 mm 
+- width of the loop conductor, ������������������������������������������������ = 0.3 mm - diameter of the optical access hole. 
+Coordinate system is indicated. For optical pathway see Fig. S8 in SI. 
+2.2 Transmission mode resonator 
+Figure 2 shows the S-parameters of the resonator in transmission mode calculated using CST 
+Microwave Studio and measured experimentally using a network vector analyzer (HP 8720A). 
+The analytical transmission and reflection coefficients of the symmetrically coupled 
+transmission resonator with unloaded quality factor ������������0 and resonance frequency ������������0 are given 
+by  
+  
+������������ = −
+1−������������������������
+1+2������������−������������������������ ;  ������������ =
+2������������
+1+2������������−������������������������ 
+(1) 
+where ������������ is the coupling parameter for both input and output and ������������ = ������������0 ∙ �
+������������0
+������������ −
+������������
+������������0� is the 
+normalized offset 16, 17. Analysis of the S-parameter traces depicted in Fig. 2 using Eqs. (1) 
+
+a
+b
+L
+★
+Wr
+G
+W
+V
+X
+X
+W
+D
+opt6 
+ 
+yields the resonator parameters ������������0 = 74, β = 11.5 and ������������0 = 73, β = 11.8 for simulation and 
+experiment, respectively. The relatively low unloaded quality factor is typical for microstrip 
+based resonators owing to relatively high conduction losses 18. The comparison of the calculated 
+coupling parameter of 11.5 with ������������ = ������������0
+������������������������
+������������������������=12.6 which can estimated analytically for the 
+resonator shows that the Ω-loop slightly increases the impedance of the resonator over the 
+impedance of the microstrip line with width ������������������������ (for more information see SI). The central 
+frequency of the experimental resonator was downshifted by about 70 MHz compared to the 
+calculation. This deviation is ascribed to slightly different real parameters of the laminate 
+(dielectric constant and thickness) as well as manufacturing tolerances. 
+ 
+Figure 2. (a) Simulated and (b) experimental S-parameters of the transmission resonator with 
+diamond: ������������11 = 20 ∙ log10 |Γ| (blue line) and ������������21 = 20 ∙ log10 |������������| (red line). The 
+dashed lines show the best fit results of the S-curves to the reflection and transmission 
+coefficients given by Eqs. (1). The disagreement at the low and high frequencies is 
+due to the frequency dependence of the characteristic impedances, which are not 
+included in Eqs. (1), and contributions from other resonator modes. 
+The resonator itself is described by the coefficient ������������2 = 1 − |Γ|2 − |������������|2, i.e. the mw power 
+transmitted to the resonator to which the magnetic field amplitude is proportional. This factor 
+can be parametrized by the power coefficient at the resonance frequency and the resonator 
+bandwidth: 
+ 
+������������(������������0) =
+2�������������
+1+2������������ ;  ������������������������1/2 =
+1+2������������
+������������0 ∙ ������������0. 
+(2) 
+Thus, the bandwidth of the resonator is 950 MHz and 990 MHz for calculation and experiment, 
+respectively, which is sufficient for most low-field applications of NV centers.  
+Figure 3 shows the spatial variation of the mw magnetic field behavior within the loop. At a 
+small elevation above the loop surface (������������ =10 µm), the loop provides a perfect magnetic field 
+directivity over the optically accessible diamond area, i.e. the magnetic field is aligned with the 
+z-axis. 
+At 
+������������ = 10 μm
+ the 
+position 
+of 
+the 
+magnetic 
+field 
+minimum 
+
+-5 F---
+5
+-10
+10
+/dB
+/dB
+-15
+-20
+-20
+-25
+-25
+a
+b
+-30
+-30
+2.4
+2.6
+2.8
+3
+3.2
+3.4
+2.4
+2.6
+2.8
+3
+3.2
+3.4
+freguency /GHz
+freguency /GHz7 
+ 
+������������(0, 70 µm, 10 µm) =1100 A/m is slightly shifted from the loop center due to the effect of the 
+gap. The magnetic field amplitude increases by about a factor of 2 to the edge of optically 
+accessible area. Above the loop the decay of the magnetic field magnitude at the loop center is 
+well described by the decay function for an ideal current loop: 
+  
+������������(0,0, ������������) = ������������(0,0,0)
+������������3
+(4������������2+������������2)
+3
+2
+ 
+(3) 
+with ������������ = 440 µm > ������������������������ due to the large width of the loop conductor (������������������������ = ������������������������/4), see Fig. 3(d). 
+Below the loop the magnetic field decays rapidly and is smaller than 70 A/m at the closest 
+position of the microscope objective surface. This guarantees the stability of the system during 
+operation as neither magnetic field distributions nor resonator parameters are influenced by the 
+microscope objective. At higher elevation above the loop the improvement of the magnetic field 
+homogeneity is accompanied by slight loss of the field directivity, see Fig S6 in SI. Within the 
+optically accessible diamond area (±x, ±y, z)=( ±150, ±150, 0+100) µm the minimum and 
+maximum of the magnetic field magnitude are 2200 A/m and 735 A/m at 1 W mw power, 
+respectively. Thus, the magnetic field of the loop compares very favorably with that of a 
+microwire, both in terms of homogeneity and directivity. 
+ 
+Figure 3. (a) Color-coded plot of the calculated mw magnetic field magnitude, |������������|, at 
+������������ = 10 µm for mw power of 1 W at the resonance frequency. The dotted and dashed 
+circles mark the position of the inner loop edge (������������������������ = 400 µm) and the optical access 
+
+a
+A/m
+G
+c
+3500
+3600
+45.2
+3000
+/ A/m
+3000
+2000
+25.2
+H(0,y,10μm) 
+2500
+2000
+1000
+12.6
+1500
+6.3
+1000
+500
+....
+0
+-0.2
+-0.15
+-0.1
+-0.05
+0
+0.05
+0.1
+0.15
+0.2
+y /mm
+1200
+3500
+b
+..........
+objective
+d
+HH
+3000
+1000
+/ A/m
+/ A/m
+HH
+2500
+800
+H,(0,0,z)
+10μm)
+2000
+600
+H(x,0,1
+1500
+400
+NV-center
+1000
+500
+200
+-0.2
+0.15
+-0.1
+-0.05
+0
+0.05
+0.1
+0.15
+0.2
+-0.2
+0
+0.2 
+0.4 
+0.6
+0.8
+/mm
+z /mm8 
+ 
+hole (������������������������������������������������ =  300 µm), respectively. (b,c,d) The ������������������������, ������������������������, ������������������������ amplitudes of the mw 
+magnetic field components as functions of the position ������������, ������������, ������������ for a mw power of 
+1 W. The black dotted lines mark the optically accessible diamond area. The 
+shadowed area in (d) shows the position region of the optical objective (������������ <  0) and 
+NV-center (������������ >  0). The dashed lines in (d) show the best fit curve of ������������������������(0,0, ������������) to 
+the function in Eq. (3). 
+2.3 Reflection mode resonator 
+The efficiency of the resonator can be improved employing reflection mode. The resonator is 
+converted to reflection mode by just disconnecting the output coax cable. The reflection 
+resonator is described by: 
+  
+������������(������������0) =
+2�������������
+1+������������ ;  ������������������������1/2 =
+1+������������
+������������0 ∙ ������������0 
+(4) 
+as derived from the reflection coefficient 16: 
+  
+������������ = −
+1−������������−������������������������
+1+������������+������������������������. 
+(5) 
+Thus, for high coupling parameters, ������������ ≫ 1, the reflection mode offers a factor 2 higher 
+magnetic field amplitudes, see SI. The bandwidth, however, becomes reduced by factor of 2. 
+The magnetic field distribution in the loop area is identical in both resonator modes. 
+Figure 4 shows the calculated and experimental S11-parameters of the resonator in reflection 
+mode. Analysis of the S-parameter traces using Eq. (5) yields the resonator parameters ������������0 =
+70, β = 8.3 and ������������0 = 60, β = 5.8 for simulation and experiment, respectively. Thus, the 
+bandwidth of the resonator is 400 MHz and 320 MHz for calculation and experiment. The 
+difference between transmission and reflection mode as well as between calculated and 
+experimental reflection parameters are mainly due to the open output line. 
+ 
+Figure 4. Simulated (a) and experimental (b) S11-parameters of the resonator operated in 
+reflection mode. The dashed line show the best fit results of the S-curves to the 
+reflection coefficient given by Eq. (5). 
+ 
+ 
+
+0
+0
+0.5
+-0.5
+-1
+/dB
+-1
+/dB
+-1.5
+S1
+-1.5
+-2
+2.5
+-2
+-3
+a
+b
+2.5
+2.4
+2.6
+2.8
+3
+3.2 
+3.4
+3.6
+2.6
+2.8
+3
+3.2
+3.4
+3.6
+frequency/GHz
+freguency/GHz9 
+ 
+3. ODMR experiments 
+The mw magnetic field behavior was verified experimentally using a previously described 
+ODMR setup capable for continuous wave and pulsed ODMR experiments, see 7 and SI. The 
+2×2×1mm3 diamond (001 cut) containing a 20 µm layer near surface that was doped with NV 
+centers was fixed to the resonator using transparent office tape, see Figs. S11 and S21 in SI. 
+The central position of the optical objective was set close to the loop center.  
+Figure 5(a) shows a cw ODMR spectrum of the NV-centers located near the focal spot, recorded 
+using mw power of 6.8 mW (the power level was calibrated at 2.87 GHz). Two intense ODMR 
+lines centered at 2.87 GHz are observed. The line splitting by 475 MHz corresponds to the 
+external magnetic field component of 8.5 mT along the NV-axis. Additional ODMR line pairs 
+with splitting of 260 MHz, 160 MHz and 60 MHz can be attributed to a set of NV-centers 
+within the sensitive volume of about 300 nm in the x-y plane and 1000 nm in the z-direction 
+(see Fig. S9 in SI). The different centers have different orientations along different [1,1,1] 
+directions.  
+To measure the precise mw magnetic field strength, we recorded Rabi oscillations at the two 
+strongest transitions, as shown in Figure 5(b). The applied mw power of 680 mW resulted in a 
+Rabi frequency of about 14 MHz. The Rabi frequency ������������������������  is proportional to the magnetic field 
+amplitude ������������: 
+  
+������������������������ = �
+2
+3 ∙ √2 ∙
+������������������������
+2������������
+������������0������������
+2 =
+������������������������
+2������������
+������������0������������
+√3 ,  
+(6) 
+where ������������������������ is the electron gyromagnetic ratio (������������������������/2������������=28 GHz/T) and ������������0 is the permeability of 
+the vacuum. The factor �2/3 accounts for the component of ������������ that is perpendicular to the NV-
+axis, (mw H-axis parallel to [0,0,1] and NV-axis parallel to [1,1,1] crystal axis). The factor √2 
+takes into account that we drive one transition of the S = 1 spin. Thus, a Rabi frequency of 
+������������������������=14 MHz at 0.68 W corresponds to a conversion efficiency of ������������ = 840
+������������
+������������ at 1 W.  
+
+10 
+ 
+ 
+Figure 5. (a) cw ODMR spectrum of the single NV-center at external magnetic field alight 
+close to [1,1,1] crystal axis recorded at 6.8 mW incident mw power. (b) The Rabi 
+oscillation traces recorded at the mw frequencies of the high frequency (upper trace; 
+marked blue in (a)) and low frequency (lower trace; red in (a)) ODMR lines and 
+680 mW mw power (+20dB increased power over the cw experiment). The solid blue 
+and red lines are the fits. 
+In the next step the cw and pulsed ODMR experiments were performed at different external 
+magnetic fields and the same input power of the mw amplifier, see Fig. 6(a). The magnetic field 
+amplitude evaluated from the Rabi frequencies overlaid with the simulation results is depicted 
+in Fig. 6(b). Both curves are in the good agreement. The variation of the mw magnetic field 
+over a frequency range of 800 MHz is less than 40%. The discrepancy at the frequencies above 
+3 GHz is attributed to gain variation of the high-power mw amplifier operated below saturation.  
+The spatial distribution of H in the x-y plane was evaluated from the Rabi experiment at 
+2.93 GHz for 10 different NV centers over the area of 70×70µm2 accessible by the 
+nanopositioner. The variation of the magnetic field of less than 20% (1000 A/m to 1250 A/m) 
+is in a good agreement with simulation predictions. 
+ 
+Figure 6. (a) cw ODMR spectra of the single NV-center at different external magnetic fields 
+at 6.8 mW incident mw power. (b) The frequency dependence of the magnetic field 
+
+ intensity I.norm.u.
+0.96
+a
+0.98
+0.92
+contrast /%
+0.88
+0.96
+473 MHz
+0.84
+f,=13.7 ±0.1 MHz
+ODMR
+0.94
+0.96
+0.92
+2.87GHz
+0.88
+0.9
+2.8
+0.84
+2.6
+2.7
+2.9
+3
+3.1
+f,=14.4 ±0.1 MHz
+Frequency/GHz
+100
+200
+300
+400
+500
+t. /ns24
+1100
+22
+A·m
+20
+18
+a
+800
+b
+16
+2.5
+2.6
+2.7
+2.8
+2.9
+3
+3.1
+3.2
+3.3
+2.4
+2.6
+2.8
+3
+3.2
+3.4
+freauency/GHz
+freguency/GHz11 
+ 
+amplitude ������������������������(0,0,10 μm) calculated for the transmission resonator (black curve, left 
+scale). Rabi nutation frequencies (dots, right scale) evaluated from experimental Rabi 
+oscillation traces recorded at the ODMR line positions in (a). The Rabi frequencies 
+are normalized to 1 W mw power calibrated at 2.87 GHz. The left and right scales 
+are adjusted according to Eq. (6). 
+These results show that the transmission resonator achieves good spatial magnetic field 
+homogeneity and the bandwidth required for most ODMR experiments on single NV-centers. 
+The transmission resonator achieves π-pulse durations of 20 ns (������������������������ = 25 MHz) using about 1 W 
+mw power. For this performance, a mw amplifier is required. This limitation can be overcome 
+by operating the resonator in reflection mode. 
+ 
+Figure 7. (a) Rabi oscillation traces recorded at the mw frequency 2.93 GHz for the resonator 
+in transmission (upper trace) and reflection (lower trace) modes at 680 mW mw 
+power. The traces were acquired subsequently at the same settings with the output 
+cable connected and disconnected. The solid-colored lines are the fits. (b) Frequency 
+dependence of the magnetic field amplitude ������������������������(0,0,10������������������������)  calculated for 
+transmission (red trace) and reflection resonator (blue trace). The amplitude ratio at 
+2.93 GHz is indicated by the arrow. 
+Figure 7(a) shows the Rabi nutation traces recorded at the high frequency ODMR line at 
+2.93 GHz with 680 mW input mw power. The upper trace was acquired using the resonator in 
+transmission mode. Subsequently, the output mw cable was disconnected converting the 
+resonator into reflection mode and the lower trace was acquired. The magnetic field amplitudes 
+evaluated from Rabi frequencies are 1190 A/m and 2070 A/m for transmission and reflection 
+settings, respectively. The experimentally observed increase of the resonator efficiency by a 
+factor of 1.74 is in a good agreement with factor 1.8 obtained from calculation, see Fig 7(b). 
+Thus, the reflection resonator is capable of producing the same magnetic field amplitude using 
+about factor of 3 smaller mw power. It achieves 20 ns π-pulses (������������������������ = 25 MHz) with as little as 
+300 mW mw power. 
+
+2200
+/.norm.u.
+b
+0.96
+tensity
+0.92
+1800
+reflection
+0.88
+x 1.8
+f.=19.9±0.1 MHz
+0.84
+A·m
+1400
+rm.u.
+.nor
+0.96
+intensity
+1000
+0.92
+transmission
+T
+0.88
+f,=34.7±0.1 MHz
+600
+0.84
+2.4
+2.6
+2.8
+3
+3.2
+3.4
+0
+40
+80
+120
+160
+200
+frequency/GHz
+t.
+/ns12 
+ 
+At this point we put two remarks: (i) The highest ������������������������ achieved in our setup allows to realize π-
+pulses with 7 ns duration and 4 ns in transmission and reflection modes, respectively. The pulse 
+profile of such short pulses is not distorted by the resonator, neither in transmission nor in 
+reflection mode, since their voltage ringing times are ������������������������ =
+1
+������������������������1/2 = 0.3 ns and 1 ns, 
+respectively. (ii) The performance of the microscope objective employed in this work, in 
+particular the efficiency of fluorescence collection, is restricted by the resonator acting as an 
+optical diaphragm, see SI. Despite this restriction, the collection of high quality ODMR data is 
+still possible within reasonable time.  
+4. Performance summary 
+The goal of this work was the design of a device for efficient control of spin quibts. For optimal 
+performance, it should have a number of properties that we specified in the introduction. Here, 
+we first summarise the quantitative performance measures of the mw and rf properties, as 
+shown in Table 1; the values shown were calculated and verified by experiments. For 
+comparison, it also includes the parameters calculated for the 50 Ω microstrip transmission line 
+with the same Ω-type loop. The resonator offers a factor of 4.8 (transmission) and 9.1 
+(reflection) higher magnetic field amplitudes at the same input mw power as compared to the 
+transmission line. Pulsed ODMR experiments with reasonably short π-pulses of 50 ns can be 
+generated using only 50 mW mw power with the resonator operated in reflection mode. Thus, 
+it allows to avoid high-power mw amplification stages or even to perform the experiments 
+directly using the output of a mw generator. The transmission mode is more advantages for 
+experiments at higher external magnetic fields owing to the larger resonator bandwidth. In 
+contrast to reflection mode, it allows to the perform experiments with combined microwave and 
+radio frequency excitations. An additional advantage of the transmission mode is the good 
+matching of the resonator input. The input return losses are <-20 dB (±50 MHz) and <-10 dB 
+(±150 MHz) around the resonance frequency, see Fig.2. Thus, in contrast to the reflection 
+resonator, the mw excitation system does not require matching elements (circulator or 
+attenuator) on the input for ODMR experiments at small external magnetic fields. 
+ 
+ 
+
+13 
+ 
+ 
+Table 1. Summary of resonator and transmission line performances. 
+Case 
+������������������������������������/������������������������������������� 
+A/m/√W 
+������������������������������������ c)  
+W 
+������������������������1/2 
+MHz 
+������������������������������������/������������������������������������� d) 
+A/m/√W 
+Transmission line 
+245 a) 
+4.0 
+full 
+252 (252) 
+Transmission resonator 
+1170 b) 
+0.18 
+950 
+265 (266) 
+Reflection resonator 
+2230 b) 
+0.05 
+400 
+10 (160) 
+a) calculated at 2.9 GHz 
+b) calculated at the resonance frequency (~2.9 GHz) 
+c) input mw power required for 50 ns π-pulse (������������������������= 10 MHz) and NV-center in 001 
+diamond 
+d) calculated at 10 MHz (200 MHz)  
+We also considered heating effects. Conduction losses in the loop can lead to a temperature 
+increase that can potentially influence the experiment. The thermal analysis of the transmission 
+mode resonator at ambient temperature shows that the temperature increase in the loop-diamond 
+region is < 0.3 K for 0.1 W continuous mw power (see Fig. S3 in SI). At this mw power the 
+produced magnetic field of 370 A/m corresponds to a Rabi frequency ������������������������=7.5 MHz which is 
+more than sufficient for cw ODMR detection with the highest contrast 6. In pulsed ODMR 
+experiments the average heat power is significantly smaller. The transmission resonator allows 
+for above 106 π-pulses at 10 W (tπ= 7 ns) with no significant heating effects. The results of the 
+thermal analysis are consistent with experimental observations. 
+The resonator design can be adapted for experiments on NV-centers at higher external magnetic 
+fields or  different types of optically active spin centers by adjusting some of the design 
+parameters. The variation of the overall resonator length, ������������, between 5 mm and 56 mm allows 
+to tune the resonance frequency between 6 GHz and 1.3 GHz without significant loss of the mw 
+performance, see Fig. S16 in SI. Further increasing or decreasing the resonance frequency is 
+possible by adjusting the dielectric constant and thickness of the substrate, and the geometry of 
+the holder. 
+Various types of microwave resonators have been proposed for experiments on optically active 
+spin centers, such as the diamond NV center, each with specific advantages and limitations. A 
+fair comparison of the performance of all these different designs would have to be made under 
+a specific set of conditions, where all designs can operate. Since this is not possible, we include 
+here a comparison with two recently reported designs with overlapping boundary conditions, 
+which are applied in several laboratories for experiments on NV-centers 10, 11, 13, 19-26. The first 
+
+14 
+ 
+system, originally reported by Bayat et al. 10, is based on a double-split-ring resonator operated 
+in reflection. In original design has an efficiency of 355 A/m/√W , with a bandwidth of 
+40 MHz. Similar numbers were reported for some modified designs 11, 25. The second design, 
+originally reported by Sasaki et al. 13, is based on a split-ring (or loop-gap) resonator in 
+reflection mode. In original design provides 240 A/m/√W with a bandwidth of 440 MHz. Better 
+parameters 485 A/m/√W and 300 MHz are reported for the modified version 20. Thus, the 
+performance of our system in both transmission and reflection modes of operation is superior 
+to both systems.  
+5. Conclusion 
+In this work we presented a novel mw excitation system based on a resonator designed for cw 
+and pulsed ODMR experiments on single NV-centers to be combined with confocal 
+microscopy. The high performance of the system was verified using numerical EM calculations 
+and ODMR experiments. The resonator can be easily and cheaply fabricated using standard 
+PCB lithography. It offers high mechanical and thermal stability. Its performance in terms of 
+mw power to magnetic field conversion efficiency and magnetic field homogeneity is superior 
+in comparison to currently used systems like the microwire-transmission line system. It also 
+offers a range of practical advantages such as simplified optical adjustment and optimization of 
+the mw power settings for all types of ODMR experiments. It also offers the possibility for 
+quick replacement of the diamond crystal because it does not have to be permanently connected 
+to the resonator. This provides the opportunity for investigation of different samples with the 
+same structure or use a resonator which is optimized for the performance of a specific type of 
+ODMR experiment.  
+While the present work focuses on applications on NV centers in diamond, the principles used 
+here are completely general and can be readily transferred to similar systems that rely on 
+efficient spin control by microwave fields. This includes not only ensembles of NV centers but 
+also other materials like semiconductors. 
+ 
+Acknowledgments 
+This project has received funding from the European Union’s Horizon 2020 research and 
+innovation program under grant agreement No 828946. The publication reflects the opinion of 
+the authors; the agency and the commission may not be held responsible for the information 
+contained in it. 
+ 
+ 
+
+15 
+ 
+Author Declarations 
+Conflict of Interest 
+The authors have no conflicts to disclose. 
+Data Availability 
+The data that support the findings of this study are available from the corresponding authors 
+upon reasonable request. 
+Author Contributions 
+Anton Savitsky: Conceptualization (equal); Formal analysis, (lead); Investigation (equal); 
+Methodology (lead); Visualization (lead); Writing – original draft (lead). Jingfu Zhang: Data 
+curation 
+(equal); 
+Investigation 
+(equal); 
+Formal 
+analysis 
+(equal); 
+Dieter 
+Suter: 
+Conceptualization (equal); Review & editing (equal); Funding acquisition (lead). 
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+Yamamoto, T. Ohshima, J. Isoya and D. Suter, Phys. Rev. Lett. 110 (24), 240501 
+(2013). 
+8. 
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+(2010). 
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+E. Abe, Rev. Sci. Instrum. 87 (5), 053904 (2016). 
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+Soc. Japan 91 (4), 044004 (2022). 
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+M. C. Sanchez, E. Martin and J. M. Zamarro, IEE Proc. Pt. H 137 (4), 209-212 (1990). 
+17. 
+J. L. Altman, Microwave circuits. (Van Nostrand, New York, 1964). 
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+(1981).
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+Kobayashi, Appl. Phys. Lett. 118 (26), 264002 (2021).
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+Suter, J. Grimmel, S. Hengesbach, M. Förtsch, K. Nakamura, H. Sumiya, S. Onoda, J.
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+IEEE Sens. J. 19 (2), 451-456 (2019).
+
+1 
+Supporting Information 
+Variable bandwidth, high efficiency microwave resonator 
+for control of spin-qubits in nitrogen-vacancy centers 
+Anton Savitsky, Jingfu Zhang and Dieter Suter 
+Faculty of Physics, Technical University Dortmund, Otto-Hahn-Str. 4a, 44227 Dortmund, 
+Germany 
+Content: 
+S1 Optimization of the resonator parameters 
+S1.1 Optimization of Ω-type loop parameters 
+S1.2 Calculation of the coupling parameter for width step coupling 
+S1.3 Performance comparison of the resonator with transmission, reflection line 
+S2. Mw magnetic field distribution at z = 100 µm 
+S3. Radio frequency performance 
+S4 Fluorescence collection efficiency and optical resolution 
+S5. Experimental details 
+S6 Frequency tuning capability 
+
+2 
+ 
+S1 Optimization of the resonator parameters 
+In this part we consider some important points for the resonator optimization based on the first 
+principals and simulations. In the first part we consider transmission line with Ω-type loop 
+which allows to optimize parameters of the loop. In the second part we compare the 
+performance of resonator with transmission and reflection line and point out the ways for 
+performance optimization. 
+S1.1 Optimization of Ω-type loop parameters 
+The loop of diameter ������������ integrated in the gap of a terminated microstrip transmission line can 
+be considered as an approximation of the ideal classical current loop. This approximation is 
+valid when (i) the loop length is much smaller than a quarter of the wavelength (������������������������ ≪ ������������������������/4); 
+(ii) the width of the loop is much smaller compared to it diameter, (iii) the shielding effects can 
+be neglected; and (iv) the loop-gap discontinuity is neglected. The EM-wave of power ������������������������������������ in 
+the microstrip line with ������������������������ generates an alternating electric current with the amplitude 
+������������������������������������������������������������ = �2������������������������������������
+������������������������
+. 
+This alternating current induces a magnetic field whose amplitude at the center of the loop is  
+������������(0,0,0) = ������������������������������������������������������������
+������������
+= 1
+������������ �2������������������������������������
+������������������������
+ . 
+The loop conversion efficiency 
+������������0 = ������������(0,0,0)
+�������������������������������������
+= 1
+������������ � 2
+������������������������
+ 
+can be increased either by decreasing the loop diameter or by lowering the line impedance. The 
+line impedance of ������������������������ = 50 ٠is, however, generally fixed to avoid mismatch with microwave 
+components. This results in ������������0 = 500 
+������������
+������������√������������ for the loop of ������������ = 400 μm used here. This value 
+defines the physical upper limit of the magnetic field amplitude reachable for a transmission 
+line.  
+The real Ω-loop differs from the ideal loop. There are several parameters which influence the 
+magnetic field amplitude and its distribution: (i) loop width, (ii) gap width and (iii) thickness 
+of the dielectric substrate. The last factor in our system is, however, defined by the necessity to 
+provide the optical access to the diamond surface and will be not considered here, i.e. we fix 
+the substrate and optical access parameters defined in the main text, see Fig. 1 (main text), and 
+consider only the first two parameters. 
+
+3 
+ 
+Loop conductor width. The width of the loop is of critical importance as it influences not only 
+magnetic field parameters but also the heating effects. Figure S1(a) shows the dependence of 
+the conversion efficiency on the loop width. As expected, the conversion efficiency decreases 
+with increasing loop width. The maximum efficiency is about a factor 1.6 smaller than in the 
+ideal loop due to the effect of the gap and metal shield on the back side of the structure. The 
+conductive power losses, ������������������������������������������������������������ =
+������������������������������������������������������������
+2
+2
+∙ ������������������������������������������������������������, in the loop also decrease, see Fig S1(b), due to 
+the decrease of the loop resistance, ������������������������������������������������������������. The optimal loop width of ������������������������ = 0.1 mm is estimated 
+by considering the maximum of the function ������������0
+2/������������������������������������������������������������  , see Fig S2. The conversion efficiency 
+drops to 245 
+������������
+������������√������������, i.e. a factor 2 smaller than for the ideal loop. The loop width of 0.1 mm also 
+guarantees better mechanical stability and stability of the system parameters to manufacturing 
+tolerances. 
+ 
+Figure S1. (a) Conversion efficiency and (b) loop ohmic losses calculated for different loop 
+widths and ������������������������ = 50 ٠transmission line feeded with 1 W mw power at 3 GHz. The 
+gap width is G =0.1 mm. The dashed line marks the value ������������������������  chosen for 
+fabrication. 
+ 
+Figure S2. Ratio of power conversion efficiency to conductive losses in the loop. 
+
+40
+35
+30
+7
+25
+20
+15
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+W./mm340
+9
+............................
+b
+a
+320
+8
+上
+300
+7
+280
+mW
+6
+260
+50
+A.m
+240
+loss
+.........................
+P
+4
+220
+200
+3
+180
+2
+160
+1
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+W. /mm
+W./mm4 
+ 
+In order to estimate the heating effects we have calculated the temperature distribution in the 
+transmission resonator continuously fed by mw power, see Fig. S3. For ������������������������ = 0.1 mm the 
+maximum temperature calculated in the system is about 0.3 K (0.1 W) and 3.5 K (1 W) starting 
+from the ambient temperature of 293 K. In contrast, the system with ������������������������ = 0.018 mm shows 
+higher temperature increase of 0.6 K (0.1 W) and 5.8 K (1 W excitation) due to the higher 
+ohmic resistance and lower thermal conductance of the loop. 
+ 
+Figure S3. Temperature distribution calculated for the transmission resonator including the 
+diamond crystal (2×2×1 mm3) with geometrical parameters given in Fig. 1 (main 
+text). CW input mw power was set to 0.1 W.  
+Gap width. Figure S4 shows the efficiency parameter as a function of the gap width. Increasing 
+the gap width from 0.005 mm to 0.3 mm (������������ = 0.75 ∙ ������������������������) has a small effect on the resonator 
+efficiency (magnetic field amplitude at the loop center) as well as the field distribution in y < 0 
+loop region. Therefore, for our resonator we have chosen G = 0.1 mm for manufacturing ease. 
+ 
+Figure S4. Conversion efficiency calculated for different gap widths G and ������������������������ = 50٠
+transmission line fed with 1 W mw power at 3 GHz. The loop width is 
+������������������������= 0.1 mm. 
+ 
+ 
+
+K
+293.3
+y=o
+293.2
+293.1
+293.0250
+246
+/ A.m".W-
+242
+238
+234
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+G /mm5 
+ 
+S1.2 Calculation of the coupling parameter for width step coupling 
+The relation between the properties of the coupling element and the resonator coupling 
+parameter can be derived in two following steps. 
+ 
+Figure S5. (a) Reflection line with Ω-type loop. (b) Transmission line with width step and 
+Ω-type loop. 
+Reflection line. The bandwidth of reflection line, Fig S5(a), can be determined considering 
+the frequency dependence of magnetic field in the loop center: 
+������������(������������)
+������������(������������0 = ������������������������
+������������ )
+= sin �2������������������������ ������������������������
+4������������� = 1
+√2
+  
+The magnetic field amplitude is reduced by √2 at ������������ = ������������0/2 which results in the bandwidth  
+������������������������1/2 = ������������0 
+The reflection line can be also considered as overcoupled resonator with the bandwidth of 
+������������������������1/2 = 1 + ������������
+������������0
+∙ ������������0 
+Thus, the coupling coefficient of the reflection resonator with ������������������������ = ������������������������ is given by 
+������������ = ������������0 − 1 ≈ ������������0 (������������������������������������ ������������0 ≫ 1) 
+The same result can be also derived from the magnetic field amplitude than considering 
+reflection line as the reflection resonator 
+2�������������
+1 + ������������ �������������0 = 2 ⟹ ������������� = �������������0 + �������������0 − 4
+2
+⟹  ������������ ≈ ������������0 
+Generally the result is quite obvious than consider the definition of the coupling coefficient 
+������������ = ������������0
+������������������������
+ 
+
+2/4
+2/4
+2/4
+2/4
+Z,ZR
+ZR ZR6 
+ 
+Thus, the external quality factor in reflection line ������������������������ = 1 or full power is dissipated in 
+external load than neglecting resonator losses, ������������0 ≫ 1. 
+Transmission line with width step. This system can be considered as transmission resonator, 
+see Fig. S5(b). The reflection coefficient at ������������ = ������������0 of this system with ������������′ = ������������0 − 1 is given 
+by 
+������������ = − 1 − ������������ + ������������′
+1 + ������������ + ������������′ = − −������������ + ������������0
+������������ + ������������0
+⇒ ������������ = ������������0
+Γ − 1
+Γ + 1 
+Taking into account that, the reflection coefficients of width step discontinuity is 
+Γ = − ������������������������ − ������������������������
+������������������������ + ������������������������
+ 
+one obtains 
+������������ = ������������0
+������������������������
+������������������������
+ or ������������������������ = ������������������������
+������������������������
+ 
+Thus, the bandwidth of the resonator can be calculated 
+������������������������������������������������������������������������������������������������������������������������: 
+������������������������1/2
+������������0
+= 1 + ������������
+������������0
+= 1
+������������0
++ ������������������������
+������������������������
+ 
+������������������������������������������������������������������������������������������������������������������������������������������������: 
+������������������������1/2
+������������0
+= 1 + 2������������
+������������0
+= 1
+������������0
++ 2 ������������������������
+������������������������
+ 
+For the resonator design in the main text (������������������������ = 50 Ω;  ������������0 = 74)  the above equation yields the 
+estimate value of the coupling parameter of ������������ = 12.6, than assuming the resonator impedance 
+to be equal to that of the transmission microstrip line with width ������������������������. The values is in a good 
+agreement with coupling parameter of 11.5 evaluated from simulated S-parameters. 
+S1.3 Performance comparison of the resonator with transmission, reflection line 
+The conversion efficiency for transmission, reflection resonator with ideal loop can be 
+calculated using following relations 
+������������������������������������������������������������������������������������������������������������������������: ������������0 = 2�������������
+1 + ������������ ∙ �������������0 ∙ 1
+������������ � 2
+������������������������
+ 
+������������������������������������������������������������������������������������������������������������������������������������������������: ������������0 = 2�������������
+1 + 2������������ ∙ �������������0 ∙ 1
+������������ � 2
+������������������������
+ 
+They take into account (i) power flow to resonator (coupling); (ii) the energy storage in the 
+resonator (quality factor); and (iii) the increase of the loop current due to decrease of the 
+resonator impedance. It is important that for high coupling coefficients (������������ = ������������0
+������������������������
+������������������������ ≫ 1) both 
+factors become independent of resonator quality factor, see Table S1. Under this condition the 
+
+7 
+ 
+increase of the resonator efficiency as compared to the transmission line can be estimated from 
+the ratio of the resonator and feed line impedances, i.e. 
+������������������������
+������������������������ for transmission and 2∙
+������������������������
+������������������������ for 
+reflection. For instance, the resonator having ������������������������ = 10 ٠is factor 5 more efficient compared to 
+transmission line having the same loop, i.e. it requires 25 times less mw power to obtain the 
+same magnetic field amplitude. We note that ������������������������ can be approximated by the impedance of the 
+microstrip line with width ������������������������ only than the discontinuity contribution to impedance can be 
+neglected. This generally holds for ������������������������ ≪ ������������ . In general case ������������������������  have to include the 
+discontinuity contribution which is best evaluated from numerical simulation. It is important to 
+note that then the heating effects become the limiting factor, the excitation mw power have to 
+be limited due to ������������������������������������������������������������ ∝ ������������������������������������������������������������
+2
+ and, as the result, the same maximum magnetic field is reachable 
+in all cases. 
+Table S1. Analytical expressions for efficiency and the bandwidth of the resonator and non-
+resonating systems containing ideal current loop. 
+Case 
+������������0 
+������������������������1/2
+������������0
+ 
+������������0 
+������������������������1/2
+������������0
+ 
+Transmission 
+line 
+1
+������������ � 2
+������������������������
+ 
+������������������������������������������������ 
+1
+������������ � 2
+������������������������
+ 
+������������������������������������������������ 
+Reflection 
+line 
+2 ∙ 1
+������������ � 2
+������������������������
+ 
+1 
+2 ∙ 1
+������������ � 2
+������������������������
+ 
+1 
+Transmission 
+resonator 
+2�������������
+1 + 2������������ �������������0
+1
+������������ � 2
+������������������������
+ 
+1 + 2������������
+������������0
+ 
+������������������������
+������������������������
+∙ 1
+������������ � 2
+������������������������
+ 
+2 ������������������������
+������������������������
+ 
+Reflection 
+resonator 
+2�������������
+1 + ������������ �������������0
+1
+������������ � 2
+������������������������
+ 
+1 + ������������
+������������0
+ 
+2������������������������
+������������������������
+∙ 1
+������������ � 2
+������������������������
+ 
+������������������������
+������������������������
+ 
+ 
+Table S2 summarizes the parameters of the transmission resonator calculated for different ������������������������ 
+and ������������ values. At large ������������������������’s the contribution of the loop and gap to resonator impedance 
+overcomes the impedance contribution of the microstrip line. Thus, it becomes the limiting 
+factor determining resonator performance, i.e. conversion factor and the bandwidth. 
+ 
+ 
+
+8 
+ 
+Table S2. Parameters of transmission resonator calculated for different ������������������������ and ������������, for definition 
+see Fig. 1 in main text. 
+Case a) 
+������������0 
+A/m/√W 
+������������0
+������������/������������0
+������������������������ b) 
+������������ c) 
+Q0 c) 
+������������������������1/2
+������������0
+ 
+������������������������
+d) 
+Ω 
+������������ e) 
+Ω 
+������������������������ = 1 ������������������������ 
+������������ = 24.5 ������������������������ 
+576 
+2.35 
+32.9  
+92  
+0.73 
+21.3 
+21.6 
+������������������������ = 2 ������������������������ 
+������������ = 22 ������������������������ 
+930 
+3.79 
+14.67 
+79 
+0.38 
+13.2 
+12.1 
+������������������������ = 3 ������������������������ 
+������������ = 17 ������������������������ 
+1170 
+4.80 
+11.50 
+74 
+0.32 
+10.4 
+8.4 
+������������������������ = 4 ������������������������ 
+������������ = 15 ������������������������ 
+1300 
+5.33 
+8.77 
+72 
+0.26 
+9.4 
+6.4 
+������������������������ = 5 ������������������������ 
+������������ = 13 ������������������������ 
+1379 
+5.65 
+7.90 
+69 
+0.24 
+8.8 
+5.2 
+������������������������ = 5.6 ������������������������ 
+������������ = 11.2 ������������������������ 
+1386 
+5.68 
+7.65 
+72 
+0.23 
+8.8 
+4.7 
+������������������������ = 6 ������������������������ 
+������������ = 10 ������������������������ 
+1422 
+5.82 
+7.54 
+62 
+0.26 
+8.6 
+4.4 
+a) ������������ was adjusted to obtain the resonance frequency of 2.95±0.05 GHz 
+b) Resonator efficiency relative to the efficiency of the transmission line (245 A/m/√W) 
+c) Evaluated from S11 and S21 curves 
+d) Resonator impedance calculated as ������������������������ = ������������������������/(������������0
+������������/������������0
+������������������������) 
+e) Impedance of microstrip line with width ������������������������. 
+ 
+S2. MW magnetic field distribution at z = 100 µm 
+Figure S6 shows the mw magnetic field distribution at ������������ = 100 µm above the loop, i.e. about 
+the highest optically accessible position with our optical objective with 340 µm working 
+distance. Compared to ������������ = 10 µm (Fig. 3 main text) the field homogeneity is improved. Over 
+the full optically accessible area the maximum and minimum field magnitudes of 1020 ������������/������������ 
+and 735 ������������/������������ are calculated, i.e. 735 ������������/������������ < |������������(������������, ������������, 100������������������������)| < 1020 ������������/������������ (for comparison 
+1070 ������������/������������ < |������������(������������, ������������, 10������������������������)| < 2270������������/������������). For the loop center and area accessible by the 
+nanopositioner (70×70µm2) the field homogeneity is even better, i.e. 832  ������������/������������  <
+|������������(������������, ������������, 100������������������������)| < 908 ������������/������������ (1085 ������������/������������ < |������������(������������, ������������, 10������������������������)| < 1256 ������������/������������). The magnetic 
+field directivity is not as perfect as at small elevations but still sufficiently high to be neglected, 
+especially in the loop center. The fields distribution compares favorably to that reported 
+previously for planar loop-gap resonator [1]. 
+
+9 
+ 
+ 
+Figure S6. (a) Calculated microwave magnetic field magnitude, |H|, at ������������ = 100 µm for a mw 
+power of 1 W. The dotted and dashed circles mark the position of the inner loop edge 
+(������������������������ =  400µ������������) and the optical access hole (������������������������������������������������ = 300 µm), respectively. (b,c,d) 
+The ������������������������, ������������������������, ������������������������ amplitudes of the mw magnetic field components as functions of 
+x,y,z for mw power of 1 W. The black dotted lines mark the optically accessible 
+diamond area. Shadowed area in (d) shows the position region of the optical objective 
+(������������ <  0) and NV-center (������������ >  0). The dashed curve in (d) shows the best fit curve 
+of ������������������������(0,0, ������������) amplitude to the function in Eq. (3) in the main text. 
+S3. Radio frequency performance 
+In experiments where Radio frequency (RF) is used to drive nuclear spins, the corresponding 
+signals can be fed into the same resonator. Figure S7 shows the power-to-field conversion 
+efficiency of the different types for RF signals for the frequency range < 200 MHz. The 
+structures (transmission line, mw resonator) are non-resonant at RF fields and therefore 
+relatively broad-band. In transmission, the efficiency does not depend on the frequency. The 
+reflection resonator performs poorly at low frequencies and should not be used under such 
+conditions. 
+
+a
+A/m
+G
+1000
+c
+1000-
+12.6
+/ A/m
+800
+500
+6.3
+H(0,y,100μm)
+600
+200-
+400
+y4
+200
+1.
+-0.15
+-0.1
+-0.05
+0
+0.05
+0.1
+0.15
+0.2
+y /mm
+1000
+1200
+b
+objective
+d
+1000
+800
+H(x,0,100μm) / A/m
+HH:
+/ A/m
+800
+..................................
+600
+HH
+(z'00)"H
+600
+400
+400
+NV-center
+200
+200
+0.15
+-0.1
+-0.05
+0
+0.05
+0.1
+0.15
+0.2
+-0.2
+0
+0.2
+0.4 
+0.6
+0.8
+/mm
+z /mm10 
+ 
+ 
+Figure S7. (a) Simulated ������������21 (upper trace) and ������������11 (lower trace) of the transmission resonator 
+with diamond for the radio frequency range. (b) The frequency dependence of the 
+magnetic field amplitude ������������������������(0,0,10 μm) calculated for the transmission line and 
+the resonator in reflection and transmission modes.  
+S4. Fluorescence collection efficiency and optical resolution 
+The fluorescence collection efficiency for an NV center as the point emitter and neglecting 
+reflections on diamond-oil interface is  
+������������������������������������ = 1 − cos ������������
+2
+, 
+The light cone opening angle, ������������, of the optical objective employed in this work (Zeiss, 
+Apochromat 100 with ������������������������ = 1.4; immersing oil Immersol 518 F with ������������ =  1.518) is ������������ =
+asin(
+������������������������
+������������ ) = 67°. Thus, the optically unrestricted objective is capable of collecting 30% of the 
+emitted photons. The diameter of the optical access hole required for this efficiency would be 
+������������������������������������������������ = 0.78 ������������������������ (loop inner diameter ������������������������������������ = 0.88 ������������������������) for NV centers close to the diamond 
+surface and a resonator thickness (substrate and copper cladding) of 0.166 mm, see Fig. S8(c). 
+In this work we decided for ������������������������������������������������ = 0.3 ������������������������ allowing for ������������ = 42°, see Fig. S8(a), in order to 
+optimize the microwave performance of the resonator. The collection efficiency, however, 
+becomes potentially reduced by factor of 2.4 as compared to the maximum possible for this 
+objective. Increasing the optical access hole diameter will improve the collection efficiency, 
+see Fig. S8(b) at the cost of (i) reduced power to magnetic field conversion efficiency, see Table 
+S1; (ii) increased heating; (iii) increased effect of the objective on the properties of the resonator 
+(resonance frequency and magnetic field distribution). These factors have to be taken into 
+
+0
+300
+a
+transmissionresonator
+-0.1
+-0.2
+250
+/dB
+transmissionline
+-0.3
+s
+-1/2
+0.4
+200
+-0.5
+150
+-15
+-20
+100
+/dB
+reflectionresonator
+-25
+S
+-30
+50
+b
+-35
+40
+0
+40
+80
+120
+160
+0
+40
+80
+120
+160
+0
+200
+200
+frequency/MHz
+frequency /MHz11 
+ 
+account when optimizing the resonator geometry for experiments which require high collection 
+efficiency. 
+ 
+Figure S8. Fluorescence optical pathway for (a) the resonator employed in this work and (c) 
+resonator configuration required for maximum fluorescence collection efficiency. 
+(b) Dependence of fluorescence collection efficiency on diameter of optical access 
+hole relative to that capable by objective. 
+Optical resolution. As pointed above the resonator acts as a diaphragm reducing numerical 
+apperture of the objective and, as the result, limiting the optical resolution. The effect is, 
+however, weaker compared to the loss of the fluorecence collection efficiency. Figure S9(a) 
+shows the fluorecence image of the single NV-center recorded with the resonator setup. 
+Evaluation of the fluorecence spatial distribution yields the FWHM of recordings about 300 nm, 
+Fig. S9(b). This number corresponds to 300������������������������
+sin (42°)
+sin(67°) = 220 ������������������������  for objective without 
+diaphragm, which is close to the number calculated for Airy pattern of a point emmiter 
+0.51������������
+������������������������ =
+0.51∙700������������������������
+1.4
+= 255 ������������������������. 
+ 
+Figure S9. Optical resolution. (a) A fluorescence image of the single NV center over a scanning 
+range of 1μm × 1 μm. (b) Fluorescence in dependence on x- and y-positions. The 
+solid lines show the fits to gaussian functions. The increase of FWHM as compared 
+to objective limit is attributed to the effect of resonator optical access hole. 
+
+a
+b
+Dent = 0.3 mm
+,=0.78mm
+0.9 
+0.8
+0.7
+0.6
+0.5
+0.4
+0.3
+0.35
+0.4
+0.45
+0.5
+0.55
+0.6
+0.65
+0.7
+0.75
+0.8
+D. /mm0.5
+a
+/rel.u.
+b
+0.4
+0.9
+0.8
+0.3
+0.8
+fluorecence/
+fluorecence /rel.u.
+0.6
+0.2-
+0.7
+0.4
+0.1 -
+0.6
+wn/
+0.2
+FWHM=290±10nm
+0
+0.5
+-0.1 -
+fluorecence /rel.u.
+0.4
+0.8
+-0.2
+0.3
+0.6
+-0.3
+0.2
+0.4
+-0.4 -
+0.1
+0.2
+0.5
+FWHM=320±10nm
+-0.5 -0.4 -0.3 -0.2 -0.1
+0
+0.1 0.20.3 0.40.5
+0.
+0.5
+-0.4
+-0.3
+0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+x /um
+x,y /um12 
+ 
+The simple geometrical approach employed here can be extended for analysis of these 
+parameters at any position within the optically accessible area. However, this approach neglects 
+additional effects like reflections on the diamond-oil interface, which should be considered for 
+a more precise analysis. 
+S5. Experimental details 
+The experiments were performed on a home-built setup, whose schematic is shown as Fig. S9. 
+Single NV centers in diamond can be optically addressed, initialized and detected with a 
+confocal microscope. Here we used a diode-pumped solid state continuous wave laser with a 
+wavelength of 532 nm (marked in green in the schematic) for the optical excitation. For pulsed 
+experiments, we use an acousto-optical modulator (AOM) to generate pulses from the 
+continuous wave laser beam or a pulsed diode laser. The microscope objective is fixed to the 
+nanopositioning system that scans the sample in three dimensions. The fluorescence light 
+(marked in red in the schematic) is also collected by this MO lens and passes through the 
+dichroic mirror to the avalanche photodiode (APD) detector, while the scattered laser light is 
+reflected by the dichroic mirror. 
+ 
+Figure S10. Schematic of the optical part of the setup. A confocal microscope is utilized for 
+initializing and detecting single NV centers. 
+ 
+
+Diamond
+MW source
+Resonator
+Objective
+xyz-Nanopositione
+Laser
+AOM
+Dichroic mirror
+APD13 
+ 
+ 
+Figure S11. Photographs of the experimental setup.  
+ 
+ 
+Figure S12. Photographs of (a) loop region of the resonator and (b) the resonator loop region 
+with a diamond mounted. 
+ 
+Figure S13. The principal microwave schemes for transmission and reflection modes. The 
+components are: (1) cw microwave source; (2) PIN-diode modulator; (3) power 
+mw amplifier; (4) circulator; (5) fast microwave diode detector for power and 
+matching control. 
+
+a
+permanent magnet
+resonancestructure
+b
+withdiamondmounted
+mwout
+objective
+mw.in
+x,y,z stage600um
+300μm
+b
+atransmission
+本
+5
+大
+?IFTV
+5
+4
+3
+2
+1
+reflection
+本
+214 
+ 
+ 
+Figure S14. Pulse sequence for measuring Rabi oscillation. The first laser pulse of 5 µs 
+initializes the NV into the ground state |g〉. The MW pulse of variable length, ������������������������, 
+flips the NV to cos��������������������������������������������������������������|������������⟩ + sin��������������������������������������������������������������|������������⟩,where |������������⟩ denotes one excited 
+state. The second laser pulse of 0.4 µs probes the population of ground state |������������⟩, 
+proportional to the FL intensity. The pulse repetition rate was generally set to 
+6400 s-1. 
+S6 Frequency tuning capability 
+In this part we consider the options for tuning the resonance frequency. This will be relevant 
+for experiments in different magnetic fields or on other types of optically active spin centers. 
+The half-wave mode (������������������������/2) resonance frequency of a microstrip of length ������������ without current 
+loop is given by the well-known expression 
+������������0 =
+������������
+2������������ ∙ �������������������������������������
+ 
+where ������������������������������������ is the effective dielectric constant of the microstrip. For a microstrip of width ������������, 
+larger than the dielectric thickness ������������, it is given by 
+������������������������������������������������ = ������������������������ + 1
+2
++ ������������������������ − 1
+2
+1
+�1 + 12 ������������
+������������
+ 
+The effective dielectric constant of ������������������������������������������������ =2.81 is calculated for microstrip with 
+������������ = 0.13 mm, ������������ = 3 mm and ������������������������ = 3 used in this work. Thus, for the fixed geometry of the 
+resonator holder (see Fig 1, main text) the resonance frequency of the microstrip can be tuned 
+from 1.6 GHz (������������ = 56mm) to 18 GHz (������������ = 5mm), see Fig. S15(a). The validity of the analytic 
+approach is also valid for the full model as the resonance frequency from EM calculations (red 
+line, Fig. S15(a)) agrees well with the analytical prediction (dashed line in Fig. S15(a)). 
+The blue line in Fig. S15(a) shows the resonance frequency of the microstrip resonator 
+including the current loop in transmission mode. The additional gap and the current loop 
+increase the intrinsic capacity and inductance of the microstrip. Thus, the resonance frequency 
+is lower than for a microstrip of the same length. The resonance frequency can be tuned between 
+1.3 GHz (������������ = 56mm) and 5.7 GHz (������������ = 5mm). It is important to note that the conversion 
+
+Laser
+532nm
+t
+MW
+FL
+readout15 
+ 
+efficiency does not strongly depend on the resonance frequency, see Fig. 15(b). A slight 
+efficiency improvement is observed at lower frequencies, accompanied by a reduction of the 
+resonator bandwidth. 
+ 
+Figure S15. (a) Dependence of the resonance frequency on the length ������������, (diamonds, blue line) 
+calculated for transmission mode. All other geometrical parameters are fixed as 
+given in the caption of Figure 1 (main text). The red line with circles shows the 
+resonance frequency of the microstrip without loop obtained from EM simulations 
+for transmission mode. The black dashed line shows the analytically calculated 
+resonance frequency of the microstrip. (b) Conversion efficiency calculated for 
+resonators of selected lengths. 
+Thus, the resonator configuration described in this work can be used for experiments with 
+microwave excitation over the frequency range of 1 GHz to 6 GHz. The range can be extended 
+to lower frequencies by further increase of the resonator length or by increasing the dielectric 
+constant of the substrate. Figure S16 shows the parameters of identical resonators with 
+L = 50 mm calculated for ε = 3 (used in this work) and ε = 10. The ratio of the resonance 
+frequencies 1.44 GHz/0.78 GHz = 1.85 agrees well with the ratio of the effective dielectric 
+constants �9.15/2.81=1.8. The large reduction in the bandwidth (300 MHz vs. 110 MHz) can 
+be  improved by decreasing the width of the resonator, ������������������������, which would lead to an increase of 
+the coupling coefficient (see section S1.3). 
+
+a
+1600
+L= 50 mm
+b
+N
+1400
+L=25mm
+frequency
+1200
+L=17mm
+1000
+A·m
+L= 8 mm
+resonance
+800
+600
+2
+400
+200
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+55
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4.5
+5
+5.5
+L /mm
+frequency /GHz16 
+ 
+ 
+Figure S16. (a,c) S-parameters of identical transmission mode resonators with length 
+������������ = 50 mm calculated for substrates having ε = 3 (a) and ε = 10 (c). All other 
+geometrical parameters are fixed as given in the caption of Figure 1 (main text). 
+(c,d) Corresponding conversion efficiencies. Resonator bandwidths and key 
+parameters are indicated. 
+An increase of the resonance frequency above 6 GHz is difficult because substrates with 
+dielectric constant < 2 are not available and a further decrease of the resonator length below 
+������������ ≤ 2 ∙ ������������������������ is not feasible, see Table S2. The resonator, however, can still be used at higher mw 
+frequencies if higher resonance modes are considered. Figure S17 shows S11 and efficiency 
+parameters of the resonator (see Fig. 1 main text) extended to higher mw frequencies, where 
+additional resonances are observed. Of particular interest is the resonance at about 11 GHz 
+which corresponds to the ~
+3
+2 ������������������������ resonance mode of the microstrip. At this mode the conversion 
+efficiency (600 A/m/√W) is reduced by factor 1.95 as compared to the main mode (1170 
+A/m/√W) but it is still about factor 2.5 higher compared to the 50 Ω microstrip with current 
+loop (245 A/m/√W). We note here that for a microstip resonator without current loop in 
+3
+2 ������������������������ 
+mode one would expect a reduction of the magnetic field amplitude in the center by a factor √3 
+compared to the fundamental 
+1
+2 ������������������������ mode. This is due to an additional resonance mode at 
+10.4 GHz, marked by * in Fig. S17. This mode generates only small magnetic fields at the 
+
+1600
+b
+1500
+5
+1400
+/dB
+-1/2
+10
+1.W
+1300
+S
+300 MHz
+-15
+1000
+-20
+a
+900
+L=50mm,W,=0.3mm,=3,r(W,=3mm)=2.81
+-25
+1.25
+1.35
+1.55
+1.65
+800
+1.2
+1.3
+1.4
+1.45
+1.5
+1.6
+1.7
+1.2
+1.25
+1.3
+1.35
+1.4
+1.45
+1.5
+1.55
+1.6
+1.65
+1.7
+frequency /GHz
+frequency /GHz
+2400
+d
+-2
+2200
+-4
+2000
+/dB
+1800
+-8
+110 MHz
+1600
+S
+-10
+-12
+-14
+1200
+-16
+c
+1000
+L = 50mm, W, = 0.1mm,  = 10, (Ws= 3mm)= 9.15
+-18
+800
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+0.65
+0.7
+0.75
+0.8
+0.85
+0.9
+freguency/GHz
+freguency/GHz17 
+ 
+position of the current loop and   is therefore not useful. Thus, resonator designs operating at 
+higher modes would require additional optimization steps in order to separate active from 
+inactive modes. 
+ 
+Figure S17. Simulated S11 parameter and conversion efficiency of the transmission mode 
+resonator described in Fig. 1 (main text) over a wider frequency range. The dashed 
+line indicates the conversion efficiency of a 50 Ω microstrip line containing a 
+current loop. The asterisk marks the frequency position of a magnetically inactive 
+mode. 
+1. 
+K. Sasaki, Y. Monnai, S. Saijo, R. Fujita, H. Watanabe, J. Ishi-Hayase, K. M. Itoh and E. 
+Abe, Rev. Sci. Instrum. 87 (2016) 053904. 
+ 
+
+1200
+1000
+-5
+800
+-10
+Mi-
+/dB
+600
+-15
+S
+400
+-20
+?
+200
+-25
+0
+-30
+1
+2
+3
+4
+5
+6
+<
+8
+9
+10
+11
+12
+frequency/GHz
\ No newline at end of file
diff --git a/ldE2T4oBgHgl3EQfeAcm/content/tmp_files/load_file.txt b/ldE2T4oBgHgl3EQfeAcm/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..ede6574d5a17d5df2951d7715bf88019f5b6a7eb
--- /dev/null
+++ b/ldE2T4oBgHgl3EQfeAcm/content/tmp_files/load_file.txt
@@ -0,0 +1,1459 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf,len=1458
+page_content='1  Title  Variable bandwidth, high efficiency microwave resonator  for control of spin-qubits in nitrogen-vacancy centers  Authors  Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Anton Savitsky, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Jingfu Zhang, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Dieter Suter  Faculty of Physics, Technical University Dortmund, Otto-Hahn-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 4a, 44227 Dortmund,  Germany  Keywords  Planar microresonator, NV center, spin qubit, microwave magnetic field  2  Abstract  Nitrogen-Vacancy (NV) centers in diamond are attractive tools for sensing and quantum  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Realisation of this potential requires effective tools for controlling the spin degree  of freedom by microwave (mw) magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' In this work we present a planar microwave  resonator optimised for microwave-optical double resonance experiments on single nitrogen-  vacancy (NV) centers in diamond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' It consists of a piece of wide microstrip line which is  symmetrically connected to two 50 Ω microstrip feed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' In the center of the resonator, an  Ω-  shaped loop focuses the current and the mw magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' It generates a relatively  homogeneous magnetic field over a volume of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='07mm2×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' It can be operated at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='9 GHz  in both transmission and reflection modes with bandwidths of 1000 MHz and 400 MHz,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The high power-to-magnetic field conversion efficiency allows to produce π-  pulses with a duration of 50 ns with only about 200 mW and 50 mW microwave power in  transmission and reflection, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The transmission mode also offers capability for  efficient radio frequency excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The resonance frequency can be tuned between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3 GHz  and 6 GHz by adjusting the length of the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This will be useful for experiments on NV-  centers at higher external magnetic fields and on different types of optically active spin centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Introduction  The NV center in diamond is used in various fields, such as quantum information, quantum  sensing, magnetometry, bioimaging, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 1-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' In all applications, efficient manipulation of the  electron spin is an essential prerequisite, both for continuous wave (cw) experiments and for  fast spin control in pulsed experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Therefore, maximising the coupling between the control  microwave (mw) magnetic field and the electron spins is a fundamental concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Here, we  consider specifically applications where single NV centers are excited by a laser and the spin  state is read out by collecting photoluminescence (PL) through a microscope objective with  large numerical aperture (NA > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Measuring a cw ODMR spectrum in such requires a mw  magnetic field of ������������ = 150 A/m (������������ = ������������0������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='19 mT) to reach the maximum fluorescence  contrast 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This field corresponds to a Rabi frequency of ������������������������ = 2 MHz if the oscillating magnetic  field is perpendicular to the NV axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' In pulsed experiments, typical π-pulse durations of ������������������������ =  50 ns are used, which correspond to an excitation bandwidth of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2/������������������������ = 24 MHz (������������������������ =  10 MHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This is sufficient for complete electron spin flip of the selected triplet spin transition  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This pulse length requires magnetic field amplitude of ������������ = 800 A/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Currently, most experimental setups rely on a wire positioned over the sample to achieve such  mw magnetic field amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' If this wire is placed in a short gap of a transmission line with  3  ������������������������ =50 ٠impedance, the electric current in the wire has an amplitude ������������ = �2������������������������������������/������������������������ =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2 A/√W ∙ �������������������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This current produces the tangential magnetic field ������������������������������������������������������������ =  ������������  2������������������������ outside the  wire of radius ������������0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' For a typical wire diameter of 20 µm, the maximum magnetic field generated  at the wire surface is ������������������������������������������������������������(������������0) = 3200  ������������  ������������ at 1W of mw power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The wire is, however, optically  opaque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Therefore, only NV-centers at a distance > √2������������0 from the center of the wire center can  be optically probed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' At this position, the magnetic field is reduced to 2600  ������������  ������������ at 1W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Even at  realistic distances from the wire of 40 µm, which still reduces the fluorescence collection  efficiency due to obstruction by the wire, pulsed experiments can be performed with 1 W mw  power, reaching �������������������������������������������������������������40 µm� = 800  ������������  ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Therefore, the microwire system is used in many  laboratories, including our laboratory7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' A major disadvantage is that the 1/������������ dependence of the  magnetic field amplitude significantly restricts the volume and the surface area available for  probing single NV-centers and requires precise initial positioning of the microscope objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Moreover, readjustment of the mw power settings is necessary for any new probed NV-center  which is not only due to the distance dependence of the magnetic field amplitude but also to the  magnetic field direction dependence on distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This problem can be overcome using a  microloop integrated into a transmission line instead of the microwire 8, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The magnetic field  amplitude in the center of the ideal loop is given by ������������������������������������������������������������ =  ������������  ������������ where ������������ is the loop diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Thus, the field of 1000  ������������  ������������ can be generated by a loop with 200 µm diameter at 1 W mw power,  which fulfills the requirements for ODMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Advantages of the loop include a good homogeneity  and directivity of the magnetic field over a 100 ×100 µm2 area, much higher than the microwire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Compared to a wire, the loop magnetic field is, however, less tolerant to the effect of the metal  case of the microscope objective, which must be positioned close to the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' We discuss  means to avoid this issue below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  There are several additional significant handicaps of loop, as well as wire, systems in  conjunction with transmission line, (i) the construction does not allow quick replacement or  reposition of the diamond spaceman;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (ii) significant heating effects (the resistance of 10 mm  copper wire with 20 µm diameter is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='25 Ω at 3 GHz, which leads to power dissipation at the  mw power level required for ODMR, but the thermal conductivity is limited due to the small  conductor cross-section);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (iii) substantial power return losses caused by the discontinuity in the  transmission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The main handicap is, however, the high mw powers required to fulfill the  magnetic field requirements for ODMR experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  4  The power limitation can be overcome using planar resonators to generate the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  In the past numerous reflection mode resonators were proposed based on different planar  structures, for instance double-split ring 10, 11, triple-split-ring 12, loop-gap 13, strip-line 14, and  several other types 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' All these resonators allow to store the mw energy for a time proportional  to the unloaded quality factor ������������0, which leads to increase of the current and, therefore, an  enhanced magnetic field amplitude proportional to �������������0  as compared to a non-resonant  structure with the same geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The resonator, however, introduces additional limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The  substantial magnetic field enhancement is achieved only within the resonator bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' For  instance, for a matched (critically coupled) reflection resonator with ������������0=100 would require 100  times less mw power at the resonance frequency of ������������0 = 3 GHz to generate the same mw  magnetic field amplitude as compared to the non-resonant structure of same geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This  effect is, however, only obtained within Δ������������1/2 = 2 ∙ ������������0  −1 ∙ ������������0= 60 MHz around the resonance  frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This would substantially limit its general applicability for ODMR on single NV-  centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The bandwidth of a matched resonator can only be increased by lowering the ������������0-value  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This, however, leads to lower mw power to magnetic field conversion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' it reduces the  usefulness of the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  The aim of this work is to develop a device that avoids these issues and can be used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=', for  ODMR spectroscopy of single NV centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' It overcomes high power requirements and  compromises limitations of transmission line systems as well as previously reported resonant  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' We design it to fulfill the following requirements: (i) easy and cheap fabrication with  reliable resonator parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (ii) compatibility with standard coaxial mw delivery system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (iii)  high mw power to magnetic field conversion efficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (iv) high magnetic field directivity and  spatial homogeneity at least in the area accessible by nanopositioners based on piezoelectric  actuator (100 ×100 µm2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (v) small mw magnetic field frequency dependence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' large  bandwidth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (vi) possibility for microwave and radio frequency excitations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (vii) high  mechanical and thermal stability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (viii) easy and reliable replacement or reposition of the  diamond specimen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' and (very important !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=') (ix) it must be compatible with high resolution  confocal objectives, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' tolerate the presence of dielectric and conductive parts in very close  vicinity (200 µm to 300 µm) of the magnetic fields of the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Results and discussion  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 Resonator design  Figure 1 depicts the design of the developed half-wave resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' It consists of a piece of wide  microstrip line symmetrically connected to two 50 Ω microstrip feed lines terminated by SMA  5  connectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' In the center of the microstrip, the Ω-loop concentrates the current and the mw  magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The geometrical parameters of the resonator investigated in this study are  summarized in the figure caption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The diamond is placed on the loop, as shown in Fig 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Thus,  the lower diamond surface is exposed to the mw magnetic field near the maximum amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  The resonator is fabricated using standard PCB lithography on low loss Rogers RO3003  laminate with 166 µm overall thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This design allows us to explore the diamond down to  least 100 µm above the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The resonator is designed for operation in transmission (the  output SMA is terminated by a 50 Ω load) or reflection (the output SMA is open to free space)  modes of operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (a) Full model of the microwave resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' It is fabricated on 60×26 mm2 Rogers  RO3003 low loss laminate ( ε������������ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='00 , tan ������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='001  at 10 GHz, dielectric  thickness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='13 mm and 18 µm copper cladding) mounted on a 60×26×1mm3 copper  holder and two standard PCB SMA flat tab connectors .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (b) Resonator design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The  geometrical parameters are: ������������������������ =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3 mm  - width of the microstrip feed line (������������������������ =  50 Ω);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' ������������������������ = 3 mm - width of the resonator, ������������ =  17 mm - length of the resonator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  ������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 mm - gap width, ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4 mm - inner diameter of the loop, ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 mm  width of the loop conductor, ������������������������������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3 mm - diameter of the optical access hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Coordinate system is indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' For optical pathway see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S8 in SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2 Transmission mode resonator  Figure 2 shows the S-parameters of the resonator in transmission mode calculated using CST  Microwave Studio and measured experimentally using a network vector analyzer (HP 8720A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  The analytical transmission and reflection coefficients of the symmetrically coupled  transmission resonator with unloaded quality factor ������������0 and resonance frequency ������������0 are given  by  ������������ = −  1−������������������������  1+2������������−������������������������ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  ������������ =  2������������  1+2������������−������������������������  (1)  where ������������ is the coupling parameter for both input and output and ������������ = ������������0 ∙ �  ������������0  ������������ −  ������������  ������������0� is the  normalized offset 16, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Analysis of the S-parameter traces depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 2 using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (1)  a  b  L  ★  Wr  G  W  V  X  X  W  D  opt6  yields the resonator parameters ������������0 = 74, β = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5 and ������������0 = 73, β = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8 for simulation and  experiment, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The relatively low unloaded quality factor is typical for microstrip  based resonators owing to relatively high conduction losses 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The comparison of the calculated  coupling parameter of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5 with ������������ = ������������0  ������������������������  ������������������������=12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6 which can estimated analytically for the  resonator shows that the Ω-loop slightly increases the impedance of the resonator over the  impedance of the microstrip line with width ������������������������ (for more information see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The central  frequency of the experimental resonator was downshifted by about 70 MHz compared to the  calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This deviation is ascribed to slightly different real parameters of the laminate  (dielectric constant and thickness) as well as manufacturing tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (a) Simulated and (b) experimental S-parameters of the transmission resonator with  diamond: ������������11 = 20 ∙ log10 |Γ| (blue line) and ������������21 = 20 ∙ log10 |������������| (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The  dashed lines show the best fit results of the S-curves to the reflection and transmission  coefficients given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The disagreement at the low and high frequencies is  due to the frequency dependence of the characteristic impedances, which are not  included in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (1), and contributions from other resonator modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  The resonator itself is described by the coefficient ������������2 = 1 − |Γ|2 − |������������|2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' the mw power  transmitted to the resonator to which the magnetic field amplitude is proportional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This factor  can be parametrized by the power coefficient at the resonance frequency and the resonator  bandwidth:  ������������(������������0) =  2�������������  1+2������������ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  ������������������������1/2 =  1+2������������  ������������0 ∙ ������������0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  (2)  Thus, the bandwidth of the resonator is 950 MHz and 990 MHz for calculation and experiment,  respectively, which is sufficient for most low-field applications of NV centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure 3 shows the spatial variation of the mw magnetic field behavior within the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' At a  small elevation above the loop surface (������������ =10 µm), the loop provides a perfect magnetic field  directivity over the optically accessible diamond area, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' the magnetic field is aligned with the  z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  At  ������������ = 10 μm  the  position  of  the  magnetic  field  minimum  5 F---  5  10  10  /dB  /dB  15  20  20  25  25  a  b  30  30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  freguency /GHz  freguency /GHz7  ������������(0, 70 µm, 10 µm) =1100 A/m is slightly shifted from the loop center due to the effect of the  gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The magnetic field amplitude increases by about a factor of 2 to the edge of optically  accessible area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Above the loop the decay of the magnetic field magnitude at the loop center is  well described by the decay function for an ideal current loop:  ������������(0,0, ������������) = ������������(0,0,0)  ������������3  (4������������2+������������2)  3  2  (3)  with ������������ = 440 µm > ������������������������ due to the large width of the loop conductor (������������������������ = ������������������������/4), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Below the loop the magnetic field decays rapidly and is smaller than 70 A/m at the closest  position of the microscope objective surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This guarantees the stability of the system during  operation as neither magnetic field distributions nor resonator parameters are influenced by the  microscope objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' At higher elevation above the loop the improvement of the magnetic field  homogeneity is accompanied by slight loss of the field directivity, see Fig S6 in SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Within the  optically accessible diamond area (±x, ±y, z)=( ±150, ±150, 0+100) µm the minimum and  maximum of the magnetic field magnitude are 2200 A/m and 735 A/m at 1 W mw power,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Thus, the magnetic field of the loop compares very favorably with that of a  microwire, both in terms of homogeneity and directivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (a) Color-coded plot of the calculated mw magnetic field magnitude, |������������|, at  ������������ = 10 µm for mw power of 1 W at the resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The dotted and dashed  circles mark the position of the inner loop edge (������������������������ = 400 µm) and the optical access  a  A/m  G  c  3500  3600  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  3000  / A/m  3000  2000  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  H(0,y,10μm)  2500  2000  1000  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  1500  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3  1000  500  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='.  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  y /mm  1200  3500  b  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='.  objective  d  HH  3000  1000  / A/m  / A/m  HH  2500  800  H,(0,0,z)  10μm)  2000  600  H(x,0,1  1500  400  NV-center  1000  500  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8  /mm  z /mm8  hole (������������������������������������������������ =  300 µm), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (b,c,d) The ������������������������, ������������������������, ������������������������ amplitudes of the mw  magnetic field components as functions of the position ������������, ������������, ������������ for a mw power of  1 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The black dotted lines mark the optically accessible diamond area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The  shadowed area in (d) shows the position region of the optical objective (������������ <  0) and  NV-center (������������ >  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The dashed lines in (d) show the best fit curve of ������������������������(0,0, ������������) to  the function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3 Reflection mode resonator  The efficiency of the resonator can be improved employing reflection mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The resonator is  converted to reflection mode by just disconnecting the output coax cable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The reflection  resonator is described by:  ������������(������������0) =  2�������������  1+������������ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  ������������������������1/2 =  1+������������  ������������0 ∙ ������������0  (4)  as derived from the reflection coefficient 16:  ������������ = −  1−������������−������������������������  1+������������+������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  (5)  Thus, for high coupling parameters, ������������ ≫ 1, the reflection mode offers a factor 2 higher  magnetic field amplitudes, see SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The bandwidth, however, becomes reduced by factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  The magnetic field distribution in the loop area is identical in both resonator modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure 4 shows the calculated and experimental S11-parameters of the resonator in reflection  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Analysis of the S-parameter traces using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (5) yields the resonator parameters ������������0 =  70, β = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3 and ������������0 = 60, β = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8 for simulation and experiment, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Thus, the  bandwidth of the resonator is 400 MHz and 320 MHz for calculation and experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The  difference between transmission and reflection mode as well as between calculated and  experimental reflection parameters are mainly due to the open output line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Simulated (a) and experimental (b) S11-parameters of the resonator operated in  reflection mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The dashed line show the best fit results of the S-curves to the  reflection coefficient given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  1  /dB  1  /dB  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  S1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  2  3  a  b  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  frequency/GHz  freguency/GHz9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' ODMR experiments  The mw magnetic field behavior was verified experimentally using a previously described  ODMR setup capable for continuous wave and pulsed ODMR experiments, see 7 and SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The  2×2×1mm3 diamond (001 cut) containing a 20 µm layer near surface that was doped with NV  centers was fixed to the resonator using transparent office tape, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S11 and S21 in SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  The central position of the optical objective was set close to the loop center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure 5(a) shows a cw ODMR spectrum of the NV-centers located near the focal spot, recorded  using mw power of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8 mW (the power level was calibrated at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='87 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Two intense ODMR  lines centered at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='87 GHz are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The line splitting by 475 MHz corresponds to the  external magnetic field component of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5 mT along the NV-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Additional ODMR line pairs  with splitting of 260 MHz, 160 MHz and 60 MHz can be attributed to a set of NV-centers  within the sensitive volume of about 300 nm in the x-y plane and 1000 nm in the z-direction  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S9 in SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The different centers have different orientations along different [1,1,1]  directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  To measure the precise mw magnetic field strength, we recorded Rabi oscillations at the two  strongest transitions, as shown in Figure 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The applied mw power of 680 mW resulted in a  Rabi frequency of about 14 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The Rabi frequency ������������������������  is proportional to the magnetic field  amplitude ������������:  ������������������������ = �  2  3 ∙ √2 ∙  ������������������������  2������������  ������������0������������  2 =  ������������������������  2������������  ������������0������������  √3 ,  (6)  where ������������������������ is the electron gyromagnetic ratio (������������������������/2������������=28 GHz/T) and ������������0 is the permeability of  the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The factor �2/3 accounts for the component of ������������ that is perpendicular to the NV-  axis, (mw H-axis parallel to [0,0,1] and NV-axis parallel to [1,1,1] crystal axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The factor √2  takes into account that we drive one transition of the S = 1 spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Thus, a Rabi frequency of  ������������������������=14 MHz at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='68 W corresponds to a conversion efficiency of ������������ = 840  ������������  ������������ at 1 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  10  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (a) cw ODMR spectrum of the single NV-center at external magnetic field alight  close to [1,1,1] crystal axis recorded at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8 mW incident mw power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (b) The Rabi  oscillation traces recorded at the mw frequencies of the high frequency (upper trace;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  marked blue in (a)) and low frequency (lower trace;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' red in (a)) ODMR lines and  680 mW mw power (+20dB increased power over the cw experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The solid blue  and red lines are the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  In the next step the cw and pulsed ODMR experiments were performed at different external  magnetic fields and the same input power of the mw amplifier, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The magnetic field  amplitude evaluated from the Rabi frequencies overlaid with the simulation results is depicted  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Both curves are in the good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The variation of the mw magnetic field  over a frequency range of 800 MHz is less than 40%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The discrepancy at the frequencies above  3 GHz is attributed to gain variation of the high-power mw amplifier operated below saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  The spatial distribution of H in the x-y plane was evaluated from the Rabi experiment at  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='93 GHz for 10 different NV centers over the area of 70×70µm2 accessible by the  nanopositioner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The variation of the magnetic field of less than 20% (1000 A/m to 1250 A/m)  is in a good agreement with simulation predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (a) cw ODMR spectra of the single NV-center at different external magnetic fields  at 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8 mW incident mw power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (b) The frequency dependence of the magnetic field  intensity I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='96  a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='92  contrast /%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='96  473 MHz  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='84  f,=13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='7 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 MHz  ODMR  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='92  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='87GHz  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='88  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='84  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='9  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  f,=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 MHz  Frequency/GHz  100  200  300  400  500  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' /ns24  1100  22  A·m  20  18  a  800  b  16  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='9  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  freauency/GHz  freguency/GHz11  amplitude ������������������������(0,0,10 μm) calculated for the transmission resonator (black curve, left  scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Rabi nutation frequencies (dots, right scale) evaluated from experimental Rabi  oscillation traces recorded at the ODMR line positions in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The Rabi frequencies  are normalized to 1 W mw power calibrated at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='87 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The left and right scales  are adjusted according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  These results show that the transmission resonator achieves good spatial magnetic field  homogeneity and the bandwidth required for most ODMR experiments on single NV-centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  The transmission resonator achieves π-pulse durations of 20 ns (������������������������ = 25 MHz) using about 1 W  mw power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' For this performance, a mw amplifier is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This limitation can be overcome  by operating the resonator in reflection mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (a) Rabi oscillation traces recorded at the mw frequency 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='93 GHz for the resonator  in transmission (upper trace) and reflection (lower trace) modes at 680 mW mw  power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The traces were acquired subsequently at the same settings with the output  cable connected and disconnected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The solid-colored lines are the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (b) Frequency  dependence of the magnetic field amplitude ������������������������(0,0,10������������������������)  calculated for  transmission (red trace) and reflection resonator (blue trace).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The amplitude ratio at  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='93 GHz is indicated by the arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure 7(a) shows the Rabi nutation traces recorded at the high frequency ODMR line at  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='93 GHz with 680 mW input mw power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The upper trace was acquired using the resonator in  transmission mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Subsequently, the output mw cable was disconnected converting the  resonator into reflection mode and the lower trace was acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The magnetic field amplitudes  evaluated from Rabi frequencies are 1190 A/m and 2070 A/m for transmission and reflection  settings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The experimentally observed increase of the resonator efficiency by a  factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='74 is in a good agreement with factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8 obtained from calculation, see Fig 7(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Thus, the reflection resonator is capable of producing the same magnetic field amplitude using  about factor of 3 smaller mw power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' It achieves 20 ns π-pulses (������������������������ = 25 MHz) with as little as  300 mW mw power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  2200  /.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='96  tensity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='92  1800  reflection  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='88  x 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8  f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='=19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 MHz  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='84  A·m  1400  rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='nor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='96  intensity  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='92  transmission  T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='88  f,=34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 MHz  600  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='84  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  0  40  80  120  160  200  frequency/GHz  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  /ns12  At this point we put two remarks: (i) The highest ������������������������ achieved in our setup allows to realize π-  pulses with 7 ns duration and 4 ns in transmission and reflection modes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The pulse  profile of such short pulses is not distorted by the resonator, neither in transmission nor in  reflection mode, since their voltage ringing times are ������������������������ =  1  ������������������������1/2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3 ns and 1 ns,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (ii) The performance of the microscope objective employed in this work, in  particular the efficiency of fluorescence collection, is restricted by the resonator acting as an  optical diaphragm, see SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Despite this restriction, the collection of high quality ODMR data is  still possible within reasonable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Performance summary  The goal of this work was the design of a device for efficient control of spin quibts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' For optimal  performance, it should have a number of properties that we specified in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Here,  we first summarise the quantitative performance measures of the mw and rf properties, as  shown in Table 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' the values shown were calculated and verified by experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' For  comparison, it also includes the parameters calculated for the 50 Ω microstrip transmission line  with the same Ω-type loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The resonator offers a factor of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8 (transmission) and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  (reflection) higher magnetic field amplitudes at the same input mw power as compared to the  transmission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Pulsed ODMR experiments with reasonably short π-pulses of 50 ns can be  generated using only 50 mW mw power with the resonator operated in reflection mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Thus,  it allows to avoid high-power mw amplification stages or even to perform the experiments  directly using the output of a mw generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The transmission mode is more advantages for  experiments at higher external magnetic fields owing to the larger resonator bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' In  contrast to reflection mode, it allows to the perform experiments with combined microwave and  radio frequency excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' An additional advantage of the transmission mode is the good  matching of the resonator input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The input return losses are <-20 dB (±50 MHz) and <-10 dB  (±150 MHz) around the resonance frequency, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Thus, in contrast to the reflection  resonator, the mw excitation system does not require matching elements (circulator or  attenuator) on the input for ODMR experiments at small external magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  13  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Summary of resonator and transmission line performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Case  ������������������������������������/�������������������������������������  A/m/√W  ������������������������������������ c)  W  ������������������������1/2  MHz  ������������������������������������/������������������������������������� d)  A/m/√W  Transmission line  245 a)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='0  full  252 (252)  Transmission resonator  1170 b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='18  950  265 (266)  Reflection resonator  2230 b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='05  400  10 (160)  a) calculated at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='9 GHz  b) calculated at the resonance frequency (~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='9 GHz)  c) input mw power required for 50 ns π-pulse (������������������������= 10 MHz) and NV-center in 001  diamond  d) calculated at 10 MHz (200 MHz)  We also considered heating effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Conduction losses in the loop can lead to a temperature  increase that can potentially influence the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The thermal analysis of the transmission  mode resonator at ambient temperature shows that the temperature increase in the loop-diamond  region is < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3 K for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 W continuous mw power (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S3 in SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' At this mw power the  produced magnetic field of 370 A/m corresponds to a Rabi frequency ������������������������=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5 MHz which is  more than sufficient for cw ODMR detection with the highest contrast 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' In pulsed ODMR  experiments the average heat power is significantly smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The transmission resonator allows  for above 106 π-pulses at 10 W (tπ= 7 ns) with no significant heating effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The results of the  thermal analysis are consistent with experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  The resonator design can be adapted for experiments on NV-centers at higher external magnetic  fields or  different types of optically active spin centers by adjusting some of the design  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The variation of the overall resonator length, ������������, between 5 mm and 56 mm allows  to tune the resonance frequency between 6 GHz and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3 GHz without significant loss of the mw  performance, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S16 in SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Further increasing or decreasing the resonance frequency is  possible by adjusting the dielectric constant and thickness of the substrate, and the geometry of  the holder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Various types of microwave resonators have been proposed for experiments on optically active  spin centers, such as the diamond NV center, each with specific advantages and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' A  fair comparison of the performance of all these different designs would have to be made under  a specific set of conditions, where all designs can operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Since this is not possible, we include  here a comparison with two recently reported designs with overlapping boundary conditions,  which are applied in several laboratories for experiments on NV-centers 10, 11, 13, 19-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The first  14  system, originally reported by Bayat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 10, is based on a double-split-ring resonator operated  in reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' In original design has an efficiency of 355 A/m/√W , with a bandwidth of  40 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Similar numbers were reported for some modified designs 11, 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The second design,  originally reported by Sasaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 13, is based on a split-ring (or loop-gap) resonator in  reflection mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' In original design provides 240 A/m/√W with a bandwidth of 440 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Better  parameters 485 A/m/√W and 300 MHz are reported for the modified version 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Thus, the  performance of our system in both transmission and reflection modes of operation is superior  to both systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Conclusion  In this work we presented a novel mw excitation system based on a resonator designed for cw  and pulsed ODMR experiments on single NV-centers to be combined with confocal  microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The high performance of the system was verified using numerical EM calculations  and ODMR experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The resonator can be easily and cheaply fabricated using standard  PCB lithography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' It offers high mechanical and thermal stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Its performance in terms of  mw power to magnetic field conversion efficiency and magnetic field homogeneity is superior  in comparison to currently used systems like the microwire-transmission line system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' It also  offers a range of practical advantages such as simplified optical adjustment and optimization of  the mw power settings for all types of ODMR experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' It also offers the possibility for  quick replacement of the diamond crystal because it does not have to be permanently connected  to the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This provides the opportunity for investigation of different samples with the  same structure or use a resonator which is optimized for the performance of a specific type of  ODMR experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  While the present work focuses on applications on NV centers in diamond, the principles used  here are completely general and can be readily transferred to similar systems that rely on  efficient spin control by microwave fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This includes not only ensembles of NV centers but  also other materials like semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Acknowledgments  This project has received funding from the European Union’s Horizon 2020 research and  innovation program under grant agreement No 828946.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The publication reflects the opinion of  the authors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' the agency and the commission may not be held responsible for the information  contained in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  15  Author Declarations  Conflict of Interest  The authors have no conflicts to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Data Availability  The data that support the findings of this study are available from the corresponding authors  upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Author Contributions  Anton Savitsky: Conceptualization (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Formal analysis, (lead);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Investigation (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Methodology (lead);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Visualization (lead);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Writing – original draft (lead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Jingfu Zhang: Data  curation  (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Investigation  (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Formal  analysis  (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Dieter  Suter:  Conceptualization (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Review & editing (equal);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Funding acquisition (lead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Suter and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Jelezko, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Reson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Spectrosc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 98-99, 50-62 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Wrachtrup and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Jelezko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Matter 18 (21), S807-S824 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Doherty, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Manson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Delaney, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Jelezko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Wrachtrup and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Hollenberg, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 528 (1), 1-45 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Barry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Schloss, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Bauch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Turner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Hart, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Pham and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Walsworth, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 92 (1), 015004 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Balasubramanian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Lazariev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Arumugam and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='-w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Duan, Curr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Opin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 20, 69-77 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Dréau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Lesik, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Rondin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Spinicelli, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Arcizet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Roch and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Jacques,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' B 84 (19), 195204 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Shim, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Niemeyer, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Taniguchi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Teraji, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Abe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Onoda, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Yamamoto, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Ohshima, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Isoya and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Suter, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 110 (24), 240501  (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Stanwix, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Pham, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Maze, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Le Sage, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Yeung, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Cappellaro, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Hemmer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Yacoby, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Lukin and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Walsworth, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' B 82 (20), 201201  (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Opaluch, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Oshnik, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Nelz and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Neu, Nanomaterials 11 (8), 2108 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Bayat, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Choy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Farrokh Baroughi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Meesala and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Loncar, Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 14 (3),  1208-1213 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Yuan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Bian, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Li and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Fang, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Express 11 (8), 086602 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Ma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Zheng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Fu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Yuan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Guo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Zhao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Shi, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Zhang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Liu,  Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 58 (5), 050919 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Sasaki, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Monnai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Saijo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Fujita, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Watanabe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Ishi-Hayase, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Itoh and  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Abe, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 87 (5), 053904 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Yap, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Negoro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Kuno, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Sakamoto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Kagawa and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Kitagawa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Japan 91 (4), 044004 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Mariani, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Nomoto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Kashiwaya and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Nomura, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 10 (1), 4813 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Sanchez, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Martin and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Zamarro, IEE Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' H 137 (4), 209-212 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Altman, Microwave circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (Van Nostrand, New York, 1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  16  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Gopinath, IEEE Transactions on Microwave Theory and Techniques 29 (2), 128-131  (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
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+page_content=' Ogawa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Ozawa, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Iwasaki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Hatano, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Sasaki and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Kobayashi, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 118 (26), 264002 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Sewani, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Vallabhapurapu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Firgau, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Adambukulam, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Johnson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Pla and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Laucht, Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 88 (12), 1156-1169 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Nomura, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Kaida, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Watanabe and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Kashiwaya, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 130 (2), 024503  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Mariani, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Umemoto and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Nomura, AIP Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 12 (6), 065321 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Reuschel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Agio and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Flatae, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Quantum Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 5 (11), 2200077  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Tsukamoto, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Ito, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Ogawa, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Ashida, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Sasaki and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Kobayashi, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 12  (1), 13942 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Stürner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Brenneis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Buck, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Kassel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Rölver, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Fuchs, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Savitsky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Suter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Grimmel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Hengesbach, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Förtsch, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Nakamura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Sumiya, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Onoda, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Isoya and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Jelezko, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Quantum Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 4 (4), 2000111 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Yuan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Zhang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Bian, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Fan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Yuan and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Fang,  IEEE Sens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 19 (2), 451-456 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  1  Supporting Information  Variable bandwidth, high efficiency microwave resonator  for control of spin-qubits in nitrogen-vacancy centers  Anton Savitsky, Jingfu Zhang and Dieter Suter  Faculty of Physics, Technical University Dortmund, Otto-Hahn-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 4a, 44227 Dortmund,  Germany  Content:  S1 Optimization of the resonator parameters  S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 Optimization of Ω-type loop parameters  S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2 Calculation of the coupling parameter for width step coupling  S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3 Performance comparison of the resonator with transmission, reflection line  S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Mw magnetic field distribution at z = 100 µm  S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Radio frequency performance  S4 Fluorescence collection efficiency and optical resolution  S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Experimental details  S6 Frequency tuning capability  2  S1 Optimization of the resonator parameters  In this part we consider some important points for the resonator optimization based on the first  principals and simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' In the first part we consider transmission line with Ω-type loop  which allows to optimize parameters of the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' In the second part we compare the  performance of resonator with transmission and reflection line and point out the ways for  performance optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 Optimization of Ω-type loop parameters  The loop of diameter ������������ integrated in the gap of a terminated microstrip transmission line can  be considered as an approximation of the ideal classical current loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This approximation is  valid when (i) the loop length is much smaller than a quarter of the wavelength (������������������������ ≪ ������������������������/4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  (ii) the width of the loop is much smaller compared to it diameter, (iii) the shielding effects can  be neglected;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' and (iv) the loop-gap discontinuity is neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The EM-wave of power ������������������������������������ in  the microstrip line with ������������������������ generates an alternating electric current with the amplitude  ������������������������������������������������������������ = �2������������������������������������  ������������������������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  This alternating current induces a magnetic field whose amplitude at the center of the loop is  ������������(0,0,0) = ������������������������������������������������������������  ������������  = 1  ������������ �2������������������������������������  ������������������������  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  The loop conversion efficiency  ������������0 = ������������(0,0,0)  �������������������������������������  = 1  ������������ � 2  ������������������������  can be increased either by decreasing the loop diameter or by lowering the line impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The  line impedance of ������������������������ = 50 ٠is, however, generally fixed to avoid mismatch with microwave  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This results in ������������0 = 500  ������������  ������������√������������ for the loop of ������������ = 400 μm used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This value  defines the physical upper limit of the magnetic field amplitude reachable for a transmission  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  The real Ω-loop differs from the ideal loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' There are several parameters which influence the  magnetic field amplitude and its distribution: (i) loop width, (ii) gap width and (iii) thickness  of the dielectric substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The last factor in our system is, however, defined by the necessity to  provide the optical access to the diamond surface and will be not considered here, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' we fix  the substrate and optical access parameters defined in the main text, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 1 (main text), and  consider only the first two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  3  Loop conductor width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The width of the loop is of critical importance as it influences not only  magnetic field parameters but also the heating effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Figure S1(a) shows the dependence of  the conversion efficiency on the loop width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' As expected, the conversion efficiency decreases  with increasing loop width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The maximum efficiency is about a factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6 smaller than in the  ideal loop due to the effect of the gap and metal shield on the back side of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The  conductive power losses, ������������������������������������������������������������ =  ������������������������������������������������������������  2  2  ������������������������������������������������������������, in the loop also decrease, see Fig S1(b), due to  the decrease of the loop resistance, ������������������������������������������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The optimal loop width of ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 mm is estimated  by considering the maximum of the function ������������0  2/������������������������������������������������������������  , see Fig S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The conversion efficiency  drops to 245  ������������  ������������√������������, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' a factor 2 smaller than for the ideal loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The loop width of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 mm also  guarantees better mechanical stability and stability of the system parameters to manufacturing  tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (a) Conversion efficiency and (b) loop ohmic losses calculated for different loop  widths and ������������������������ = 50 ٠transmission line feeded with 1 W mw power at 3 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The  gap width is G =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The dashed line marks the value ������������������������  chosen for  fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Ratio of power conversion efficiency to conductive losses in the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  40  35  30  7  25  20  15  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='/mm340  9  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='.  b  a  320  8  上  300  7  280  mW  6  260  50  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='m  240  loss  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  P  4  220  200  3  180  2  160  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' /mm  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='/mm4  In order to estimate the heating effects we have calculated the temperature distribution in the  transmission resonator continuously fed by mw power, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' For ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 mm the  maximum temperature calculated in the system is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3 K (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 W) and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5 K (1 W) starting  from the ambient temperature of 293 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' In contrast, the system with ������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='018 mm shows  higher temperature increase of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6 K (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 W) and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8 K (1 W excitation) due to the higher  ohmic resistance and lower thermal conductance of the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Temperature distribution calculated for the transmission resonator including the  diamond crystal (2×2×1 mm3) with geometrical parameters given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 1 (main  text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' CW input mw power was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Gap width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Figure S4 shows the efficiency parameter as a function of the gap width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Increasing  the gap width from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='005 mm to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3 mm (������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='75 ∙ ������������������������) has a small effect on the resonator  efficiency (magnetic field amplitude at the loop center) as well as the field distribution in y < 0  loop region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Therefore, for our resonator we have chosen G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 mm for manufacturing ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Conversion efficiency calculated for different gap widths G and ������������������������ = 50٠ transmission line fed with 1 W mw power at 3 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The loop width is  ������������������������= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  K  293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3  y=o  293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='0250  246  / A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='m".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='W-  242  238  234  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3  G /mm5  S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2 Calculation of the coupling parameter for width step coupling  The relation between the properties of the coupling element and the resonator coupling  parameter can be derived in two following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (a) Reflection line with Ω-type loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (b) Transmission line with width step and  Ω-type loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Reflection line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The bandwidth of reflection line,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Fig S5(a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' can be determined considering  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='the frequency dependence of magnetic field in the loop center:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������(������������)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������(������������0 = ������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������ )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='= sin �2������������������������ ������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4������������� = 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='√2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='The magnetic field amplitude is reduced by √2 at ������������ = ������������0/2 which results in the bandwidth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������1/2 = ������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='The reflection line can be also considered as overcoupled resonator with the bandwidth of  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������1/2 = 1 + ������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' the coupling coefficient of the reflection resonator with ������������������������ = ������������������������ is given by  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������ = ������������0 − 1 ≈ ������������0 (������������������������������������ ������������0 ≫ 1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='The same result can be also derived from the magnetic field amplitude than considering  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='reflection line as the reflection resonator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2�������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 + ������������ �������������0 = 2 ⟹ ������������� = �������������0 + �������������0 − 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='⟹  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������ ≈ ������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='Generally the result is quite obvious than consider the definition of the coupling coefficient  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������ = ������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2/4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2/4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2/4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2/4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='ZR  ZR ZR6  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' the external quality factor in reflection line ������������������������ = 1 or full power is dissipated in  external load than neglecting resonator losses,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' ������������0 ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Transmission line with width step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This system can be considered as transmission resonator,  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The reflection coefficient at ������������ = ������������0 of this system with ������������′ = ������������0 − 1 is given  by  ������������ = − 1 − ������������ + ������������′  1 + ������������ + ������������′ = − −������������ + ������������0  ������������ + ������������0  ⇒ ������������ = ������������0  Γ − 1  Γ + 1  Taking into account that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' the reflection coefficients of width step discontinuity is  Γ = − ������������������������ − ������������������������  ������������������������ + ������������������������  one obtains  ������������ = ������������0  ������������������������  ������������������������  or ������������������������ = ������������������������  ������������������������  Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' the bandwidth of the resonator can be calculated  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������������������������������������������������������������������������������������������������������:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='= 1 + ������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='= 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='+ ������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������������������������������������������������������������������������������������������������������������������������������:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='= 1 + 2������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='= 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='+ 2 ������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='For the resonator design in the main text (������������������������ = 50 Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  ������������0 = 74)  the above equation yields the  estimate value of the coupling parameter of ������������ = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6, than assuming the resonator impedance  to be equal to that of the transmission microstrip line with width ������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The values is in a good  agreement with coupling parameter of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5 evaluated from simulated S-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3 Performance comparison of the resonator with transmission,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' reflection line  The conversion efficiency for transmission,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' reflection resonator with ideal loop can be  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='calculated using following relations  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������������������������������������������������������������������������������������������������������: ������������0 = 2�������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 + ������������ ∙ �������������0 ∙ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������ � 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������������������������������������������������������������������������������������������������������������������������������: ������������0 = 2�������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 + 2������������ ∙ �������������0 ∙ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������ � 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='They take into account (i) power flow to resonator (coupling);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (ii) the energy storage in the  resonator (quality factor);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' and (iii) the increase of the loop current due to decrease of the  resonator impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' It is important that for high coupling coefficients (������������ = ������������0  ������������������������  ������������������������ ≫ 1) both  factors become independent of resonator quality factor, see Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Under this condition the  7  increase of the resonator efficiency as compared to the transmission line can be estimated from  the ratio of the resonator and feed line impedances, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  ������������������������  ������������������������ for transmission and 2∙  ������������������������  ������������������������ for  reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' For instance, the resonator having ������������������������ = 10 ٠is factor 5 more efficient compared to  transmission line having the same loop, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' it requires 25 times less mw power to obtain the  same magnetic field amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' We note that ������������������������ can be approximated by the impedance of the  microstrip line with width ������������������������ only than the discontinuity contribution to impedance can be  neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This generally holds for ������������������������ ≪ ������������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' In general case ������������������������  have to include the  discontinuity contribution which is best evaluated from numerical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' It is important to  note that then the heating effects become the limiting factor, the excitation mw power have to  be limited due to ������������������������������������������������������������ ∝ ������������������������������������������������������������  2  and, as the result, the same maximum magnetic field is reachable  in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Analytical expressions for efficiency and the bandwidth of the resonator and non-  resonating systems containing ideal current loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='Case  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������1/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='Transmission  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='line  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������ � 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������ � 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='Reflection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='line  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2 ∙ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������ � 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2 ∙ 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������ � 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='Transmission  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='resonator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2�������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 + 2������������ �������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������ � 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 + 2������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������ � 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2 ������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='Reflection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='resonator  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2�������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 + ������������ �������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������ � 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1 + ������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������ � 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='Table S2 summarizes the parameters of the transmission resonator calculated for different ������������������������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='and ������������ values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' At large ������������������������’s the contribution of the loop and gap to resonator impedance  overcomes the impedance contribution of the microstrip line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Thus, it becomes the limiting  factor determining resonator performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' conversion factor and the bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  8  Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Parameters of transmission resonator calculated for different ������������������������ and ������������, for definition  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 1 in main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Case a)  ������������0  A/m/√W  ������������0  ������������/������������0  ������������������������ b)  ������������ c)  Q0 c)  ������������������������1/2  ������������0  ������������������������  d)  Ω  ������������ e)  Ω  ������������������������ = 1 ������������������������  ������������ = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5 ������������������������  576  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='35  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='9  92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='73  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  ������������������������ = 2 ������������������������  ������������ = 22 ������������������������  930  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='79  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='67  79  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='38  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  ������������������������ = 3 ������������������������  ������������ = 17 ������������������������  1170  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='80  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='50  74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='32  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  ������������������������ = 4 ������������������������  ������������ = 15 ������������������������  1300  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='33  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='77  72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='26  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  ������������������������ = 5 ������������������������  ������������ = 13 ������������������������  1379  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='65  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='90  69  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='24  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  ������������������������ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6 ������������������������  ������������ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2 ������������������������  1386  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='68  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='65  72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='23  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='7  ������������������������ = 6 ������������������������  ������������ = 10 ������������������������  1422  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='82  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='54  62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='26  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  a) ������������ was adjusted to obtain the resonance frequency of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='95±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='05 GHz  b) Resonator efficiency relative to the efficiency of the transmission line (245 A/m/√W)  c) Evaluated from S11 and S21 curves  d) Resonator impedance calculated as ������������������������ = ������������������������/(������������0  ������������/������������0  ������������������������)  e) Impedance of microstrip line with width ������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' MW magnetic field distribution at z = 100 µm  Figure S6 shows the mw magnetic field distribution at ������������ = 100 µm above the loop, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' about  the highest optically accessible position with our optical objective with 340 µm working  distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Compared to ������������ = 10 µm (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 3 main text) the field homogeneity is improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Over  the full optically accessible area the maximum and minimum field magnitudes of 1020 ������������/������������  and 735 ������������/������������ are calculated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 735 ������������/������������ < |������������(������������, ������������, 100������������������������)| < 1020 ������������/������������ (for comparison  1070 ������������/������������ < |������������(������������, ������������, 10������������������������)| < 2270������������/������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' For the loop center and area accessible by the  nanopositioner (70×70µm2) the field homogeneity is even better, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 832  ������������/������������  <  |������������(������������, ������������, 100������������������������)| < 908 ������������/������������ (1085 ������������/������������ < |������������(������������, ������������, 10������������������������)| < 1256 ������������/������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The magnetic  field directivity is not as perfect as at small elevations but still sufficiently high to be neglected,  especially in the loop center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The fields distribution compares favorably to that reported  previously for planar loop-gap resonator [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  9  Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (a) Calculated microwave magnetic field magnitude, |H|, at ������������ = 100 µm for a mw  power of 1 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The dotted and dashed circles mark the position of the inner loop edge  (������������������������ =  400µ������������) and the optical access hole (������������������������������������������������ = 300 µm), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (b,c,d)  The ������������������������, ������������������������, ������������������������ amplitudes of the mw magnetic field components as functions of  x,y,z for mw power of 1 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The black dotted lines mark the optically accessible  diamond area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Shadowed area in (d) shows the position region of the optical objective  (������������ <  0) and NV-center (������������ >  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The dashed curve in (d) shows the best fit curve  of ������������������������(0,0, ������������) amplitude to the function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (3) in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Radio frequency performance  In experiments where Radio frequency (RF) is used to drive nuclear spins, the corresponding  signals can be fed into the same resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Figure S7 shows the power-to-field conversion  efficiency of the different types for RF signals for the frequency range < 200 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The  structures (transmission line, mw resonator) are non-resonant at RF fields and therefore  relatively broad-band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' In transmission, the efficiency does not depend on the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The  reflection resonator performs poorly at low frequencies and should not be used under such  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  a  A/m  G  1000  c  1000-  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  / A/m  800  500  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3  H(0,y,100μm)  600  200-  400  y4  200  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  y /mm  1000  1200  b  objective  d  1000  800  H(x,0,100μm) / A/m  HH:  / A/m  800  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='.  600  HH  (z\'00)"H  600  400  400  NV-center  200  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='05  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8  /mm  z /mm10  Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (a) Simulated ������������21 (upper trace) and ������������11 (lower trace) of the transmission resonator  with diamond for the radio frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (b) The frequency dependence of the  magnetic field amplitude ������������������������(0,0,10 μm) calculated for the transmission line and  the resonator in reflection and transmission modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Fluorescence collection efficiency and optical resolution  The fluorescence collection efficiency for an NV center as the point emitter and neglecting  reflections on diamond-oil interface is  ������������������������������������ = 1 − cos ������������  2  ,  The light cone opening angle, ������������, of the optical objective employed in this work (Zeiss,  Apochromat 100 with ������������������������ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' immersing oil Immersol 518 F with ������������ =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='518) is ������������ =  asin(  ������������������������  ������������ ) = 67°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Thus, the optically unrestricted objective is capable of collecting 30% of the  emitted photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The diameter of the optical access hole required for this efficiency would be  ������������������������������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='78 ������������������������ (loop inner diameter ������������������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='88 ������������������������) for NV centers close to the diamond  surface and a resonator thickness (substrate and copper cladding) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='166 mm, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S8(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  In this work we decided for ������������������������������������������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3 ������������������������ allowing for ������������ = 42°, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S8(a), in order to  optimize the microwave performance of the resonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The collection efficiency, however,  becomes potentially reduced by factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4 as compared to the maximum possible for this  objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Increasing the optical access hole diameter will improve the collection efficiency,  see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S8(b) at the cost of (i) reduced power to magnetic field conversion efficiency, see Table  S1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (ii) increased heating;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (iii) increased effect of the objective on the properties of the resonator  (resonance frequency and magnetic field distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' These factors have to be taken into  0  300  a  transmissionresonator  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  250  /dB  transmissionline  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3  s  1/2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  150  15  20  100  /dB  reflectionresonator  25  S  30  50  b  35  40  0  40  80  120  160  0  40  80  120  160  0  200  200  frequency/MHz  frequency /MHz11  account when optimizing the resonator geometry for experiments which require high collection  efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Fluorescence optical pathway for (a) the resonator employed in this work and (c)  resonator configuration required for maximum fluorescence collection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  (b) Dependence of fluorescence collection efficiency on diameter of optical access  hole relative to that capable by objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Optical resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' As pointed above the resonator acts as a diaphragm reducing numerical  apperture of the objective and, as the result, limiting the optical resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The effect is,  however, weaker compared to the loss of the fluorecence collection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Figure S9(a)  shows the fluorecence image of the single NV-center recorded with the resonator setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Evaluation of the fluorecence spatial distribution yields the FWHM of recordings about 300 nm,  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S9(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This number corresponds to 300������������������������  sin (42°)  sin(67°) = 220 ������������������������  for objective without  diaphragm, which is close to the number calculated for Airy pattern of a point emmiter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='51������������  ������������������������ =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='51∙700������������������������  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  = 255 ������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Optical resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (a) A fluorescence image of the single NV center over a scanning  range of 1μm × 1 μm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (b) Fluorescence in dependence on x- and y-positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The  solid lines show the fits to gaussian functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The increase of FWHM as compared  to objective limit is attributed to the effect of resonator optical access hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  a  b  Dent = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3 mm  ,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='78mm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
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+page_content='5  x /um  x,y /um12  The simple geometrical approach employed here can be extended for analysis of these  parameters at any position within the optically accessible area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' However, this approach neglects  additional effects like reflections on the diamond-oil interface, which should be considered for  a more precise analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Experimental details  The experiments were performed on a home-built setup, whose schematic is shown as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Single NV centers in diamond can be optically addressed, initialized and detected with a  confocal microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Here we used a diode-pumped solid state continuous wave laser with a  wavelength of 532 nm (marked in green in the schematic) for the optical excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' For pulsed  experiments, we use an acousto-optical modulator (AOM) to generate pulses from the  continuous wave laser beam or a pulsed diode laser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The microscope objective is fixed to the  nanopositioning system that scans the sample in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The fluorescence light  (marked in red in the schematic) is also collected by this MO lens and passes through the  dichroic mirror to the avalanche photodiode (APD) detector, while the scattered laser light is  reflected by the dichroic mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Schematic of the optical part of the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' A confocal microscope is utilized for  initializing and detecting single NV centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Diamond  MW source  Resonator  Objective  xyz-Nanopositione  Laser  AOM  Dichroic mirror  APD13  Figure S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Photographs of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Photographs of (a) loop region of the resonator and (b) the resonator loop region  with a diamond mounted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The principal microwave schemes for transmission and reflection modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The  components are: (1) cw microwave source;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (2) PIN-diode modulator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (3) power  mw amplifier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (4) circulator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (5) fast microwave diode detector for power and  matching control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  a  permanent magnet  resonancestructure  b  withdiamondmounted  mwout  objective  mw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='in  x,y,z stage600um  300μm  b  atransmission  本  5  大  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='IFTV  5  4  3  2  1  reflection  本  214  Figure S14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Pulse sequence for measuring Rabi oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The first laser pulse of 5 µs  initializes the NV into the ground state |g〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The MW pulse of variable length, ������������������������,  flips the NV to cos��������������������������������������������������������������|������������⟩ + sin��������������������������������������������������������������|������������⟩,where |������������⟩ denotes one excited  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The second laser pulse of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4 µs probes the population of ground state |������������⟩,  proportional to the FL intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The pulse repetition rate was generally set to  6400 s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  S6 Frequency tuning capability  In this part we consider the options for tuning the resonance frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This will be relevant  for experiments in different magnetic fields or on other types of optically active spin centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  The half-wave mode (������������������������/2) resonance frequency of a microstrip of length ������������ without current  loop is given by the well-known expression  ������������0 =  ������������  2������������ ∙ �������������������������������������  where ������������������������������������ is the effective dielectric constant of the microstrip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' For a microstrip of width ������������,  larger than the dielectric thickness ������������, it is given by  ������������������������������������������������ = ������������������������ + 1  2  + ������������������������ − 1  2  1  �1 + 12 ������������  ������������  The effective dielectric constant of ������������������������������������������������ =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='81 is calculated for microstrip with  ������������ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='13 mm, ������������ = 3 mm and ������������������������ = 3 used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Thus, for the fixed geometry of the  resonator holder (see Fig 1, main text) the resonance frequency of the microstrip can be tuned  from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6 GHz (������������ = 56mm) to 18 GHz (������������ = 5mm), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S15(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The validity of the analytic  approach is also valid for the full model as the resonance frequency from EM calculations (red  line, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S15(a)) agrees well with the analytical prediction (dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S15(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  The blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S15(a) shows the resonance frequency of the microstrip resonator  including the current loop in transmission mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The additional gap and the current loop  increase the intrinsic capacity and inductance of the microstrip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Thus, the resonance frequency  is lower than for a microstrip of the same length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The resonance frequency can be tuned between  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3 GHz (������������ = 56mm) and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='7 GHz (������������ = 5mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' It is important to note that the conversion  Laser  532nm  t  MW  FL  readout15  efficiency does not strongly depend on the resonance frequency, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 15(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' A slight  efficiency improvement is observed at lower frequencies, accompanied by a reduction of the  resonator bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure S15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (a) Dependence of the resonance frequency on the length ������������, (diamonds, blue line)  calculated for transmission mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' All other geometrical parameters are fixed as  given in the caption of Figure 1 (main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The red line with circles shows the  resonance frequency of the microstrip without loop obtained from EM simulations  for transmission mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The black dashed line shows the analytically calculated  resonance frequency of the microstrip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (b) Conversion efficiency calculated for  resonators of selected lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Thus, the resonator configuration described in this work can be used for experiments with  microwave excitation over the frequency range of 1 GHz to 6 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The range can be extended  to lower frequencies by further increase of the resonator length or by increasing the dielectric  constant of the substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Figure S16 shows the parameters of identical resonators with  L = 50 mm calculated for ε = 3 (used in this work) and ε = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The ratio of the resonance  frequencies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='44 GHz/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='78 GHz = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='85 agrees well with the ratio of the effective dielectric  constants �9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='15/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='81=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The large reduction in the bandwidth (300 MHz vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 110 MHz) can  be  improved by decreasing the width of the resonator, ������������������������, which would lead to an increase of  the coupling coefficient (see section S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  a  1600  L= 50 mm  b  N  1400  L=25mm  frequency  1200  L=17mm  1000  A·m  L= 8 mm  resonance  800  600  2  400  200  5  10  15  20  25  30  35  40  45  50  55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  L /mm  frequency /GHz16  Figure S16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' (a,c) S-parameters of identical transmission mode resonators with length  ������������ = 50 mm calculated for substrates having ε = 3 (a) and ε = 10 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' All other  geometrical parameters are fixed as given in the caption of Figure 1 (main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  (c,d) Corresponding conversion efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Resonator bandwidths and key  parameters are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  An increase of the resonance frequency above 6 GHz is difficult because substrates with  dielectric constant < 2 are not available and a further decrease of the resonator length below  ������������ ≤ 2 ∙ ������������������������ is not feasible, see Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The resonator, however, can still be used at higher mw  frequencies if higher resonance modes are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Figure S17 shows S11 and efficiency  parameters of the resonator (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 1 main text) extended to higher mw frequencies, where  additional resonances are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Of particular interest is the resonance at about 11 GHz  which corresponds to the ~  3  2 ������������������������ resonance mode of the microstrip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' At this mode the conversion  efficiency (600 A/m/√W) is reduced by factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='95 as compared to the main mode (1170  A/m/√W) but it is still about factor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5 higher compared to the 50 Ω microstrip with current  loop (245 A/m/√W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' We note here that for a microstip resonator without current loop in  3  2 ������������������������  mode one would expect a reduction of the magnetic field amplitude in the center by a factor √3  compared to the fundamental  1  2 ������������������������ mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This is due to an additional resonance mode at  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4 GHz, marked by * in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' S17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' This mode generates only small magnetic fields at the  1600  b  1500  5  1400  /dB  1/2  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='W  1300  S  300 MHz  15  1000  20  a  900  L=50mm,W,=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3mm,=3,r(W,=3mm)=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='81  25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='65  800  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='55  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='7  frequency /GHz  frequency /GHz  2400  d  2  2200  4  2000  /dB  1800  8  110 MHz  1600  S  10  12  14  1200  16  c  1000  L = 50mm, W, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='1mm,  = 10, (Ws= 3mm)= 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='15  18  800  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='9  freguency/GHz  freguency/GHz17  position of the current loop and   is therefore not useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Thus, resonator designs operating at  higher modes would require additional optimization steps in order to separate active from  inactive modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Figure S17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Simulated S11 parameter and conversion efficiency of the transmission mode  resonator described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' 1 (main text) over a wider frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The dashed  line indicates the conversion efficiency of a 50 Ω microstrip line containing a  current loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' The asterisk marks the frequency position of a magnetically inactive  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Sasaki, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Monnai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Saijo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Fujita, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Watanabe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Ishi-Hayase, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content=' Itoh and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
+page_content='  Abe, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
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+page_content=' 87 (2016) 053904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ldE2T4oBgHgl3EQfeAcm/content/2301.03911v1.pdf'}
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+Blazar boosted Dark Matter constraining model dependent σeχ :
+Role of energy dependent cross sections
+Supritha Bhowmick,1, ∗ Diptimoy Ghosh,1, † and Divya Sachdeva2, ‡
+1Department of Physics, Indian Institute of Science Education and Research Pune, India
+2Laboratoire de Physique de l’Ecole Normale Sup´erieure, CNRS,
+Universit´e PSL, Sorbonne Universit´es, 24 rue Lhomond, 75005 Paris, France
+Elastic collisions with relativistic electrons from the blazar’s jet can accelerate dark matter (DM)
+particles in the DM spike surrounding the supermassive black hole at its center. This can allow one to
+set limits on the DM-electron scattering cross section (¯σeχ) for DM masses less than 100 MeV, which
+has been found to be orders of magnitude stronger than the equivalent results from cosmic rays for
+energy-independent cross-section [1]. We also consider DM particles boosted by energetic electrons
+in the jets of the blazars TXS 0506+056 and BL Lacertae. In this study, we consider both vector
+and scalar mediators for the scattering of electron and electrophilic fermionic DM. We highlight
+that the ensuing energy dependency of the S-matrix for the corresponding Lorentz structure of the
+vertex significantly modifies the constraints. We found that the revised exclusion limits are atleast
+three orders of magnitude stronger than the conclusions drawn from simple constant crosssection
+assumption. Our limits are also assessed for the less cuspy spike.
+I.
+INTRODUCTION
+The Cold Dark Matter (CDM) provides a compelling
+explanation for a broad range of observations, including
+rotation curves in spiral galaxies, gravitational micro-
+lensing, cluster collisions (the Bullet Cluster), and tem-
+perature anisotropy in the spectrum of cosmic mi-
+crowave background radiation. To that end, a variety of
+particle physics models predict a feeble interaction be-
+tween SM and DM, which can be investigated using Di-
+rect detection (DD) experiments. The DD experiments
+identify the nuclear or electronic recoils produced by the
+scattering between DM and the detector’s (target) nu-
+clei or electron. The average velocity of DM particles
+in the solar vicinity, however, restricts the amount of
+energy that may be deposited in a detector.
+For ex-
+ample: detectors like
+Xenon1T can detect DM mass
+mχ ∼ O(1 MeV), corresponding to electronic recoil of
+∼ O(1 keV). The neutrino detectors like Super-K are
+sensitive to recoil energy threshold of ∼ O(1 MeV) lead-
+ing to the smallest accessible DM mass of O(1 GeV).
+Thus, these detectors appear to have a limited range for
+detecting lighter DM particles. Since these observations
+have been negative about the sign of DM, it is critical
+to develop methods for probing the sub-GeV/MeV mass
+range.
+The reach of these experiments has been extended to
+DM masses well below 1 GeV in recent years, thanks
+to the novel idea of boosting the halo DM through its
+interaction with the SM particles via cosmic rays
+[2–
+17], primordial black holes, diffuse Supernova Neutrino
+Background (DSNB) [9–13], and blazars [1, 18].
+De-
+spite the fact that the boosted DM flux is substantial
+and DM particles are (semi)relativistic, the sensitiv-
+∗ supritha.bhowmick@students.iiserpune.ac.in
+† diptimoy.ghosh@iiserpune.ac.in
+‡ divya.sachdeva@phys.ens.fr
+ity is achieved at larger cross sections because the up-
+scattered subcomponent flux is significantly lower than
+the galactic DM population.
+In this paper, we consider the blazar boosted DM,
+which was proposed in Ref. [1, 18] in the context of
+electrophilic fermionic DM. The presence of a super-
+massive Black Hole (BH) at the blazar center, which
+provides a dense DM population compensates for the
+blazar’s large distance from Earth by producing DM
+flux that is stronger than that from galactic CRs. The
+existing literature, however, assumes that DM interac-
+tion cross-sections are independent of DM energy. Al-
+though this simple assumption makes calculations eas-
+ier, it’s not physically realistic.
+It would also be an
+incorrect approximation to make, notably in scenarios
+where a significant DM flux becomes relativistic after
+being scattered by energetic particles. Some of the pre-
+viously mentioned works [6, 15–17, 19] for cosmic ray
+boosted DM, have already discovered that the limits for
+energy-dependent cross-section differ by orders of mag-
+nitude from those obtained under the assumption of a
+constant cross section.
+The notion of an energy-dependent scattering cross-
+section is thus primarily investigated in the present work
+by taking into account electrophilic fermionic DM that
+has been boosted by energetic electrons from blazars.
+To constrain the scattering cross-section, we use elec-
+tron recoil measurements in Super-Kamiokande. This
+work is organized as follows. We discuss the spectrum
+of energetic particles in the blazar jets in Section II, and
+describe DM density profiles in Section III. In Section
+IV, we estimate the Blazar boosted DM (BBDM) flux
+and compute the event rate. In Section V, we present
+simplified DM models. We present the main results of
+our paper in Section VI, i.e., the energy dependent ex-
+clusion bound from BBDM-electron scattering, and in
+Section VII, we summarize and conclude.
+arXiv:2301.00209v1  [hep-ph]  31 Dec 2022
+
+2
+II.
+BLAZAR JET SPECTRUM
+Blazars are characterized by a non-thermal spectral
+energy distribution (SED). This spectrum has a low en-
+ergy peak in the infra-red or X-ray region, which has
+been accepted to be due to synchrotron emission of elec-
+trons in the jet. Another peak at γ-ray frequencies could
+be due to highly relativistic protons [20–24], as moti-
+vated by the recent IceCube detection [25–27] of a high
+energy neutrino from TXS 0506+056 blazar. Since DM
+considered in this work is electrophilic, at tree level it
+can only interact with electrons. Therefore, we are only
+concerned with the blazar jets’ electron spectrum.
+We follow the procedure laid out in Ref. [1, 18] to
+compute the spectrum of the energetic electrons in the
+blazar jets, assuming “Blob geometry” model [28]. In
+this model, the energetic particles in the blazar jets
+move isotropically in a “blob” frame, as the blob tra-
+verses outwards along the jet axis. The Lorentz boost
+factor of the blob is given by ΓB = (1 − β2
+B)−1/2, where
+βB is the blob’s propagation speed. The inclination of
+the jet axis with respect to the line of sight (LOS) is
+taken to be θLOS.
+In the blob frame, the energetic electrons follow a
+power law distribution with a high and a low energy
+cutoff (γ′
+max,e and γ′
+min,e respectively). This spectrum
+can then be frame transformed to the observer’s rest
+frame (for details of the derivation, see [18]), given by :
+dΓe
+dTedΩ = ce
+4π Γ−αe
+B
+�
+1 + Te
+me
+�−αe
+×
+βe(1 − βeβBµ)−αe
+�
+(1 − βeβBµ)2 − (1 − β2e) (1 − β2
+B)
+(1)
+where me and Te is the mass and kinetic energy of
+the electron respectively. Speed of the electrons is given
+by βe =
+�
+1 − m2
+e/(Te + me)2�1/2.
+Doppler factor for
+the blob frame is D = (ΓB(1 − βB cos θLOS))−1. αe is
+the power index of the electron spectrum in the blob
+frame. µ is the cosine of the angle between direction of
+the electron’s motion and the jet axis. It is related to
+the scattering angle in the blob frame (¯µs) by [2, 18] :
+µ(¯µs, φs) = ¯µs cos θLOS + sin φs sin θLOS
+�
+1 − ¯µ2s (2)
+where φs is the azimuth with respect to the LOS. ¯µs
+is related to the kinetic energy of the blazar jet electron
+and the kinetic energy (Tχ) transferred to the DM, as
+follows :
+¯µs(Te, Tχ) =
+�
+1 + T max
+χ
+− Tχ
+Tχ
+(me + mχ)2 + 2mχTe
+(Te + me + mχ)2
+�−1/2
+(3)
+Now, ce is a normalisation constant which is deter-
+mined from the blazar jet electron luminosity (Le),
+where the latter depends on ce as [18, 29] :
+Le = cem2
+eΓ2
+B
+� γ′
+max,e
+γ′
+min,e
+(γ′
+e)1−αe dγ′
+e ,
+(4)
+and thus ce is simply given by :
+ce =
+Le
+m2eΓ2
+B
+×
+�
+�
+�
+�
+�
+(2 − αe) /
+��
+γ′
+max,e
+�2−αe −
+�
+γ′
+min,e
+�2−αe�
+if αe ̸= 2 ;
+1/ log
+�
+γ′
+max,e/γ′
+min,e
+�
+if αe = 2.
+(5)
+The parameters γ′
+min,e, γ′
+max,e, αe, Le and D are fit-
+ted to the SED of a blazar. The doppler factor is as-
+sumed to be either 2ΓB or ΓB. These two cases corre-
+spond to TXS 0506+056 (θLOS = 0) and BL Lacertae
+(θLOS ∼ 3.82◦). All the parameters required to find the
+blazar jet spectrum of TXS 0506+056 and BL Lacertae,
+along with the blazar redshift and luminosity distance
+(dL), are mentioned in Table I. The electron spectrum
+is plotted in Fig. 1.
+III.
+DM DENSITY PROFILE
+N-body simulations and observations are not sensi-
+tive at subparsec sizes, thus, the DM distribution near
+Galactic center is not well known. The central super-
+massive black hole (SMBH) can have a considerable
+impact on DM density if the SMBH grows adiabati-
+cally, i.e., on a timescale much longer than its dynami-
+cal timescale. The DM density in a region correspond-
+ing to the sphere of gravitational influence of the black
+hole (BH) is expected to be significantly enhanced [31].
+This results in a morphological feature known as a DM
+spike, which corresponds to a DM profile with a power
+law scaling ρ(r) ∝ r−γsp [31]. Here γsp =
+9−2γ
+4−γ com-
+monly ranges from 2.25 to 2.5, depending on the slope
+of the initial DM halo distribution, γ. In this work, we
+assume the initial central DM profile is Navarro-Frenk-
+White, γ = 1. Also, for DM annihilating with cross-
+section ⟨σv⟩ann., the innermost region of the DM spike is
+
+3
+Parameter
+TXS 0506+056
+BL Lacertae
+Redshift
+0.337
+0.069
+dL
+1835.4 Mpc
+322.7 Mpc
+MBH
+3.09 × 108 M⊙
+8.65 × 107 M⊙
+ΓB
+20
+15
+θLOS
+0◦
+3.82◦
+αe
+2
+3.5
+�
+γ′
+min,e, γ′
+max,e
+�
+(500, 1.3 × 104) (700, 1.5 × 104)
+Le (erg/s)
+1.32 × 1044
+8.7 × 1042
+TABLE I: Model parameters for TXS 0506+056 [21] and
+BL Lacertae blazars [30].
+FIG. 1: The electron spectrum in the observer’s frame is
+plotted above, for the blazars TXS 0506+056 (solid lines)
+and BL Lacertae (dashed lines). The spectrum is shown for
+two different polar angles : θ = 0◦ (in red), θ = 10◦
+(purple). For larger kinetic energies (Te ≳ 10 GeV), the
+electron flux from TXS 0506+056 blazar exceeds the flux
+from BL Lacertae.
+depleted because DM annihilate efficiently on account
+of high DM density, leading to “annihilation plateau”
+density given by
+ρsat =
+mχ
+⟨σv⟩ann.tBH
+,
+(6)
+where tBH ∼ 109 yrs is the age of the BH. The DM
+density profile in such a spike is given by
+ρ(r) =
+�
+�
+�
+�
+�
+0
+r < 4RS
+ρsat
+4RS ≤ r < Rsat
+N1r−γsp
+Rsat ≤ r < Rsp
+N2r−γ
+r ≥ Rsp
+,
+(7)
+where RS = 2GM/c2 is the Schwarzchild radius of the
+BH, Rsp = 105 Rs is the radius of the spike [29] and
+ρ(r) goes to zero in the region r < 4Rs due to DM
+particles being captured by the SMBH. The saturation
+density ρsat and the saturation radius Rsat are related
+by the equality ρ(Rsat) = ρsat.The normalization N1 of
+ρ is determined by observing that the mass of the spike
+is of the same order as MBH within the spike radius [32]
+and N2 is determined by observing the continuity of the
+profile at Rsp.
+For a pre-existent DM halo with γ = 1, the final DM
+profile near BH corresponds to γsp = 7/3. A more re-
+alistic model was obtained in Ref. [33], where the time-
+evolution of dark matter distribution was investigated
+on sub-parsec scales. This implied softening of the DM
+density spike, due to scattering of DM by stars and cap-
+ture of DM particles by the SMBH, dampening it to
+γsp = 3/2. Thus, in this work, we consider both the
+DM profile parameters γsp = 7/3 and γsp = 3/2 along
+with two extreme values of ⟨σv⟩ann.. We define these as
+Profile 1: ρ(Rsat ≤ r < Rsp) = N1r−7/3
+Profile 2: ρ(Rsat ≤ r < Rsp) = N1r−3/2.
+For each of these profiles, the two benchmark points
+(BMPs) are defined as:
+BMP 1: No DM annihilation, i.e., ⟨σv⟩ann. = 0
+BMP 2: ⟨σv⟩ann. = 3 × 10−26 cm3s−1, thermal
+relic cross-section.
+Another quantity relevant to the computation of the
+BBDM flux is the line of sight (LOS) integral of DM
+density around the blazar. This provides a measure of
+the number of DM particles being boosted by the blazar.
+At a certain distance r from the blazar, it is defined as
+ΣLOS(r) =
+� r
+rmin
+ρ(r′)dr′
+(8)
+where rmin is the distance from the SMBH from where
+the blazar jet starts. To get a measure of all boosted DM
+particles, we want the LOS integral at large distances
+(r >> 105RS), and we define Σtot
+LOS = ΣLOS(r >>
+105RS).
+In this work, we will study BBDM flux from TXS
+0506+056 and BL Lacertae, and for these blazars, rmin
+lies within 100RS [21, 29, 30]. We take rmin = 4RS,
+noting that Σtot
+LOS is independent of choice of rmin for
+models which allow for DM pair annihilation. For no
+DM annihilation, Σtot
+LOS decreases by an order of magni-
+tude when rmin is changed to 100RS.
+The DM density and L.O.S. integral profiles are plot-
+ted in Fig 2 for two benchmark points. For both DM
+density profiles, BMP1 yields a larger spike and a larger
+LOS integral (Σtot
+LOS). This results in a larger BBDM
+flux and consequently a stronger exclusion bound on
+DM-electron interaction cross section. Hence, one can
+expect models with no DM annihilation to yield bet-
+ter bounds. Moreover, even though the LOS integral
+for the idealistic spike (Profile 1) is very large and thus
+results in substantial BBDM flux, the more realistic pro-
+file (Profile 2) with a softer spike would lead to a much
+smaller LOS integral, and hence a much weaker exclu-
+sion bound.
+
+Blazar Jet Electron Flux
+TXS 0506+056, 0 = 0°
+TXS 0506+056, 0 = 10°
+BL Lacertae,  = 0°
+1056
+BL Lacertae,  = 10°
+1053
+1038
+10-4
+10-3
+10-1
+100
+101
+10-2
+102
+Te (GeV)4
+(a) DM Density Profile
+(b) LOS integral profile
+FIG. 2: The profiles of ρDM (Fig. 2a) and ΣDM (Fig. 2b)
+are plotted above for TXS 0506+056 blazar parameters.
+The DM mass chosen for these figures is mχ = 1 MeV.
+The profile 1 (red) and profile 2 (blue) are plotted for BMP
+1 (solid curve) and BMP 2 (dashed curve). BL Lacertae,
+on the other hand, is less massive than TXS 0506+056,
+and yields a larger spike at a smaller distance from the BH.
+IV.
+BLAZAR BOOSTED DARK MATTER
+FLUX AND EVENT RATE
+DM particles are boosted via elastic collisions with
+the relativistic electrons in the blazar jet. The DM dif-
+ferential flux resulting out of collision with the electrons
+is obtained as follows :
+dφχ
+dTχ
+=
+Σtot
+DM
+2πmχd2
+L
+� 2π
+0
+dφs
+� T max
+e
+(Tχ,φs)
+T min
+e
+(Tχ,φs)
+dTe
+×dσχe
+dTχ
+dΓe
+dTedΩ ,
+(9)
+where σχe is the DM-electron interaction cross section.
+The integration over φs becomes trivial in case of TXS
+0506+056, where the system is symmetric about LOS,
+and we can simply set µ = ¯µs (from Eqn. (2)).
+The
+maximal
+kinetic
+energy
+of
+the
+blazar
+jet
+electrons
+along
+LOS
+is
+given
+by
+T max
+e,jet
+=
+me
+�
+γ′
+max,e Γ−1
+B (1 − βB cos θLOS)−1 − 1
+�
+.
+This
+is
+set as the upper bound of the integral on Te in
+Eqn. (9).
+The lower bound is set by the minimum
+kinetic energy required for scattering, given by
+T min
+e
+=
+�Tχ
+2 − me
+� �
+1 ±
+�
+1 + 2Tχ(me + mχ)2
+mχ(Tχ − 2me)2
+�
+,
+(10)
+with + and − applicable for Tχ > 2me and Tχ <
+2me respectively.
+However, the kinetic energy of the
+slowest electrons in the blazar jets could be larger
+than T min
+e
+.
+In such a case, the kinetic energy of the
+least energetic electron in the jet, given by T min
+e,jet =
+me
+�
+γ′
+min,e Γ−1
+B (1 − βB cos θLOS)−1 − 1
+�
+, sets the lower
+bound of the integral in Eqn. (9).
+The differential cross section ( dσχe/dTχ ) of the DM-
+blazar jet electron interaction is given by,
+dσχe
+dTχ
+= |M|2
+16πse
+1
+T max
+χ
+(11)
+where M is the interaction matrix element, a function
+of Tχ and Te. se is the centre of momentum energy for
+the electron-DM collision given by :
+se = (mχ + me)2 + 2mχTe ,
+(12)
+and T max
+χ
+is the maximum kinetic energy that can be
+imparted to a DM particle by a blazar jet electron of
+energy Te is given by :
+T max
+χ
+=
+T 2
+e + 2meTe
+Te + (me + mχ)2 / (2mχ)
+(13)
+The effect of including energy dependence in DM-
+electron interaction can be seen from the BBDM flux
+plots, given in Figs. 3 and 4. Profile 1 of DM density
+clearly gives larger BBDM flux as compared to Pro-
+file 2, as expected from larger DM spike for profile 1
+shown in Fig. 2 in Section III. For DM to register event
+at Super-K, kinetic energies greater than ∼ 0.1 GeV
+are relevant.
+In this energy range, the heavy media-
+tor scenario gives much larger BBDM flux as compared
+to the constant cross section case, while on the other
+hand, the light mediator regime yields a much smaller
+BBDM flux. From this, we expect the exclusion limit
+on DM-electron interaction, arising from light mediator
+regime, to be extremely weak. Since the vector medi-
+ator case gives slightly larger BBDM flux as compared
+to the scalar mediator scenario, we hope for moderately
+better bounds from vector mediators.
+Also, since for
+any given DM Profile or BMP, the BBDM flux is larger
+for smaller mass DM particles, we can expect bounds
+to grow stronger for lighter DM particle. Finally, the
+BBDM flux plots terminate at a certain value of Tχ
+because the blazar jet electrons boosting the DM par-
+ticles have an upper cutoff on their energies (for TXS
+0506+056 jets, T max
+e,jet ∼ 260 GeV and for BL Lacertae
+jets, T max
+e,jet ∼ 225 GeV).
+
+TXS 0506+056
+1018
+sp = 7/3，BMP1
+sp = 3/2,BMP1
+sp = 7/3,BMP2
+sp = 3/2,BMP2
+1015
+pDM (GeV/cm3)
+1012
+109
+106
+103
+100
+100
+101
+102
+103
+104
+105
+106
+r (Rs)TXS 0506+056
+1032
+1030
+(GeV /cm²)
+1026
+ZDM (
+1024
+sp = 7/3,BMP1
+sp = 3/2, BMP1
+1022
+sp = 7/3,BMP2
+sp = 3/2, BMP2
+100
+101
+102
+103
+104
+105
+106
+r (Rs)5
+TXS 0506+056 blazar is further away from us as
+compared to BL Lacertae, and is more massive, lead-
+ing to a larger DM density spike and hence a larger
+ΣLOS. The contribution to the DM flux coming from
+the factors outside the integrals in Eqn. (9) is hence
+much larger for BL Lacertae than TXS 0506+056 (i.e.,
+�
+ΣLOS/d2
+L
+�
+BL Lac = 103 �
+ΣLOS/d2
+L
+�
+TXS ).
+Inspite of
+this, we note in Figs. 3 and 4 that the flux of DM
+particles boosted by TXS 0506+056 is larger than the
+BBDM flux of BL Lacertae, for more energetic DM par-
+ticles (Tχ ≳ 10 GeV). This is because, the kinetic en-
+ergy range of the electron responsible for boosting the
+DM particle to energies greater than 10 GeV is roughly
+Te ≳ 10 GeV.
+For this energy range, the electron
+spectrum in TXS blazar is larger than that of BL Lac
+(Fig. 1). As a result of this, we expect stronger bounds
+to arise from TXS blazar.
+The obtained boosted DM flux will yield the following
+rate of electron recoil events in Super-K
+dR
+dER
+= ℵ
+� ∞
+T min
+χ
+dTχ
+dφχ
+dTχ
+dσχe
+dER
+(14)
+where ℵ = 7.5 × 1033 is the effective number of target
+electrons in Super-K, and dσχe/dER is the differential
+DM-target electron interaction cross section, given by
+dσχe
+dER
+= |M|2
+16πsχ
+1
+Emax
+R
+(15)
+where sχ is centre of momentum energy for the DM-
+target electron collision which can be obtained from
+Eqn. (12) under the substitution :
+mχ ↔ me and
+Te → Tχ . Emax
+R
+is the maximum possible recoil in the
+detector, that can be imparted by a DM particle with
+kinetic energy Tχ, and can be obtained from Eqn. (13)
+with the appropriate substitutions mentioned before.
+To get total number of expected recoil events (Neχ)
+in a certain energy bin, Eqn. (14) needs to be integrated
+over ER, as follows
+Neχ = ℵ Texp
+� ER,max
+ER,min
+dER
+� ∞
+T min
+χ
+dTχ
+dφχ
+dTχ
+dσχe
+dER
+(16)
+where Texp = 2628.1 days is the exposure time, and
+[ER,min, ER,max] is the recoil energy range of each bin.
+V.
+SIMPLIFIED DM MODEL
+We assume that a fermionic DM particle χ, of mass
+mχ, only interacts with electrons. This scenario is pos-
+sible in several leptophilic particle DM models [34–43].
+Additionally, the electron-DM interaction is mediated
+by a scalar or vector particle, given as
+L = gχφφ¯χχ + geφφ¯ee
+or
+(17)
+= gχA′A′
+µ ¯χγµχ + geA′A′
+µ¯eγµe
+(18)
+For simplicity, we will drop A′ and φ from subscripts in
+the coupling constants such that gχ (ge) is the coupling
+(a) BBDM flux for heavy mediator scenario
+(b) BBDM flux for light mediator scenario
+FIG. 3: Flux of DM particles, boosted by energetic
+electrons in the jets of TXS 0506+056 blazar, is plotted
+above, for heavy (3a) and light (3b) mediators. The
+parameters chosen for the above plots are ¯σeχ = 10−30 cm2
+and BMP1. The vector and the scalar mediator cases have
+been plotted in solid and dashed lines respectively. For
+comparision, DM flux for constant cross section scenario
+have also been plotted in dotted lines. Two DM masses
+have been considered, mχ = 1 keV (plotted in black) and
+mχ = 1 MeV (plotted in red) for DM density Profile 1 (i.e.
+γsp = 7/3). To avoid overcrowding, only vector mediator
+case is considered for Profile 2 (i.e. γsp = 3/2), and BBDM
+flux is plotted (in blue) corresponding to DM mass
+mχ = 1 MeV. Clearly, Profile 2 yields a smaller BBDM
+flux as compared to Profile 1.
+constant of the dark mediator to the DM particle (elec-
+tron). Next, we provide the differential cross-section for
+different operators and inspect the effect of the Lorentz
+structure. For that, we define the following quantities :
+M2 = 16g2
+eg2
+χm2
+em2
+χ
+(q2
+ref − m2
+i )2
+(19)
+¯σeχ =
+µ2
+χe
+16πm2em2χ
+M2
+(20)
+
+TXS 0506+056, Heavy Mediator
+Vector Mediator
+mx = 1 keV,sp=
+Scalar Mediator
+mx =1 MeV, sp = 
+105
+Constant CS
+102
+10-1
+10-4
+10-7
+10-10
+10-2
+10-1
+100
+101
+102
+103
+Tx (GeV)TXS 0506+056, Light Mediator
+103
+Vector Mediator
+mx=1 keV,sp=
+ Scalar Mediator
+mx =1 MeV,sp= 
+10-1
+Constant CS
+10-5
+10
+:10-13
+10-17
+10-21
+25
+10-
+100
+10-2
+10-1
+101
+102
+103
+Tx (GeV)6
+(a) BBDM flux for heavy mediator scenario
+(b) BBDM flux for light mediator scenario
+FIG. 4: Flux of DM particles, boosted by energetic
+electrons in the jets of BL Lacertae blazar, is plotted above,
+for heavy (4a) and light (4b) mediators. The parameters
+chosen for the above plots are ¯σeχ = 10−30 cm2 and BMP1.
+The vector and the scalar mediator cases have been plotted
+in solid and dashed lines respectively. For comparision, DM
+flux for constant cross section scenario have also been
+plotted in dotted lines. Two DM masses have been
+considered, mχ = 1 keV (plotted in black) and mχ = 1 MeV
+(plotted in red) for DM density Profile 1 (i.e. γsp = 7/3).
+To avoid overcrowding, only vector mediator case is
+considered for Profile 2 (i.e. γsp = 3/2), and BBDM flux is
+plotted (in blue) corresponding to DM mass mχ = 1 MeV.
+Clearly, Profile 2 yields a smaller BBDM flux as compared
+to Profile 1.
+where qref = αme is the reference momentum trans-
+ferred.
+Here mi is the mass of the dark mediator
+(i = A′, φ for vector, scalar mediator) and µeχ is the
+reduced mass of the DM-electron system. We also de-
+fine a form factor,
+F 2
+DM(q2) = |M|2/M2
+(21)
+This factor contains the energy dependence arising in
+the differential cross section dσχe/dTχ due to the blazar
+jet electrons boosting the DM particles and the Lorentz
+structure of the interaction. The explicit form of FDM
+depends on the model of DM and mediator considered.
+A similar form factor, Frec, contains energy depen-
+dence in the differential cross section dσχe/dER arising
+due to interaction of relativistic DM particles with the
+electrons in Super-K, and can be obtained from the
+form factor FDM of Eqn. (21) by making the substitu-
+tions : me ↔ mχ, Tχ → ER and Te → Tχ.
+Hence the differential cross sections, dσχe/dTχ and
+dσχe/dER, relevant in the DM-blazar jet electron scat-
+tering and DM scattering at the detector end respec-
+tively, are given by :
+dσχe
+dTχ
+= ¯σeχ
+m2
+em2
+χ
+µ2eχ
+F 2
+DM(q2)
+sCRT max
+χ
+(22)
+and,
+dσχe
+dER
+= ¯σeχ
+m2
+em2
+χ
+µ2eχ
+F 2
+rec(q2)
+sχEmax
+R
+(23)
+Under the energy independent approximation for the
+cross section, the differential cross section would simply
+be :
+dσχe
+dTχ
+=
+¯σeχ
+T max
+χ
+, dσχe
+dER
+=
+¯σeχ
+Emax
+R
+(24)
+A.
+Scalar Mediator
+Considering a scalar mediator (denoted as φ), one can
+calculate F 2
+DM for the interaction between electrons in
+blazar jets and non-relativistic DM, using Eqn. (21) to
+obtain
+F 2
+DM(q) =
+�
+q2
+ref − m2
+φ
+�2
+�
+q2 − m2
+φ
+�2
+(2mχ + Tχ)
+�
+2m2
+e + mχTχ
+�
+4mχm2e
+(25)
+The differential cross section (dσχe/dTχ) w.r.t. the
+DM energy (Tχ), is :
+dσχe
+dTχ
+= ˜σeχ
+(q2
+ref − m2
+φ)2
+(q2 − m2
+φ)2
+� mχ
+4µ2eχ
+(2mχ + Tχ)
+�
+2m2
+e + mχTχ
+�
+sCRT max
+χ
+�
+(26)
+The form factor Frec and the differential cross-section
+w.r.t. the recoil energy of the detector (dσχe/dER) are
+obtained from Eqn. (25) and Eqn. (26) by performing
+the substitutions prescribed in the previous section, viz.
+me ↔ mχ, Tχ → ER, Te → Tχ, se → sχ .
+
+BL Lacertae, Heavy Mediator
+Vector Mediator
+mx = 1 keV，sp =
+Scalar Mediator
+105
+Constant CS
+102
+10-1
+10-4
+10-7
+10-10
+10-2
+10-1
+100
+101
+102
+103
+Tx (GeV)BL Lacertae, Light Mediator
+103
+Vector Mediator
+mx=1 keV，sp=
+Scalar Mediator
+mx =1 MeV,sp= 
+10-1
+Constant CS
+= 1 MeV...sp-= 2
+10-5
+10
+10-13
+10-17
+10-21
+25
+10-
+100
+10-2
+10-1
+101
+102
+103
+Tx (GeV)7
+B.
+Vector Mediator
+Using a similar treatment for the vector mediator (de-
+noted by A′), we find that
+F 2
+DM(q2) =
+�
+q2
+ref − m2
+A′
+�2
+(q2 − m2
+A′)2
+1
+2mχm2e
+�
+2mχ (me + Te)2 −
+Tχ
+�
+(me + mχ)2 + 2mχTe
+�
++ mχT 2
+χ
+�
+(27)
+and,
+dσχe
+dTχ
+= ¯σeχ
+�
+q2
+ref − m2
+A′
+�2
+(q2 − m2
+A′)2
+mχ
+2µ2eχsCRT max
+χ
+�
+2mχ(me + Te)2
+−Tχ{(me + mχ)2 + 2mχTe} + mχT 2
+χ
+�
+(28)
+VI.
+RESULTS
+Taking into account the signal efficiency of each recoil
+bin (ϵsig), the exclusion limit on ¯σeχ is obtained by
+Neχϵsig < NB,
+(29)
+where Neχ, obtained from Eqn. (16), is the number of
+expected recoil events arising out of collision of target
+electron with DM particles boosted by the blazars. NB
+(B = TXS, BL for TXS 0506+056 and BL Lacertae)
+is 95% CL upper limits on number of events from the
+blazars.
+Three energy bins were considered in the analysis
+released by Super-K collaboration [44].
+The total
+number of events, the Monte Carlo simulation of the
+background, signal efficiency and spatial distribution of
+events were provided for each bin.
+One can use this
+data to select signals from a certain “searching cone” in
+the direction of the blazar. This removes the majority
+of the background from the data, increasing sensitivity.
+The selected signal is then used in the standard Poisson
+method [? ] to yield 95% CL upper limit on expected
+number of events (NB) for each of the three bins. This
+analysis was performed by the authors of Ref. [1], and
+we use their results (i.e. NB), summarised in Table II.
+(For details of the analysis, see [1, 4, 5, 44]). This gives
+us all the numbers relevant to finding exclusion limits
+on ¯σeχ using Eqn. (29).
+Bins
+ER(GeV)
+ϵsig
+NTXS
+NBL
+Bin 1
+(0.1, 1.33) 93.0%
+19.39
+17.27
+Bin 2
+(1.33, 20) 91.3%
+3.42
+6.27
+Bin 3
+(20, 103)
+81.1%
+2.98
+2.98
+TABLE II: Signal efficiency (ϵsig) and 95% CL upper
+limits on number of events (NTXS and NBL) from the
+blazars, provided for the three recoil bins of Super-K.
+Super-K is located deep underground to reduce
+background. As a result, the DM flux entering the de-
+tector is significantly attenuated, primarily as a result of
+its interaction with electrons on the Earth’s surface, and
+this gives rise to the attenuation bound in the exclusion
+plot. We provide an approximation of the attenuation
+bound, which is the cross section for which the DM par-
+ticle with Tχ ∼ 10 GeV can impart the threshold recoil
+energy in the detector. For this, we solve the following
+equation to calculate the energy (Tr) lost by the DM
+dTχ
+dx = −
+�
+T
+nT
+� T max
+r
+0
+dσ
+dTr
+TrdTr
+(30)
+and estimate ¯σeχ so that kinetic energy of the DM parti-
+cle at depth z, denoted by T z
+χ, is the detector threshold
+Eth (we consider Eth = 100 MeV corresponding to Bin
+1), for an initial kinetic energy Tχ,in ∼ 10 GeV. The
+area bounded by the attenuation bound and the exclu-
+sion bound is ruled out by our analysis. In this work, we
+limit ourselves to elastic scattering and ignore backscat-
+tering of light DM particles into the atmosphere. Note
+that the attenuation limits exist only for the heavy me-
+diators. There is no attenuation bound shown for the
+light mediator scenario with elastic scatterings and the
+attenuation bound shown for heavy mediator may also
+vary once a more elaborate study is performed, which
+we leave for future work.
+The exclusion bound arising from Super-K data is
+shown in Figs. 5 and 6 in the heavy mediator regime
+for scalar and vector operators. Taking energy depen-
+dence into account, the exclusion bound is significantly
+different compared to the bound obtained from constant
+cross section assumption. BMP1 sets a stronger bound
+as compared to BMP2, and DM density Profile 1 yields
+a better bound as compared to Profile 2.
+This is in
+agreement with what we expected from the density pro-
+files (Fig. 2) in Section III and the BBDM flux plots
+(Figs. 3a and 4a) in Section IV.
+Amongst the three recoil bins in Super-K, the
+strongest bound is set by Bin 3 ( Bin 2 ) for heavy
+mediator (constant cross section) cases.
+This result
+can be explained from the BBDM flux plots for TXS
+(Fig. 3a), by observing that the BBDM flux for heavy
+mediator is largest for Tχ ∼ (20 GeV, 103 GeV), which
+is nearly the DM energy range relevant to produce re-
+coil in the third bin of Super-K Similarly, for con-
+stant cross section case, the BBDM flux is largest for
+Tχ ∼ (1 GeV, 20 GeV), which is roughly the DM en-
+ergy range relevant to the second recoil energy bin of
+Super-K. For BL Lacertae, even though the BBDM
+flux is largest Tχ ∼ (1 GeV, 20 GeV) , which is the DM
+energy range relevant for Bin 2 (Fig. 4a), the event rate
+is largest for Bin 3 due to the larger size of the Bin.
+We also note that the constraints from TXS 0506+056
+is stronger than constraints from BL Lacertae, which
+is what we expected from Fig. 1 and the discussion in
+Section IV.
+In the exclusion bound plots (Figs. 5 and 6), we plot
+the most stringent limit on ¯σeχ coming from the three
+bins. Note that the bounds arising from scalar and vec-
+tor mediators are almost same. The reason for this can
+
+8
+(a) Exclusion Bound for TXS 0506+056
+(b) Exclusion Bound for BL Lacertae
+FIG. 5: The exclusion bound is plotted for the blazars
+TXS 0506+056 (5a) and BL Lacertae (5b), corresponding
+to BMP1. The cases considered are heavy vector mediator
+(plotted in solid lines), heavy scalar mediator (plotted in
+dashed lines) and constant cross section case (plotted in
+dotted lines). The various DM density profiles considered
+are Profile 1 (in red) and Profile 2 (in blue). The direct
+detection bounds from Xenon10, Xenon100,
+SENSEI [45–47] and DarkSide-50 [48] are also plotted.
+The bound arising due to DM attenuation is also given for
+heavy mediator scenario (plotted in grey). The area bounded
+by the attenuation bound and the exclusion bound is ruled
+out by our analysis. Exclusion limit from Superconducting
+Nanowires is provided in cyan color. Constraint due to
+solar reflection of DM [15, 49] is shown in amber color.
+The exclusion bound given by Cosmic Ray electron (CRe)
+boosted DM [50] is plotted in red (See text for more details).
+be understood from Figs. 3 and 4, where we see that
+the BBDM flux for DM energies relevant to recoil Bin
+3 of Super-K differs by ∼ O(10) for the two operators,
+which results in very little difference in the exclusion
+limits. However, the limit coming from the three dif-
+ferent bins can differ by ∼ 2 or 3 orders of magnitude
+for certain mχ, so a combined analysis of the three bins
+(a) Exclusion Bound for TXS 0506+056
+(b) Exclusion Bound for BL Lacertae
+FIG. 6: The exclusion bound is plotted for the blazars
+TXS 0506+056 (6a) and BL Lacertae (6b), corresponding
+to BMP2. The cases considered are heavy vector mediator
+(plotted in solid lines), heavy scalar mediator (plotted in
+dashed lines) and constant cross section case (plotted in
+dotted lines). The various DM density profiles considered
+are Profile 1 (in red) and Profile 2 (in blue). The direct
+detection bounds from Xenon10, Xenon100,
+SENSEI [45–47] and DarkSide-50 [48] are also plotted.
+The bound arising due to DM attenuation is also given for
+heavy mediator scenario (plotted in grey). The area bounded
+by the attenuation bound and the exclusion bound is ruled
+out by our analysis. Exclusion limit from Superconducting
+Nanowires is provided in cyan color. Constraint due to
+solar reflection of DM [15, 49] is shown in amber color.
+The exclusion bound given by Cosmic Ray electron (CRe)
+boosted DM [50] is plotted in red (See text for more details).
+might change the bounds. We, however, leave out such
+an analysis from this work.
+Since “heavy” and “light” mediator regimes are the
+convenient extremes of the actual DM model, the true
+exclusion bound would lie somewhere in between the
+bound set by these two regimes. Thus we compare the
+exclusion bound corresponding to various masses of the
+
+TXS 0506+056, Heavy Mediator, BMP2
+Solar Reflection
+Superconducting Nanowires
+10-27
+SENSEI
+10-30
+Bound
+XENON1
+XENON10
+10-33
+2
+cm
+50
+a
+1b
+CRe Boosted DM
+10-39
+10-42
+Vector Mediator
+sp = 7/3
+Scalar Mediator
+sp = 3/2
+Constant CS
+10-45
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+mx (GeV)BL Lacertae, Heavy Mediator, BMP2
+Solar Reflection
+Superconducting Nanowires
+10-27
+ENSEI
+10-30
+XENON1
+Super-K Attn. Boune
+ON10
+10-33
+2
+cm
+CRe Boosted DM
+1b
+10-39
+10-42
+Vector Mediator
+sp = 7/3
+Scalar Mediator
+sp = 3/2
+Constant CS
+10-45
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+mx (GeV)TXS 0506+056, Heavy Mediator, BMP1
+Solar Reflection
+Superconducting Nanowires
+10-27
+SENSEI
+10-30
+Bound
+Super-K Attn.
+2
+cm
+XEN
+50
+10-36
+sted DM
+1b
+CRe Boost
+S
+10-39
+10-42
+Vector Mediator
+sp = 7/3
+Scalar Mediator
+sp = 3/2
+Constant CS
+10-45
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+mx (GeV)BL Lacertae, Heavy Mediator, BMP1
+Solar Reflection
+Superconducting Nanowires
+10-27
+SENSEI
+10-30
+Bound
+Super-K Attn.
+10-33
+2
+cm
+CRe Boosted DM
+10-36
+1b
+10-39
+10-42
+Vector Mediator
+sp = 7/3
+Scalar Mediator
+sp = 3/2
+Constant CS
+10-45
+10-6
+10-5
+10-4
+10-3
+10-2
+10-1
+mx (GeV)9
+FIG. 7: The exclusion bound is plotted for the blazar TXS
+0506+056 for various mediator masses, corresponding to
+the vector mediator scenario. The mediator masses chosen
+are 10 GeV, 1 MeV, 10−2 MeV and 10−4 MeV, all plotted
+in different linestyles. The bound for heavy (in black) and
+light (in grey) mediator regime is also plotted. The profiles
+chosen are DM density Profile 1, BMP1.
+vector mediator for Profile 1 and BMP1 in Fig. 7. A
+similar comparison and scaling exists for Profile 2 and
+BMP2. Clearly, a mediator of mass 10 GeV corresponds
+to the heavy regime, and a mediator of mass 10−4 MeV
+reproduces the exclusion limit set by the light mediator
+regime.
+Apart from blazars jets, Cosmic Ray electrons (CRe)
+provide yet another environment to produce boosted
+DM particles. Exclusion bound from CRe boosted DM,
+using Super-K data, is plotted alongwith our bounds in
+Figs. 5 and 6. Furthermore, Refs. [51, 52] propose a DM
+detection device with extremely low recoil trigger made
+using Superconducting Nanowires.
+The best bounds
+from such a prototype device is also shown. Currently
+our results are much stronger, but proposed devices with
+materials like NbN and Al might give better exclusions
+in the near future. Constraints from other direct detec-
+tion experiments, such as Xenon10, Xenon100, SEN-
+SEI and DarkSide-50 are also shown. Exclusion limits
+from solar reflection of DM [15, 49] are important in the
+heavy mediator case, and are provided as well.
+The cosmological constraints from Big Bang Nucle-
+osynthesis (BBN) rule out thermal DM of mχ ≲ 10 MeV
+stringently [53, 54].
+However, these bounds can be
+relaxed in DM models with couplings to both neutri-
+noes as well as electrons [55]. The CMB observations
+similarly constrain DM annihilating to an e−e+ pair
+severely [56].
+An elaborate dark sector associated in
+these models can relax these BBN and CMB constraints,
+so that DM mostly annihilate to other dark sector par-
+ticles [57].
+VII.
+SUMMARY & OUTLOOK
+Blazars, in addition to being a key source of high en-
+ergy electrons, are projected to have a DM density spike
+in their core due to DM accretion onto their SMBH.
+Despite large uncertainties from astrophysics and the
+unknown annihilation properties of dark matter in the
+density of the succeeding DM spike, strong bounds on
+the elastic scattering cross section for DM-electron scat-
+tering have been obtained in Ref. [1]. Here, we demon-
+strate how these limits change when the resulting energy
+dependence of the S-matrix for the associated vertex
+Lorentz structure is taken into consideration. We re-
+main agnostic of the relic abundance mechanism since
+DM models might include an extended dark sector that
+has a significant impact on how much DM is there in
+the Universe right now. To that end, we derived lim-
+its using Super-K data. And, we found that the con-
+straints on such energy-dependent scattering cross sec-
+tions, which mostly depend on mediator mass, are at
+least several orders of magnitude tighter than the cur-
+rent limits from Blazars in the literature for the constant
+cross section assumption. Though the constant cross-
+section is a practical and meaningful way to explain a
+concept, in reality it corresponds to a small parameter
+space of a DM model. Our bounds are, however, weak-
+ened if the mediator mass is sufficiently small.
+This
+is because the BBDM flux is orders of magnitude less
+than the constant cross-section in the relevant energy
+bin (see fig. 3b,4b). We also studied the less cuspy pro-
+file of the DM spike and realised that the constraints on
+σeχ from BBDM are still significant compared to Cos-
+mic ray boosted DM.
+Another subtlety is that, in addition to relativistic
+electrons, blazars also contain energetic protons, which
+may contribute to the BBDM flux. However, the contri-
+bution is insignificant since the coupling with the proton
+is loop-suppressed if there is just a tree-level interaction
+of DM with charged electrons. Another natural assump-
+tion, inspired by the standard model’s SU(2)L gauge
+symmetry, is that neutrino should have the same cross
+section with DM as charged leptons, allowing us to com-
+pare the current σeχσνχ limits for Cosmic ray boosted
+DM [14].
+We intend to investigate this possibility in
+next work.
+ACKNOWLEDGEMENTS
+D.G. acknowledges support through the Ramanujan
+Fellowship and MATRICS Grant of the Department of
+Science and Technology, Government of India. D.S. has
+received funding from the European Union’s Horizon
+2020 research and innovation programme under grant
+agreement No 101002846, ERC CoG “CosmoChart”.
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+TXS 0506+056, Vector Mediator, BMP1, sp = 7/3
+10-17
+mA = 10-4 MeV
+10-21
+MeV
+mA^ = 10
+10-25
+2
+cm
+MeV
+mA
+Xa
+16
+10-33
+10-37
+mA' = 10 GeV
+10-41
+Heavy Mediator
+mA* = 1 MeV
+mA = 10-4 MeV
+= 10 GeV
+mA^ = 10-2 MeV
+Light Mediator
+mA'
+10-45
+10-6
+10-5
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+10-3
+10-2
+10-1
+mx
+(GeV)10
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diff --git a/m9AyT4oBgHgl3EQfYvdT/content/tmp_files/load_file.txt b/m9AyT4oBgHgl3EQfYvdT/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..93a41630c4f6f03d686d6b6429809ac3cd655a5e
--- /dev/null
+++ b/m9AyT4oBgHgl3EQfYvdT/content/tmp_files/load_file.txt
@@ -0,0 +1,868 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf,len=867
+page_content='Blazar boosted Dark Matter constraining model dependent σeχ :  Role of energy dependent cross sections  Supritha Bhowmick,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' ∗ Diptimoy Ghosh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' † and Divya Sachdeva2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' ‡  1Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Indian Institute of Science Education and Research Pune,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' India  2Laboratoire de Physique de l’Ecole Normale Sup´erieure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Universit´e PSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Sorbonne Universit´es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 24 rue Lhomond,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 75005 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' France  Elastic collisions with relativistic electrons from the blazar’s jet can accelerate dark matter (DM)  particles in the DM spike surrounding the supermassive black hole at its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' This can allow one to  set limits on the DM-electron scattering cross section (¯σeχ) for DM masses less than 100 MeV, which  has been found to be orders of magnitude stronger than the equivalent results from cosmic rays for  energy-independent cross-section [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' We also consider DM particles boosted by energetic electrons  in the jets of the blazars TXS 0506+056 and BL Lacertae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' In this study, we consider both vector  and scalar mediators for the scattering of electron and electrophilic fermionic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' We highlight  that the ensuing energy dependency of the S-matrix for the corresponding Lorentz structure of the  vertex significantly modifies the constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' We found that the revised exclusion limits are atleast  three orders of magnitude stronger than the conclusions drawn from simple constant crosssection  assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Our limits are also assessed for the less cuspy spike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  INTRODUCTION  The Cold Dark Matter (CDM) provides a compelling  explanation for a broad range of observations, including  rotation curves in spiral galaxies, gravitational micro-  lensing, cluster collisions (the Bullet Cluster), and tem-  perature anisotropy in the spectrum of cosmic mi-  crowave background radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' To that end, a variety of  particle physics models predict a feeble interaction be-  tween SM and DM, which can be investigated using Di-  rect detection (DD) experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The DD experiments  identify the nuclear or electronic recoils produced by the  scattering between DM and the detector’s (target) nu-  clei or electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The average velocity of DM particles  in the solar vicinity, however, restricts the amount of  energy that may be deposited in a detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  For ex-  ample: detectors like  Xenon1T can detect DM mass  mχ ∼ O(1 MeV), corresponding to electronic recoil of  ∼ O(1 keV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The neutrino detectors like Super-K are  sensitive to recoil energy threshold of ∼ O(1 MeV) lead-  ing to the smallest accessible DM mass of O(1 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Thus, these detectors appear to have a limited range for  detecting lighter DM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Since these observations  have been negative about the sign of DM, it is critical  to develop methods for probing the sub-GeV/MeV mass  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The reach of these experiments has been extended to  DM masses well below 1 GeV in recent years, thanks  to the novel idea of boosting the halo DM through its  interaction with the SM particles via cosmic rays  [2–  17], primordial black holes, diffuse Supernova Neutrino  Background (DSNB) [9–13], and blazars [1, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  De-  spite the fact that the boosted DM flux is substantial  and DM particles are (semi)relativistic, the sensitiv-  ∗ supritha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='bhowmick@students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='iiserpune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='in  † diptimoy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='ghosh@iiserpune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='in  ‡ divya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='sachdeva@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='ens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='fr  ity is achieved at larger cross sections because the up-  scattered subcomponent flux is significantly lower than  the galactic DM population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  In this paper, we consider the blazar boosted DM,  which was proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' [1, 18] in the context of  electrophilic fermionic DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The presence of a super-  massive Black Hole (BH) at the blazar center, which  provides a dense DM population compensates for the  blazar’s large distance from Earth by producing DM  flux that is stronger than that from galactic CRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The  existing literature, however, assumes that DM interac-  tion cross-sections are independent of DM energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Al-  though this simple assumption makes calculations eas-  ier, it’s not physically realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  It would also be an  incorrect approximation to make, notably in scenarios  where a significant DM flux becomes relativistic after  being scattered by energetic particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Some of the pre-  viously mentioned works [6, 15–17, 19] for cosmic ray  boosted DM, have already discovered that the limits for  energy-dependent cross-section differ by orders of mag-  nitude from those obtained under the assumption of a  constant cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The notion of an energy-dependent scattering cross-  section is thus primarily investigated in the present work  by taking into account electrophilic fermionic DM that  has been boosted by energetic electrons from blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  To constrain the scattering cross-section, we use elec-  tron recoil measurements in Super-Kamiokande.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' This  work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' We discuss the spectrum  of energetic particles in the blazar jets in Section II, and  describe DM density profiles in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' In Section  IV, we estimate the Blazar boosted DM (BBDM) flux  and compute the event rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' In Section V, we present  simplified DM models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' We present the main results of  our paper in Section VI, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=', the energy dependent ex-  clusion bound from BBDM-electron scattering, and in  Section VII, we summarize and conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='00209v1  [hep-ph]  31 Dec 2022  2  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  BLAZAR JET SPECTRUM  Blazars are characterized by a non-thermal spectral  energy distribution (SED).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' This spectrum has a low en-  ergy peak in the infra-red or X-ray region, which has  been accepted to be due to synchrotron emission of elec-  trons in the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Another peak at γ-ray frequencies could  be due to highly relativistic protons [20–24], as moti-  vated by the recent IceCube detection [25–27] of a high  energy neutrino from TXS 0506+056 blazar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Since DM  considered in this work is electrophilic, at tree level it  can only interact with electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Therefore, we are only  concerned with the blazar jets’ electron spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  We follow the procedure laid out in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' [1, 18] to  compute the spectrum of the energetic electrons in the  blazar jets, assuming “Blob geometry” model [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' In  this model, the energetic particles in the blazar jets  move isotropically in a “blob” frame, as the blob tra-  verses outwards along the jet axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The Lorentz boost  factor of the blob is given by ΓB = (1 − β2  B)−1/2, where  βB is the blob’s propagation speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The inclination of  the jet axis with respect to the line of sight (LOS) is  taken to be θLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  In the blob frame, the energetic electrons follow a  power law distribution with a high and a low energy  cutoff (γ′  max,e and γ′  min,e respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' This spectrum  can then be frame transformed to the observer’s rest  frame (for details of the derivation, see [18]), given by :  dΓe  dTedΩ = ce  4π Γ−αe  B  �  1 + Te  me  �−αe  ×  βe(1 − βeβBµ)−αe  �  (1 − βeβBµ)2 − (1 − β2e) (1 − β2  B)  (1)  where me and Te is the mass and kinetic energy of  the electron respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Speed of the electrons is given  by βe =  �  1 − m2  e/(Te + me)2�1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Doppler factor for  the blob frame is D = (ΓB(1 − βB cos θLOS))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' αe is  the power index of the electron spectrum in the blob  frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' µ is the cosine of the angle between direction of  the electron’s motion and the jet axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' It is related to  the scattering angle in the blob frame (¯µs) by [2, 18] :  µ(¯µs, φs) = ¯µs cos θLOS + sin φs sin θLOS  �  1 − ¯µ2s (2)  where φs is the azimuth with respect to the LOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' ¯µs  is related to the kinetic energy of the blazar jet electron  and the kinetic energy (Tχ) transferred to the DM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' as  follows :  ¯µs(Te,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Tχ) =  �  1 + T max  χ  − Tχ  Tχ  (me + mχ)2 + 2mχTe  (Te + me + mχ)2  �−1/2  (3)  Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' ce is a normalisation constant which is deter-  mined from the blazar jet electron luminosity (Le),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  where the latter depends on ce as [18,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 29] :  Le = cem2  eΓ2  B  � γ′  max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='e  γ′  min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='e  (γ′  e)1−αe dγ′  e ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  (4)  and thus ce is simply given by :  ce =  Le  m2eΓ2  B  ×  �  �  �  �  �  (2 − αe) /  ��  γ′  max,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='e  �2−αe −  �  γ′  min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='e  �2−αe�  if αe ̸= 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  1/ log  �  γ′  max,e/γ′  min,e  �  if αe = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  (5)  The parameters γ′  min,e, γ′  max,e, αe, Le and D are fit-  ted to the SED of a blazar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The doppler factor is as-  sumed to be either 2ΓB or ΓB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' These two cases corre-  spond to TXS 0506+056 (θLOS = 0) and BL Lacertae  (θLOS ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='82◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' All the parameters required to find the  blazar jet spectrum of TXS 0506+056 and BL Lacertae,  along with the blazar redshift and luminosity distance  (dL), are mentioned in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The electron spectrum  is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  DM DENSITY PROFILE  N-body simulations and observations are not sensi-  tive at subparsec sizes, thus, the DM distribution near  Galactic center is not well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The central super-  massive black hole (SMBH) can have a considerable  impact on DM density if the SMBH grows adiabati-  cally, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=', on a timescale much longer than its dynami-  cal timescale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The DM density in a region correspond-  ing to the sphere of gravitational influence of the black  hole (BH) is expected to be significantly enhanced [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  This results in a morphological feature known as a DM  spike, which corresponds to a DM profile with a power  law scaling ρ(r) ∝ r−γsp [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Here γsp =  9−2γ  4−γ com-  monly ranges from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='25 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='5, depending on the slope  of the initial DM halo distribution, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' In this work, we  assume the initial central DM profile is Navarro-Frenk-  White, γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Also, for DM annihilating with cross-  section ⟨σv⟩ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=', the innermost region of the DM spike is  3  Parameter  TXS 0506+056  BL Lacertae  Redshift  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='337  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='069  dL  1835.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='4 Mpc  322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='7 Mpc  MBH  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='09 × 108 M⊙  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='65 × 107 M⊙  ΓB  20  15  θLOS  0◦  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='82◦  αe  2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='5  �  γ′  min,e, γ′  max,e  �  (500, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='3 × 104) (700, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='5 × 104)  Le (erg/s)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='32 × 1044  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='7 × 1042  TABLE I: Model parameters for TXS 0506+056 [21] and  BL Lacertae blazars [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 1: The electron spectrum in the observer’s frame is  plotted above, for the blazars TXS 0506+056 (solid lines)  and BL Lacertae (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The spectrum is shown for  two different polar angles : θ = 0◦ (in red), θ = 10◦  (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' For larger kinetic energies (Te ≳ 10 GeV), the  electron flux from TXS 0506+056 blazar exceeds the flux  from BL Lacertae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  depleted because DM annihilate efficiently on account  of high DM density, leading to “annihilation plateau”  density given by  ρsat =  mχ  ⟨σv⟩ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='tBH  ,  (6)  where tBH ∼ 109 yrs is the age of the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The DM  density profile in such a spike is given by  ρ(r) =  �  �  �  �  �  0  r < 4RS  ρsat  4RS ≤ r < Rsat  N1r−γsp  Rsat ≤ r < Rsp  N2r−γ  r ≥ Rsp  ,  (7)  where RS = 2GM/c2 is the Schwarzchild radius of the  BH, Rsp = 105 Rs is the radius of the spike [29] and  ρ(r) goes to zero in the region r < 4Rs due to DM  particles being captured by the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The saturation  density ρsat and the saturation radius Rsat are related  by the equality ρ(Rsat) = ρsat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='The normalization N1 of  ρ is determined by observing that the mass of the spike  is of the same order as MBH within the spike radius [32]  and N2 is determined by observing the continuity of the  profile at Rsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  For a pre-existent DM halo with γ = 1, the final DM  profile near BH corresponds to γsp = 7/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' A more re-  alistic model was obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' [33], where the time-  evolution of dark matter distribution was investigated  on sub-parsec scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' This implied softening of the DM  density spike, due to scattering of DM by stars and cap-  ture of DM particles by the SMBH, dampening it to  γsp = 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Thus, in this work, we consider both the  DM profile parameters γsp = 7/3 and γsp = 3/2 along  with two extreme values of ⟨σv⟩ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='. We define these as  Profile 1: ρ(Rsat ≤ r < Rsp) = N1r−7/3  Profile 2: ρ(Rsat ≤ r < Rsp) = N1r−3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  For each of these profiles, the two benchmark points  (BMPs) are defined as:  BMP 1: No DM annihilation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=', ⟨σv⟩ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' = 0  BMP 2: ⟨σv⟩ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' = 3 × 10−26 cm3s−1, thermal  relic cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Another quantity relevant to the computation of the  BBDM flux is the line of sight (LOS) integral of DM  density around the blazar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' This provides a measure of  the number of DM particles being boosted by the blazar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  At a certain distance r from the blazar, it is defined as  ΣLOS(r) =  � r  rmin  ρ(r′)dr′  (8)  where rmin is the distance from the SMBH from where  the blazar jet starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' To get a measure of all boosted DM  particles, we want the LOS integral at large distances  (r >> 105RS), and we define Σtot  LOS = ΣLOS(r >>  105RS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  In this work, we will study BBDM flux from TXS  0506+056 and BL Lacertae, and for these blazars, rmin  lies within 100RS [21, 29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' We take rmin = 4RS,  noting that Σtot  LOS is independent of choice of rmin for  models which allow for DM pair annihilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' For no  DM annihilation, Σtot  LOS decreases by an order of magni-  tude when rmin is changed to 100RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The DM density and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' integral profiles are plot-  ted in Fig 2 for two benchmark points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' For both DM  density profiles, BMP1 yields a larger spike and a larger  LOS integral (Σtot  LOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' This results in a larger BBDM  flux and consequently a stronger exclusion bound on  DM-electron interaction cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Hence, one can  expect models with no DM annihilation to yield bet-  ter bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Moreover, even though the LOS integral  for the idealistic spike (Profile 1) is very large and thus  results in substantial BBDM flux, the more realistic pro-  file (Profile 2) with a softer spike would lead to a much  smaller LOS integral, and hence a much weaker exclu-  sion bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Blazar Jet Electron Flux  TXS 0506+056, 0 = 0°  TXS 0506+056, 0 = 10°  BL Lacertae,  = 0°  1056  BL Lacertae,  = 10°  1053  1038  10-4  10-3  10-1  100  101  10-2  102  Te (GeV)4  (a) DM Density Profile  (b) LOS integral profile  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 2: The profiles of ρDM (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 2a) and ΣDM (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 2b)  are plotted above for TXS 0506+056 blazar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The DM mass chosen for these figures is mχ = 1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The profile 1 (red) and profile 2 (blue) are plotted for BMP  1 (solid curve) and BMP 2 (dashed curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' BL Lacertae,  on the other hand, is less massive than TXS 0506+056,  and yields a larger spike at a smaller distance from the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  BLAZAR BOOSTED DARK MATTER  FLUX AND EVENT RATE  DM particles are boosted via elastic collisions with  the relativistic electrons in the blazar jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The DM dif-  ferential flux resulting out of collision with the electrons  is obtained as follows :  dφχ  dTχ  =  Σtot  DM  2πmχd2  L  � 2π  0  dφs  � T max  e  (Tχ,φs)  T min  e  (Tχ,φs)  dTe  ×dσχe  dTχ  dΓe  dTedΩ ,  (9)  where σχe is the DM-electron interaction cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The integration over φs becomes trivial in case of TXS  0506+056, where the system is symmetric about LOS,  and we can simply set µ = ¯µs (from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' (2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The  maximal  kinetic  energy  of  the  blazar  jet  electrons  along  LOS  is  given  by  T max  e,jet  =  me  �  γ′  max,e Γ−1  B (1 − βB cos θLOS)−1 − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  This  is  set as the upper bound of the integral on Te in  Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The lower bound is set by the minimum  kinetic energy required for scattering, given by  T min  e  =  �Tχ  2 − me  � �  1 ±  �  1 + 2Tχ(me + mχ)2  mχ(Tχ − 2me)2  �  ,  (10)  with + and − applicable for Tχ > 2me and Tχ <  2me respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  However, the kinetic energy of the  slowest electrons in the blazar jets could be larger  than T min  e  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  In such a case, the kinetic energy of the  least energetic electron in the jet, given by T min  e,jet =  me  �  γ′  min,e Γ−1  B (1 − βB cos θLOS)−1 − 1  �  , sets the lower  bound of the integral in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The differential cross section ( dσχe/dTχ ) of the DM-  blazar jet electron interaction is given by,  dσχe  dTχ  = |M|2  16πse  1  T max  χ  (11)  where M is the interaction matrix element, a function  of Tχ and Te.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' se is the centre of momentum energy for  the electron-DM collision given by :  se = (mχ + me)2 + 2mχTe ,  (12)  and T max  χ  is the maximum kinetic energy that can be  imparted to a DM particle by a blazar jet electron of  energy Te is given by :  T max  χ  =  T 2  e + 2meTe  Te + (me + mχ)2 / (2mχ)  (13)  The effect of including energy dependence in DM-  electron interaction can be seen from the BBDM flux  plots, given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Profile 1 of DM density  clearly gives larger BBDM flux as compared to Pro-  file 2, as expected from larger DM spike for profile 1  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 2 in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' For DM to register event  at Super-K, kinetic energies greater than ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='1 GeV  are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  In this energy range, the heavy media-  tor scenario gives much larger BBDM flux as compared  to the constant cross section case, while on the other  hand, the light mediator regime yields a much smaller  BBDM flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' From this, we expect the exclusion limit  on DM-electron interaction, arising from light mediator  regime, to be extremely weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Since the vector medi-  ator case gives slightly larger BBDM flux as compared  to the scalar mediator scenario, we hope for moderately  better bounds from vector mediators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Also, since for  any given DM Profile or BMP, the BBDM flux is larger  for smaller mass DM particles, we can expect bounds  to grow stronger for lighter DM particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Finally, the  BBDM flux plots terminate at a certain value of Tχ  because the blazar jet electrons boosting the DM par-  ticles have an upper cutoff on their energies (for TXS  0506+056 jets, T max  e,jet ∼ 260 GeV and for BL Lacertae  jets, T max  e,jet ∼ 225 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  TXS 0506+056  1018  sp = 7/3，BMP1  sp = 3/2,BMP1  sp = 7/3,BMP2  sp = 3/2,BMP2  1015  pDM (GeV/cm3)  1012  109  106  103  100  100  101  102  103  104  105  106  r (Rs)TXS 0506+056  1032  1030  (GeV /cm²)  1026  ZDM (  1024  sp = 7/3,BMP1  sp = 3/2, BMP1  1022  sp = 7/3,BMP2  sp = 3/2, BMP2  100  101  102  103  104  105  106  r (Rs)5  TXS 0506+056 blazar is further away from us as  compared to BL Lacertae, and is more massive, lead-  ing to a larger DM density spike and hence a larger  ΣLOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The contribution to the DM flux coming from  the factors outside the integrals in Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' (9) is hence  much larger for BL Lacertae than TXS 0506+056 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=',  �  ΣLOS/d2  L  �  BL Lac = 103 �  ΣLOS/d2  L  �  TXS ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Inspite of  this, we note in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 3 and 4 that the flux of DM  particles boosted by TXS 0506+056 is larger than the  BBDM flux of BL Lacertae, for more energetic DM par-  ticles (Tχ ≳ 10 GeV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' This is because, the kinetic en-  ergy range of the electron responsible for boosting the  DM particle to energies greater than 10 GeV is roughly  Te ≳ 10 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  For this energy range, the electron  spectrum in TXS blazar is larger than that of BL Lac  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' As a result of this, we expect stronger bounds  to arise from TXS blazar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The obtained boosted DM flux will yield the following  rate of electron recoil events in Super-K  dR  dER  = ℵ  � ∞  T min  χ  dTχ  dφχ  dTχ  dσχe  dER  (14)  where ℵ = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='5 × 1033 is the effective number of target  electrons in Super-K, and dσχe/dER is the differential  DM-target electron interaction cross section, given by  dσχe  dER  = |M|2  16πsχ  1  Emax  R  (15)  where sχ is centre of momentum energy for the DM-  target electron collision which can be obtained from  Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' (12) under the substitution :  mχ ↔ me and  Te → Tχ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Emax  R  is the maximum possible recoil in the  detector, that can be imparted by a DM particle with  kinetic energy Tχ, and can be obtained from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' (13)  with the appropriate substitutions mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  To get total number of expected recoil events (Neχ)  in a certain energy bin, Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' (14) needs to be integrated  over ER, as follows  Neχ = ℵ Texp  � ER,max  ER,min  dER  � ∞  T min  χ  dTχ  dφχ  dTχ  dσχe  dER  (16)  where Texp = 2628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='1 days is the exposure time, and  [ER,min, ER,max] is the recoil energy range of each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  SIMPLIFIED DM MODEL  We assume that a fermionic DM particle χ, of mass  mχ, only interacts with electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' This scenario is pos-  sible in several leptophilic particle DM models [34–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Additionally, the electron-DM interaction is mediated  by a scalar or vector particle, given as  L = gχφφ¯χχ + geφφ¯ee  or  (17)  = gχA′A′  µ ¯χγµχ + geA′A′  µ¯eγµe  (18)  For simplicity, we will drop A′ and φ from subscripts in  the coupling constants such that gχ (ge) is the coupling  (a) BBDM flux for heavy mediator scenario  (b) BBDM flux for light mediator scenario  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 3: Flux of DM particles, boosted by energetic  electrons in the jets of TXS 0506+056 blazar, is plotted  above, for heavy (3a) and light (3b) mediators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The  parameters chosen for the above plots are ¯σeχ = 10−30 cm2  and BMP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The vector and the scalar mediator cases have  been plotted in solid and dashed lines respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' For  comparision, DM flux for constant cross section scenario  have also been plotted in dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Two DM masses  have been considered, mχ = 1 keV (plotted in black) and  mχ = 1 MeV (plotted in red) for DM density Profile 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  γsp = 7/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' To avoid overcrowding, only vector mediator  case is considered for Profile 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' γsp = 3/2), and BBDM  flux is plotted (in blue) corresponding to DM mass  mχ = 1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Clearly, Profile 2 yields a smaller BBDM  flux as compared to Profile 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  constant of the dark mediator to the DM particle (elec-  tron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Next, we provide the differential cross-section for  different operators and inspect the effect of the Lorentz  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' For that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' we define the following quantities :  M2 = 16g2  eg2  χm2  em2  χ  (q2  ref − m2  i )2  (19)  ¯σeχ =  µ2  χe  16πm2em2χ  M2  (20)  TXS 0506+056,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Heavy Mediator  Vector Mediator  mx = 1 keV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='sp=  Scalar Mediator  mx =1 MeV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' sp =  105  Constant CS  102  10-1  10-4  10-7  10-10  10-2  10-1  100  101  102  103  Tx (GeV)TXS 0506+056,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Light Mediator  103  Vector Mediator  mx=1 keV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='sp=  Scalar Mediator  mx =1 MeV,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='sp=  10-1  Constant CS  10-5  10  :10-13  10-17  10-21  25  10-  100  10-2  10-1  101  102  103  Tx (GeV)6  (a) BBDM flux for heavy mediator scenario  (b) BBDM flux for light mediator scenario  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 4: Flux of DM particles, boosted by energetic  electrons in the jets of BL Lacertae blazar, is plotted above,  for heavy (4a) and light (4b) mediators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The parameters  chosen for the above plots are ¯σeχ = 10−30 cm2 and BMP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The vector and the scalar mediator cases have been plotted  in solid and dashed lines respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' For comparision, DM  flux for constant cross section scenario have also been  plotted in dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Two DM masses have been  considered, mχ = 1 keV (plotted in black) and mχ = 1 MeV  (plotted in red) for DM density Profile 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' γsp = 7/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  To avoid overcrowding, only vector mediator case is  considered for Profile 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' γsp = 3/2), and BBDM flux is  plotted (in blue) corresponding to DM mass mχ = 1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Clearly, Profile 2 yields a smaller BBDM flux as compared  to Profile 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  where qref = αme is the reference momentum trans-  ferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Here mi is the mass of the dark mediator  (i = A′, φ for vector, scalar mediator) and µeχ is the  reduced mass of the DM-electron system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' We also de-  fine a form factor,  F 2  DM(q2) = |M|2/M2  (21)  This factor contains the energy dependence arising in  the differential cross section dσχe/dTχ due to the blazar  jet electrons boosting the DM particles and the Lorentz  structure of the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The explicit form of FDM  depends on the model of DM and mediator considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  A similar form factor, Frec, contains energy depen-  dence in the differential cross section dσχe/dER arising  due to interaction of relativistic DM particles with the  electrons in Super-K, and can be obtained from the  form factor FDM of Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' (21) by making the substitu-  tions : me ↔ mχ, Tχ → ER and Te → Tχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Hence the differential cross sections,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' dσχe/dTχ and  dσχe/dER,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' relevant in the DM-blazar jet electron scat-  tering and DM scattering at the detector end respec-  tively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' are given by :  dσχe  dTχ  = ¯σeχ  m2  em2  χ  µ2eχ  F 2  DM(q2)  sCRT max  χ  (22)  and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  dσχe  dER  = ¯σeχ  m2  em2  χ  µ2eχ  F 2  rec(q2)  sχEmax  R  (23)  Under the energy independent approximation for the  cross section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' the differential cross section would simply  be :  dσχe  dTχ  =  ¯σeχ  T max  χ  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' dσχe  dER  =  ¯σeχ  Emax  R  (24)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Scalar Mediator  Considering a scalar mediator (denoted as φ), one can  calculate F 2  DM for the interaction between electrons in  blazar jets and non-relativistic DM, using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' (21) to  obtain  F 2  DM(q) =  �  q2  ref − m2  φ  �2  �  q2 − m2  φ  �2  (2mχ + Tχ)  �  2m2  e + mχTχ  �  4mχm2e  (25)  The differential cross section (dσχe/dTχ) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' the  DM energy (Tχ), is :  dσχe  dTχ  = ˜σeχ  (q2  ref − m2  φ)2  (q2 − m2  φ)2  � mχ  4µ2eχ  (2mχ + Tχ)  �  2m2  e + mχTχ  �  sCRT max  χ  �  (26)  The form factor Frec and the differential cross-section  w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' the recoil energy of the detector (dσχe/dER) are  obtained from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' (25) and Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' (26) by performing  the substitutions prescribed in the previous section, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  me ↔ mχ, Tχ → ER, Te → Tχ, se → sχ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  BL Lacertae, Heavy Mediator  Vector Mediator  mx = 1 keV，sp =  Scalar Mediator  105  Constant CS  102  10-1  10-4  10-7  10-10  10-2  10-1  100  101  102  103  Tx (GeV)BL Lacertae, Light Mediator  103  Vector Mediator  mx=1 keV，sp=  Scalar Mediator  mx =1 MeV,sp=  10-1  Constant CS  = 1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='sp-= 2  10-5  10  10-13  10-17  10-21  25  10-  100  10-2  10-1  101  102  103  Tx (GeV)7  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Vector Mediator  Using a similar treatment for the vector mediator (de-  noted by A′), we find that  F 2  DM(q2) =  �  q2  ref − m2  A′  �2  (q2 − m2  A′)2  1  2mχm2e  �  2mχ (me + Te)2 −  Tχ  �  (me + mχ)2 + 2mχTe  �  + mχT 2  χ  �  (27)  and,  dσχe  dTχ  = ¯σeχ  �  q2  ref − m2  A′  �2  (q2 − m2  A′)2  mχ  2µ2eχsCRT max  χ  �  2mχ(me + Te)2  −Tχ{(me + mχ)2 + 2mχTe} + mχT 2  χ  �  (28)  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  RESULTS  Taking into account the signal efficiency of each recoil  bin (ϵsig), the exclusion limit on ¯σeχ is obtained by  Neχϵsig < NB,  (29)  where Neχ, obtained from Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' (16), is the number of  expected recoil events arising out of collision of target  electron with DM particles boosted by the blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' NB  (B = TXS, BL for TXS 0506+056 and BL Lacertae)  is 95% CL upper limits on number of events from the  blazars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Three energy bins were considered in the analysis  released by Super-K collaboration [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The total  number of events, the Monte Carlo simulation of the  background, signal efficiency and spatial distribution of  events were provided for each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  One can use this  data to select signals from a certain “searching cone” in  the direction of the blazar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' This removes the majority  of the background from the data, increasing sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The selected signal is then used in the standard Poisson  method [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' ] to yield 95% CL upper limit on expected  number of events (NB) for each of the three bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' This  analysis was performed by the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' [1], and  we use their results (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' NB), summarised in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  (For details of the analysis, see [1, 4, 5, 44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' This gives  us all the numbers relevant to finding exclusion limits  on ¯σeχ using Eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Bins  ER(GeV)  ϵsig  NTXS  NBL  Bin 1  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='33) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='0%  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='39  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='27  Bin 2  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='33, 20) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='3%  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='42  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='27  Bin 3  (20, 103)  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='1%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='98  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='98  TABLE II: Signal efficiency (ϵsig) and 95% CL upper  limits on number of events (NTXS and NBL) from the  blazars, provided for the three recoil bins of Super-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Super-K is located deep underground to reduce  background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' As a result, the DM flux entering the de-  tector is significantly attenuated, primarily as a result of  its interaction with electrons on the Earth’s surface, and  this gives rise to the attenuation bound in the exclusion  plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' We provide an approximation of the attenuation  bound, which is the cross section for which the DM par-  ticle with Tχ ∼ 10 GeV can impart the threshold recoil  energy in the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' For this, we solve the following  equation to calculate the energy (Tr) lost by the DM  dTχ  dx = −  �  T  nT  � T max  r  0  dσ  dTr  TrdTr  (30)  and estimate ¯σeχ so that kinetic energy of the DM parti-  cle at depth z, denoted by T z  χ, is the detector threshold  Eth (we consider Eth = 100 MeV corresponding to Bin  1), for an initial kinetic energy Tχ,in ∼ 10 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The  area bounded by the attenuation bound and the exclu-  sion bound is ruled out by our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' In this work, we  limit ourselves to elastic scattering and ignore backscat-  tering of light DM particles into the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Note  that the attenuation limits exist only for the heavy me-  diators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' There is no attenuation bound shown for the  light mediator scenario with elastic scatterings and the  attenuation bound shown for heavy mediator may also  vary once a more elaborate study is performed, which  we leave for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The exclusion bound arising from Super-K data is  shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 5 and 6 in the heavy mediator regime  for scalar and vector operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Taking energy depen-  dence into account, the exclusion bound is significantly  different compared to the bound obtained from constant  cross section assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' BMP1 sets a stronger bound  as compared to BMP2, and DM density Profile 1 yields  a better bound as compared to Profile 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  This is in  agreement with what we expected from the density pro-  files (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 2) in Section III and the BBDM flux plots  (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 3a and 4a) in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Amongst the three recoil bins in Super-K, the  strongest bound is set by Bin 3 ( Bin 2 ) for heavy  mediator (constant cross section) cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  This result  can be explained from the BBDM flux plots for TXS  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 3a), by observing that the BBDM flux for heavy  mediator is largest for Tχ ∼ (20 GeV, 103 GeV), which  is nearly the DM energy range relevant to produce re-  coil in the third bin of Super-K Similarly, for con-  stant cross section case, the BBDM flux is largest for  Tχ ∼ (1 GeV, 20 GeV), which is roughly the DM en-  ergy range relevant to the second recoil energy bin of  Super-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' For BL Lacertae, even though the BBDM  flux is largest Tχ ∼ (1 GeV, 20 GeV) , which is the DM  energy range relevant for Bin 2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 4a), the event rate  is largest for Bin 3 due to the larger size of the Bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  We also note that the constraints from TXS 0506+056  is stronger than constraints from BL Lacertae, which  is what we expected from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 1 and the discussion in  Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  In the exclusion bound plots (Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 5 and 6), we plot  the most stringent limit on ¯σeχ coming from the three  bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Note that the bounds arising from scalar and vec-  tor mediators are almost same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The reason for this can  8  (a) Exclusion Bound for TXS 0506+056  (b) Exclusion Bound for BL Lacertae  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 5: The exclusion bound is plotted for the blazars  TXS 0506+056 (5a) and BL Lacertae (5b), corresponding  to BMP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The cases considered are heavy vector mediator  (plotted in solid lines), heavy scalar mediator (plotted in  dashed lines) and constant cross section case (plotted in  dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The various DM density profiles considered  are Profile 1 (in red) and Profile 2 (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The direct  detection bounds from Xenon10, Xenon100,  SENSEI [45–47] and DarkSide-50 [48] are also plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The bound arising due to DM attenuation is also given for  heavy mediator scenario (plotted in grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The area bounded  by the attenuation bound and the exclusion bound is ruled  out by our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Exclusion limit from Superconducting  Nanowires is provided in cyan color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Constraint due to  solar reflection of DM [15, 49] is shown in amber color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The exclusion bound given by Cosmic Ray electron (CRe)  boosted DM [50] is plotted in red (See text for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  be understood from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 3 and 4, where we see that  the BBDM flux for DM energies relevant to recoil Bin  3 of Super-K differs by ∼ O(10) for the two operators,  which results in very little difference in the exclusion  limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' However, the limit coming from the three dif-  ferent bins can differ by ∼ 2 or 3 orders of magnitude  for certain mχ, so a combined analysis of the three bins  (a) Exclusion Bound for TXS 0506+056  (b) Exclusion Bound for BL Lacertae  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 6: The exclusion bound is plotted for the blazars  TXS 0506+056 (6a) and BL Lacertae (6b), corresponding  to BMP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The cases considered are heavy vector mediator  (plotted in solid lines), heavy scalar mediator (plotted in  dashed lines) and constant cross section case (plotted in  dotted lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The various DM density profiles considered  are Profile 1 (in red) and Profile 2 (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The direct  detection bounds from Xenon10, Xenon100,  SENSEI [45–47] and DarkSide-50 [48] are also plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The bound arising due to DM attenuation is also given for  heavy mediator scenario (plotted in grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The area bounded  by the attenuation bound and the exclusion bound is ruled  out by our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Exclusion limit from Superconducting  Nanowires is provided in cyan color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Constraint due to  solar reflection of DM [15, 49] is shown in amber color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The exclusion bound given by Cosmic Ray electron (CRe)  boosted DM [50] is plotted in red (See text for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  might change the bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' We, however, leave out such  an analysis from this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Since “heavy” and “light” mediator regimes are the  convenient extremes of the actual DM model, the true  exclusion bound would lie somewhere in between the  bound set by these two regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Thus we compare the  exclusion bound corresponding to various masses of the  TXS 0506+056, Heavy Mediator, BMP2  Solar Reflection  Superconducting Nanowires  10-27  SENSEI  10-30  Bound  XENON1  XENON10  10-33  2  cm  50  a  1b  CRe Boosted DM  10-39  10-42  Vector Mediator  sp = 7/3  Scalar Mediator  sp = 3/2  Constant CS  10-45  10-6  10-5  10-4  10-3  10-2  10-1  mx (GeV)BL Lacertae, Heavy Mediator, BMP2  Solar Reflection  Superconducting Nanowires  10-27  ENSEI  10-30  XENON1  Super-K Attn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Boune  ON10  10-33  2  cm  CRe Boosted DM  1b  10-39  10-42  Vector Mediator  sp = 7/3  Scalar Mediator  sp = 3/2  Constant CS  10-45  10-6  10-5  10-4  10-3  10-2  10-1  mx (GeV)TXS 0506+056, Heavy Mediator, BMP1  Solar Reflection  Superconducting Nanowires  10-27  SENSEI  10-30  Bound  Super-K Attn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  2  cm  XEN  50  10-36  sted DM  1b  CRe Boost  S  10-39  10-42  Vector Mediator  sp = 7/3  Scalar Mediator  sp = 3/2  Constant CS  10-45  10-6  10-5  10-4  10-3  10-2  10-1  mx (GeV)BL Lacertae, Heavy Mediator, BMP1  Solar Reflection  Superconducting Nanowires  10-27  SENSEI  10-30  Bound  Super-K Attn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  10-33  2  cm  CRe Boosted DM  10-36  1b  10-39  10-42  Vector Mediator  sp = 7/3  Scalar Mediator  sp = 3/2  Constant CS  10-45  10-6  10-5  10-4  10-3  10-2  10-1  mx (GeV)9  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 7: The exclusion bound is plotted for the blazar TXS  0506+056 for various mediator masses, corresponding to  the vector mediator scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The mediator masses chosen  are 10 GeV, 1 MeV, 10−2 MeV and 10−4 MeV, all plotted  in different linestyles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The bound for heavy (in black) and  light (in grey) mediator regime is also plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The profiles  chosen are DM density Profile 1, BMP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  vector mediator for Profile 1 and BMP1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' A  similar comparison and scaling exists for Profile 2 and  BMP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Clearly, a mediator of mass 10 GeV corresponds  to the heavy regime, and a mediator of mass 10−4 MeV  reproduces the exclusion limit set by the light mediator  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Apart from blazars jets, Cosmic Ray electrons (CRe)  provide yet another environment to produce boosted  DM particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Exclusion bound from CRe boosted DM,  using Super-K data, is plotted alongwith our bounds in  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Furthermore, Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' [51, 52] propose a DM  detection device with extremely low recoil trigger made  using Superconducting Nanowires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The best bounds  from such a prototype device is also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Currently  our results are much stronger, but proposed devices with  materials like NbN and Al might give better exclusions  in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Constraints from other direct detec-  tion experiments, such as Xenon10, Xenon100, SEN-  SEI and DarkSide-50 are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Exclusion limits  from solar reflection of DM [15, 49] are important in the  heavy mediator case, and are provided as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  The cosmological constraints from Big Bang Nucle-  osynthesis (BBN) rule out thermal DM of mχ ≲ 10 MeV  stringently [53, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  However, these bounds can be  relaxed in DM models with couplings to both neutri-  noes as well as electrons [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' The CMB observations  similarly constrain DM annihilating to an e−e+ pair  severely [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  An elaborate dark sector associated in  these models can relax these BBN and CMB constraints,  so that DM mostly annihilate to other dark sector par-  ticles [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  SUMMARY & OUTLOOK  Blazars, in addition to being a key source of high en-  ergy electrons, are projected to have a DM density spike  in their core due to DM accretion onto their SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Despite large uncertainties from astrophysics and the  unknown annihilation properties of dark matter in the  density of the succeeding DM spike, strong bounds on  the elastic scattering cross section for DM-electron scat-  tering have been obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Here, we demon-  strate how these limits change when the resulting energy  dependence of the S-matrix for the associated vertex  Lorentz structure is taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' We re-  main agnostic of the relic abundance mechanism since  DM models might include an extended dark sector that  has a significant impact on how much DM is there in  the Universe right now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' To that end, we derived lim-  its using Super-K data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' And, we found that the con-  straints on such energy-dependent scattering cross sec-  tions, which mostly depend on mediator mass, are at  least several orders of magnitude tighter than the cur-  rent limits from Blazars in the literature for the constant  cross section assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Though the constant cross-  section is a practical and meaningful way to explain a  concept, in reality it corresponds to a small parameter  space of a DM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Our bounds are, however, weak-  ened if the mediator mass is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  This  is because the BBDM flux is orders of magnitude less  than the constant cross-section in the relevant energy  bin (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' 3b,4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' We also studied the less cuspy pro-  file of the DM spike and realised that the constraints on  σeχ from BBDM are still significant compared to Cos-  mic ray boosted DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  Another subtlety is that, in addition to relativistic  electrons, blazars also contain energetic protons, which  may contribute to the BBDM flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' However, the contri-  bution is insignificant since the coupling with the proton  is loop-suppressed if there is just a tree-level interaction  of DM with charged electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Another natural assump-  tion, inspired by the standard model’s SU(2)L gauge  symmetry, is that neutrino should have the same cross  section with DM as charged leptons, allowing us to com-  pare the current σeχσνχ limits for Cosmic ray boosted  DM [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  We intend to investigate this possibility in  next work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' acknowledges support through the Ramanujan  Fellowship and MATRICS Grant of the Department of  Science and Technology, Government of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' has  received funding from the European Union’s Horizon  2020 research and innovation programme under grant  agreement No 101002846, ERC CoG “CosmoChart”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Granelli, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Ullio, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
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+page_content=' Sahdev, Dark matter, muon anomalous magnetic mo-  ment, and the XENON1T excess, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content=' D 103,  015006 (2021), arXiv:2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
+page_content='08205 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/m9AyT4oBgHgl3EQfYvdT/content/2301.00209v1.pdf'}
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+Option pricing under the normal SABR model
+with Gaussian quadratures
+Jaehyuk Choia,∗, Byoung Ki Seob
+aPeking University HSBC Business School,
+University Town, Nanshan, Shenzhen 518055, China
+bSchool of Business Administration, Ulsan National Institute of Science and Technology
+Abstract
+The stochastic-alpha-beta-rho (SABR) model has been widely adopted in options trading. In particular,
+the normal (β = 0) SABR model is a popular model choice for interest rates because it allows negative
+asset values. The option price and delta under the SABR model are typically obtained via asymptotic
+implied volatility approximation, but these are often inaccurate and arbitrage-able. Using a recently
+discovered price transition law, we propose a Gaussian quadrature integration scheme for price options
+under the normal SABR model.
+The compound Gaussian quadrature sum over only 49 points can
+calculate a very accurate price and delta that are arbitrage-free.
+Keywords:
+Gaussian quadrature, normal model, SABR model, stochastic volatility
+1. Introduction
+The stochastic-alpha-beta-rho (SABR) model (Hagan et al., 2002) has been widely adopted in the fi-
+nancial industry for the pricing and risk management of European options, owing to its intuitive and
+parsimonious parameterization. It has been a standard practice for practitioners to obtain the option
+price and delta from the asymptotic approximation of implied volatility (Hagan et al., 2002), but the
+approximation loses accuracy and allows arbitrage as the variance of volatility becomes large. Despite
+numerous attempts to improve implied volatility approximation (Ob�l´oj, 2007; Paulot, 2015; Lorig et al.,
+2017; Yang et al., 2017; Choi and Wu, 2021a), it does not seem possible to obtain an approximation accu-
+rate for all parameter ranges. To date, there are several full-scale methods for pricing the SABR model:
+Monte–Carlo simulations (Chen et al., 2012; Leitao et al., 2017a,b; Cai et al., 2017; Choi et al., 2019; Cui
+et al., 2021), finite difference methods (Park, 2014; von Sydow et al., 2019), and continuous-time Markov
+chains (Cui et al., 2018). These approaches require heavy computation and complex implementation
+compared with the approximation approach. Therefore, they are difficult to implement in practice.
+We contribute to the literature by proposing a novel and efficient pricing scheme under the normal
+∗Corresponding author Tel: +86-755-2603-0568; Address: Rm 755, Peking University HSBC Business School, University
+Town, Nanshan, Shenzhen 518055, China
+Email addresses: jaehyuk@phbs.pku.edu.cn (Jaehyuk Choi), bkseo@unist.ac.kr (Byoung Ki Seo)
+Preprint submitted to arXiv
+November 16, 2022
+arXiv:2301.02797v1  [q-fin.PR]  7 Jan 2023
+
+(β = 0) SABR model. Despite being a special case, the normal SABR model has gained attention for
+modeling interest rates owing to its flexibility to allow negative asset prices. The normal model based
+on the arithmetic Brownian motion (BM) has long been used in fixed income markets, as opposed to
+the Black–Scholes model based on geometric BM.1 The normal SABR model is a natural extension of
+the normal model with stochastic volatility that exhibits a volatility smile. Antonov et al. (2015) adopt
+the normal SABR model as a key component of the mixture approach for modeling a low-interest-rate
+environment. Choi et al. (2019) propose the hyperbolic normal stochastic volatility (NSVh) model as a
+broader class of normal stochastic volatility models.
+Based on the price transition law of Choi et al. (2019), we express the option price as a double sum
+of compound Gaussian quadratures consisting of Gauss–Hermite and Gauss–Laguerre quadratures. Our
+scheme is highly efficient in the sense that quadrature points with only 49 (i.e., 7×7) nodes can produce
+very accurate option values. Unlike with the asymptotic implied volatility approach, the resulting prices
+obtained with our approach are arbitrage-free.
+Using a similar algorithm, we can also evaluate the
+option delta (equivalently, the probability distribution). Our study extends the previous literature on
+the normal SABR model. It compliments Choi et al. (2019) by showing that the price transition law is
+useful for deterministic pricing as well as Monte–Carlo simulation. It extends the integral representations
+of the normal SABR model previously observed by Korn and Tang (2013) and Antonov et al. (2015),
+as we provide an alternative representation and a practical numerical scheme together. The study also
+extends the stochastic volatility benchmark proposed by von Sydow et al. (2019). Given the availability
+of highly accurate option values, our method can serve as a testing benchmark for general SABR pricing
+methods proposed in the future.
+The remainder of this paper is organized as follows. Section 2 reviews the normal SABR model.
+Section 3 introduces the proposed quadrature scheme. Section 4 presents the numerical results. Finally,
+Section 5 concludes the paper.
+2. Normal SABR model
+The SABR model (Hagan et al., 2002) is a stochastic volatility model specified by
+dFt
+F β
+t
+= σt ρdWt
+and
+dσt
+σt
+= ν dZt,
+(1)
+where Ft and σt are the processes for the forward price and volatility, respectively; ν is the volatility of
+volatility; β ∈ [0, 1] is the elasticity parameter; and Wt and Zt are the BMs correlated by dWtdZt = ρ dt.
+1See Choi et al. (2022) for a survey of the normal (Bachelier) model.
+2
+
+To simplify the notations for the remainder of this paper, we also denote
+ρ∗ =
+�
+1 − ρ2
+and
+ξ = 1
+2ν
+√
+T,
+(2)
+where T is the option expiry.
+The normal SABR model is the case with β = 0. Unlike when β > 0, the asset price Ft can freely
+become negative in the normal SABR model, requiring no boundary condition at the origin.2 Here, we
+are concerned with the undiscounted price and delta of the call option with strike price K and expiry T:
+C(K) = E([FT − K]+)
+and
+D(K) = ∂C(K)
+∂F0
+.
+Since F0 and K affect the option value only through moneyness, F0 −K, under the normal SABR model,
+the option delta is also equal to the complementary probability distribution of FT :
+D(K) = ∂C(K)
+∂F0
+= −∂C(K)
+∂K
+= Prob(FT ≥ K).
+2.1. Hagan et al. (2002)’s normal volatility approximation
+Hagan et al. (2002) present the celebrated implied Black–Scholes volatility of the SABR model, from
+which the option price can be quickly computed using the Black–Scholes formula. In their study, Hagan
+et al. (2002) first derive the implied normal volatility, even for all β ∈ [0, 1], because they use perturbation
+from normal diffusion. They then convert the derived volatility to the equivalent Black–Scholes volatility
+using another approximation. However, for the normal SABR model, normal volatility is more relevant,
+as has been discussed. Therefore, the normal volatility approximation in the intermediate step (Hagan
+et al., 2002, (A.59)) is preferred among practitioners, as it avoids further approximation error. The
+implied normal volatility for β = 0 is given by
+σn(K) = σ0
+z
+x(z)
+�
+1 + 2 − 3ρ2
+6
+ξ2
+�
+for
+z = ν
+σ0
+(K − F0)
+(3)
+where K is the strike price, ξ is given in Eq. (2), and x(z) is defined by3
+x(z) = log
+�V (z) + z + ρ
+1 + ρ
+�
+for
+V (z) =
+�
+1 + 2ρz + z2.
+(4)
+2In some studies (Antonov et al., 2015), the model is explicitly referred to as the normal free-boundary SABR. When
+0 < β < 1, the absorbing boundary condition is imposed at the origin, resulting in a mass at zero. See Gulisashvili et al.
+(2018); Chen and Yang (2019); Choi and Wu (2021b) for details.
+3At and near z = 0, z/x(z) should be evaluated as 1 from Taylor’s expansion near z = 0:
+x(z)
+z
+= 1 − ρ
+2 z + 3ρ2 − 1
+6
+z2 − (5ρ2 − 3)ρ
+8
+z3 + · · · .
+3
+
+The option price is obtained by plugging σn(K) into the Bachelier option formula:
+C(K) ≈ (F0 − K)N(dn) + σn
+√
+T n(dn)
+for
+dn =
+F0 − K
+σn(K)
+√
+T
+,
+(5)
+where n(z) and N(z) are the probability density function (PDF) and cumulative distribution function
+(CDF), respectively, of the standard normal distribution.
+This volatility approximation, Eq. (3), is the leading and first-order terms of the asymptotic expansion
+around a small ξ. Therefore, the accuracy of the approximation noticeably deteriorates when ξ is not
+small. Despite various attempts to improve approximation (Ob�l´oj, 2007; Paulot, 2015; Lorig et al., 2017),
+the intrinsic limitation arising from asymptotic approximation is difficult to overcome. Inaccurate option
+prices can also lead to inaccurate risk management. The option delta D(K) implied from the volatility
+approximation,
+D(K) = −∂C(K)
+∂K
+≈ C(K − h) − C(K + h)
+2h
+for small h,
+(6)
+can deviate from the true delta significantly.
+2.2. Existing option price representation
+Based on the heat kernel on hyperbolic geometry (i.e., Poincar´e half-plane), Henry-Labord`ere (2005,
+2008) derives a two-dimensional integral representation of the option price under the normal SABR
+model, and Korn and Tang (2013) correct mistakes in the formula. Their formula, with the simplification
+of Antonov et al. (2019, § 3.5.2), is as follows:
+C(K) = [F0 − K]+ +
+1
+2π
+√
+2
+σ0
+ρ∗ν
+� ∞
+s0
+Q(s)ds,
+(7)
+where ρ∗ is given in Eq. (2), and
+s0 = cosh−1
+��
+k2 + ρ2∗ − ρk
+ρ2∗
+�
+with
+k = z + ρ = ν
+σ0
+(K − F0) + ρ.
+The integrand Q(s) is given by
+Q(s) =
+�
+e− s
+2 N
+�
+− s
+2ξ + ξ
+�
++ e
+s
+2 N
+�
+− s
+2ξ − ξ
+�� � α+(s)
+α−(s)
+dα
+�
+cosh s − cosh d(α)
+where d(α) and α±(s) are
+d(α) = cosh−1
+�1
+2
+�
+α + 1
+α
+�
++ (k − ρα)2
+2ρ2∗α
+�
+,
+α±(s) = ρk + ρ2
+∗ cosh s ±
+�
+sinh2 s − (k − ρ cosh s)2.
+4
+
+Antonov et al. (2015, § 3.5.1) also derive a different two-dimensional integral representation:
+C(K) = [F0 − K]+ + σ0
+πν
+� ∞
+s0
+G(4ξ2, s)
+sinh s
+�
+sinh2 s − (k − ρ cosh s)2 ds
+(8)
+where G(t, s) is the CDF of the heat kernel (Antonov et al., 2019, § 3.4.5):
+G(t, s) = 2e−t/8
+t
+√
+πt
+� ∞
+s
+du ue−u2/2t√
+cosh u − cosh s.
+This representation is also based on the hyperbolic geometry heat kernel; however, the outcome differs
+from that of Korn and Tang (2013) because the integration is performed in a different order among the
+variables.
+To evaluate these integral representations, Eqs. (7) and (8), one must resort to numerical integra-
+tion. However, the use of generic numerical integrals over two-dimensional indefinite regions can be an
+extremely slow process because it requires evaluations of dense points. Korn and Tang (2013) present
+the option prices evaluated for two test cases, but they do not provide the implementation details of the
+numerical integration.
+3. Gaussian quadrature integration scheme
+Using the results of Choi et al. (2019), we present a new integral representation of the option price, which
+differs from those in Section 2.2. This new representation is evidently easier to evaluate using Gaussian
+quadratures. First, we review Choi et al. (2019)’s price transition law.
+3.1. NSVh model and the closed-form price transition formula
+Choi et al. (2019) introduce the NSVh model as
+dFt = σt
+�
+ρ dZ[λν/2]
+t
++ ρ∗ dXt
+�
+and
+dσt
+σt
+= ν dZ[λν/2]
+t
+,
+(9)
+where Zt and Xt are two independent BMs, and Z[µ]
+t
+= Zt +µ t is the BM with a drift µ. By introducing
+the drift term of Zt, the NSVh model generalizes the normal SABR; the normal SABR model is the
+NSVh model with λ = 0. Based on Bougerol’s identity in hyperbolic geometry (Alili and Gruet, 1997),
+the NSVh model admits the following closed-form transition law for σT and FT (Choi et al., 2019,
+Corollary 1):
+σT = σ0 exp
+�
+ν ¯ZT
+�
+,
+FT
+d= F0 + σ0ρ
+ν
+�
+eν ¯
+ZT − eλν2T/2�
++ σ0ρ∗
+ν
+cos θ φ
+�
+ν ¯ZT , ν
+�
+R2
+T + ¯Z2
+T
+�
+,
+(10)
+5
+
+where
+¯ZT = Z[(λ−1)ν/2]
+T
+and
+φ(Z, D) = eZ/2√
+2 cosh D − 2 cosh Z
+(Z ≤ D)
+and RT is the two-dimensional squared Bessel process (i.e., RT = X2
+T + Y 2
+T for two independent BMs,
+XT and YT ), and θ ∈ [0, π] is a uniformly distributed random angle. As these random variables can be
+easily sampled, the transition law serves as a closed-form exact simulation scheme for the normal SABR
+model, which outperforms the method of Cai et al. (2017) for the β = 0 case. See Choi et al. (2019) for
+further details.
+3.2. New integral representation of the option price
+The closed-form transition law in Eq. (10) also enables an efficient pricing method for vanilla options.
+The random variables ZT , RT , and θ follow normal, exponential, and uniform distributions, respectively,
+whose PDFs are simple.
+Let us introduce the integral variables u and v, which represent ZT and RT , respectively:
+u = Z[−ν/2]
+T√
+T
+= ZT
+√
+T
+− ξ
+�
+uλ =
+¯ZT
+√
+T
+= u + λξ
+�
+and
+v = R2
+T
+T .
+Here, uλ is introduced to maintain the generality of the NSVh model. Under the normal SABR model
+(i.e., λ = 0), however, uλ is equal to u. The probability densities around variables u, v, and θ are
+respectively given by
+fu(u) = e−ξu− ξ2
+2 n(u),
+fv(v) = 1
+2e− 1
+2 v,
+and
+fθ(θ) = 1
+π .
+Note that the term e−ξu− ξ2
+2 is the Radon-Nikodym derivative that arises from our definition of u. We
+define u as such because the term makes the integrand more suitable for numerical integration, as opposed
+to naively defining u = ZT /
+√
+T. For example, e−ξu offsets eZ/2 from φ(Z, D), mitigating the exponential
+growth as shown below.
+The terminal asset price FT is expressed as a function of u, v, and θ,
+FT (u, v, θ) = F0 + ρσ0
+ν e2λξ2 �
+e2ξu − 1
+�
++ ρ∗σ0
+ν
+cosh θ φ
+�
+2ξuλ, 2ξ
+�
+v + u2
+λ
+�
+Therefore, the undiscounted price of the European option struck at K under the normal SABR model
+is expressed as an expectation over the three variables:
+C(K) =
+� ∞
+−∞
+du e−ξu− ξ2
+2 n(u)
+� ∞
+0
+dv
+2 e− v
+2
+� π
+0
+dθ
+π [FT (u, v, θ) − K]+.
+6
+
+We further decompose the payout by introducing
+k(u) = ν
+σ0
+e−ξuλ(K − F0) − 2ρeλξ2 sinh (ξu) ,
+(11)
+h(u, v) = ρ∗
+�
+2 cosh
+�
+2ξ
+�
+v + u2
+λ
+�
+− 2 cosh(2ξuλ)
+�1/2
+.
+(12)
+We can express the payout and option price as
+FT − K = σ0
+ν eξuλ (h(u, v) cos θ − k(u)) ,
+(13)
+C(K) = σ0
+ν e(λ−1/2)ξ2 � ∞
+−∞
+du n(u)
+� ∞
+0
+dv
+2 e− v
+2
+� π
+0
+dθ
+π [h(u, v) cos θ − k(u)]+.
+(14)
+Next, we identify the integration region of positive payout. Note that, because h(u, v) is a monoton-
+ically increasing function of v, taking values from 0 to ∞ for all values of u, it is always possible to find
+v∗(u) such that h(u, v∗) = |k(u)|:
+v∗(u) =
+1
+4ξ2 acosh2
+�
+cosh (2ξuλ) + k2(u)
+2ρ2∗
+�
+− u2
+λ.
+(15)
+Only when k(u) = 0, v∗(u) = 0. As long as v ≥ v∗, it is also possible to find θ∗ such that h(u, v) cos θ∗ =
+|k(u)|:
+θ∗(u, v) = arccos
+� |k(u)|
+h(u, v)
+�
+�
+0 ≤ θ∗ ≤ π
+2
+�
+.
+(16)
+When k(u) = 0 or v → ∞, θ∗(u, v) = π/2.
+Using v∗(u) and θ∗(u, v), we express the region of (v, θ), where the payout, h(u, v) cos θ − k(u), is
+positive or negative, depending on the sign of k(u). When k(u) ≥ 0, the payout is positive, if
+{(v, θ) : 0 ≤ v ≤ v∗(u)
+and
+0 ≤ θ ≤ θ∗(u, v)}.
+Therefore,
+� ∞
+0
+dv
+2 e− v
+2
+� π
+0
+dθ
+π [h(u, v) cos θ − k(u)]+
+=
+� ∞
+v∗
+dv
+2π e− v
+2
+� θ∗
+0
+dθ (h(u, v) cos θ − k(u))
+=
+� ∞
+v∗
+dv
+2π e− v
+2
+��
+h2(u, v) − k2(u) − θ∗(u, v)k(u)
+�
+When k(u) < 0, it is easier to identify the region of negative payout:
+{(v, θ) : 0 ≤ v ≤ v∗(u)
+and
+π − θ∗(u, v) ≤ θ ≤ π}.
+7
+
+Using the identity [x]+ = x − [x]−, where x = h(u, v) cos θ − k(u), the integration along θ and v becomes
+� ∞
+0
+dv
+2 e− v
+2
+� π
+0
+dθ
+π [h(u, v) cos θ − k(u)]+
+= −k(u) −
+� ∞
+0
+dv
+2 e− v
+2
+� π
+0
+dθ
+π [h(u, v) cos θ − k(u)]−
+= −k(u) −
+� ∞
+v∗
+dv
+2π e− v
+2
+� π
+π−θ∗ dθ (h(u, v) cos θ − k(u))
+= −k(u) +
+� ∞
+v∗
+dv
+2π e− v
+2
+��
+h2(u, v) − k2(u) + θ∗(u, v)k(u)
+�
+Combining the two cases, we finally represent the option price as
+C(K) =σ0
+ν e(λ−1/2)ξ2 � ∞
+−∞
+du n(u)
+�
+[−k(u)]+ +
+� ∞
+v∗
+dv
+2π e− v
+2
+��
+h2(u, v) − k2(u) − θ∗(u, v)|k(u)|
+��
+=σ0
+ν e(λ−1/2)ξ2 � ∞
+−∞
+du n(u)
+�
+[−k(u)]+ + e− v∗
+2
+� ∞
+0
+dv
+2π e− v
+2
+��
+h2(u, v∗ + v) − k2(u) − θ∗(u, v∗ + v)|k(u)|
+��
+,
+(17)
+where k(u), h(u, v), v∗(u), and θ∗(u, v) are defined in Eqs. (11), (12), (15), and (16), respectively.
+The option delta, D(K) = Prob(FT ≥ K), can be similarly obtained as
+D(K) =
+� ∞
+−∞
+du e−ξu− ξ2
+2 n(u)
+� ∞
+0
+dv
+2π e− v
+2
+� π
+0
+dθ 1h(u,v) cos θ>k(u)
+=
+� ∞
+−∞
+du e−ξu− ξ2
+2 n(u)
+�
+1−k(u) + sgn(k(u))
+� ∞
+v∗
+dv
+2π e− v
+2 θ∗(u, v)
+�
+=
+� ∞
+−∞
+du e−ξu− ξ2
+2 n(u)
+�
+1−k(u) + sgn(k(u))e− v∗
+2
+� ∞
+0
+dv
+2π e− v
+2 θ∗(u, v∗ + v)
+�
+,
+(18)
+where sgn(x) is the sign function and 1x is the indicator function (i.e., 1x = 1 if x > 0 or 0 otherwise).
+From option delta D(K), the CDF at K can also be obtained as F(K) = 1 − D(K).
+3.3. Integration with Gaussian quadratures
+The integral representations in the previous section allow for efficient evaluations because Gaussian
+quadratures exist for the probability densities of the variables. While we integrate θ analytically, we
+use the Gauss-Hermite quadrature for u and the Gauss-Laguerre quadrature for v. Let {ui} and {wi}
+for i = 1, . . . , N be the points and weights, respectively, of the Gauss-Hermite quadrature with respect
+to the weight function n(u), and let {vj} and { ¯wj} for j = 1, . . . , M be those of the Gauss–Laguerre
+quadrature associated with the weight function e− v
+2 (v ≥ 0). The integrations with respect to n(u) and
+e−v/2 are then accurately approximated by the weighted sums,
+� ∞
+−∞
+du n(u)f(u) ≈
+N
+�
+i=1
+wif(ui)
+and
+� ∞
+v∗ dv e− v
+2 g(v) ≈ e− v∗
+2
+M
+�
+k=1
+¯wjg(v∗ + vj),
+8
+
+Table 1: Parameter sets used in the simulations
+Case
+σ0
+ν
+ρ
+T
+F0
+1
+100
+0.5
+0
+30
+350
+2
+100
+0.5
+-0.3
+30
+350
+3
+100
+0.5
+-0.6
+30
+350
+for some functions f(u) and g(v), respectively. The points and weights of the Gaussian quadrature are
+optimally chosen from orthogonal polynomials, in the sense that the quadrature of size N can precisely
+evaluate the moments of the probability density up to the order 2N − 1. The quadrature points and
+weights can be easily generated using public numerical libraries.4 Because we perform two-dimensional
+integration, we construct the compound quadrature (ui, vj) with weight wi ¯wj, whose size is NM.
+Let us define the indexed function values at the quadrature points by
+ki = k(ui),
+v∗
+i = v∗(ui),
+hij = h(ui, v∗
+i + vj),
+θ∗
+ij = θ∗(ui, v∗
+i + vj).
+Then, the option price in Eq. (17) is evaluated as a double sum over the composite Gaussian quadratures:
+C(K) = σ0
+ν e(λ−1/2)ξ2
+N
+�
+i=1
+wi
+�
+�[−ki]+ + e−
+v∗
+i
+2
+M
+�
+j=1
+¯wj
+2π
+��
+h2
+ij − k2
+i − θ∗
+ij |ki|
+�
+�
+� .
+(19)
+The calculation can be performed simply as a vector–matrix operation:
+C(K) = σ0
+ν e(λ−1/2)ξ2wT (b + A ¯
+w) ,
+where w is the size N vector of wi, ¯
+w is the size M vector of ¯wj, b is the size N vector of [−ki]+, and
+A is the N × M matrix of
+Aij = 1
+2π e−
+v∗
+i
+2
+��
+h2
+ij − k2
+i − θ∗
+ij ki
+�
+.
+The option delta in Eq. (18) can be similarly evaluated using the Gaussian quadrature:
+D(K) =
+N
+�
+i=1
+�
+�wie−ξui− ξ2
+2
+�
+�1−ki + sgn(ki)e−
+v∗
+i
+2
+M
+�
+j=1
+¯wj
+2π θ∗
+ij
+�
+�
+�
+� .
+4. Numerical Experiments
+We test the accuracy of our Gaussian quadrature method for calculating the option price and delta. We
+use the three parameter sets shown in Table 1. Case 1 is one of the parameter sets tested by Korn and
+4We use the python functions, scipy.special.eval hermitenorm and scipy.special.roots genlaguerre, for Gauss–
+Hermite and Gauss–Laguerre quadratures, respectively.
+9
+
+Tang (2013, Tables 3–4 and Figure 4), but scaled by 104. This case is understood as a parameter set for
+the swaption volatility smile in the unit of basis point. We price this case again because it is a challenging
+example. Given a large value of ξ = 1.37, the error from Hagan’s normal volatility approximation in
+Eq. (3) is significantly large, as shown by Korn and Tang (2013). From Case 1, we vary correlation ρ
+from 0% to −30% in Case 2 and −60% in Case 3.
+Table 2:
+The call option price and delta for Case 1 obtained from N × M Gaussian quadrature points and Hagan’s
+volatility approximation. The exact values are obtained with 90 × 180 points. The delta value and error are in % unit.
+Case 1
+Option price
+Option delta (%)
+Error from the exact value
+Exact
+Error from the exact value
+Exact
+Strike
+7×7
+10×10
+14×14
+Hagan
+value
+7×7
+10×10
+14×14
+Hagan
+value
+0
+0.16
+0.14
+0.07
+92.28
+572.02
+-0.19
+-0.10
+-0.06
+21.45
+84.47
+100
+0.21
+0.15
+0.08
+70.49
+489.88
+-0.26
+-0.14
+-0.09
+21.85
+79.42
+200
+0.30
+0.17
+0.10
+49.61
+414.24
+-0.41
+-0.23
+-0.14
+19.03
+71.16
+300
+0.45
+0.27
+0.16
+35.07
+349.19
+-0.64
+-0.44
+-0.31
+8.43
+58.06
+350
+-0.01
+0.00
+0.00
+32.92
+322.16
+-0.00
+-0.00
+0.00
+-0.00
+50.00
+400
+0.45
+0.27
+0.16
+35.07
+299.19
+0.64
+0.44
+0.31
+-8.43
+41.94
+500
+0.30
+0.17
+0.10
+49.61
+264.24
+0.41
+0.23
+0.14
+-19.03
+28.84
+600
+0.21
+0.15
+0.08
+70.49
+239.88
+0.26
+0.14
+0.09
+-21.85
+20.58
+700
+0.16
+0.14
+0.07
+92.28
+222.02
+0.19
+0.10
+0.06
+-21.45
+15.53
+Table 3:
+The call option price and delta for Case 2 obtained from N × M Gaussian quadrature points and Hagan’s
+volatility approximation. The exact values are obtained with 90 × 180 points. The delta value and error are in % unit.
+Case 2
+Option price
+Option delta (%)
+Error from the exact value
+Exact
+Error from the exact value
+Exact
+Strike
+7×7
+10×10
+14×14
+Hagan
+value
+7×7
+10×10
+14×14
+Hagan
+value
+0
+0.22
+0.21
+0.10
+105.57
+580.55
+-0.13
+-0.09
+-0.06
+22.48
+86.50
+100
+0.28
+0.19
+0.11
+82.14
+495.84
+-0.20
+-0.09
+-0.07
+24.33
+82.66
+200
+0.39
+0.14
+0.08
+57.25
+415.99
+-0.30
+-0.12
+-0.08
+25.09
+76.51
+300
+0.48
+0.27
+0.14
+33.37
+344.19
+-0.45
+-0.24
+-0.15
+21.48
+66.23
+350
+0.46
+0.36
+0.19
+23.83
+312.82
+-0.53
+-0.36
+-0.24
+16.24
+59.01
+400
+0.52
+0.29
+0.18
+17.55
+285.36
+-0.13
+-0.10
+-0.08
+8.54
+50.68
+500
+0.53
+0.26
+0.12
+17.29
+243.03
+0.56
+0.35
+0.22
+-7.05
+34.52
+600
+0.34
+0.16
+0.07
+28.63
+214.53
+0.31
+0.17
+0.10
+-14.24
+23.41
+700
+0.25
+0.13
+0.05
+43.88
+194.70
+0.20
+0.11
+0.06
+-15.70
+16.83
+Tables 2–4 present the results for the three cases. We first obtain the exact values of option price
+and delta from the Gaussian quadrature integration (Eqs. (17) and (18), respectively) with a very dense
+quadrature set with N = 90 and M = 180. The option prices for K = 300, 350, and 400 in Case 1 is
+consistent with the values reported by Korn and Tang (2013, Table 3). Then, we report the errors of the
+option price and delta from a small quadrature size and from Hagan’s volatility approximation (Eqs. (5)
+and (6)). We increase the quadrature size N = M from 7, to 10, to 14, roughly doubling the total
+number of points from 49, to 100, to 196. The tables show that our quadrature integration accurately
+10
+
+Table 4:
+The call option price and delta for Case 3 obtained from N × M Gaussian quadrature points and Hagan’s
+volatility approximation. The exact values are obtained with 90 × 180 points. The delta value and error are in % unit.
+Case 3
+Option price
+Option delta (%)
+Error from the exact value
+Exact
+Error from the exact value
+Exact
+Strike
+7×7
+10×10
+14×14
+Hagan
+value
+7×7
+10×10
+14×14
+Hagan
+value
+0
+-0.04
+0.04
+0.03
+72.67
+569.45
+-0.03
+-0.09
+-0.07
+16.99
+89.20
+100
+-0.04
+0.13
+0.01
+54.56
+481.52
+-0.13
+-0.04
+-0.04
+19.30
+86.47
+200
+0.04
+0.14
+0.06
+33.96
+397.03
+-0.25
+-0.03
+-0.02
+21.92
+82.16
+300
+0.13
+0.13
+0.06
+10.93
+318.23
+-0.33
+-0.09
+-0.06
+23.80
+74.72
+350
+0.05
+0.14
+0.07
+-0.91
+282.24
+-0.32
+-0.18
+-0.11
+23.26
+68.93
+400
+0.30
+0.18
+0.09
+-11.93
+249.61
+-0.39
+-0.30
+-0.19
+20.32
+61.23
+500
+0.40
+0.22
+0.11
+-25.71
+198.02
+0.67
+0.48
+0.33
+5.97
+41.57
+600
+0.14
+0.05
+0.02
+-25.13
+165.13
+0.33
+0.19
+0.11
+-5.36
+25.56
+700
+0.07
+0.02
+0.01
+-17.86
+144.45
+0.18
+0.10
+0.06
+-8.32
+16.74
+evaluates the price and delta. With only 7 × 7 points, the price error is less than one basis point and
+the delta error is within 1%. These errors are negligible for practical purposes. Figure 1 displays the
+implied normal volatility (left) and delta (right) for the three test cases. The result with 7 × 7 points
+is visually indistinguishable from the exact value with 90 × 180 points. By contrast, Hagan’s volatility
+approximation shows significant deviations from the exact value. Moreover, the option delta incorrectly
+goes below 0 or above 100%. Given that the option delta is equal to the complementary CDF under the
+normal SABR model, this implies an arbitrage opportunity. It illustrates a possible danger of using a
+volatility approximation approach.
+5. Conclusion
+The normal (β = 0) SABR model is an important special case of the popular SABR model for modeling
+interest rates as it allows negative asset prices. Although Hagan’s implied volatility approximation is
+easy to use, its accuracy deteriorates and arbitrage is possible. This study provides an efficient numerical
+scheme for option prices and deltas. Based on the new price transition law, our proposed scheme adopts a
+compound Gaussian quadrature. Numerical tests show that a quadrature with only 7×7 nodes accurately
+evaluates European options without arbitrage. Given that the SABR model still requires better pricing
+methods for general cases (0 ≤ β ≤ 1), our new scheme will serve to provide the benchmarks for testing
+the methods proposed in the future.
+Declarations of Interest
+The authors report no conflicts of interest. The authors alone are responsible for the content and writing
+of the paper.
+11
+
+Figure 1:
+The volatility smile (left) and delta (right) as functions of strike price for the three test cases in Table 1.
+0
+100
+200
+300
+400
+500
+600
+700
+Strike price ( K )
+150
+160
+170
+180
+190
+200
+210
+Implied normal volatility ( 
+N )
+Case 1
+Quad (7x7)
+Hagan
+Exact
+0
+100
+200
+300
+400
+500
+600
+700
+Strike price ( K )
+0
+20
+40
+60
+80
+100
+Delta (%)
+Case 1
+Quad (7x7)
+Hagan
+Exact
+0
+100
+200
+300
+400
+500
+600
+700
+Strike price ( K )
+140
+160
+180
+200
+220
+Implied normal volatility ( 
+N )
+Case 2
+Quad (7x7)
+Hagan
+Exact
+0
+100
+200
+300
+400
+500
+600
+700
+Strike price ( K )
+0
+20
+40
+60
+80
+100
+Delta (%)
+Case 2
+Quad (7x7)
+Hagan
+Exact
+0
+100
+200
+300
+400
+500
+600
+700
+Strike price ( K )
+120
+140
+160
+180
+200
+Implied normal volatility ( 
+N )
+Case 3
+Quad (7x7)
+Hagan
+Exact
+0
+100
+200
+300
+400
+500
+600
+700
+Strike price ( K )
+20
+40
+60
+80
+100
+Delta (%)
+Case 3
+Quad (7x7)
+Hagan
+Exact
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+14
+
diff --git a/pdE0T4oBgHgl3EQf9AL7/content/tmp_files/load_file.txt b/pdE0T4oBgHgl3EQf9AL7/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..bd248566facbfd89f0bf3cdb7308f57dd4553238
--- /dev/null
+++ b/pdE0T4oBgHgl3EQf9AL7/content/tmp_files/load_file.txt
@@ -0,0 +1,925 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf,len=924
+page_content='Option pricing under the normal SABR model  with Gaussian quadratures  Jaehyuk Choia,∗, Byoung Ki Seob  aPeking University HSBC Business School,  University Town, Nanshan, Shenzhen 518055, China  bSchool of Business Administration, Ulsan National Institute of Science and Technology  Abstract  The stochastic-alpha-beta-rho (SABR) model has been widely adopted in options trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' In particular,  the normal (β = 0) SABR model is a popular model choice for interest rates because it allows negative  asset values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The option price and delta under the SABR model are typically obtained via asymptotic  implied volatility approximation, but these are often inaccurate and arbitrage-able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Using a recently  discovered price transition law, we propose a Gaussian quadrature integration scheme for price options  under the normal SABR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  The compound Gaussian quadrature sum over only 49 points can  calculate a very accurate price and delta that are arbitrage-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Keywords:  Gaussian quadrature, normal model, SABR model, stochastic volatility  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Introduction  The stochastic-alpha-beta-rho (SABR) model (Hagan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2002) has been widely adopted in the fi-  nancial industry for the pricing and risk management of European options, owing to its intuitive and  parsimonious parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' It has been a standard practice for practitioners to obtain the option  price and delta from the asymptotic approximation of implied volatility (Hagan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2002), but the  approximation loses accuracy and allows arbitrage as the variance of volatility becomes large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Despite  numerous attempts to improve implied volatility approximation (Ob�l´oj, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Paulot, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Lorig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=',  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Choi and Wu, 2021a), it does not seem possible to obtain an approximation accu-  rate for all parameter ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' To date, there are several full-scale methods for pricing the SABR model:  Monte–Carlo simulations (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Leitao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2017a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Cui  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2021), finite difference methods (Park, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' von Sydow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2019), and continuous-time Markov  chains (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' These approaches require heavy computation and complex implementation  compared with the approximation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Therefore, they are difficult to implement in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  We contribute to the literature by proposing a novel and efficient pricing scheme under the normal  ∗Corresponding author Tel: +86-755-2603-0568;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Address: Rm 755, Peking University HSBC Business School, University  Town, Nanshan, Shenzhen 518055, China  Email addresses: jaehyuk@phbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='cn (Jaehyuk Choi), bkseo@unist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='kr (Byoung Ki Seo)  Preprint submitted to arXiv  November 16, 2022  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='02797v1  [q-fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='PR]  7 Jan 2023  (β = 0) SABR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Despite being a special case, the normal SABR model has gained attention for  modeling interest rates owing to its flexibility to allow negative asset prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The normal model based  on the arithmetic Brownian motion (BM) has long been used in fixed income markets, as opposed to  the Black–Scholes model based on geometric BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='1 The normal SABR model is a natural extension of  the normal model with stochastic volatility that exhibits a volatility smile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Antonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2015) adopt  the normal SABR model as a key component of the mixture approach for modeling a low-interest-rate  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2019) propose the hyperbolic normal stochastic volatility (NSVh) model as a  broader class of normal stochastic volatility models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Based on the price transition law of Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2019), we express the option price as a double sum  of compound Gaussian quadratures consisting of Gauss–Hermite and Gauss–Laguerre quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Our  scheme is highly efficient in the sense that quadrature points with only 49 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 7×7) nodes can produce  very accurate option values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Unlike with the asymptotic implied volatility approach, the resulting prices  obtained with our approach are arbitrage-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Using a similar algorithm, we can also evaluate the  option delta (equivalently, the probability distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Our study extends the previous literature on  the normal SABR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' It compliments Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2019) by showing that the price transition law is  useful for deterministic pricing as well as Monte–Carlo simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' It extends the integral representations  of the normal SABR model previously observed by Korn and Tang (2013) and Antonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2015),  as we provide an alternative representation and a practical numerical scheme together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The study also  extends the stochastic volatility benchmark proposed by von Sydow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Given the availability  of highly accurate option values, our method can serve as a testing benchmark for general SABR pricing  methods proposed in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Section 2 reviews the normal SABR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Section 3 introduces the proposed quadrature scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Section 4 presents the numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Finally,  Section 5 concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Normal SABR model  The SABR model (Hagan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2002) is a stochastic volatility model specified by  dFt  F β  t  = σt ρdWt  and  dσt  σt  = ν dZt,  (1)  where Ft and σt are the processes for the forward price and volatility, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' ν is the volatility of  volatility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' β ∈ [0, 1] is the elasticity parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' and Wt and Zt are the BMs correlated by dWtdZt = ρ dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  1See Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2022) for a survey of the normal (Bachelier) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  2  To simplify the notations for the remainder of this paper, we also denote  ρ∗ =  �  1 − ρ2  and  ξ = 1  2ν  √  T,  (2)  where T is the option expiry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  The normal SABR model is the case with β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Unlike when β > 0, the asset price Ft can freely  become negative in the normal SABR model, requiring no boundary condition at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='2 Here, we  are concerned with the undiscounted price and delta of the call option with strike price K and expiry T:  C(K) = E([FT − K]+)  and  D(K) = ∂C(K)  ∂F0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Since F0 and K affect the option value only through moneyness, F0 −K, under the normal SABR model,  the option delta is also equal to the complementary probability distribution of FT :  D(K) = ∂C(K)  ∂F0  = −∂C(K)  ∂K  = Prob(FT ≥ K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Hagan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2002)’s normal volatility approximation  Hagan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2002) present the celebrated implied Black–Scholes volatility of the SABR model, from  which the option price can be quickly computed using the Black–Scholes formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' In their study, Hagan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2002) first derive the implied normal volatility, even for all β ∈ [0, 1], because they use perturbation  from normal diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' They then convert the derived volatility to the equivalent Black–Scholes volatility  using another approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' However, for the normal SABR model, normal volatility is more relevant,  as has been discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Therefore, the normal volatility approximation in the intermediate step (Hagan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2002, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='59)) is preferred among practitioners, as it avoids further approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The  implied normal volatility for β = 0 is given by  σn(K) = σ0  z  x(z)  �  1 + 2 − 3ρ2  6  ξ2  �  for  z = ν  σ0  (K − F0)  (3)  where K is the strike price, ξ is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2), and x(z) is defined by3  x(z) = log  �V (z) + z + ρ  1 + ρ  �  for  V (z) =  �  1 + 2ρz + z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  (4)  2In some studies (Antonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2015), the model is explicitly referred to as the normal free-boundary SABR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' When  0 < β < 1, the absorbing boundary condition is imposed at the origin, resulting in a mass at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' See Gulisashvili et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Chen and Yang (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Choi and Wu (2021b) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  3At and near z = 0, z/x(z) should be evaluated as 1 from Taylor’s expansion near z = 0:  x(z)  z  = 1 − ρ  2 z + 3ρ2 − 1  6  z2 − (5ρ2 − 3)ρ  8  z3 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  3  The option price is obtained by plugging σn(K) into the Bachelier option formula:  C(K) ≈ (F0 − K)N(dn) + σn  √  T n(dn)  for  dn =  F0 − K  σn(K)  √  T  ,  (5)  where n(z) and N(z) are the probability density function (PDF) and cumulative distribution function  (CDF), respectively, of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  This volatility approximation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (3), is the leading and first-order terms of the asymptotic expansion  around a small ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Therefore, the accuracy of the approximation noticeably deteriorates when ξ is not  small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Despite various attempts to improve approximation (Ob�l´oj, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Paulot, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Lorig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2017),  the intrinsic limitation arising from asymptotic approximation is difficult to overcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Inaccurate option  prices can also lead to inaccurate risk management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The option delta D(K) implied from the volatility  approximation,  D(K) = −∂C(K)  ∂K  ≈ C(K − h) − C(K + h)  2h  for small h,  (6)  can deviate from the true delta significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Existing option price representation  Based on the heat kernel on hyperbolic geometry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', Poincar´e half-plane), Henry-Labord`ere (2005,  2008) derives a two-dimensional integral representation of the option price under the normal SABR  model, and Korn and Tang (2013) correct mistakes in the formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Their formula, with the simplification  of Antonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2019, § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='2), is as follows:  C(K) = [F0 − K]+ +  1  2π  √  2  σ0  ρ∗ν  � ∞  s0  Q(s)ds,  (7)  where ρ∗ is given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2), and  s0 = cosh−1  ��  k2 + ρ2∗ − ρk  ρ2∗  �  with  k = z + ρ = ν  σ0  (K − F0) + ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  The integrand Q(s) is given by  Q(s) =  �  e− s  2 N  �  − s  2ξ + ξ  �  + e  s  2 N  �  − s  2ξ − ξ  �� � α+(s)  α−(s)  dα  �  cosh s − cosh d(α)  where d(α) and α±(s) are  d(α) = cosh−1  �1  2  �  α + 1  α  �  + (k − ρα)2  2ρ2∗α  �  ,  α±(s) = ρk + ρ2  ∗ cosh s ±  �  sinh2 s − (k − ρ cosh s)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  4  Antonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2015, § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='1) also derive a different two-dimensional integral representation:  C(K) = [F0 − K]+ + σ0  πν  � ∞  s0  G(4ξ2, s)  sinh s  �  sinh2 s − (k − ρ cosh s)2 ds  (8)  where G(t, s) is the CDF of the heat kernel (Antonov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2019, § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='5):  G(t, s) = 2e−t/8  t  √  πt  � ∞  s  du ue−u2/2t√  cosh u − cosh s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  This representation is also based on the hyperbolic geometry heat kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' however, the outcome differs  from that of Korn and Tang (2013) because the integration is performed in a different order among the  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  To evaluate these integral representations, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (7) and (8), one must resort to numerical integra-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' However, the use of generic numerical integrals over two-dimensional indefinite regions can be an  extremely slow process because it requires evaluations of dense points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Korn and Tang (2013) present  the option prices evaluated for two test cases, but they do not provide the implementation details of the  numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Gaussian quadrature integration scheme  Using the results of Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2019), we present a new integral representation of the option price, which  differs from those in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' This new representation is evidently easier to evaluate using Gaussian  quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' First, we review Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2019)’s price transition law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' NSVh model and the closed-form price transition formula  Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2019) introduce the NSVh model as  dFt = σt  �  ρ dZ[λν/2]  t  + ρ∗ dXt  �  and  dσt  σt  = ν dZ[λν/2]  t  ,  (9)  where Zt and Xt are two independent BMs, and Z[µ]  t  = Zt +µ t is the BM with a drift µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' By introducing  the drift term of Zt, the NSVh model generalizes the normal SABR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' the normal SABR model is the  NSVh model with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Based on Bougerol’s identity in hyperbolic geometry (Alili and Gruet, 1997),  the NSVh model admits the following closed-form transition law for σT and FT (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2019,  Corollary 1):  σT = σ0 exp  �  ν ¯ZT  �  ,  FT  d= F0 + σ0ρ  ν  �  eν ¯  ZT − eλν2T/2�  + σ0ρ∗  ν  cos θ φ  �  ν ¯ZT , ν  �  R2  T + ¯Z2  T  �  ,  (10)  5  where  ¯ZT = Z[(λ−1)ν/2]  T  and  φ(Z, D) = eZ/2√  2 cosh D − 2 cosh Z  (Z ≤ D)  and RT is the two-dimensional squared Bessel process (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', RT = X2  T + Y 2  T for two independent BMs,  XT and YT ), and θ ∈ [0, π] is a uniformly distributed random angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' As these random variables can be  easily sampled, the transition law serves as a closed-form exact simulation scheme for the normal SABR  model, which outperforms the method of Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2017) for the β = 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' See Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (2019) for  further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' New integral representation of the option price  The closed-form transition law in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (10) also enables an efficient pricing method for vanilla options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  The random variables ZT , RT , and θ follow normal, exponential, and uniform distributions, respectively,  whose PDFs are simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Let us introduce the integral variables u and v, which represent ZT and RT , respectively:  u = Z[−ν/2]  T√  T  = ZT  √  T  − ξ  �  uλ =  ¯ZT  √  T  = u + λξ  �  and  v = R2  T  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Here, uλ is introduced to maintain the generality of the NSVh model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Under the normal SABR model  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', λ = 0), however, uλ is equal to u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The probability densities around variables u, v, and θ are  respectively given by  fu(u) = e−ξu− ξ2  2 n(u),  fv(v) = 1  2e− 1  2 v,  and  fθ(θ) = 1  π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Note that the term e−ξu− ξ2  2 is the Radon-Nikodym derivative that arises from our definition of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' We  define u as such because the term makes the integrand more suitable for numerical integration, as opposed  to naively defining u = ZT /  √  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' For example, e−ξu offsets eZ/2 from φ(Z, D), mitigating the exponential  growth as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  The terminal asset price FT is expressed as a function of u, v, and θ,  FT (u, v, θ) = F0 + ρσ0  ν e2λξ2 �  e2ξu − 1  �  + ρ∗σ0  ν  cosh θ φ  �  2ξuλ, 2ξ  �  v + u2  λ  �  Therefore, the undiscounted price of the European option struck at K under the normal SABR model  is expressed as an expectation over the three variables:  C(K) =  � ∞  −∞  du e−ξu− ξ2  2 n(u)  � ∞  0  dv  2 e− v  2  � π  0  dθ  π [FT (u, v, θ) − K]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  6  We further decompose the payout by introducing  k(u) = ν  σ0  e−ξuλ(K − F0) − 2ρeλξ2 sinh (ξu) ,  (11)  h(u, v) = ρ∗  �  2 cosh  �  2ξ  �  v + u2  λ  �  − 2 cosh(2ξuλ)  �1/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  (12)  We can express the payout and option price as  FT − K = σ0  ν eξuλ (h(u, v) cos θ − k(u)) ,  (13)  C(K) = σ0  ν e(λ−1/2)ξ2 � ∞  −∞  du n(u)  � ∞  0  dv  2 e− v  2  � π  0  dθ  π [h(u, v) cos θ − k(u)]+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  (14)  Next, we identify the integration region of positive payout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Note that, because h(u, v) is a monoton-  ically increasing function of v, taking values from 0 to ∞ for all values of u, it is always possible to find  v∗(u) such that h(u, v∗) = |k(u)|:  v∗(u) =  1  4ξ2 acosh2  �  cosh (2ξuλ) + k2(u)  2ρ2∗  �  − u2  λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  (15)  Only when k(u) = 0, v∗(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' As long as v ≥ v∗, it is also possible to find θ∗ such that h(u, v) cos θ∗ =  |k(u)|:  θ∗(u, v) = arccos  � |k(u)|  h(u, v)  �  �  0 ≤ θ∗ ≤ π  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  (16)  When k(u) = 0 or v → ∞, θ∗(u, v) = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Using v∗(u) and θ∗(u, v), we express the region of (v, θ), where the payout, h(u, v) cos θ − k(u), is  positive or negative, depending on the sign of k(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' When k(u) ≥ 0, the payout is positive, if  {(v, θ) : 0 ≤ v ≤ v∗(u)  and  0 ≤ θ ≤ θ∗(u, v)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Therefore,  � ∞  0  dv  2 e− v  2  � π  0  dθ  π [h(u, v) cos θ − k(u)]+  =  � ∞  v∗  dv  2π e− v  2  � θ∗  0  dθ (h(u, v) cos θ − k(u))  =  � ∞  v∗  dv  2π e− v  2  ��  h2(u, v) − k2(u) − θ∗(u, v)k(u)  �  When k(u) < 0, it is easier to identify the region of negative payout:  {(v, θ) : 0 ≤ v ≤ v∗(u)  and  π − θ∗(u, v) ≤ θ ≤ π}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  7  Using the identity [x]+ = x − [x]−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' where x = h(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' v) cos θ − k(u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' the integration along θ and v becomes  � ∞  0  dv  2 e− v  2  � π  0  dθ  π [h(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' v) cos θ − k(u)]+  = −k(u) −  � ∞  0  dv  2 e− v  2  � π  0  dθ  π [h(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' v) cos θ − k(u)]−  = −k(u) −  � ∞  v∗  dv  2π e− v  2  � π  π−θ∗ dθ (h(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' v) cos θ − k(u))  = −k(u) +  � ∞  v∗  dv  2π e− v  2  ��  h2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' v) − k2(u) + θ∗(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' v)k(u)  �  Combining the two cases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' we finally represent the option price as  C(K) =σ0  ν e(λ−1/2)ξ2 � ∞  −∞  du n(u)  �  [−k(u)]+ +  � ∞  v∗  dv  2π e− v  2  ��  h2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' v) − k2(u) − θ∗(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' v)|k(u)|  ��  =σ0  ν e(λ−1/2)ξ2 � ∞  −∞  du n(u)  �  [−k(u)]+ + e− v∗  2  � ∞  0  dv  2π e− v  2  ��  h2(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' v∗ + v) − k2(u) − θ∗(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' v∗ + v)|k(u)|  ��  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  (17)  where k(u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' h(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' v),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' v∗(u),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' and θ∗(u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' v) are defined in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (11), (12), (15), and (16), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  The option delta, D(K) = Prob(FT ≥ K), can be similarly obtained as  D(K) =  � ∞  −∞  du e−ξu− ξ2  2 n(u)  � ∞  0  dv  2π e− v  2  � π  0  dθ 1h(u,v) cos θ>k(u)  =  � ∞  −∞  du e−ξu− ξ2  2 n(u)  �  1−k(u) + sgn(k(u))  � ∞  v∗  dv  2π e− v  2 θ∗(u, v)  �  =  � ∞  −∞  du e−ξu− ξ2  2 n(u)  �  1−k(u) + sgn(k(u))e− v∗  2  � ∞  0  dv  2π e− v  2 θ∗(u, v∗ + v)  �  ,  (18)  where sgn(x) is the sign function and 1x is the indicator function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 1x = 1 if x > 0 or 0 otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  From option delta D(K), the CDF at K can also be obtained as F(K) = 1 − D(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Integration with Gaussian quadratures  The integral representations in the previous section allow for efficient evaluations because Gaussian  quadratures exist for the probability densities of the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' While we integrate θ analytically, we  use the Gauss-Hermite quadrature for u and the Gauss-Laguerre quadrature for v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Let {ui} and {wi}  for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' , N be the points and weights, respectively, of the Gauss-Hermite quadrature with respect  to the weight function n(u), and let {vj} and { ¯wj} for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' , M be those of the Gauss–Laguerre  quadrature associated with the weight function e− v  2 (v ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The integrations with respect to n(u) and  e−v/2 are then accurately approximated by the weighted sums,  � ∞  −∞  du n(u)f(u) ≈  N  �  i=1  wif(ui)  and  � ∞  v∗ dv e− v  2 g(v) ≈ e− v∗  2  M  �  k=1  ¯wjg(v∗ + vj),  8  Table 1: Parameter sets used in the simulations  Case  σ0  ν  ρ  T  F0  1  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='5  0  30  350  2  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='3  30  350  3  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='6  30  350  for some functions f(u) and g(v), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The points and weights of the Gaussian quadrature are  optimally chosen from orthogonal polynomials, in the sense that the quadrature of size N can precisely  evaluate the moments of the probability density up to the order 2N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The quadrature points and  weights can be easily generated using public numerical libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='4 Because we perform two-dimensional  integration, we construct the compound quadrature (ui, vj) with weight wi ¯wj, whose size is NM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Let us define the indexed function values at the quadrature points by  ki = k(ui),  v∗  i = v∗(ui),  hij = h(ui, v∗  i + vj),  θ∗  ij = θ∗(ui, v∗  i + vj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Then, the option price in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (17) is evaluated as a double sum over the composite Gaussian quadratures:  C(K) = σ0  ν e(λ−1/2)ξ2  N  �  i=1  wi  �  �[−ki]+ + e−  v∗  i  2  M  �  j=1  ¯wj  2π  ��  h2  ij − k2  i − θ∗  ij |ki|  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  (19)  The calculation can be performed simply as a vector–matrix operation:  C(K) = σ0  ν e(λ−1/2)ξ2wT (b + A ¯  w) ,  where w is the size N vector of wi, ¯  w is the size M vector of ¯wj, b is the size N vector of [−ki]+, and  A is the N × M matrix of  Aij = 1  2π e−  v∗  i  2  ��  h2  ij − k2  i − θ∗  ij ki  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  The option delta in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (18) can be similarly evaluated using the Gaussian quadrature:  D(K) =  N  �  i=1  �  �wie−ξui− ξ2  2  �  �1−ki + sgn(ki)e−  v∗  i  2  M  �  j=1  ¯wj  2π θ∗  ij  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Numerical Experiments  We test the accuracy of our Gaussian quadrature method for calculating the option price and delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' We  use the three parameter sets shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Case 1 is one of the parameter sets tested by Korn and  4We use the python functions, scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='eval hermitenorm and scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='special.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='roots genlaguerre, for Gauss–  Hermite and Gauss–Laguerre quadratures, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  9  Tang (2013, Tables 3–4 and Figure 4), but scaled by 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' This case is understood as a parameter set for  the swaption volatility smile in the unit of basis point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' We price this case again because it is a challenging  example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Given a large value of ξ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='37, the error from Hagan’s normal volatility approximation in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (3) is significantly large, as shown by Korn and Tang (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' From Case 1, we vary correlation ρ  from 0% to −30% in Case 2 and −60% in Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Table 2:  The call option price and delta for Case 1 obtained from N × M Gaussian quadrature points and Hagan’s  volatility approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The exact values are obtained with 90 × 180 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The delta value and error are in % unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Case 1  Option price  Option delta (%)  Error from the exact value  Exact  Error from the exact value  Exact  Strike  7×7  10×10  14×14  Hagan  value  7×7  10×10  14×14  Hagan  value  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='07  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='28  572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
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+page_content='06  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='45  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='47  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
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+page_content='83  Tables 2–4 present the results for the three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' We first obtain the exact values of option price  and delta from the Gaussian quadrature integration (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (17) and (18), respectively) with a very dense  quadrature set with N = 90 and M = 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The option prices for K = 300, 350, and 400 in Case 1 is  consistent with the values reported by Korn and Tang (2013, Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Then, we report the errors of the  option price and delta from a small quadrature size and from Hagan’s volatility approximation (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' (5)  and (6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' We increase the quadrature size N = M from 7, to 10, to 14, roughly doubling the total  number of points from 49, to 100, to 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The tables show that our quadrature integration accurately  10  Table 4:  The call option price and delta for Case 3 obtained from N × M Gaussian quadrature points and Hagan’s  volatility approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The exact values are obtained with 90 × 180 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
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+page_content='  Case 3  Option price  Option delta (%)  Error from the exact value  Exact  Error from the exact value  Exact  Strike  7×7  10×10  14×14  Hagan  value  7×7  10×10  14×14  Hagan  value  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
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+page_content='74  evaluates the price and delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' With only 7 × 7 points, the price error is less than one basis point and  the delta error is within 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' These errors are negligible for practical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Figure 1 displays the  implied normal volatility (left) and delta (right) for the three test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The result with 7 × 7 points  is visually indistinguishable from the exact value with 90 × 180 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' By contrast, Hagan’s volatility  approximation shows significant deviations from the exact value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Moreover, the option delta incorrectly  goes below 0 or above 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Given that the option delta is equal to the complementary CDF under the  normal SABR model, this implies an arbitrage opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' It illustrates a possible danger of using a  volatility approximation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Conclusion  The normal (β = 0) SABR model is an important special case of the popular SABR model for modeling  interest rates as it allows negative asset prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Although Hagan’s implied volatility approximation is  easy to use, its accuracy deteriorates and arbitrage is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' This study provides an efficient numerical  scheme for option prices and deltas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Based on the new price transition law, our proposed scheme adopts a  compound Gaussian quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Numerical tests show that a quadrature with only 7×7 nodes accurately  evaluates European options without arbitrage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Given that the SABR model still requires better pricing  methods for general cases (0 ≤ β ≤ 1), our new scheme will serve to provide the benchmarks for testing  the methods proposed in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Declarations of Interest  The authors report no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' The authors alone are responsible for the content and writing  of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  11  Figure 1:  The volatility smile (left) and delta (right) as functions of strike price for the three test cases in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='700  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Strike price ( K )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='150  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='160  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='170  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='180  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='190  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='210  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Implied normal volatility (  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='N )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Case 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Quad (7x7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Hagan  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Exact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='700  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Strike price ( K )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Delta (%)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Case 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Quad (7x7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Hagan  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Exact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='700  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Strike price ( K )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='140  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='160  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='180  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='220  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Implied normal volatility (  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='N )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Case 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Quad (7x7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Hagan  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Exact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='700  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Strike price ( K )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='40  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='60  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='80  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Delta (%)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Case 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Quad (7x7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Hagan  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Exact  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='700  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Strike price ( K )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='120  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='140  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='160  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='180  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Implied normal volatility (  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='N )  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Case 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Quad (7x7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='Hagan  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
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+page_content=' International Journal of Theoretical and Applied Finance 15, 1250016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
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+page_content=' The equivalent constant-elasticity-of-variance (CEV) volatility of the stochastic-  alpha-beta-rho (SABR) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Journal of Economic Dynamics and Control 128, 104143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
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+page_content='  Choi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
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+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
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+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='1876908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
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+page_content=' A general valuation framework for SABR and stochastic local  volatility models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
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+page_content=', Leitao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
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+page_content=', Haentjens, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', Wald´en, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
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+page_content=' BENCHOP - SLV:  The BENCHmarking project in option pricing - Stochastic and local volatility problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' International  Journal of Computer Mathematics 96, 1910–1923.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='1080/00207160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='1544368.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  Yang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', Chen, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', Wan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=', 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Approximate arbitrage-free option pricing under the SABR  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' Journal of Economic Dynamics and Control 83, 198–214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='jedc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
+page_content='  14' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pdE0T4oBgHgl3EQf9AL7/content/2301.02797v1.pdf'}
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+Combinatorial Causal Bandits without Graph Skeleton
+Shi Feng * 1 2 Nuoya Xiong * 1 Wei Chen 2
+Abstract
+In combinatorial causal bandits (CCB), the learn-
+ing agent chooses a subset of variables in each
+round to intervene and collects feedback from
+the observed variables to minimize expected re-
+gret or sample complexity. Previous works study
+this problem in both general causal models and
+binary generalized linear models (BGLMs). How-
+ever, all of them require prior knowledge of causal
+graph structure. This paper studies the CCB prob-
+lem without the graph structure on binary general
+causal models and BGLMs. We first provide an
+exponential lower bound of cumulative regrets
+for the CCB problem on general causal models.
+To overcome the exponentially large space of pa-
+rameters, we then consider the CCB problem on
+BGLMs. We design a regret minimization algo-
+rithm for BGLMs even without the graph skeleton
+and show that it still achieves O(
+√
+T ln T) ex-
+pected regret. This asymptotic regret is the same
+as the state-of-art algorithms relying on the graph
+structure. Moreover, we sacrifice the regret to
+O(T
+2
+3 ln T) to remove the weight gap covered by
+the asymptotic notation. At last, we give some
+discussions and algorithms for pure exploration
+of the CCB problem without the graph structure.
+1. Introduction
+The multi-armed bandits (MAB) problem is a classical
+model in sequential decision-making (Robbins, 1952; Auer
+et al., 2002; Bubeck et al., 2012). In each round, the learning
+agent chooses an arm and observes the feedback reward cor-
+responding to that arm, with the goal of either maximizing
+the cumulative reward over T rounds (regret minimization),
+or minimizing the sample complexity to find the interven-
+tion closest to the optimal one (pure exploration). MAB
+can be extended to have more structures among arms and
+1Institute for Interdisciplinary Information Sciences, Tsinghua
+University, Beijing, China 2Microsoft Research, Beijing, China.
+Correspondence to:
+Shi Feng <shifeng-thu@outlook.com>,
+Nuoya Xiong <xiongny20@mails.tsinghua.edu.cn>, Wei Chen
+<weic@microsoft.com>.
+Preprint. Under review.
+reward functions, which leads to more advanced learning
+techniques. Such structured bandit problems include com-
+binatorial causal bandits (Chen et al., 2013; 2016), linear
+bandits (Abbasi-Yadkori et al., 2011; Agrawal & Goyal,
+2013; Li et al., 2017), and sparse linear bandits (Abbasi-
+Yadkori et al., 2012).
+In this paper, we study another structured bandit problem
+called causal bandits, which is first proposed by (Lattimore
+et al., 2016). It consists of a causal graph G = (X∪{Y }, E)
+indicating the causal relationship among the observed vari-
+ables. In each round, the learning agent selects one or a few
+variables in X to intervene, gains the reward as the output of
+Y , and observes the values of all variables in X ∪{Y }. The
+use of causal bandits is possible in a variety of contexts that
+involve causal relationships, including medical drug testing,
+performance tuning, policy making, scientific experimental
+process, etc.
+In all previous literature except (Lu et al., 2021), the struc-
+ture of the causal graph is known, but the underlying proba-
+bility distributions governing the causal model are unknown.
+Lu et al. (2021) further assumes that the graph structure
+is unknown and the learning agent can only see the graph
+skeleton. Here, graph skeleton is also called essential graph
+(G´amez et al., 2013) and means all the edges in G without
+direction information. In our paper, we further consider that
+the graph skeleton is unknown and remove the unrealistic
+assumption |Pa(Y )| = 1 in (Lu et al., 2021). In many
+scenarios, the learning agent needs to learn the causal re-
+lationships between variables and thus needs to learn the
+graph without any prior information. For example, in poli-
+cymaking for combating COVID-19, many possible factors
+like food supply, medical resources, vaccine research, pub-
+lic security, and public opinion may consequently impact
+the mortality rate. However, the causal relationships among
+these factors are not readily known and need to be clarified
+during the sequential decision-making process. Learning
+the causal graph from scratch while identifying the optimal
+intervention raises a new challenge to the learning problem.
+For regret minimization, we study combinatorial causal
+bandits (CCB) under the binary generalized linear mod-
+els (BGLMs) as (Feng & Chen, 2023; Xiong & Chen,
+2023). Using a novel initialization phase, we could de-
+termine the ancestor structure of the causal graph for the
+arXiv:2301.13392v1  [cs.LG]  31 Jan 2023
+
+Combinatorial Causal Bandits without Graph Skeleton
+BGLM when the minimum edge weight in the model satis-
+fies a weight gap assumption. This is enough to perform a
+CCB algorithm based on maximum likelihood estimation
+on it (Feng & Chen, 2023). The resulting algorithm BGLM-
+OFU-Unknown achieves O(
+√
+T log T) regret, where T is
+the time horizon. The big O notation only holds for T larger
+than a threshold so the weight gap assumption is hidden by
+the asymptotic notation. For binary linear models (BLMs),
+one could sacrifice O(T
+1
+6 ) regret to remove the weight gap
+assumption. The algorithms we design for BLMs allow
+hidden variables and use linear regression instead of MLE
+to remove an assumption on parameters.
+For pure exploration, we give some discussions on general
+causal models. If we allow the weight gap, a trivial solu-
+tion exists. Without the weight gap, we give an adaptive
+algorithm for general causal model in the atomic setting.
+In summary, our contribution includes: (a) providing an ex-
+ponential lower bound of cumulative regret for CCB on gen-
+eral causal model, (b) proposing an O(
+√
+T ln T) cumulative
+regret CCB algorithm BGLM-OFU-Unknown for BGLMs
+without graph skeleton, (c) proposing an O(T
+2
+3 ln T) cu-
+mulative regret CCB algorithm for BLMs without graph
+skeleton and the weight gap assumption, (d) giving the first
+discussion including algorithms and lower bounds on pure
+exploration of causal bandits on general causal models and
+atomic intervention without knowing the graph structure.
+2. Related Works
+Causal Bandits. The causal bandits problem is first pro-
+posed by Lattimore et al. (2016). They discuss the simple
+regret for parallel graphs and general graphs with known
+probability distributions P(Pa(Y )|a) for any action a. Sen
+et al. (2017); Nair et al. (2021); Maiti et al. (2021) generalize
+the simple regret study for causal bandits to more general
+causal graphs and soft interventions. Lu et al. (2020); Nair
+et al. (2021); Maiti et al. (2021) consider cumulative regret
+for causal bandits problem. However, all of these studies
+are not designed for combinatorial action set and has expo-
+nentially large regret or sample complexity with respect to
+the graph size if the actions are combinatorial. Yabe et al.
+(2018); Feng & Chen (2023); Xiong & Chen (2023); Varici
+et al. (2022) consider combinatorial action set for causal
+bandits problem. Among them, Feng & Chen (2023) are
+the first to remove the requirement of T > �
+X∈X 2|Pa(X)|
+and proposes practical CCB algorithms on BGLMs with
+O(
+√
+T ln T) regret. Xiong & Chen (2023) simultaneously
+propose CCB algorithms on BGLMs as well as general
+causal models with polynomial sample complexity with re-
+spect to the graph size. Varici et al. (2022) further include
+soft interventions in the CCB problem, but their work is
+on Linear Structural Equation Models. Lee & Bareinboim
+(2018; 2019; 2020) propose several CCB algorithms on gen-
+eral causal bandits problem, but they focus on empirical
+studies while we provide theoretical regret analysis. All
+of the above works require the learning agent to know the
+graph structure in advance. Lu et al. (2021) is the first and
+only work on causal bandits without graph structure. How-
+ever, their algorithm is limited to the case of |Pa(Y )| = 1
+for the atomic setting, and thus the main technical issue
+degenerates to finding the particular parent of Y so that one
+could intervene on this node for the optimal reward.
+Social Network and Causality. Causal models have intrin-
+sic connections with influence propagation in social net-
+works. Feng & Chen (2021) study the identifiability in the
+Independent Cascade (IC) propagation model as a causal
+model. The BGLM studied in this paper contains the IC
+model and linear threshold (LT) model in a DAG as special
+cases, and is also related to the general threshold model
+(Kempe et al., 2003). Moreover, Feng & Chen (2023);
+Xiong & Chen (2023) also study causal bandits on BGLMs
+to avoid the exponentially large parameter space of general
+causal models. These papers borrow some techniques and
+ideas from influence maximization literature, including (Li
+et al., 2020) and (Zhang et al., 2022). However, in our
+BGLM CCB problem, the graph skeleton is unknown, and
+we need adaptation and integration of previous techniques
+together with some new ingredients.
+3. Model
+We utilize capital letters (U, X, Y . . .) to represent variables
+and their corresponding lower-case letters to indicate their
+values, as was frequently done in earlier causal inference
+literatures (see, for example, (Pearl, 2009; Pearl & Macken-
+zie, 2018)). To express a group or a vector of variables or
+values, we use boldface characters like X and x.
+Causal Models. A causal graph G = (X ∪ {Y }, E) is a
+directed acyclic graph consisting of intervenable variables
+X, a special target node Y without outgoing edges, and
+the set of directed edges E connecting nodes in X ∪ {Y }.
+Denote n = |X| as the number of nodes in X. For sim-
+plicity, in this paper we consider all variables in X ∪ {Y }
+are (0, 1)-binary random variables. In our main text, all
+the variables in X ∪ {Y } are known and their values can
+be observed but the edges in E are unknown and cannot
+be directly observed. We refer to the in-neighbor nodes of
+a node X in G as the parents of X, denoted by Pa(X),
+and the values of these parent random variables as pa(X).
+According to the definition of causal Bayesian model (Pearl,
+2009), the probability distribution P(X|Pa(X)) is used to
+represent the causal relationship between X and its parents
+for every conceivable value combination of Pa(X). More-
+over, we define the ancestors of a node X ∈ X ∪ {Y } by
+Anc(X).
+
+Combinatorial Causal Bandits without Graph Skeleton
+We mainly study the Markovian causal graph G in this
+paper, which means that there are no hidden variables in
+G and every observed variable X has some randomness
+that is not brought on by any other variables. In this study,
+we dedicate random variable X1 to be a special variable
+that always takes the value 1 and is a parent of all other
+observed random variables in order to model this effect of
+the Markovian model.
+In this paper, we study a special causal model called bi-
+nary generalized linear model (BGLM). Specifically, in
+BGLM, we have P(X = 1|Pa(X) = pa(X)) = fX(θ∗
+X ·
+pa(X)) + εX, where fX is a monotone increasing func-
+tion, θ∗
+X is an unknown weight vector in [0, 1]|Pa(X)|, and
+εX is a zero-mean sub-Gaussian noise that ensures that the
+probability does not exceed 1. We use the notation θ∗
+X′,X
+to denote the entry in vector θ∗
+X that corresponds to node
+X′ ∈ Pa(X), θ∗ to denote the vector of all the weights,
+and Θ to denote the feasible domain for the weights. We
+also use notation ε to represent all noise random variables
+(εX)X∈X∪Y .
+We also study binary linear model (BLM) and linear model
+in this paper. In BLMs, all fX’s are identity functions,
+so P(X = 1|Pa(X) = pa(X)) = θ∗
+X · pa(X) + εX.
+When we remove the noise variable εX, BLM coincides
+with the linear threshold (LT) model for influence cas-
+cades (Kempe et al., 2003) in a DAG. In linear models,
+we remove the randomness of conditional probabilities, so
+X = θ∗
+X · pa(X) + εX. For a node X and one of its parent
+X′, the corresponding weight are denoted as θ∗
+X′,X.
+For the unknown causal graph, there is an important param-
+eter θ∗
+min = min(X′,X)∈E θ∗
+X′,X, which represents the min-
+imum weight gap for all edges. Intuitively, this minimum
+gap measures the difficulty for the algorithm to discover the
+edge and its correct direction. When the gap is relatively
+large, we can expect to discover the whole graph accurately
+during the learning process; When the gap is very small,
+we cannot guarantee to discover the graph directly and we
+must come up with another way to solve the causal bandit
+problem on an inaccurate model.
+Combinatorial Causal Bandits. The problem of combi-
+natorial causal bandits (CCB) is first introduced in (Feng
+& Chen, 2023) and describes the following setting and the
+online learning task. The intervention can be done on no
+all variables except X1 and Y . The action set is defined
+by A ⊆ {do(S) = s}S⊆X\{X1},s∈{0,1}|S|. The expected
+reward under intervention on S ⊆ X\{X1} is denoted as
+E[Y |do(S = s)]. A learning agent runs an algorithm π for
+T rounds. In particular, an atomic intervention only inter-
+venes one node, i.e. |S| = 1. In our paper, we assume the
+observation do() and atomic interventions do(X = x) are
+always in our action set, because they are needed to discover
+the graph structure.
+The performance of the agent could be measured by the
+regret of the algorithm π. The regret Rπ(T) in our context
+is the difference between the cumulative reward using algo-
+rithm π and the expected cumulative reward of choosing best
+action S∗. Here, S∗ ∈ argmaxdo(S=s)∈A E[Y |do(S)].
+Formally, we have
+Rπ(T) = E
+� T
+�
+t=1
+(E[Y |do(S∗ = s∗)] − E[Y |do(Sπ
+t = sπ
+t )])
+�
+,
+(1)
+where Sπ
+t is the intervention set selected by algorithm π
+in round t. The expectation is from the randomness of the
+causal model and the algorithm π.
+In this paper, we mainly focus on the regret minimization
+problem, and we will discuss the pure exploration problem
+and its sample complexity in the Section 7. We defer the
+definition of sample complexity to that section.
+4. Lower Bound on General Binary Causal
+Model
+In this section, we explain why we only consider BGLM
+and BLM instead of the general binary causal model in
+the combinatorial causal bandit setting. Note that in the
+general case both the number of actions and the number of
+parameters of the causal model are exponentially large to
+the size of the graph. The following theorem shows that in
+the general binary causal model, the regret bound must be
+exponential to the size of the graph when T is sufficiently
+large, or simply linear to T when T is not large enough.
+This means that we cannot avoid the exponential factor for
+the general case, and thus justify our consideration of the
+BGLM and BLM settings with only a linear number of
+parameters.
+Theorem 1 (Binary Model Lower Bound). Recall that n =
+|X|. For any algorithm, when T ≥ 16(2n−1)
+3
+, there exists a
+bandit instance T such that
+ET [R(T)] ≥
+√
+2nT
+8e
+.
+Moreover, when T ≤
+16(2n−1)
+3
+, there exists a bandit in-
+stance T that
+ET [R(T)] ≥ T
+16e.
+The lower bound contains two parts. The first part shows
+that the asymptotic regret cannot avoid an exponential term
+2n when T is large. The second part states that if T is not
+exponentially large, the regret will be linear at the worst
+case. The proof technique of this lower bound is similar
+to but not the same as previous classical bandit, because
+the existence of observation do() and atomic intervention
+
+Combinatorial Causal Bandits without Graph Skeleton
+do(Xi = 1) may provide more information. To our best
+knowledge, this result is the first regret lower bound on
+the general causal model considering the potential role of
+observation and atomic intervention. The result shows that
+in the general binary causal model setting, it is impossible
+to avoid the exponential term in the cumulative regret even
+with the observations on null and atomic interventions. The
+proof of lower bound is provided in Appendix C.5.
+The main idea is to consider the action set A
+=
+{do(), do(X = x), do(X = x)} for all node X, x ∈
+{0, 1}, x ∈ {0, 1}n be the null intervention, atomic in-
+terventions and actions that intervene all nodes. The causal
+graph we use is a parallel graph where all nodes in X di-
+rectly points to Y with no other edges in the graph, and each
+node Xi ∈ X has probability P(Xi = 1) = P(Xi = 0) =
+0.5. Intuitively, under this condition the null intervention
+and atomic interventions can provide limited information
+to the agent. This fact shows that observations and atomic
+interventions may not be conducive to our learning process
+in the worst case on the general binary causal model.
+5. BGLM CCB without Graph Skeleton but
+with Minimum Weight Gap
+In this section, we propose an algorithm for causal bandits
+on Markovian BGLMs based on maximum likelihood es-
+timation (MLE) without any prior knowledge of the graph
+skeleton.
+Our idea is to try to discover the causal graph structure and
+then apply the recent CCB algorithm with known graph
+structure (Feng & Chen, 2023). We discover the graph struc-
+ture by using atomic interventions in individual variables.
+However, there are a few challenges we need to face on
+graph discovery. First, it could be very difficult to exactly
+identify all parent-child relationships, since some grand-
+parent nodes may also have strong causal influence to its
+grand-child nodes. Fortunately, we find that it is enough to
+identify ancestor-descendant relationships instead of parent-
+child relationships, since we can artificially add an edge
+with 0 weight between each pair of ancestor and descendant
+without impacting the causal propagation results. Another
+challenge is the minimum weight gap. When the weight of
+an edge is very small, we need to perform more atomic inter-
+ventions to identify its existence and its direction. Hence, we
+design an initialization phase with the number of rounds pos-
+itively correlated to the total round number T and promise
+that the ancestor-descendant relationship can always be iden-
+tified correctly with a large probability when T is sufficiently
+large.
+Following (Li et al., 2017; Feng & Chen, 2023; Xiong &
+Chen, 2023), we have three assumptions:
+Assumption 1. For every X ∈ X ∪ {Y }, fX is twice
+differentiable. Its first and second order derivatives are
+upper-bounded by L(1)
+fX > 0 and L(2)
+fX > 0.
+Let κ = infX∈X∪{Y },v∈[0,1]|Pa(X)|,||θ−θ∗
+X||≤1 ˙fX(v · θ).
+Assumption 2. We have κ > 0.
+Assumption 3. There exists a constant ζ > 0 such that for
+any X ∈ X ∪ {Y } and X′ ∈ Anc(X), for any value vec-
+tor v ∈ {0, 1}|Anc(X)\{X′,X1}|, the following inequalities
+hold:
+Pr
+ε,X,Y {X′ = 1|Anc(X) \ {X′, X1} = v} ≥ ζ,
+(2)
+Pr
+ε,X,Y {X′ = 0|Anc(X) \ {X′, X1} = v} ≥ ζ.
+(3)
+Assumptions 1 and 2 are the classical assumptions in gener-
+alized linear model (Li et al., 2017). Assumption 3 makes
+sure that each ancestor node of X has some freedom to
+become 0 and 1 with a non-zero probability, even when
+the values of all other ancestors of X are fixed, and it is
+originally given in (Feng & Chen, 2023) with additional
+justifications. For BLMs and continuous linear models, we
+propose an algorithm based on linear regression without the
+need of this assumption in Appendix B.
+To discover the ancestors of all variables, we need to per-
+form an extra initialization phase (see Algorithm 1). We
+denote the total number of rounds by T and arbitrary con-
+stants c0, c1 to make sure that c0T 1/2 ∈ N+. In the initial-
+ization phase, from X1 to Xn, we intervene each of them to
+1 and 0 for c0T 1/2 times respectively. We denote the value
+of X in the tth round by X(t). For every two variables
+Xi, Xj ∈ X\{X1}, if
+1
+c0
+√
+T
+c0
+√
+T
+�
+k=1
+�
+X(2ic0
+√
+T +k)
+j
+− X((2i+1)c0
+√
+T +k)
+j
+�
+> c1T − 1
+5 ,
+(4)
+we set Xi as an ancestor of Xj. Here, X(2ic0
+√
+T +k)
+j
+’s
+with k ∈ [c0
+√
+T] are the values of Xj in the rounds that
+do(Xi = 1) is chosen; X((2i+1)c0
+√
+T +k)
+j
+, k ∈ [c0
+√
+T]
+are the values of Xj in the rounds that do(Xi = 0) is
+chosen. Specifically, if Xi is not an ancestor of Xj, the
+value of Xj is not impacted by intervention on Xi. Si-
+multaneously, if Xi ∈ Pa(Xj), the value of Xj is no-
+tably impacted by do(Xi) so the difference of Xj under
+do(Xi = 1), do(Xi = 0) can be used as a discriminator for
+the ancestor-descendant relationship between Xi and Xj.
+This is formally shown by Lemma 1.
+Lemma 1. Let G be a BGLM with parameter θ∗ that satis-
+fies Assumption 2. Recall that θ∗
+min = min(X′,X)∈E θ∗
+X′,X.
+If Xi
+∈
+Pa(Xj), we have E[Xj|do(Xi
+=
+1)] −
+E[Xj|do(Xi
+= 0)] ≥ κθ∗
+Xi,Xj
+≥ κθ∗
+min; if Xi is
+not an ancestor of Xj, we have E[Xj|do(Xi = 1)] =
+E[Xj|do(Xi = 0)].
+
+Combinatorial Causal Bandits without Graph Skeleton
+Algorithm 1 BGLM-OFU-Unknown for BGLM CCB Prob-
+lem
+1: Input: Graph G = (X ∪{Y }, E), action set A, param-
+eters L(1)
+fX, L(2)
+fX, κ, ζ in Assumption 1, 2 and 3, positive
+constants c0 and c1 for initialization phase such that
+c0
+√
+T ∈ N+.
+2: /* Initialization Phase: */
+3: Do each intervention among do(X2 = 1), do(X2 =
+0), · · · , do(Xn = 1), do(Xn = 0) for c0T 1/2 times in
+order and observe the feedback (Xt, Yt), 1 ≤ t ≤ T0.
+4: Compute the ancestors �
+Anc(X), X ∈ X ∪ {Y } by
+BGLM-Ancestors((X1, Y1), · · · , (XT0, YT0), c0, c1)
+(see Algorithm 2).
+5: Initialize M0,X ← 0 ∈ R|�
+Anc(X)|×|�
+Anc(X)| for all
+X ∈ X ∪ {Y }, δ ←
+1
+3n
+√
+T , R ← ⌈
+512n(L(2)
+fX )2
+κ4
+(n2 +
+ln 1
+δ )⌉, T0
+←
+2(n − 1)c0T 1/2, T1
+←
+T0 +
+max
+�
+c
+ζ2 ln 1
+δ , (8n2−6)R
+ζ
+�
+and ρ ← 3
+κ
+�
+log(1/δ).
+6: Do no intervention on BGLM G for T1 − T0 rounds
+and observe feedback (Xt, Yt), T0 + 1 ≤ t ≤ T1.
+7: /* Iterative Phase: */
+8: for t = T1 + 1, T1 + 2, · · · , T do
+9:
+{ˆθt−1,X, Mt−1,X}X∈X∪{Y }
+=
+BGLM-Estimate((X1, Y1), · · · , (Xt−1, Yt−1))
+(see Algorithm 3).
+10:
+Compute the confidence ellipsoid Ct,X = {θ′
+X ∈
+[0, 1]|�
+Anc(X)| :
+���θ′
+X − ˆθt−1,X
+���
+Mt−1,X
+≤ ρ} for
+any node X ∈ X ∪ {Y }.
+11:
+Adopt argmaxdo(S=s)∈A,θ′
+t,X∈Ct,X E[Y |do(S
+=
+s)] as (St, st, ˜θt).
+12:
+Intervene all the nodes in St to st and observe the
+feedback (Xt, Yt).
+13: end for
+We use the above idea to implement the procedure in Algo-
+rithm 2, and then put this procedure in the initial phase and
+integrate this step into BGLM-OFU proposed by (Feng &
+Chen, 2023), to obtain our main algorithm, BGLM-OFU-
+Unknown (Algorithm 1).
+Notice that each term in Eq. (4) is a random sample of
+E[Xj|do(Xi = 1)] − E[Xj|do(Xi = 0)], which means
+that the left-hand side of Eq. (4) is just an estimation of
+E[Xj|do(Xi = 1)] − E[Xj|do(Xi = 0)]. Such expression
+can be bounded by concentration inequalities. Hence we
+can prove that Algorithm 2 identifies Xi ∈ Anc(Xj) with
+false positive rate and false negative rate both no more than
+exp
+�
+− c0c2
+1T 1/10
+2
+�
+when θ∗
+min ≥ 2c1κ−1T −1/5. Formally,
+we have the following lemma that shows the probability of
+correctness for Algorithm 2. For completeness, the proof of
+Lemma 2 is put in appendix.
+Algorithm 2 BGLM-Ancestors
+1: Input:
+Observations ((X1, Y1), · · · , (XT0, YT0)),
+positive constants c0 and c1.
+2: Output: �
+Anc(X), ancestors of X, X ∈ X ∪ {Y }.
+3: For all X ∈ X, �
+Anc(X) = ∅, �
+Anc(Y ) = X.
+4: for i ∈ {2, 3, · · · , n} do
+5:
+for j ∈ {2, 3, · · · , n}\{i} do
+6:
+if �c0
+√
+T
+k=1
+�
+X(2ic0
+√
+T +k)
+j
+− X((2i+1)c0
+√
+T +k)
+j
+�
+>
+c0c1T 3/10 then
+7:
+Add Xi into �
+Anc(Xj).
+8:
+end if
+9:
+end for
+10: end for
+11: Recompute the transitive closure of �
+Anc(·), i.e., if
+Xi ∈ �
+Anc(Xj) and Xj ∈ �
+Anc(Xℓ), then add Xi to
+�
+Anc(Xℓ).
+Algorithm 3 BGLM-Estimate
+1: Input: All observations ((X1, Y1), · · · , (Xt, Yt)) un-
+til round t.
+2: Output: {ˆθt,X, Mt,X}X∈X∪{Y }
+3: For each X ∈ X ∪ {Y }, i ∈ [t], construct data pair
+(V i,X, X(i)) with V i,X the vector of ancestors of X in
+round i, and X(i) the value of X in round i if X ̸∈ Si.
+4: for X ∈ X ∪ {Y } do
+5:
+Calculate
+the
+maximum-likelihood
+es-
+timator
+ˆθt,X
+by
+solving
+the
+equation
+�t
+i=1(X(i) − fX(V ⊺
+i,XθX))V i,X = 0.
+6:
+Mt,X = �t
+i=1 V i,XV ⊺
+i,X.
+7: end for
+Lemma 2 (Positive Rate of BGLM-Order). Suppose As-
+sumption 2 holds for the BGLM G.
+In the initializa-
+tion phase of Algorithm 1, Algorithm 2 finds a consistent
+ancestor-descendant relationship for the BGLM G with
+probability no less than 1−2
+�n−1
+2
+�
+exp
+�
+− c0c2
+1T 1/10
+2
+�
+when
+θ∗
+min ≥ 2c1κ−1T −1/5.
+We refer to the condition θ∗
+min ≥ 2c1κ−1T −1/5 in this
+lemma as weight gap assumption. The number of initial-
+ization rounds in Algorithm 1 is O(
+√
+T). According to
+Lemma 2, the expected regret contributed by incorrect-
+ness of the ancestor-descendant relationship does not ex-
+ceed O
+�
+T exp
+�
+− c0c2
+1T 1/10
+2
+��
+= o(
+√
+T). Therefore, after
+adding the initialization, the expected regret of BGLM-OFU-
+Unknown increases by no more than o(
+√
+T) over BGLM-
+OFU (Algorithm 1 in (Feng & Chen, 2023)). Thus we have
+the following theorem to show the regret of Algorithm 2,
+which is formally proved in appendix.
+
+Combinatorial Causal Bandits without Graph Skeleton
+Theorem 2 (Regret Bound of BGLM-OFU-Unknown). De-
+note L(1)
+max = maxX∈X∪{Y } L(1)
+fX. Under Assumptions 1, 2
+and 3, the regret of BGLM-OFU-Unknown (Algorithms 1, 2
+and 3) is bounded as
+R(T) = O
+� 1
+κn
+3
+2 L(1)
+max
+√
+T log T
+�
+,
+(5)
+where the terms of o(
+√
+T ln T) are omitted, and the big O
+notation holds for T ≥ 32
+�
+c1
+κθ∗
+min
+�5
+.
+Compared to (Feng & Chen, 2023), Theorem 5 has the
+same asymptotic regret, The only additional assumption is
+T ≥ 32 (c1/(κθ∗
+min))5. Intuitively, this extra assumption
+guarantees that we can discover the ancestor-descendant
+relationship consistent with the true graph. Our result indi-
+cates that not knowing the causal graph does not provide
+substantial difficulty with the weight gap assumption.
+Remark 1. Because Lemma 2 requires weight gap assump-
+tion, in the proof of this regret bound, we only consider
+the case of T ≥ 32 (c1/(κθ∗
+min))5. This limitation does not
+impact the asymptotic big O notation in our regret bound.
+However, when the round number T is not that large, the
+regret can be linear with respect to T. We remove this
+weight gap assumption in Section 6 for the linear model
+setting. The c0 and c1 are two adjustable constants in prac-
+tice. When T is small, one could try a small c0 to shorten
+the initialization phase, i.e., to make sure that T0 ≪ T, and
+a small c1 to satisfy the weight gap assumption. When T
+is large, one could consider larger c0 and c1 for a more ac-
+curate ancestor-descendant relationship. However, because
+θ∗
+min is unknown, one cannot promise that the weight gap
+assumption holds by manipulating c1, i.e., θ∗
+min may be too
+small for any practical T given c1.
+6. BLM CCB without Graph Skeleton and
+Weight Gap Assumption
+In the previous section, we find that if T > O((θ∗
+min)−5),
+we can get a valid upper bound. However, in reality, we
+have two challenges: 1) We do not know the real value
+of θ∗
+min, and this makes it hard to know when an edge’s
+direction is identified. 2) When θ∗
+min → 0, it makes it
+very difficult to estimate the graph accurately. To solve
+these challenges, we must both eliminate the dependence of
+θ∗
+min in our analysis, and think about how the result will be
+influenced by an inaccurate model. In this section, we give
+a causal bandit algorithm and show that the algorithm can
+always give ˜O(T 2/3) regret. This sub-linear regret result
+shows that the challenge can be solved by some additional
+techniques.
+In this section, we consider a special case of BGLM called
+Binary Linear Model (BLM), where fX becomes identity
+Algorithm 4 BLM-LR-Unknown for BLM CCB Problem
+without Weight Gap
+1: Input: Graph G = (X ∪ {Y }, E), action set A, posi-
+tive constants c0 and c1 for initialization phase.
+2: Initialize δ ←
+1
+n
+√
+T , ρt ←
+�
+n log(1 + tn) + 2 log 1
+δ
++√n for t
+=
+0, 1, 2, · · · , T and T0
+←
+2(n −
+1)c0T 2/3 log(T).
+3: /* Initialization Phase: */
+4: Do each intervention among do(X2 = 1), do(X2 =
+0), · · · , do(Xn = 1), do(Xn = 0) for c0T 2/3 times in
+order and observe the feedback (Xt, Yt) for 1 ≤ t ≤ T0.
+5: Compute the ancestors �
+Anc(X), X ∈ X ∪ {Y } by
+Nogap-BLM-Ancestors((X1, Y1), · · · , (XT0, YT0), c0
+, c1) (see Algorithm 5).
+6: /* Iterative Phase: */
+7: Initialize M0,X ← I ∈ R|�
+Anc(X)|×|�
+Anc(X)|, b0,X ←
+0|�
+Anc(X)| for all X ∈ X ∪ {Y }, ˆθ0,X
+← 0 ∈
+R|�
+Anc(X)| for all X ∈ X ∪ {Y }.
+8: for t = T0 + 1, T0 + 2, · · · , T do
+9:
+Compute the confidence ellipsoid Ct,X = {θ′
+X ∈
+[0, 1]|�
+Anc(X)| :
+���θ′
+X − ˆθt−1,X
+���
+Mt−1,X
+≤ ρt−1} for
+any node X ∈ X ∪ {Y }.
+10:
+Adopt argmaxdo(S=s)⊆A,θ′
+t,X∈Ct,X E[Y |do(S
+=
+s)] as (St, st, ˜θt).
+11:
+Intervene all the nodes in St to st and observe the
+feedback (Xt, Yt).
+12:
+for X ∈ X ∪ {Y } do
+13:
+Construct data pair (V t,X, X(t)) with V t,X the
+vector of ancestors of X in round t, and X(t) the
+value of X in round t if X ̸∈ St.
+14:
+Mt,X = Mt−1,X + V t,XV ⊺
+t,X, bt,X = bt−1,X +
+X(t)V t,X, ˆθt,X = M −1
+t,Xbt,X.
+15:
+end for
+16: end for
+function. The linear structure allows us to release the As-
+sumption 1-3 (Feng & Chen, 2023) and analyze the influ-
+ence of an inaccurate model.
+The main algorithm follows the BLM-LR algorithm in (Feng
+& Chen, 2023), which uses linear regression to estimate the
+weight θ∗, and the pseudocode is provided in Algorithm 4.
+We add a graph discovery process (Algorithm 5) in the
+initialization phase using O(nT 2/3 log T) times rather than
+O(nT 1/2) in the previous section. For any edge X′ →
+X with weight θ∗
+X′,X ≥ T −1/3, with probability at least
+1 − 1/T 2, we expect to identify the edge’s direction within
+O(nT 2/3 log(T)) samples for do(X′ = 1) and do(X′ =
+0) by checking whether the difference P(X | do(X′ =
+1)) − P(X | do(X′ = 0)) is large than T −1/3. Since
+
+Combinatorial Causal Bandits without Graph Skeleton
+Algorithm 5 Nogap-BLM-Ancestors
+1: Input:
+Observations ((X1, Y1), · · · , (XT0, YT0)),
+positive constants c0 and c1.
+2: Output: For all X ∈ X ∪ {Y }, �
+Anc(X).
+3: For all X ∈ X, �
+Anc(X) = ∅, �
+Anc(Y ) = X.
+4: for i ∈ {2, 3, · · · , n} do
+5:
+for j ∈ {2, 3, · · · , n}\{i} do
+6:
+if �c0T 2/3
+k=1
+�
+X(c0(2i)T 2/3+k)
+j
+− X(c0(2i+1)T 2/3+k)
+j
+�
+> c0c1T 1/3 log(T 2) then
+7:
+Add Xi into �
+Anc(Xj).
+8:
+end if
+9:
+end for
+10: end for
+11: Recompute the transitive closure of �
+Anc(·).
+the above difference is always larger than θ∗
+X′,X, after the
+initialization phase, the edge X′ → X will be added to the
+graph if θ∗
+X′,X ≥ T −1/3.
+Moreover, if X′ is not an ancestor of X, we claim that it
+cannot be a estimated as an ancestor after the initialization
+phase. This is because in this case P(X | do(X′ = 1)) =
+P(X) = P(X | do(X′ = 0)). Denote the estimated graph
+G′ as the graph with edge X′ → X for all X′ ∈ �
+Anc(X).
+We then have the following lemma.
+Lemma 3. In Algorithm 4, if the constants c0 and c1 satisfy
+that c0 ≥ max{ 1
+c2
+1 ,
+1
+(1−c1)2 }, with probability at least 1 −
+(n − 1)(n − 2)
+1
+T 1/3 , after the initialization phase we have
+1).If X′ is a true parent of X in G with weight θ∗
+X′,X ≥
+T −1/3, the edge X′ → X will be identified and added to
+the estimated graph G′.
+2).If X′ is not an ancestor of X in G, X′ → X will not be
+added into G′.
+The properties above together provide the analytic basis for
+the following observation, which plays a key role in our
+further analysis. Denote the estimated accuracy r = T −1/3.
+We know the linear regression for X will be performed
+on X and all its possible ancestoers �
+Anc(X) we esti-
+mated.
+For the true parent node X′ in G that is not
+contained in �
+Anc(X), we have θ∗
+X′,X ≤ r.
+Suppose
+�
+Anc(X) = {X1, X2, · · · , Xm}, and true parents which
+is not contained in �
+Anc(X) are Xm+1, · · · , Xm+k. Thus
+θ∗
+Xm+i,X ≤ r for all 1 ≤ i ≤ k.
+Also, assume X1, · · · , Xt(t < m) are true parents of X in
+G. For Xm+i, by law of total expectation, the expectation
+of X can be rewritten as
+E[X | X1, · · · , Xt]
+= EXm+1,··· ,Xm+k[E[X | X1, · · · , Xt, Xm+1, · · · , Xm+k]]
+= EXm+1,··· ,Xm+k
+�
+t
+�
+i=1
+θ∗
+Xi,XXi +
+m+k
+�
+i=m+1
+θ∗
+Xi,XXi
+�
+=
+t
+�
+i=1
+θ∗
+Xi,XXi +
+m+k
+�
+i=m+1
+θ∗
+Xi,XE[Xi] =
+t
+�
+i=1
+θ∗′
+Xi,XXi,
+where
+θ∗′
+Xi,X = θ∗
+Xi,X,
+i ≥ 2,
+(6)
+θ∗′
+X1,X = θ∗
+X1,X +
+m+k
+�
+i=m+1
+θ∗
+Xi,XE[Xi].
+(7)
+Eq.(7) is because X1 = 1 always holds. Then we have
+|θ∗′
+Xi,X − θ∗
+Xi,X| ≤ �m+k
+i=m+1 θ∗
+Xi,X ≤ kr ≤ nr, which
+shows that the difference between θ′ and θ is small if ac-
+curacy r is small. Let model M ′ represent the model with
+graph G′ with weights θ∗′ defined above. The following
+lemma shows the key observation:
+Lemma 4. The linear regression performed on graph G′ in
+Algorithm 4 (lines 12–15) gives the estimation ˆθ′ such that
+∥( ˆθ′t,X − θ∗′
+X)∥Mt,X ≤
+�
+n log(1 + tn) + 2 log(1/δ) + √n,
+where Mt,X is defined in Algorithm 4.
+This lemma shows that, the linear regression performed
+on the inaccurate estimated linear model M ′ is equivalent
+to the regression for θ∗′. Note that this regression only
+gives us the approximation in some direction with respect
+to elliptical norm, allowing the variables to be dependent.
+Based on claim above, we only need to measure the dif-
+ference for E[Y | do(S = 1)] on model M and M ′. The
+following lemma shows that the difference between two
+models can be bounded by our estimated accuracy r:
+Lemma 5. |EM[Y | do(S = 1)]−EM ′[Y | do(S = 1)]| ≤
+n2(n + 1)r, where r is the estimated accuracy defined in
+the start of this section.
+The Lemma 5 gives us a way to bound our linear regres-
+sion performance on the estimated model M ′. Suppose
+our linear regression achieves O(
+√
+T) regret comparing to
+maxS EM ′[Y | do(S = 1)], based on our estimated ac-
+curacy r = O(T −1/3), the regret for optimization error is
+O(T 2/3), which is the same order as the initialization phase.
+Moreover, it implies that we cannot set r to a larger gap,
+such as r = O(T −1/2), because it would lead to the regret
+of optimization error linear to T.
+From these two lemmas, we can measure the error for
+initialization phase. Motivated by Explore-then-Commit
+
+Combinatorial Causal Bandits without Graph Skeleton
+framework, we can achieve sublinear regret without the
+weight gap assumption. The detailed proof is provided in
+Appendix C.2 and Appendix C.3.
+Theorem 3. If c0 ≥ max{ 1
+c2
+1 ,
+1
+(1−c1)2 }, the regret of Algo-
+rithm 4 running on BLM is upper bounded as
+R(T) = O((n3T 2/3) log T).
+Theorem 3 states the regret of our algorithm without weight
+gap. The leading term of the result is O(T 2/3 log T), which
+has higher order than O(
+√
+T log T), the regret of the pre-
+vious Algorithm 2 and the BLM-LR algorithm in (Feng
+& Chen, 2023). This degradation in regret bound can be
+viewed as the cost of removing the weight gap assumption,
+which makes the accurate discovery of the causal graph ex-
+tremely difficult. How to devise a O(
+√
+T log T) algorithm
+without weight gap assumption is still an open problem.
+Using the transformation in Section 5.1 in (Feng & Chen,
+2023), this algorithm can also work with hidden variables.
+7. Pure Exploration of Causal Bandits
+without Graph Structure
+Another performance measure for bandit algorithms is
+called sample complexity. In this setting, the agent aims
+to find an action with the maximum expected reward using
+as small number of rounds as possible. This setting is also
+called pure exploration. To be more specific, the agent is
+willing to find ε-optimal arm with probability at least 1 − δ
+by sampling as few rounds as possible for fixed parameter ε
+and δ. For pure exploration, we consider the general binary
+causal model with only null and atomic interventions, and
+study the gap-dependent bounds, meaning that the sample
+complexity depends on the reward gap between the optimal
+and suboptimal actions. Moreover, let a∗ be one of the
+optimal actions. For each action a = do(Xi = x), define
+µa = E[Y | a] and the gap for action a to be
+∆a =
+� µa∗ − maxa∈A\{a∗}{µa},
+a = a∗;
+µa∗ − µa,
+a ̸= a∗.
+(8)
+According to the causal discovery literature (Pearl, 2009), by
+passive observations alone one can obtain an essential graph
+of the causal graph, with some edge directions unidentified.
+We assume that the essential graph is known but the exact
+graph structure is unknown, which is also considered by (Lu
+et al., 2021), with additional assumptions on the graph.
+One naive solution for this problem is to first identify the
+graph structure and then performed the pure exploration
+algorithm of causal bandits with known graph (Xiong &
+Chen, 2023).
+Define ce = |P(X | do(X′ = 1)) −
+P(X′ | do(X = 0))| for each edge e = (X, X′) and
+cX = mine:X→X′
+1
+c2e . Then this naive solution admits a
+sample complexity about
+˜O
+��
+a∈S
+1
+max{∆a, ε/2}2 +
+�
+x∈X
+1
+c2
+X
+�
+,
+(9)
+where S is a particular set defined following the previous
+work (Xiong & Chen, 2023) and the definition is provided
+in Appendix D. The first term is the sample complexity in
+(Xiong & Chen, 2023), while the second term is the cost for
+identifying the directions of all edges in the essential graph.
+This naive solution separates the causal discovery phase
+and learning phase, so it cannot discover the directions
+adaptively. In the Appendix D, we propose an adaptive
+algorithm to discover the edges’ directions and learn the
+reward distribution in parallel, which can provide a lower
+sample complexity for some cases.
+However, when the ∆a and cX is small, both the naive al-
+gorithm and our algorithms provided in Appendix D suffers
+Ω( n
+ε2 log(1/δ)) sample complexity. We claim that pure ex-
+ploration for the general binary causal model is intrinsically
+hard due to unknown graph structure. To show this, we state
+a negative result for pure exploration of causal bandits on
+unknown graph structure with atomic intervention. It states
+that even if we have all observation distribution P(X, Y ) as
+prior knowledge, we still cannot achieve better sample com-
+plexity result than the result in the classical pure exploration
+problem for the multi-armed bandit O( n
+ε2 log(1/δ)).
+Theorem 4 (Lower bound). Consider causal bandits with
+only essential graph and atomic intervention, for any algo-
+rithm which can output ε-optimal action with probability at
+least 1 − δ, there is a bandit instance with expected sample
+complexity Ω( n
+ε2 log(1/δ)) even if we have all observational
+distribution P(X, Y ).
+Note that if we know distribution P(X, Y ) and the ex-
+act graph structure, we can compute each intervention
+P(Y | do(X = x)) by do-calculus because the absence
+of hidden variables. So Theorem 4 shows the intrinsic hard-
+ness provided by unknown graph structure. The detailed
+proof can be found in Appendix D.
+8. Future Work
+This paper is the first theoretical study on causal bandits
+without the graph skeleton. There are many future direc-
+tions to extend this work. We believe that similar initial-
+ization methods and proof techniques can be used to de-
+sign causal bandits algorithms for other parametric models
+without the skeleton, like linear structural equation models
+(SEM). Moreover, how to provide a algorithm with �O(
+√
+T)
+regret without weight gap assumption is interesting and still
+open. For pure exploration, the combinatorial setting needs
+more research.
+
+Combinatorial Causal Bandits without Graph Skeleton
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+
+Combinatorial Causal Bandits without Graph Skeleton
+A. Proofs for Propositions in Section 5
+In this section, we give proofs that are omitted in Section 5 of our main text.
+A.1. Proof of Lemma 1
+Lemma 1. Let G be a BGLM with parameter θ∗ that satisfies Assumption 2. Recall that θ∗
+min = min(X′,X)∈E θ∗
+X′,X. If
+Xi ∈ Pa(Xj), we have E[Xj|do(Xi = 1)] − E[Xj|do(Xi = 0)] ≥ κθ∗
+Xi,Xj ≥ κθ∗
+min; if Xi is not an ancestor of Xj, we
+have E[Xj|do(Xi = 1)] = E[Xj|do(Xi = 0)].
+Proof. At first, we define an equivalent threshold model form of the BGLM as follows. For each node X, we randomly
+sample a threshold γX uniformly from [0, 1], i.e., γX ∼ U[0, 1]. Then if fX(Pa(X)·θ∗
+X)+εX ≥ γX, X is activated, i.e., X
+is set to 1; otherwise, X is not activated, i.e., X is set to 0. Therefore, if we ignore ε, the BGLM model belongs to the family
+of general threshold models (Kempe et al., 2003). For convenience, we denote the vector of all γX, X ∈ X ∪ {Y }\{X1}
+by γ. The vector of fixing all entries in γ except γX is denoted by γ−X.
+Now we prove the first part of this lemma: E[Xj|do(Xi = 1)] − E[Xj|do(Xi = 0)] ≥ κθ∗
+Xi,Xj ≥ κθ∗
+min if Xi ∈ Pa(Xj).
+By the definition of our equivalent threshold model, we know that after fixing all the thresholds γX’s and noises εX’s, the
+propagation result is completely determined merely by the intervention. Therefore, we have
+E[Xj|do(Xi = 1)] = Eγ∈(U[0,1])n,ε[Xj|do(Xi = 1)]
+= Eγ−Xj ∈(U[0,1])n−1,ε
+�
+Pr
+γXj ∼U[0,1]
+�
+Xj = 1|do(Xi = 1), γ−Xj, ε
+��
+,
+and
+E[Xj|do(Xi = 0)] = Eγ−Xj ∈(U[0,1])n−1,ε
+�
+Pr
+γXj ∼U[0,1]
+�
+Xj = 1|do(Xi = 0), γ−Xj, ε
+��
+.
+Hence, in order to prove E[Xj|do(Xi = 1)] − E[Xj|do(Xi = 0)] ≥ κθ∗
+Xi,Xj ≥ κθ∗
+min, we only need to prove
+Pr
+γXj ∼U[0,1]
+�
+Xj = 1|do(Xi = 1), γ−Xj, ε
+�
+−
+Pr
+γXj ∼U[0,1]
+�
+Xj = 0|do(Xi = 0), γ−Xj, ε
+�
+≥ κθ∗
+min.
+When γ−Xj and ε are fixed, all the nodes in X ∪ {Y }\ ({Xj} ∪ {Des(Xj)}) are already fixed given an arbitrarily fixed
+intervention. Here, Des(Xj) is used to represent the descendants of Xj. Suppose under do(Xi = 1), γ−Xj and ε, the value
+vector of parents of Xj is pa1(Xj); under do(Xi = 0), γ−Xj and ε, the value vector of parents of Xj is pa0(Xj). By
+induction along the topological order, nodes in X ∪ {Y }\ ({Xj} ∪ {Des(Xj)}) that is activated under do(Xi = 0), γ−Xj
+and ε must be also activated under do(Xi = 1), γ−Xj and ε. Therefore, entries in pa1(Xj)−pa0(Xj) are all non-negative
+and the entry in pa1(Xj) − pa0(Xj) for the value of Xj is 1. From this observation, we can deduce that
+fXj(pa1(Xj) · θ∗
+Xj) − fXj(pa0(Xj) · θ∗
+Xj) ≥ κ
+�
+pa1(Xj) · θ∗
+Xj − pa0(Xj) · θ∗
+Xj
+�
+≥ κθ∗
+Xi,Xj.
+Hence, we have
+Pr
+γXj ∼U[0,1]
+�
+Xj = 1|do(Xi = 1), γ−Xj, ε
+�
+−
+Pr
+γXj ∼U[0,1]
+�
+Xj = 0|do(Xi = 0), γ−Xj, ε
+�
+=
+Pr
+γXj ∼U[0,1]
+�
+fXj(pa1(Xj) · θ∗
+Xj) ≥ γXj + εXj|εXj
+�
+−
+Pr
+γXj ∼U[0,1]
+�
+fXj(pa0(Xj) · θ∗
+Xj) ≥ γXj + εXj|εXj
+�
+=
+�
+fXj(pa1(Xj) · θ∗
+Xj) − εXj
+�
+−
+�
+fXj(pa0(Xj) · θ∗
+Xj) − εXj
+�
+≥ κθ∗
+Xi,Xj ≥ κθ∗
+min,
+which is what we want. Until now, the first part of Lemma 1 has been proved.
+
+Combinatorial Causal Bandits without Graph Skeleton
+Then we prove the second part of this lemma: E[Xj|do(Xi = 1)] = E[Xj|do(Xi = 0)] if Xj is not a descendant of Xi. In
+this situation, we know from the graph structure that (Xj ⊥⊥ Xi)G{Xi}, where G{Xi} is the graph obtained by deleting from
+G all arrows pointing to Xi. According to the third law of do-calculus (Pearl, 2012), we deduce that
+E[Xj|do(Xi = 1)] = Pr{Xj = 1|do(Xi = 1)} = Pr{Xj = 1|} = Pr{Xj = 1|do(Xi = 0)} = E[Xj|do(Xi = 0)].
+Now Lemma 1 is completely proved.
+Corollary 1 (An Extension of Lemma 1). Suppose G is a BGLM with parameter θ∗ that satisfying Assumption 2 and
+do(S = s) is an intervention such that Xi, Xj /∈ S. If Xi ∈ Pa(Xj), we have E[Xj|do(Xi = 1), do(S = s)] −
+E[Xj|do(Xi = 0), do(S = s)] ≥ κθ∗
+Xi,Xj ≥ κθ∗
+min; if Xi is not an ancestor of Xj, we have E[Xj|do(Xi = 1), do(S =
+s)] = E[Xj|do(Xi = 0), do(S = s)].
+Proof. According to (Pearl, 2012), Pr{Xj|do(Xi), do(S)} is equivalent to Pr{Xj|do(Xi)} in a new model G′ such that
+all in-edges of S are deleted and all nodes in S are fixed by s. We know that Lemma 1 holds in G′, so this corollary holds
+in G.
+A.2. Proof of Lemma 2
+Lemma 2 (Positive Rate of BGLM-Order). Suppose Assumption 2 holds for the BGLM G. In the initialization phase of
+Algorithm 1, Algorithm 2 finds a consistent ancestor-descendant relationship for the BGLM G with probability no less than
+1 − 2
+�n−1
+2
+�
+exp
+�
+− c0c2
+1T 1/10
+2
+�
+when θ∗
+min ≥ 2c1κ−1T −1/5.
+Proof. We first assume that for every pair of nodes if Xi ∈ Pa(Xj), Algorithm 2 puts Xj as a descendant of Xi in the
+ancestor-descendant relationship; if Xj is not a descendant of Xi, Algorithm 2 do not put Xj as an descendant of Xi in the
+ancestor-descendant relationship. This event is denoted by E for simplicity. We prove that when event E does occur, the
+ancestor-descendant relationship we find is absolutely consistent with the true graph structure of G. Otherwise, suppose there
+is a mistake in the ancestor-descendant relationship such that Xi is an ancestor of Xj but not put in �
+Anc(Xj). We denote a
+directed path from Xi to Xj by Xi → Xk1 → Xk2 → · · · → Xkp → Xj. Therefore, Xk1 must be put in �
+Anc(Xi), Xk2
+must be put in �
+Anc(Xk1), ..., Xj must be put in �
+Anc(Xkp). In conclusion, Xj should be put in �
+Anc(Xi), which is a
+contradiction. Hence, there is no mistake in the ancestor-descendant relationship given event E.
+Now we only prove that using Algorithm 2, with probability no less than 1 − 2
+�n−1
+2
+�
+exp
+�
+− c0c2
+1T 1/10
+2
+�
+, event E defined
+in the paragraph above occurs. For a pair of nodes Xi, Xj ∈ X\{X1}, if Xi ∈ Pa(Xj), we know from Lemma 1
+that E[Xj|do(Xi = 1)] − E[Xj|do(Xi = 0)] ≥ κθ∗
+min. We denote the difference between random variable Xj given
+do(Xi = 1) and random variable Xj given do(Xi = 0) by Z. In �c0T 1/2
+k=1
+�
+X(2ic0T 1/2+k)
+j
+− X((2i+1)c0T 1/2+k)
+j
+�
+, each
+term X(2ic0T 1/2+k)
+j
+− X((2i+1)c0T 1/2+k)
+j
+is an i.i.d. sample of Z. We denote X(2ic0T 1/2+k)
+j
+− X((2i+1)c0T 1/2+k)
+j
+by Zk.
+
+Combinatorial Causal Bandits without Graph Skeleton
+We know that Zk ∈ [−1, 1] and E[Zk] ≥ κθ∗
+min, so according to Hoeffding’s inequality (Hoeffding, 1994), we have
+Pr
+�
+�
+�
+c0T 1/2
+�
+k=1
+�
+X(2ic0T 1/2+k)
+j
+− X((2i+1)c0T 1/2+k)
+j
+�
+> c0c1T 3/10
+�
+�
+�
+= Pr
+�
+�
+�
+c0T 1/2
+�
+k=1
+Zk > c0c1T 3/10
+�
+�
+�
+= 1 − Pr
+�
+�
+�
+c0T 1/2
+�
+k=1
+Zk ≤ c0c1T 3/10
+�
+�
+�
+≥ 1 − exp
+�
+−2
+�
+c0T 1/2κθ∗
+min − c0c1T 3/10�2
+4c0T 1/2
+�
+= 1 − exp
+�
+−c0
+�
+T 1/4κθ∗
+min − c1T 1/20�2
+2
+�
+≥ 1 − exp
+�
+−c2
+1c0T 1/10
+2
+�
+.
+(because T ≥ 32
+�
+c1
+κθ∗
+min
+�5
+)
+Similarly, if Xj is not a descendant of Xi, we do not put Xi in �
+Anc(Xj) in the ancestor-descendant relationship if and
+only if �c0T 1/2
+k=1
+�
+X(2ic0T 1/2+k)
+j
+− X((2i+1)c0T 1/2+k)
+j
+�
+≤ c0c1T 3/10. Now we still have Zk ∈ [−1, 1] but E[Zk] = 0.
+Therefore, according to Hoeffding’s inequality (Hoeffding, 1994), we have
+Pr
+�
+�
+�
+c0T 1/2
+�
+k=1
+�
+X(2ic0T 1/2+k)
+j
+− X((2i+1)c0T 1/2+k)
+j
+�
+≤ c0c1T 3/10
+�
+�
+�
+= 1 − Pr
+�
+�
+�
+c0T 1/2
+�
+k=1
+Zk > c0c1T 3/10
+�
+�
+�
+> 1 − exp
+�
+−2
+�
+c0c1T 3/10�2
+4c0T 1/2
+�
+= 1 − exp
+�
+−c2
+1c0T 1/10
+2
+�
+.
+Hence, by union bound (Boole’s inequality (Bonferroni, 1936)), the probability of E is no less than 1 −
+2
+�n−1
+2
+�
+exp
+�
+− c2
+1c0T 1/10
+2
+�
+. This is because when Xi, Xj ∈ X\{X1}, there are 2
+�n−1
+2
+�
+possible choices of them that
+are tested by Algorithm 2. When E happens, Algorithm 2 gets the ancestor-descendant relationship correct, so Lemma 2 is
+proved.
+A.3. Proof of Theorem 2
+In the following proofs on a BGLM G, when X′ ∈ Anc(X) but X′ /∈ Pa(X), we add an edge X′ → X with weight
+θX′,X = 0 into G and this does not impact the propagation results of G. Let D = maxX∈X∪{Y } |Pa(X)| be the maximum
+in-degree. After doing this transformation, D = n and Anc(X) = Pa(X) for all X ∈ X ∪ {Y } in this subsection.
+Before the proof of this theorem, we introduce several lemmas at first. The first component is based on the result of
+maximum-likelihood estimation (MLE). It gives a theoretical measurement for the accuracy of estimated ˆθ computed by
+MLE. One who is interested could find the proof of this lemma in Appendix C.2 of (Feng & Chen, 2022).
+Lemma 6 (Lemma 1 in (Feng & Chen, 2023)). Suppose that Assumptions 1 and 2 hold. Moreover, given δ ∈ (0, 1), assume
+that
+λmin(Mt,X) ≥
+512|Pa(X)|
+�
+L(2)
+fX
+�2
+κ4
+�
+|Pa(X)|2 + ln 1
+δ
+�
+.
+(10)
+Then with probability at least 1 − 3δ, the maximum-likelihood estimator satisfies , for any v ∈ R|Pa(X)|,
+���v⊺(ˆθt,X − θ∗
+X)
+��� ≤ 3
+κ
+�
+log(1/δ) ∥v∥M −1
+t,X ,
+
+Combinatorial Causal Bandits without Graph Skeleton
+where the probability is taken from the randomness of all data collected from round 1 to round t.
+The second component is called the group observation modulated (GOM) bounded smoothness property (Li et al., 2020). It
+shows that a small change in parameters θ leads to a small change in the reward. Under our BGLM setting, this lemma is
+proved in Appendix C.3 of (Feng & Chen, 2022).
+Lemma 7 (Lemma 2 in (Feng & Chen, 2023)). For any two weight vectors θ1, θ2 ∈ Θ for a BGLM G, the difference of
+their expected reward for any intervened set S can be bounded as
+��σ(S, θ1) − σ(S, θ2)
+�� ≤ Eε,γ
+�
+�
+�
+X∈XS,Y
+��V ⊺
+X(θ1
+X − θ2
+X)
+�� L(1)
+fX
+�
+� ,
+(11)
+where XS,Y is the set of nodes in paths from S to Y excluding S, and V X is the propagation result of the parents of X
+under parameter θ2. The expectation is taken over the randomness of the thresholds γ and the noises ε.
+Thirdly, we propose a lemma in order to bound the sum of ∥V t,X∥M −1
+t−1,X at first. This lemma is proved in Appendix C.4 of
+(Feng & Chen, 2022).
+Lemma 8 (Lemma 9 in (Feng & Chen, 2022)). Let {W t}∞
+t=1 be a sequence in Rd satisfying ∥W t∥ ≤
+√
+d. Define W 0 = 0
+and Mt = �t
+i=0 W iW ⊺
+i . Suppose there is an integer t1 such that λmin(Mt1+1) ≥ 1, then for all t2 > 0,
+t1+t2
+�
+t=t1
+∥W t∥M −1
+t−1 ≤
+�
+2t2d log(t2d + t1).
+At last, in order to show that λmin(MT1,X) ≥ R after the initialization phase of Algorithm 1 and thus satisfy the condition
+of Lemma 6, we introduce Lemma 9. This lemma is improved upon Lemma 7 in (Feng & Chen, 2022) and enables us to use
+Lecu´e and Mendelson’s inequality (Nie, 2022) in our later theoretical regret analysis.
+Let Sphere(d) denote the sphere of the d-dimensional unit ball.
+Lemma 9. For any v = (v1, v2, · · · , v|Pa(X)|) ∈ Sphere(|Pa(X)|) and any X ∈ X ∪ {Y } in a BGLM that satisfies
+Assumption 3, we have
+Pr
+ε,X,Y
+�
+|Pa(X) · v| ≥
+1
+√
+4D2 − 3
+�
+≥ ζ,
+where Pa(X) is the random vector generated by the natural Bayesian propagation in BGLM G with no interventions
+(except for setting X1 to 1).
+Proof. The lemma is similarly proved as Lemma 7 in (Feng & Chen, 2022) using the idea of Pigeonhole principle. Let
+Pa(X) = (Xi1 = X1, Xi2, Xi3, · · · , Xi|Pa(X)|) as the random vector and pa(X) = (x1 = 1, xi1, xi2, xi3, · · · , xi|Pa(X)|)
+as a possible valuation of Pa(X). Without loss of generality, we suppose that |v2| ≥ |v3| ≥ · · · ≥
+��v|Pa(X)|
+��. For
+simplicity, we denote D0 =
+√
+D − 1 +
+1
+2√D−1. If |v1| ≥
+D0
+√
+D2
+0+1, we can deduce that
+|pa(X) · v| ≥ |v1| − |v2| − |v3| − · · · −
+��v|Pa(X)|
+��
+≥
+D0
+�
+D2
+0 + 1
+−
+�
+(D − 1)
+�
+|v2|2 + |v3|2 + · · ·
+��v|Pa(X)|
+��2�
+(12)
+≥
+D0
+�
+D2
+0 + 1
+−
+�
+(D − 1)
+�
+1 −
+D2
+0
+D2
+0 + 1
+�
+(13)
+=
+1
+2
+�
+(D2
+0 + 1)(D − 1)
+=
+1
+√
+4D2 − 3
+,
+where Inequality (12) is by the Cauchy-Schwarz inequality and the fact that |Pa(X)| ≤ D, and Inequality (13) uses the
+fact that v ∈ Sphere(|Pa(X)|). Thus, when |v1| ≥
+D0
+√
+D2
+0+1, the event |Pa(X) · v| ≥
+1
+√
+4D2−3 holds deterministically.
+
+Combinatorial Causal Bandits without Graph Skeleton
+Otherwise, when |v1| <
+D0
+√
+D2
+0+1, we use the fact that |v2| is the largest among |v2|, |v3|, . . . and deduce that
+|v2| ≥
+1
+√n − 1
+�
+|v2|2 + |v3|2 + · · · ≥
+�
+1 −
+�
+D0
+√
+D2
+0+1
+�2
+√n − 1
+=
+2
+√
+4D2 − 3
+.
+(14)
+Therefore, using the fact that
+Pr
+ε,X,Y {Xi1 = 1, Xi2 = xi2, Xi3 = xi3, · · · } =
+Pr
+ε,X,Y {Xi2 = xi2|Xi1 = 1, Xi3 = xi3, · · · } ·
+Pr
+ε,X,Y {(Xi1 = 1, Xi3 = xi3, · · · } ≥ ζ
+Pr
+ε,X,Y {Xi1 = 1, Xi3 = xi3, · · · }
+and �
+xi3,xi4,··· Prε,X,Y {Xi1 = 1, Xi3 = xi3, · · · } = 1, we have
+Pr
+ε,X,Y
+�
+|Pa(X) · v| ≥
+1
+√
+4D2 − 3
+�
+=
+�
+xi3,xi4,···
+Pr{Xi1 = 1, Xi2 = 1, Xi3 = xi3, · · · } · I
+�
+|(1, 1, xi3, xi4, · · · ) · (v1, v2, v3, · · · )| ≥
+1
+√
+4D2 − 3
+�
++
+�
+xi3,xi4,···
+Pr{Xi1 = 1, Xi2 = 0, Xi3 = xi3, · · · } · I
+�
+|(1, 0, xi3, xi4, · · · ) · (v1, v2, v3, · · · )| ≥
+1
+√
+4D2 − 3
+�
+≥
+�
+xi3,xi4,···
+ζ Pr{Xi1 = 1, Xi3 = xi3, Xi4 = xi4 · · · } · I
+�
+|(1, 1, xi3, xi4, · · · ) · (v1, v2, v3, · · · )| ≥
+1
+√
+4D2 − 3
+�
++
+�
+xi3,xi4,···
+ζ Pr{Xi1 = 1, Xi3 = xi3, Xi4 = xi4, · · · } · I
+�
+|(1, 0, xi3, xi4, · · · ) · (v1, v2, v3, · · · )| ≥
+1
+√
+4D2 − 3
+�
+= ζ ·
+�
+xi3,xi4,···
+Pr{Xi1 = 1, Xi3 = xi3, Xi4 = xi4, · · · }
+�
+I
+�
+|(1, 1, xi3, xi4, · · · ) · (v1, v2, v3, · · · )| ≥
+1
+√
+4D2 − 3
+�
++I
+�
+|(1, 0, xi3, xi4, · · · ) · (v1, v2, v3, · · · )| ≥
+1
+√
+4D2 − 3
+��
+≥ ζ
+�
+xi3,xi4,···
+Pr{Xi1 = 1, Xi3 = xi3, Xi4 = xi4, · · · }
+(15)
+= ζ,
+which is exactly what we want to prove. Inequality (15) holds because otherwise, at least for some xi3, xi4, . . ., both
+indicators on the left-hand side of the inequality have to be 0, which implies that
+|(1, 1, xi3, xi4, · · · ) · (v1, v2, v3, · · · ) − (1, 0, xi3, xi4, · · · ) · (v1, v2, v3, · · · )| = |v2| <
+2
+√
+4D2 − 3
+,
+(16)
+but this contradicts to Inequality (14).
+Having these four lemmas above together with Lemma 2 proved in Appendix A.2, we are finally able to prove the regret
+bound of BGLM-OFU-Unknown algorithm (Theorem 2) as below.
+Theorem 2 (Regret Bound of BGLM-OFU-Unknown). Denote L(1)
+max = maxX∈X∪{Y } L(1)
+fX. Under Assumptions 1, 2 and
+3, the regret of BGLM-OFU-Unknown (Algorithms 1, 2 and 3) is bounded as
+R(T) = O
+� 1
+κn
+3
+2 L(1)
+max
+√
+T log T
+�
+,
+(5)
+where the terms of o(
+√
+T ln T) are omitted, and the big O notation holds for T ≥ 32
+�
+c1
+κθ∗
+min
+�5
+.
+
+Combinatorial Causal Bandits without Graph Skeleton
+Proof. We only consider the case of T ≥ 32
+�
+c1
+κθ∗
+min
+�5
+in this proof because the big O notation is asymptotic.
+Let Ht be the history of the first t rounds and Rt be the regret in the tth round. Because the reward node Y is in interval
+[0, 1], we can deduce that for any t ≤ T1, Rt ≤ 1. Now we consider the case of t > T1. According to Lemma 2, with
+probability at least 1 − 2
+�n−1
+2
+�
+exp
+�
+− c0c2
+1T 1/10
+2
+�
+, Algorithm 2 returns a correct ancestor-descendant relationship, i.e.,
+�
+Anc(X) = Anc(X) for X ∈ X ∪ {Y }. Next we bound the regret conditioned on the correct ancestor-descendant
+relationship. When t > T1, we have
+E[Rt|Ht−1] = E[σ(Sopt, θ∗) − σ(St, θ∗)|Ht−1],
+(17)
+where the expectation is taken over the randomness of St.
+Then for T1
+<
+t
+≤
+T, we define ξt−1,X for
+X
+∈ X ∪ {Y } as ξt−1,X
+=
+����vT (ˆθt−1,X − θ∗
+X)
+��� ≤ ρ · ∥v∥M −1
+t−1,X , ∀v ∈ R|Pa(X)|�
+.
+According to the defini-
+tion of Algorithm 1, we can deduce that λmin(Mt−1,X) ≥ λmin(MT1,X).
+By Lecu´e and Mendelson’s inequal-
+ity (Nie, 2022; Feng & Chen, 2022) (conditions of this inequality satisfied according to Lemma 9), we have
+Pr {λmin(MT1,X) < R} ≤ Pr {λmin(MT1,X − MT0,X) < R} ≤ exp
+�
+− (T1−T0)ζ2
+c
+�
+where c, ζ are constants. Then
+we can define ξt−1 = ∧X∈X∪{Y }ξt−1,X and let ξt−1 be its complement.
+By Lemma 6, we have Pr
+�
+ξt−1
+�
+≤
+�
+3δ + exp
+�
+− (T1−T0)ζ2
+c
+�
++ 3δ exp
+�
+− (T1−T0)ζ2
+c
+��
+n ≜ perror.
+Because under ξt−1, for any X ∈ X ∪ {Y } and v ∈ R|Pa(X)|, we have
+���vT (ˆθt−1,X − θ∗
+X)
+��� ≤ ρ · ∥v∥M −1
+t−1,X. Therefore,
+by the definition of ˜θt, we have σ(St, ˜θt) ≥ σ(Sopt, θ∗) because θ∗ is in our confidence ellipsoid. Hence,
+E[Rt] ≤ Pr {ξt−1} · E[σ(Sopt, θ∗) − σ(St, θ∗)] + Pr(ξt−1)
+≤ E[σ(Sopt, θ∗) − σ(St, θ∗)] + perror
+≤ E[σ(St, ˜θt) − σ(St, θ∗)] + perror.
+Then we need to bound σ(St, ˜θt) − σ(St, θ∗) carefully.
+Therefore, according to Lemma 6 and Lemma 7, we can deduce that
+E[Rt] ≤ E
+�
+�
+�
+X∈XSt,Y
+���V t,X(˜θt,X − θ∗
+X)
+��� L(1)
+fX
+�
+� + perror
+≤ E
+�
+�
+�
+X∈XSt,Y
+∥V t,X∥M −1
+t−1,X
+���˜θt,X − θ∗
+X
+���
+Mt−1,X
+L(1)
+fX
+�
+� + perror
+≤ 2ρ · E
+�
+�
+�
+X∈XSt,Y
+∥V t,X∥M −1
+t−1,X L(1)
+fX
+�
+� + perror.
+The last inequality holds because
+���˜θt,X − θ∗
+X
+���
+Mt−1,X
+≤
+���˜θt,X − ˆθt−1,X
+���
+Mt−1,X
++
+���ˆθt−1,X − θ∗
+X
+���
+Mt−1,X
+≤ 2ρ.
+Therefore, conditioned on the correct ancestor-descendant relationship, the total regret can be bounded as
+R(T) ≤ 2ρ · E
+�
+�
+T
+�
+t=T0+1
+�
+X∈XSt,Y
+∥V t,X∥M −1
+t−1,X L(1)
+fX
+�
+� + perror(T − T1) + T1.
+For convenience, we define W t,X as a vector such that if X ∈ St, W t,X = 0|Pa(X)|; if X ̸∈ St, W t,X = V t,X. Using
+
+Combinatorial Causal Bandits without Graph Skeleton
+Lemma 8, we can get the result:
+R(T) ≤
+�
+�2ρE
+�
+�
+T
+�
+t=T0+1
+�
+X∈XSt,Y
+∥V t,X∥M −1
+t−1,X L(1)
+fX
+�
+� + perror(T − T1) + T1
+�
+�
+�
+1 − 2
+�n − 1
+2
+�
+exp
+�
+−c0c2
+1T 1/10
+2
+��
++ 2
+�n − 1
+2
+�
+exp
+�
+−c0c2
+1T 1/10
+2
+�
+T
+≤ 2ρE
+�
+�
+T
+�
+t=T0+1
+�
+X∈X∪{Y }
+∥W t,X∥M −1
+t−1,X L(1)
+fX
+�
+� + perror(T − T1) + T1 + 2
+�n − 1
+2
+�
+exp
+�
+−c0c2
+1T 1/10
+2
+�
+T
+≤ 2ρ ·
+max
+X∈X∪{Y }
+�
+L(1)
+fX
+�
+E
+�
+�
+�
+X∈X∪{Y }
+�
+2(T − T0)|Pa(X)| log ((T − T0)|Pa(X)| + T0)
+�
+�
++ perror(T − T1) + T1 + 2
+�n − 1
+2
+�
+exp
+�
+−c0c2
+1T 1/10
+2
+�
+T
+= O
+� 1
+κn
+3
+2 √
+TL(1)
+max ln T
+�
+= ˜O
+� 1
+κn
+3
+2 √
+TL(1)
+max
+�
+because ρ = 3
+κ
+�
+log(1/δ), exp
+�
+− c0c2
+1T 1/10
+2
+�
+T = o(
+√
+T) and perrorT = o(
+√
+T).
+B. A BLM CCB Algorithm with Safety Gap Based on Linear Regression
+As BLM is a special case of BGLM, the initialization phase in BGLM-OFU-Unknown to determine the ancestor-descendant
+relationship can also be used on BLMs. Feng & Chen (2023) propose a CCB algorithm for BLMs using linear regression
+instead of MLE to remove the requirement of Assumption 3. Furthermore, BLM takes the identity function as fX’s, so
+Assumptions 1 and 2 is neither required. The specific algorithm BLM-LR-Unknown-SG (BLM-LR-Unknown algorithm
+with safety gap) is demonstrated in Algorithm 6.
+The following theorem shows the regret bound of BLM-LR-Unknown-SG. It is not surprising that this algorithm could also
+work on linear models with continuous variables as Appendix F in (Feng & Chen, 2022). The dominant term in the expected
+regret does not increase compared to BLM-LR in (Feng & Chen, 2023).
+Theorem 5 (Regret Bound of BLM-LR-Unknown-SG). The regret of BLM-LR-Unknown-SG running on BLM or linear
+model is bounded as
+R(T) = O
+�
+n
+5
+2 √
+T log T
+�
+,
+where the terms of o(
+√
+T ln T) are omitted, and the big O notation holds for T ≥ 32
+�
+c1
+κθ∗
+min
+�5
+.
+Proof. In the following proof on G, when X′ ∈ Anc(X) but X′ /∈ Pa(X), we add an edge X′ → X with weight
+θX′,X = 0 into G and this does not impact the propagation results of G. After doing this transformation, D = n and
+Anc(X) = Pa(X) for all X ∈ X ∪ {Y }.
+According to Lemma 2, with probability at least 1 − 2
+�n−1
+2
+�
+exp
+�
+− c0c2
+1T 1/10
+2
+�
+, Algorithm 2 returns a correct ancestor-
+descendant relationship, i.e., Anc(X) = �
+Anc(X) for X ∈ X ∪ {Y }. Moreover, by Lemma 11 in (Feng & Chen,
+2022), with probability at most nδ, event
+�
+∃T0 < t ≤ T, x ∈ X ∪ {Y } :
+���θ∗′
+X − ˆθt,X
+��� > ρt
+�
+occurs. Now we bound
+the expected regret conditioned on the absence of this event and finding a correct ancestor-descendant relationship. For
+
+Combinatorial Causal Bandits without Graph Skeleton
+Algorithm 6 BLM-LR-Unknown-SG for BLM and Linear Model CCB Problem
+1: Input: Graph G = (X ∪ {Y }, E), action set A, positive constants c0 and c1 for initialization phase such that
+c0
+√
+T ∈ N+.
+2: /* Initialization Phase: */
+3: Do each intervention among do(X2 = 1), do(X2 = 0), · · · , do(Xn = 1), do(Xn = 0) for c0T 1/2 times in order and
+observe the feedback (Xt, Yt) for 1 ≤ t ≤ T0.
+4: Determine
+a
+feasible
+ancestor-descendant
+relationship
+�
+Anc(X)’s
+for
+X
+∈
+X
+∪
+{Y }
+by
+BGLM-Ancestors((X1, Y1), · · · , (XT0, YT0), c1) (see Algorithm 2).
+5: Initialize M0,X ← I ∈ R|�
+Anc(X)|×|�
+Anc(X)|, b0,X ← 0|�
+Anc(X)| for all X ∈ X ∪ {Y }, ˆθ0,X ← 0 ∈ R|�
+Anc(X)| for all
+X ∈ X ∪ {Y }, δ ←
+1
+n
+√
+T , ρt ←
+�
+n log(1 + tn) + 2 log 1
+δ + √n for t = 0, 1, 2, · · · , T and T0 ← 2(n − 1)c0T 1/2.
+6: /* Iterative Phase: */
+7: for t = T0 + 1, T0 + 2, · · · , T do
+8:
+Compute the confidence ellipsoid Ct,X = {θ′
+X ∈ [0, 1]|�
+Anc(X)| :
+���θ′
+X − ˆθt−1,X
+���
+Mt−1,X
+≤ ρt−1} for any node
+X ∈ X ∪ {Y }.
+9:
+(St, st, ˜θt) = argmaxdo(S=s)∈A,θ′
+t,X∈Ct,X E[Y |do(S = s)].
+10:
+Intervene all the nodes in St to st and observe the feedback (Xt, Yt).
+11:
+for X ∈ X ∪ {Y } do
+12:
+Construct data pair (V t,X, X(t)) with V t,X the vector of ancestors of X in round t, and X(t) the value of X in
+round t if X ̸∈ St.
+13:
+Mt,X = Mt−1,X + V t,XV ⊺
+t,X, bt,X = bt−1,X + X(t)V t,X, ˆθt,X = M −1
+t,Xbt,X.
+14:
+end for
+15: end for
+T0 < t ≤ T, according to Theorem 1 in (Li et al., 2020) and Theorem 7, we can deduce that
+E [Rt] = E
+�
+σ′(Sopt, θ∗′) − σ′(St, θ∗′)
+�
+≤ E
+�
+σ′(St, ˜θt) − σ′(St, θ∗′)
+�
+≤ E
+�
+�
+�
+X∈XSt,Y
+���V ⊺
+t,X(˜θt,X − θ∗′
+X)
+���
+�
+�
+≤ E
+�
+�
+�
+X∈XSt,Y
+∥V t,X∥M −1
+t−1,X
+���˜θt,X − θ∗′
+X
+���
+Mt−1,X
+�
+�
+≤ E
+�
+�
+�
+X∈XSt,Y
+2ρt−1 ∥V t,X∥M −1
+t−1,X
+�
+� ,
+since ˜θt,X, θ∗
+X are both in the confidence set. Thus, we have
+R(T) = E
+� T
+�
+t=1
+Rt
+�
+≤ E
+�
+T
+�
+t=T0+1
+Rt
+�
++ T0
+≤ 2ρT · E
+�
+�
+T
+�
+t=T0+1
+�
+X∈XSt,Y
+∥V t,X∥M −1
+t−1,X
+�
+� + T0.
+For convenience, we define W t,X as a vector such that if X ∈ St, W t,X = 0|Pa(X)|; if X ̸∈ St, W t,X = V t,X.
+
+Combinatorial Causal Bandits without Graph Skeleton
+According to Cauchy-Schwarz inequality, we have
+R(T) ≤ 2ρT · E
+�
+�
+T
+�
+t=T0+1
+�
+X∈X∪{Y }
+∥W t,X∥M −1
+t−1,X
+�
+� + T0
+≤ 2ρT · E
+�
+�√
+T ·
+�
+X∈X∪{Y }
+�
+�
+�
+�
+T
+�
+t=T0+1
+∥W t,X∥2
+M −1
+t−1,X
+�
+� + T0
+≤ 2ρT · E
+�
+�√
+T ·
+�
+X∈X∪{Y }
+�
+�
+�
+�
+T
+�
+t=1
+∥W t,X∥2
+M −1
+t−1,X
+�
+� + 2(n − 1)c0T 1/2.
+Note that Mt,X = Mt−1,X + W t,XW ⊺
+t,X and therefore, det (Mt,X) = det(Mt−1,X)
+�
+1 + ∥W t,X∥2
+M −1
+t−1,X
+�
+, we have
+T
+�
+t=1
+∥W t,X∥2
+M −1
+t−1,X ≤
+T
+�
+t=1
+n
+log(n + 1) · log
+�
+1 + ∥W t,X∥2
+M −1
+t−1,X
+�
+≤
+n
+log(n + 1) · log det(MT,X)
+det(I)
+≤ n|Pa(X)|
+log(n + 1) · log tr(MT,X)
+|Pa(X)|
+≤ n|Pa(X)|
+log(n + 1) · log
+�
+1 +
+T
+�
+t=1
+∥W t,X∥2
+2
+|Pa(X)|
+�
+≤
+nD
+log(n + 1) log(1 + T).
+Therefore, the final conditional regret R(T) is bounded by
+R(T) ≤ 2ρT n
+�
+T
+nD
+log(n + 1) log(1 + T) + 2(n − 1)c0T 1/2,
+because ρT =
+�
+D log(1 + TD) + 2 log 1
+δ +
+√
+D. When
+�
+∃t ∈ (T0, T], x ∈ X ∪ {Y } :
+���θ∗′
+X − ˆθt,X
+��� > ρt
+�
+does occur
+or Algorithm 2 finds an incorrect order, the regret is no more than T. Therefore, the total regret is no more than
+�
+2ρT n
+�
+T
+nD
+log(n + 1) log(1 + T) + 2(n − 1)c0T 1/2
+� �
+1 − nδ − 2
+�n − 1
+2
+�
+exp
+�
+−c0c2
+1T 1/10
+2
+��
++ T
+�
+nδ + 2
+�n − 1
+2
+�
+exp
+�
+−c0c2
+1T 1/10
+2
+��
+≤ 2ρT n
+�
+T
+nD
+log(n + 1) log(1 + T) + o(
+√
+T ln T)
+= O
+�
+n
+5
+2 √
+T log T
+�
+,
+which is exactly what we want.
+Replacing Lemma 11 in (Feng & Chen, 2022) by Lemma 12 in (Feng & Chen, 2022), the above proof for BLMs is still
+feasible for the regret on linear models without any other modification.
+Remark 2. According to the transformation in Section 5.1 of (Feng & Chen, 2023), this algorithm also works for some
+BLMs with hidden variables. Using that transformation, running BLM-LR-Unknown-SG on G is equivalent to running on
+a Markovian BLM or linear model G′, where parameter θ∗ is also transformed to a new set of parameters θ∗′. Here, we
+disallow the graph structure where a hidden node has two paths to Xi and Xi’s descendant Xj and the paths contain only
+hidden nodes except the end points Xi and Xj.
+
+Combinatorial Causal Bandits without Graph Skeleton
+C. Proofs for Propositions in Section 6
+C.1. Proof of Lemma 3
+Lemma 3. In Algorithm 4, if the constants c0 and c1 satisfy that c0 ≥ max{ 1
+c2
+1 ,
+1
+(1−c1)2 }, with probability at least
+1 − (n − 1)(n − 2)
+1
+T 1/3 , after the initialization phase we have
+1).If X′ is a true parent of X in G with weight θ∗
+X′,X ≥ T −1/3, the edge X′ → X will be identified and added to the
+estimated graph G′.
+2).If X′ is not an ancestor of X in G, X′ → X will not be added into G′.
+Proof. First, for each node Xj and its parent Xi with weight θ∗
+Xi,Xj ≥ T −1/3, by Lemma 1, we can have
+E[Xj | do(Xi = 1)] − E[Xj | do(Xi = 0)] ≥ θ∗
+Xi,Xj
+Then each element X(c0(2i)T 2/3+k)
+j
+− X(c0(2i+1)T 2/3+k)
+j
+is an i.i.d sample of Z = Xj |do(Xi=1) −Xj |do(Xi=0) with
+E[Z] ≥ θ∗
+Xi,Xj ≥ T −1/3. By the Hoeffding’s inequality, if we choose c1 < 1 and c0(1 − c1)2 > 1
+3, we have
+Pr
+�
+�
+�
+c0T 2/3
+�
+k=1
+�
+X(c0(2i)T 2/3+k)
+j
+− X(c0(2i+1)T 2/3+k)
+j
+�
+> c0c1T 1/3 log(T 2)
+�
+�
+�
+≥ 1 − exp
+�
+−2 log(T 2)
+�
+c0T 2/3E[Z] − c0c1T 1/3�2
+4c0T 2/3
+�
+≥ 1 − exp
+�
+−2 log(T 2)
+�
+c0T 1/3 − c0c1T 1/3�2
+4c0T 2/3
+�
+≥ 1 − exp
+�
+−c0(1 − c1)2 log(T 2)
+2
+�
+≥ 1 − T −c0(1−c1)2
+≥ 1 − 1
+T .
+Taking the union bound for all X and X′, with probability at least 1 −
+�n−1
+2
+� 1
+T 2 , the edge X′ → X with θ∗
+X′,X will be
+identified and added to the estimated graph G′. Also, assume Xi is not an ancestor of Xj, then
+E[Xj | do(Xi = 1)] − E[Xi | do(Xi = 0)] = 0.
+Thus the element X(c0(2i)T 2/3+k)
+j
+− X(c0(2i+1)T 2/3+k)
+j
+is an i.i.d sample of Z′ = Xj |do(Xi=1) −Xj |do(Xi=0) with
+E[Z′] = 0. Thus by Hoeffding’s inequality,
+Pr
+�
+�
+�
+c0T 2/3
+�
+k=1
+�
+X(c0(2i)T 2/3+k)
+j
+− X(c0(2i+1)T 2/3+k)
+j
+�
+> c0c1T 1/3 log(T 2)
+�
+�
+�
+≤ exp
+�
+−2 log(T 2)
+�
+c0T 2/3E[Z] − c0c1T 1/3�2
+4c0T 2/3
+�
+≤ exp
+�
+−c0c2
+1 log T
+�
+≤ T −c0c2
+1
+≤ 1
+T .
+and then with probability at least 1 −
+�n−1
+2
+� 1
+T , we will not add the edge X′ → X in the graph G. Combining these two
+facts, we complete the proof.
+
+Combinatorial Causal Bandits without Graph Skeleton
+C.2. Proof of Lemma 4
+For each node X, consider the estimated possible parent Pa′(X), then our observation V t,X ∈ {0, 1}Pa′(X) are the values
+of Pa′(X). Since we have θ′ that
+E[Xt | V t,X] = θT
+t,XV t,X.
+(18)
+Thus applying Lemma 1 in (Li et al., 2020), we can have
+|θ′
+X − θ′
+t,X|Mt,X ≤
+�
+n log(1 + tn) + 2 log(1/δ) + √n.
+(19)
+C.3. Proof of Lemma 5
+Note that M represents the model with true graph G and true weights θ, and M ′ represents the model with estimated graph
+G′ and estimated weights M ′, then difference
+|θ′
+Xi,X − θXi,X| ≤ nr
+(20)
+Now we construct a auxillary model M ′′, which has graph G′ and weights θ on it. The parent of X in model M Pa′′(X) is
+equivalent to Pa′(X). Then we prove the following two claims:
+Claim 1. |EM[Y | do(S = 1)] − EM ′′[Y | do(S = 1)]| ≤ n2r.
+Proof. Let the topological order be X1, X2, · · · , Xn. First, EM[X1 | do(S)] − EM ′′[X1 | do(S)] = 0 ≤ nr because
+X1 is always 1. Assume Xq+1 /∈ S EM[Xi | do(S)] − EM ′′[Xi | do(S)] ≤ qnr for all i ≤ q, then if Xq+1 ∈ S,
+EM[Xq+1 | do(S)] − EM ′′[Xq+1 | do(S)] = 0 ≤ (q + 1)nr holds trivially. Thus now we assume Xq+1 /∈ S.
+EM[Xq+1 | do(S)] − EM ′′[Xq+1 | do(S)]
+= EM
+�
+�
+�
+Xi∈Pa(Xq+1)
+θXi,Xq+1Xi
+�����do(S)
+�
+� − EM ′′
+�
+�
+�
+Xi∈Pa′′(Xq+1)
+θXi,Xq+1Xi
+�����do(S)
+�
+�
+=
+�
+Xi∈Pa′′(Xq+1)
+θXi,Xq+1(EM[Xi | do(S)] − EM ′′[Xi | do(S)])+
+�
+Xi∈Pa(Xq+1)\Pa′′(Xq+1)
+θXi,Xq+1EM[Xi | do(S)]
+≤
+�
+Xi∈Pa(Xq+1)
+θXi,Xq+1qnr + rn
+≤ (q + 1)nr
+where the first equality follows the definition of linear model, the second equality is because θ′
+X′,X = 0 if X′ is not a true
+parent of X in G. The third inequality is derived by induction, and the last inequality is because ∥θX′,Xq+1∥1 ≤ 1.
+Claim 2. |EM ′[Y | do(S = 1)] − EM ′′[Y | do(S = 1)]| ≤ n3r.
+Proof. First, EM[X1 | do(S)] − EM ′′[X1 | do(S)] = 0 ≤ n2r Then similarly, assume EM[Xi | do(S)] − EM ′′[Xi |
+
+Combinatorial Causal Bandits without Graph Skeleton
+do(S)] ≤ qn2r for all i ≤ q and Xq+1 /∈ S. Then
+EM ′[Xq+1 | do(S)] − EM ′′[Xq+1 | do(S)]
+= EM ′
+�
+�
+�
+Xi∈Pa′(Xq+1)
+θ′
+Xi,Xq+1Xi
+�����do(S)
+�
+� − EM ′′
+�
+�
+�
+Xi∈Pa′′(Xq+1)
+θXi,Xq+1Xi
+�����do(S)
+�
+�
+=
+�
+Xi∈Pa′′(Xq+1)
+θ′
+Xi,Xq+1EM ′[Xi | do(S)] − θXi,Xq+1EM ′′[Xi | do(S)]
+=
+�
+Xi∈Pa′′(Xq+1)
+(θ′
+Xi,Xq+1 − θXi,Xq+1)EM ′[Xi | do(S)]+
+�
+Xi∈Pa′′(Xq+1)
+θXi,Xq+1(EM ′[Xi | do(S)] − EM ′′[Xi | do(S)])
+= n2r + n2qr
+≤ (q + 1)n2r.
+where the first equality follows the definition, the second equality is because Pa′(X) = Pa′′(X) for any node X. The
+fourth inequality derived from induction , inequality (20) and Xi ∈ [0, 1]. By induction, we complete the proof.
+Now we prove the Lemma 5:
+Proof. Combining Claim 1 and Claim 2, we have
+EM[Y | do(S)] − EM ′[Y | do(S)] ≤ n2(n + 1)r.
+(21)
+C.4. Proof of Theorem 3
+Proof. Denote the original model and estimated model as M and M ′ The initialization phase will lead to regret at most
+T0 = 16(n − 1)T 2/3. At Iterative phase, denote the optimal action to be do(S∗ = 1), by Lemma 3 and the guarantee of
+BLM-LR, with probability at least 1 − (n − 1)(n − 2) 1
+T
+T
+�
+t=1
+EM[Y | do(S∗ = 1)] − EM[Y | do(St = 1)]
+=
+T
+�
+t=1
+((EM[Y | do(S∗ = 1)] − EM ′[Y | do(S∗ = 1)]) + (EM ′[Y | do(S∗ = 1)] − EM ′[Y | do(St = 1)]))
+≤ T0 +
+T
+�
+t=T0+1
+n2(n + 1)T −1/3 +
+T
+�
+t=T0+1
+(EM ′[Y | do(S∗ = 1)] − EM ′[Y | do(St = 1)])
+≤ T0 + n2(n + 1)T 2/3 + cn2�
+nT log T
+= O((n3T 2/3 + n3√
+T) log T)
+= O(n3T 2/3 log T),
+where the first inequality is derived from Lemma 5, and the second inequality is the guarantee of BLM-LR in Theorem 3 of
+(Feng & Chen, 2023).
+Thus the total regret will be bounded by
+R(T) ≤ (n − 1)(n − 2)
+T
+· T + O((n3T 2/3) log T)
+= O((n3T 2/3) log T).
+The first inequality is because our regret have an upper bound T.
+
+Combinatorial Causal Bandits without Graph Skeleton
+C.5. Proof of Theorem 1
+Proof. Consider the causal bandit instances Ti with parallel graph (E = {Xi → Y, 1 ≤ i ≤ n}.) and A = {do(), do(X =
+x), do(X = x)} for all node X, x ∈ {0, 1}, x ∈ {0, 1}n be all observation, atomic intervention and actions that intervene
+all nodes.
+For T1, we assume Xi are independent with each other and P(Xi = 1) = P(Xi = 0) = 0.5. Define
+P(Y = 1) =
+�
+0.5 + ∆
+if X1 = X2 = · · · = Xn = 0
+0.5
+otherwise
+Then for Ti, 2 ≤ i ≤ 2n, consider the binary representation of i − 1 as
+¯
+b1b2 · · · bn. Then assume Xi are independent with
+each other and P(Xi = 1) = 0.5, and define
+P(Y = 1) =
+�
+�
+�
+�
+�
+0.5 + ∆
+if X1 = X2 = · · · = Xn = 0
+0.5 + 2∆
+if Xj = bj for all 1 ≤ j ≤ n
+0.5
+otherwise
+Now in Ti, do
+�
+X = b1b2 · · · bn
+�
+is the best action, and other actions will lead to at least ∆ regret.
+Denote Ta(t) for action a ∈ A as the number of times taking a until time t. To simplify the notation, we denote ai as
+do(X = x), where x is the binary representatino of i − 1, {b1, b2, · · · , bn}. Then for instances T1 and Ti, we have
+ET1[R(t)] ≥ PT1(Ta1(t) ≤ t/2)t∆
+2 ,
+ETi[R(t)] ≥ PTi(Ta1(t) > t/2)t∆
+2
+Thus
+ET1[R(t)] + ETi[R(t)] > t∆
+2 (PT1(Ta1(t) ≤ t/2) + PTi(Ta1(t) > t/2))
+≥ t∆
+4 exp (−KL(PT1, PTi))
+Now we need to bound KL(PTi, PT1).
+KL(PT1, PTi) ≤
+�
+a∈A
+ET1[Ta(t)]KL(PT1(X, Y | a)∥PTi(X, Y | a))
+(22)
+=
+�
+a∈A
+ET1[Ta(t)]KL(PT1(Y | a)∥PTi(Y | a))
+(23)
+≤ ET1[Tai(t)] · KL(0.5∥0.5 + 2∆) +
+�
+a=do(Xi=x),do()
+ET1[Ta(t)] · KL(0.5∥0.5 +
+∆
+2n−2 )
+(24)
+≤ ET1[Tai(n)] · 2∆2 + t ·
+∆2
+22n−3 .
+(25)
+where (24) is because for a = do(Xi = x) or a = do(), P(Y | do(a)) ≥ 0.5 in T1, and P(Y | do(a)) ≤ 0.5 +
+2∆
+2n−1 =
+0.5 +
+∆
+2n−2 in Ti. Now we choose
+i = argmin
+j>1
+ET1[Taj(t)],
+(26)
+then we have
+ET1[Ta1(t)] ≤
+T
+2n − 1.
+(27)
+
+Combinatorial Causal Bandits without Graph Skeleton
+Then by (25), choosing ∆ =
+�
+2n−1
+3t , we have
+KL(PT1, PTi) ≤ 2t∆2
+2n − 1 + t∆2
+22n−3 ≤ t∆2 ·
+3
+2n − 1 = 1
+(28)
+Thus
+ET1[R(t)] + ETi[R(t)] ≥ t∆
+4 exp (−KL(PT1, PTi))
+≥ t∆
+4e
+≥
+�
+(2n − 1)t
+4
+√
+3e
+≥
+√
+2nt
+8e .
+Then max{ET1[R(t)], ETi[R(t)]} ≥
+√
+2nt
+16e . We complete the proof when t ≥ 16(2n−1)
+3
+.
+Now suppose t ≤ 16(2n−1)
+3
+, choose ∆ = 1
+4, then based on (25) and (27), we have
+KL(PT1, PTi) ≤
+t
+8(2n − 1) +
+t
+22n+1
+≤ 2
+3 + 16
+3 · 2n − 1
+22n+1
+≤ 1.
+Then we have
+ET1[R(t)] + ETi[R(t)] ≥ t∆
+4 exp (−KL(PT1, PTi))
+≥ t∆
+4e
+≥
+t
+16e,
+and max{ET1[R(t)], ETi[R(t)]} ≥
+t
+32e.
+D. General Causal Bandits without Graph Structure
+In this section, we only consider the atomic intervention, and provide an algorithm to solve causal bandits with the graph
+skeleton on binary model. We only consider the atomic intervention setting. An atomic intervention is do(X = x), where
+X is a node of graph G and x ∈ {0, 1}.
+D.1. General Causal Bandit Algorithms
+We first provide the positive results, which provides an algorithm to improve the sample complexity comparing to applying
+the multi-armed bandit approach directly.
+At each iteration we try to recover the edges’ direction in parallel using sub-procedure ”RECOVER-EDGE(a)” for a ∈ A.
+For action a = do(X = x), this sub-procedure first performs two interventions do(X = 1) and do(X = 0), then chooses
+an undirected edge (X, X′) corresponding to X (if exists), and then perform do(X′ = 1), do(X′ = 0). The goal of these
+operations is to estimate the difference between P(X = 1 | do(X′ = 0)) and P(X = 1 | do(X′ = 1)), and also the
+difference between P(X′ = 1 | do(X = 0)), P(X′ = 1 | do(X = 0)). which decides whether X′ → X or X → X′.
+By this sub-procedure in parallel, the algorithm estimate the model and recover the edges’ direction simultaneously and
+adaptively. To measure the difficulty for identified the direction of edges, for e : X → X′ we define
+ce = P(X′ = 1 | do(X = 1)) − P(X′ = 1 | do(X = 0))
+(29)
+ca = cX =
+min
+e:X→X′ ce.
+(30)
+
+Combinatorial Causal Bandits without Graph Skeleton
+Algorithm 7 Causal-PE-unknown(G, A, ε, δ)
+1: Initialize t = 1, Ta(0) = 0, ˆµa = 0 for all arms a ∈ A, Aknown = ∅
+2: for t = 1, 2, · · · , do
+3:
+at−1
+h
+= argmaxa∈A ˆµt−1
+a
+4:
+at−1
+l
+= argmaxa∈A\at−1
+h
+(U t−1
+a
+)
+5:
+if Uat−1
+l
+≤ Lat−1
+h
++ ε then
+6:
+Return at−1
+h
+7:
+end if
+8:
+Perform do() operation and observe Xt and Yt. For a = do(), Ta(t) = Ta(t−1)+1, Da(t) = Da(t−1), ra,∅(t) =
+1
+Ta(t)
+�t
+j=1 Yj, pa,∅(t) = 1.
+9:
+for a = do(X = x) ∈ Aknown do
+10:
+Ta,z(t) = Ta,z(t − 1) + I{Xt = x, P = z}, Ta(t) = minz{Ta,z(t)}, where P = Pa(X). Da(t) = Da(t − 1).
+11:
+Update ra,z(t) =
+1
+Ta,z(t)
+�t
+j=1 I{Xj = x, P j = z}Yj.
+12:
+Update pa,z(t) = 1
+t
+�t
+j=1 I{P j = z}.
+13:
+Estimate ˆµO,a(t) = �
+z ra,z(t)pa,z(t) and calculate [Lt
+O,a, U t
+O,a]by (34) and (35).
+14:
+end for
+15:
+RECOVER-EDGE(at−1
+h
+).
+16:
+RECOVER-EDGE(at−1
+l
+).
+17:
+Update empirical mean ˆµI,a(t) using interventional dataand interventional confidence bound [Lt
+I,a, U t
+I,a]
+18:
+Update confidence bound [Lt
+a, U t
+a] by (33), ˆµa = (Lt
+a + U t
+a)/2, for each arm a.
+19: end for
+ce measure the difficulty for distinguishing the direction for an edge, and ca = cX represents the hardness for discovering
+all directions corresponding to X and its childs.
+The main Algorithm 7 is followed from (Xiong & Chen, 2023). During the algorithm, we add ”RECOVER-EDGE”
+sub-procedure to identify the directions of the unknown edges. This sub-procedure first perform intervention do(X = 0)
+and do(X = 1) on the node X. Then if there is an edge (X′, X) which direction has not been identified, it chooses one such
+edge and perform do(X′ = 1) and do(X′ = 0). Then it constructs the confidence bound for all P(X′ = 1 | do(X = 1)),
+P(X′ = 1 | do(X = 0)), P(X = 1 | do(X′ = 1)) and P(X = 1 | do(X′ = 0)) based on Hoeffding’s concentration
+bound. In fact, assume there are Da(t) samples for a = do(X′ = x), x ∈ {0, 1} until round t, then the confidence bound
+for X conditioning on do(X′ = x) is defined by
+[LX|do(X′=x), UX|do(X′=x)] =
+�
+ˆP(X = 1 | do(X′ = x)) −
+�
+2
+Da(t) log 4n2t2
+δ
+, ˆP(X = 1 | do(X′ = x)) +
+�
+2
+Da(t) log 4n2t2
+δ
+�
+,
+(31)
+where n is the number of nodes, and ˆP(X = 1 | do(X′ = x)) are the empirical mean of P(X = 1 | do(X′ = x)) using
+all these Da(t) samples for do(X′ = x). Other confidence bounds define in this way similarly.
+Moreover, at iteration t, Line 4-Line 6 first choose two actions at−1
+h
+and at−1
+l
+through LUCB1 algorithm. Then, we use
+Aknown to represent all nodes actions do(X = x) where all the edges corresponding to X are identified. In fact, if all the
+edges corresponding to X are identified, we can find the true parent set Pa(X). Then we can use do-calculus to estimate
+the causal effect:
+E[Y | do(X = x)] =
+�
+z
+P(Y | X = x, Z = z)P(Z = z).
+(32)
+Line 9-14 enmurates all these actions, and calculate corresponding confidence bound. The confidence bound is calculated by
+[Lt
+a, U t
+a] = [Lt
+O,a, U t
+O,a] ∩ [Lt
+I,a, U t
+I,a],
+(33)
+where the first term [Lt
+O,a, U t
+O,a] = (−∞, ∞) for a = do(X = x) if the parents of X are not sure at time t. In fact, if we
+do not discover all the edges corresponding to X, we cannot estimate the causal effect E[Y | do(X = x)] using do-calculus.
+
+Combinatorial Causal Bandits without Graph Skeleton
+Algorithm 8 RECOVER-EDGE(a)
+1: if a = do() then
+2:
+Return.
+3: else
+4:
+Assume a = do(X = x). Sample action do(X = 1), do(X = 0).
+5:
+Da′(t) = Da′(t) + 1 for a′ = do(X = 1) and a′ = do(X = 0).
+6:
+Estimate P(X′ = 1 | do(X = 1)) and P(X′ = 1 | do(X = 0)) using interventional data for neighbor X′, where
+the direction of (X′, X) is unknown.
+7:
+Update the confidence bound [LX′|do(X=1), UX′|do(X=1)] and [LX′|do(X=0), UX′|do(X=0)] by (31).
+8:
+if [LX′|do(X=1), UX′|do(X=1)] ∩ [LX′|do(X=0), UX′|do(X=0)] = ∅ then
+9:
+recover X → Xi.
+10:
+end if
+11:
+if ∃X′ such that (X′, X) is unknown then
+12:
+Choose one such X′ and perform do(X′ = 1) and do(X′ = 0).
+13:
+Estimate P(X = 1 | do(X′ = 0)) and P(X = 1 | do(X′ = 1)) using interventional data.
+14:
+Update the confidence bound [LX|do(X′=1), UX|do(X′=1)] and [LX|do(X′=0), UX|do(X′=0)] by (31).
+15:
+if [LX|do(X′=1), UX|do(X′=1)] ∩ [LX|do(X′=0), UX|do(X′=0)] = ∅ then
+16:
+recover X → Xi.
+17:
+end if
+18:
+Da′(t) = Da′(t) + 1 for a′ = do(X′ = 1) and a′ = do(X′ = 0).
+19:
+end if
+20: end if
+For nodes which parent set is identified, we calculate
+[Lt
+O,a, U t
+O,a] = [ˆµO,a(t) − βO,a(t), ˆµO,a(t) + βO,a(t)], [Lt
+I,a, U t
+I,a] = [ˆµI,a(t) − βI,a(t), ˆµI,a(t) + βI,a(t)]
+(34)
+The term ˆµO,a is calculated by estimating all terms at the right side of (32) empirically, and confidence radius is given by
+βO,a(t) =
+�
+12
+Ta(t) log 16n2Zat3
+δ
+, βI,a(t) = 2
+�
+1
+Da(t) log 2n log(2t)
+δ
+(35)
+Similar to the (Xiong & Chen, 2023), we can prove it is a valid confidence radius, which means that the true effect µO,a will
+fall into the confidence bound [Lt
+O,a, U t
+O,a] with a high probability.
+Line 15-16 try to recover the edge for action chosen by LUCB1 algorithm. At the end of this iteration, the algorithm updates
+all parameters and confidence bounds.
+To represent the complexity result, we first provide the definition of gap-dependent threshold in (Xiong & Chen, 2023): For
+a = do(X = x) and one possible configuration of the parent z ∈ {0, 1}|Pa(X)|, define qa,z = P(X = x, Pa(X) = z)
+and qa = minz{qa,z}. Then sort the arm set as qa1 · max{∆a1, ε/2}2 ≤ qa2 · max{∆a2, ε/2}2 ≤ · · · ≤ qa|A| ·
+max{∆a|A|, ε/2}2. Recall that ∆a = µ∗ − µa is the reward gap between the optimal reward and the reward of action a.
+Then Hr is defined by
+Hr =
+r
+�
+i=1
+1
+max{∆ai, ε/2}2 .
+(36)
+Definition 1 (Gap-dependent observation threshold((Xiong & Chen, 2023))). For a given causal graph G and its associated
+qa’s and ∆a’s, the gap-dependent observation threshold mε,∆ is defined as:
+mε,∆ = min
+�
+τ :
+�����
+�
+a ∈ A
+�����qa max {∆a, ε/2}2 <
+1
+Hτ
+������ ≤ τ
+�
+.
+Denote action set S = {a ∈ A : qa max{∆a, ε/2}2 <
+1
+Hmε,∆ } are all actions which qa is relatively small, then |S| ≤ mε,∆.
+Intuitively, action a with smaller qa are harder to be estimated by observation: If we assume qa = qa,z for a fixed vector z,
+
+Combinatorial Causal Bandits without Graph Skeleton
+then P(X = x, Pa(X) = z) is hard to observe and estimate by empirical estimation. Thus S contains all actions that are
+relatively hard to observe, so it is more efficient to estimate µa by intervention for a ∈ S. Based on this definition, we can
+provide the final sample complexity result:
+Theorem 6. Denote H = �
+a∈S
+1
+max{∆a,ε/2}2 + �
+a/∈S min{
+1
+max{∆a,ε/2}2 , 1
+c2a + �
+e:X′→X
+1
+c2e }. With probability 1 − 4δ,
+Algorithm 7 will return a ε-optimal arm with sample complexity bound at most
+T = O
+�
+H log
+�nZH
+δ
+��
+,
+where ce, ca is defined in (29) and (30).
+The result can be explained in an intuitive way. The first term of H is the summation of all actions in S. As we discussed
+above, it is more efficient to estimate the µa with intervention for a ∈ S. Thus, this summation can be regarded as the
+sample complexity applying multi-armed bandit algorithm (e.g. LUCB1) directly. The second term is to estimate the actions
+by observation. For each action a = do(X = x) with larger qa, we can first identify the edge’s direction corresponding to
+the node X, and then using do-calculus to estimate the reward. The term
+1
+c2a + �
+e:X′→X
+1
+c2e represents the complextity
+to identify the directions, and the complexity for using do-calculus can be contained in the first term �
+a∈S
+1
+max{∆a,ε/2}2
+because of the definition of gap-dependent observation threshold. Also, the term min{
+1
+max{∆a,ε/2}2 , 1
+c2a + �
+e:X′→X
+1
+c2e }
+is because when we are discovering the edges’ direction, if the reward can be estimated by intervention accurately, we
+turn to use interventional estimation and give up the causal discovery for this node. The detailed proof can be found in the
+Section D.2.
+Even if these two mechanisms can reduce the sample complexity, at the worst case the complexity also degenerates to
+O(n/ε2), which is equal to the complexity for multi-armed bandit. We provide a lower bound to show that this problem
+cannot be avoided.
+Theorem 7 (Lower bound). Consider causal bandits with only essential graph and atomic intervention, for any (ε, δ)−PAC
+algorithm, there is a bandit instance with expected sample complexity Ω( n
+ε2 log(1/δ)) even if we have all observational
+distribution P(X, Y ).
+Theorem 7 states that even if we receive all observational distribution, which shows the intrinsic hardness for unknown
+graph. Indeed, the proof of lower bound shows that the unknown direction will lead to different interventional effects even
+when the observational distribution are the same, leading to a unavoidable hardness.
+D.2. Proof of Theorem 6
+First, fixed an action a = do(Xi = x), z ∈ {0, 1}|Pa(X)| , then Ta,z(t) = �t
+j=1 I{Xj,i = x, Pa(Xi)j = z} and the
+empirical mean ˆqa,z(t) = Ta,z(t)/t. Then denote 2|Pa(X)| = Za, if qa,z(t) ≥ 6
+t log(2nZa/δ), with probability at least
+1 −
+δ
+2nZa , we can have
+|ˆqa,z(t) − qa,z(t)| <
+�
+6qa,z(t)
+t
+log
+�2nZa
+δ
+�
+Hence
+ˆqa(t) = min
+z {ˆqa,z(t)} ≤ min
+z {qa,z +
+�
+6qa,z
+t
+log 2nZa
+δ
+} = qa +
+�
+6qa
+t
+log 2nZa
+δ
+.
+(37)
+When qa ≥ 3
+t log 2nZa
+δ
+, f(x) = x −
+�
+6x
+t log 2nZa
+δ
+is a increasing function.
+ˆqa(t) ≥ min
+z {qa,z −
+�
+6qa,z
+t
+log 2nZa
+δ
+} = qa −
+�
+6qa
+t
+log 2nZa
+δ
+.
+(38)
+So define the event as
+E1(t) =
+�
+∀a ∈ A with t ≥ 6
+qa
+log
+�2nZa
+δ
+�
+, |ˆqa(t) − qa| ≤
+�
+6qa
+t
+log
+�2nZa
+δ
+��
+
+Combinatorial Causal Bandits without Graph Skeleton
+then Pr{Ec
+1(t)} ≤ δ, where Ec means the complement of the event E.
+Now we consider the concentration bound. First, by classical anytime confidence bound, with probability at least 1 −
+δ
+2n,
+for any time Da(t) ≥ 1
+|ˆµI,a(t) − µI,a| < 2
+�
+1
+Da(t) log
+�2n log(2Da(t))
+δ
+�
+≤ 2
+�
+1
+Da(t) log
+�2n log(2t)
+δ
+�
+Thus define the event as
+E2 =
+�
+∀t, a, |ˆµI,a(t) − µI,a| < 2
+�
+1
+Da(t) log
+�2n log(2t)
+δ
+��
+,
+then Pr{Ec
+2} ≤ δ.
+Consider the observational confidence bound. First, if a /∈ Aknown, [Lt
+O,a, U t
+O,a] = (−∞, ∞) and then the ˆµO,a(t) ∈
+[Lt
+O,a, U t
+O,a]. Now we consider that if a = do(X = x) ∈ Aknown and the parent of X is P . By Hoeffding’s inequality,
+with probability at least 1 − δ/16n2Zat3, for a = do(X = x),
+|ra,z(t) − P(Y = 1 | X = x, P = z)| >
+�
+1
+2Ta,z(t) log 16n2Zat3
+δ
+(39)
+Also, by Chernoff’s inequality, since qa ≤ P(P = z) for all z ∈ {0, 1}|P |, when t ≥
+6
+qa log
+�
+16n2Zat3
+δ
+�
+with probability
+at least 1 − δ/16n2Zat3 we will have
+|pa,z(t) − P(P = z)| >
+�
+6P(P = z)
+t
+log 16n2Zat3
+δ
+,
+(40)
+then
+ˆµO,a =
+�
+z
+ra,z(t) · pa,z(t)
+≤
+�
+z
+P(Y = 1 | X = x, P = z)pa,z(t) +
+�
+z
+pa,z(t)
+�
+1
+2Ta,z(t) log 16n2Zat3
+δ
+≤
+�
+z
+P(Y = 1 | X = x, P = z)pa,z(t) +
+�
+1
+2Ta(t) log 16n2Zat3
+δ
+≤
+�
+z
+P(Y = 1 | X = x, P = z)P(P = z) +
+�
+z
+�
+6P(P = z)
+t
+log 16n2Zat3
+δ
++
+�
+1
+2Ta(t) log 16n2Zat3
+δ
+≤ µa +
+�
+6Z
+t log 16n2Zat3
+δ
++
+�
+1
+2Ta(t) log 16n2Zat3
+δ
+≤ µa +
+�
+6
+Ta(t) log 16n2Zat3
+δ
++
+�
+1
+2Ta(t) log 16n2Zat3
+δ
+,
+= µa +
+�
+8
+Ta(t) log 16n2Zat3
+δ
+.
+Also, if t ≤
+6
+qa log 16n2Zat3
+δ
+, first by Chernoff inequality, set Q =
+6
+qa log 16n2Zat3
+δ
+, then with probability at least 1 −
+δ/16n2Zat3, we have
+ˆqa(Q) ≤ 2qa.
+(41)
+
+Combinatorial Causal Bandits without Graph Skeleton
+by E1(Q).
+Ta(t) ≤ Ta(Q) ≤ ˆqa(Q) · Q ≤ 2qa · Q = 12
+qa
+log 16n2Zat3
+δ
+.
+Then
+�
+12
+Ta(t) log 16n2Zat3
+δ
+≥ 1 and the inequality
+|ˆµO,a(t) − µO,a| ≤
+�
+12
+Ta(t) log 16n2Zat3
+δ
+also holds. Thus we define the event
+E3 =
+�
+∀a, t, |ˆµO,a(t) − µO,a| ≤
+�
+12
+Ta(t) log 16n2Zat3
+δ
+�
+then by taking the union bound of (39), (40) and (41),
+Pr{Ec
+3} ≤
+∞
+�
+t=1
+�
+a∈A
+�
+z
+3 ·
+δ
+16n2Zat3
+≤
+∞
+�
+t=1
+δ
+4t3
+≤ δ.
+Now we consider how to bound our sample complexity based on events E1, E2 and E3. First, we provide the following
+lemma in (Xiong & Chen, 2023):
+Lemma 10 (Lemma 6 in (Xiong & Chen, 2023)). Under the event E1, E2 and E3, at round t, if we have
+βat
+h(t) ≤
+max{∆at
+h, ε/2}
+4
+, βat
+l(t) ≤
+max{∆at
+l, ε/2}
+4
+,
+where at
+h, at
+l are the actions performed by algorithm at round t. then the algorithm will stop at round t + 1.
+Now assume the algorithm does not terminate at T1 = 192H log(nZT 3
+1 /δ), where Z = maxa Za. For a ∈ S, Da(t). Note
+that H ≥ Hmε,∆. Thus at round T1, for action a with qa ≥
+1
+Hmε,∆·max{∆a,ε/2}2 ≥ 192
+T1 log 16nZaT 3
+1
+δ
+, if a ∈ Aknown, then
+under event E1(T1), we have
+ˆqa(T1) ≥ qa −
+�
+6qa
+T1
+log 16nZaT 3
+1
+δ
+≥ qa
+2 .
+Then
+βa(T1) ≤ βO,a(T1) =
+�
+12
+Ta(T1) log 16n2Zat3
+δ
+≤
+�
+12qa
+2T1
+≤ max{∆a, ε/2}2
+4
+.
+Now we prove that if Da(t) is large for some a, then a ∈ Aknown.
+Lemma 11. With probability at least 1 − δ, denote Ca =
+1
+c2a + �
+e:X′→X
+1
+c2e . If Da(t) ≥ 32Ca log(4n2t2/δ), a ∈ Aknown.
+Proof. If Da(t) ≥ 8Ca log t, we have called sub-procedure RECOVER-EDGE(a) for Da(t) times. Then, for each edge
+e : X → X′, we will perform intervention do(X = 1), do(X = 0) for at least Da(t) times and observe the empirical
+
+Combinatorial Causal Bandits without Graph Skeleton
+difference | ˆP(X′ | do(X = 1)) − ˆP(X′ | do(X = 0))|. By Hoeffding’s inequality and union bound on all time t and the
+�n−1
+2
+�
+ordered-pair (X′, X), with probability at least 1 − δ, for all t ∈ [T] and all X′, X we have
+| ˆP(X′ | do(X = 1)) − P(X′ | do(X = 1))| ≤
+�
+2
+Da(t) log 4n2t2
+δ
+| ˆP(X′ | do(X = 0) − P(X′ | do(X = 0))| ≤
+�
+2
+Da(t) log 4n2t2
+δ
+Then for the confidence bounds
+[LX′|do(X=1), UX′|do(X=1)] =
+�
+ˆP(X′ | do(X = 1)) −
+�
+2
+Da(t) log 4n2t2
+δ
+, ˆP(X′ | do(X = 1)) +
+�
+2
+Da(t) log 4n2t2
+δ
+�
+[LX′|do(X=0), UX′|do(X=0)] =
+�
+ˆP(X′ | do(X = 0)) −
+�
+2
+Da(t) log 4n2t2
+δ
+, ˆP(X′ | do(X = 0)) +
+�
+2
+Da(t) log 4n2t2
+δ
+�
+the intersection
+[LX′|do(X=1), UX′|do(X=1)] ∩ [LX′|do(X=0), UX′|do(X=0)] = ∅
+since
+| ˆP(X′ | do(X = 1) − ˆP(X′ | do(X = 0))|
+≥|P(X′ | do(X = 1) − P(X′ | do(X = 0))| − |P(X′ | do(X = 1) − ˆP(X′ | do(X = 1))|
+− |P(X′ | do(X = 0) − ˆP(X′ | do(X = 0))|
+≥ca − 2
+�
+2
+Da(t) log 4n2t2
+δ
+≥2
+�
+2
+Da(t) log 4n2t2
+δ
+.
+where we use Da(t) ≥
+1
+c2a log 4n2t2
+δ
+. Then the edge’s direction will be identified correctly.
+Consider the edge e : X′ → X, then if we sample do(X′ = 1) and do(X′ = 0) for 1
+c2e log 4n2t2
+δ
+times within sub-procedures
+RECOVER-EDGE(a), similarly we will identify the edge X′ → X. Then because the RECOVER-EDGE(a) will perform
+intervention do(X′ = 0) and do(X′ = 1) for the X′ that the direction of (X′, X) has not been discovered each time, after
+�
+e:X′→X
+1
+c2e log 4n2t2
+δ
+.
+Then we define
+E4 = {Lemma 11 holds}
+Then Pr{Ec
+4} ≤ δ. Also, under the event E2, the following lemma shows that if Da(t) is really large, we can estimate the
+µa accurately.
+Lemma 12. Under event E2, if Da(t) ≥
+64
+max{∆a,ε/2}2 log 16n2Zat3
+δ
+, then
+βa(T1) ≤ max{∆a, ε/2}2
+4
+.
+Proof. In fact,
+βa(t) ≤ βI,a(t) = 2
+�
+1
+Da(t) log
+�2n log(2t)
+δ
+�
+≤ 2
+�
+1
+Da(t) log 16n2Zat3
+δ
+≤ max{∆a, ε/2}2
+4
+.
+
+Combinatorial Causal Bandits without Graph Skeleton
+Now we turn to our main result. From the Lemma 10, at least one arm a with βa(t) ≥ max{∆a,ε/2}
+4
+will be performed an
+intervention at each round t ≥ T1. Under the event E1, E2, E3 and E4, these interventions will only performed in two types
+of action a:
+• qa ≤
+1
+Hmε,∆·max{∆a,ε/2}2 and Da(t) ≤
+64
+max{∆a,ε/2}2 log 16n2Zat3
+δ
+.
+• Da(t) ≤ min{MCa log(t),
+64
+max{∆a,ε/2}2 log 16n2Zat3
+δ
+}.
+Note that qa ≤
+1
+Hmε,∆·max{∆a,ε/2}2 implies that a ∈ S, then after at most T2 rounds, where
+T2 = 64
+��
+a∈S
+1
+max{∆a, ε/2}2 +
+�
+a/∈S
+min
+�
+1
+max{∆a, ε/2}2 , 1
+c2a
++
+�
+e:X′→X
+1
+c2e
+��
+log 16n2ZT 3
+2
+δ
+= 64H log 16n2ZT 3
+2
+δ
+the algorithm should terminates. The fist term is the summation of all actions in S, and the second term is for the second
+type of actions, which Da(t) ≤ min{MCa log(t),
+64
+max{∆a,ε/2}2 log 16n2Zat3
+δ
+}. Denote T = T1 + T2, then
+T = T1 + T2 ≤ 256H log 16n2ZT 3
+δ
+≤ 768H log 16nZT
+δ
+Then by the Lemma 13, with probability at least 1 − 4δ, the sample complexity has the upper bound
+T = O
+�
+H log
+�nZH
+δ
+��
+Replace δ to δ/4, we derive the sample complexity in the Theorem 6. The correctness of algorithm can be derived by
+LUCB1 algorithm. We provide a short argument here. Because the stopping rule is ˆµt
+at
+l + βat
+l(t) ≤ ˆµt
+at
+h − βat
+h(t) + ε, if
+a∗ ̸= at
+h, we have
+µat
+h + ε ≥ ˆµat
+h − βat
+h(t) + ε ≥ ˆµat
+l + βat
+l(t) ≥ ˆµa∗ + βa∗(t) ≥ µa∗.
+Hence either a∗ = at
+h or at
+h is ε-optimal arm.
+D.3. Proof of Lemma 10
+For completeness, we provide the proof in (Xiong & Chen, 2023).
+Proof. If the optimal arm a∗ = at
+h,
+ˆµat
+l + βat
+l(t) ≤ µat
+l + 2βat
+l(t)
+≤ µat
+l +
+max{∆at
+l, ε/2}
+2
+≤ µat
+h − ∆at
+l +
+max{∆at
+l, ε/2}
+2
+≤ ˆµat
+h + βa∗(Ta∗(t)) − ∆at
+l +
+max{∆at
+l, ε/2}
+2
+≤ ˆµat
+h − βa∗(Ta∗(t)) +
+max{∆a∗, ε/2} + max{∆at
+l, ε/2}
+2
+− ∆at
+l
+≤ ˆµat
+h − βa∗(Ta∗(t)) +
+∆a∗ + ε/2 + ∆at
+l + ε/2
+2
+− ∆at
+l
+≤ ˆµat
+h − βa∗(Ta∗(t)) + ε.
+
+Combinatorial Causal Bandits without Graph Skeleton
+If optimal arm a∗ ̸= at
+h, and the algorithm doesn’t stop at round t + 1, then we prove a∗ ̸= at
+l. Otherwise, assume a∗ = at
+l
+ˆµt
+at
+h ≤ µt
+at
+h +
+max{∆at
+h, ε/2}
+4
+(42)
+= µt
+at
+l − ∆at
+h +
+max{∆at
+h, ε/2}
+4
+(43)
+≤ µt
+at
+l −
+3∆at
+h
+4
++ ε/4
+(44)
+≤ ˆµt
+at
+l + max{∆a∗, ε/2}
+4
+−
+3∆at
+h
+4
++ ε/4
+(45)
+≤ ˆµt
+at
+l + ε/2 −
+∆at
+h
+2 .
+(46)
+From the definition of at
+h, we know ε > ∆at
+h ≥ ∆a∗, βat
+h(t) ≤ ε/4, βat
+l(t) ≤ ε/4. Then ˆµt
+at
+l + βat
+l(t) + βat
+h(t) ≤
+ˆµat
+l + ε/2 ≤ ˆµt
+at
+h + ε, which means the algorithm stops at round t + 1.
+Now we can assume a∗ ̸= at
+l, a∗ ̸= at
+h. Then
+µat
+l + 2βat
+l(t) ≥ ˆµat
+l + βat
+l(t) ≥ ˆµa∗ + βa∗(Ta∗(t)) ≥ µa∗ = µat
+l + ∆at
+l.
+(47)
+Thus
+∆at
+l ≤ 2βat
+l(t) ≤
+max{∆at
+l, ε/2}
+2
+,
+(48)
+which leads to ∆at
+l ≤ ε/2, βat
+l(t) ≤ ε/8. Since
+Also,
+µat
+h + βat
+h(t) ≥ ˆµat
+h ≥ ˆµat
+l ≥ µa∗ − βat
+l(t) = µat
+h + ∆at
+h − βat
+l(t),
+(49)
+which leads to
+max{∆at
+h, ε/2}
+4
+≥ ∆at
+h − ε/8,
+(50)
+and ∆at
+h ≤ ε/2, βat
+h(t) ≤ ε/8. Hence ˆµt
+at
+l + βat
+l(t) + βat
+h(t) ≤ ˆµat
+l + ε/2 ≤ ˆµt
+at
+h + ε, which means the algorithm stops at
+round t + 1.
+D.4. Proof of Theorem 7
+Proof. We construct n − 1 graphs with the same distribution P(X, Y ) but different causal graph. Indeed, We construct
+the bandit instances {ξi}2≤i≤n as follows. For instance ξ2, the graph structure contains edge X1 → Y, X2 → X1, X1 →
+Xi(3 ≤ i ≤ n) and X2 → Xi(3 ≤ i ≤ n). For instances ξi(3 ≤ i ≤ n), we change X1 → Xi to Xi → X1. The graph
+structure are shown in the Figure 1 and Figure 2.
+The observational distribution for all instance is:
+P(X, Y ) = p1p2 · · · pn,
+(51)
+where
+p1 = 0.5,
+(52)
+p2 =
+� 0.5 + ε
+x2 = x1
+0.5 − ε
+x2 ̸= x1
+(53)
+pi =
+� 0.5 + 4ε
+xi = x1
+0.5 − 4ε
+xi ̸= x1
+.
+(54)
+
+Combinatorial Causal Bandits without Graph Skeleton
+Figure 1. Causal Bandits Instance τ2
+Figure 2. Causal Bandits Instance τi(i = 3)
+It is easy to check that �
+x,y P(X = x, Y = y) = 1 and P(Xi = 1) = 0.5. The action set is do(), do(Xi = 1), do(Xi = 0)
+where 2 ≤ i ≤ n, which means the action set does not contain do(X1 = x) for x = 0, 1.
+Now in ξ2, we consider P(Y = 1 | do(X2 = 1)). Actually, it is easy to show that P(Y = 1 | do(X2 = 1)) = P(X1 = 1 |
+do(X2 = 1)) = 0.5 + ε. Similarly, P(Y = 1 | do(X2 = 0)) = 0.5 − ε. For other actions, P(Y = 1 | a) = P(X1 = 1 |
+a) = 0.5 since other actions a will not influence the value of X1.
+Now consider instance ξi for 3 ≤ i ≤ n. For action do() and do(Xj = x) with j ̸= 2, i, it will not influence the value of X1
+and then P(Y = 1 | a) = 0.5. Now consider action a = do(X2 = 1), we have
+P(Y = 1 | do(X2 = 1)) = P(X1 = 1 | do(X2 = 1))
+= P(X1 = 1 | X2 = 1) = 0.5 + ε.
+Similarly, P(Y = 1 | do(X2 = 0)) = 0.5 − ε.
+
+X. ..Combinatorial Causal Bandits without Graph Skeleton
+Now we calculate P(Y = 1 | do(Xi = 1)) in instance ξi. In fact, denote q = 0.5 + 4ε and by do-calculus,
+P(X1 = 1 | do(Xi = 1)) =
+�
+x=0,1
+P(X1 = 1 | Xi = 1, X2 = x)P(X2 = x)
+= 0.5(P(X1 = 1 | Xi = 1, X2 = 0) + P(X1 = 1 | Xi = 1, X2 = 1)
+= 0.5
+�P(X1 = 1, Xi = 1, X2 = 0)
+P(Xi = 1, X2 = 0)
++ P(X1 = 1, Xi = 1, X2 = 1)
+P(Xi = 1, X2 = 1)
+�
+= 0.5
+�
+(0.5 + 4ε)(0.5 − ε)
+(0.5 + 4ε)(0.5 − ε) + (0.5 − 4ε)(0.5 + ε) +
+(0.5 + 4ε)(0.5 + ε)
+(0.5 + 4ε)(0.5 + ε) + (0.5 − 4ε)(0.5 − ε)
+�
+= 0.5
+�
+q(0.5 − ε)
+q(0.5 − ε) + (1 − q)(0.5 + ε) +
+q(0.5 + ε)
+q(0.5 + ε) + (1 − q)(0.5 − ε)
+�
+= 0.5
+�
+q(0.5 − ε)
+0.5 − (2q − 1)ε +
+q(0.5 + ε)
+0.5 + (2q − 1)ε
+�
+= q
+� 0.52 − (2q − 1)ε2
+0.52 − (2q − 1)2ε2
+�
+≤ q = 0.5 + 4ε.
+Also, we prove that
+q
+� 0.52 − (2q − 1)ε2
+0.52 − (2q − 1)2ε2
+�
+≥ 0.5 + 2ε.
+Actually, this inequality is equal to
+(0.5 + 4ε)(0.52 − 8ε3) ≥ (0.5 + 2ε)(0.52 − 8ε4)
+⇐⇒ 1 ≥ 56ε3 + 8ε2 − 32ε4.
+When ε is small enough, this inequality holds. In summary, we have
+P(X1 = 1 | do(Xi = 1)) ∈ [0.5 + 2ε, 0.5 + 4ε].
+Similarly, we can get
+P(X1 = 1 | do(Xi = 0)) = 0.5(P(X1 = 1 | Xi = 0, X2 = 1) + P(X1 = 1 | Xi = 0, X2 = 0))
+= (1 − q)
+� 0.52 − (1 − 2q)ε2
+0.52 − (1 − 2q)2ε2
+�
+∈ [0.5 − 4ε, 0.5].
+Now in instance ξ2, the output action should be do(X2 = 1), while in instance ξi, the output action should be do(Xi = 1).
+Now by Pinkser’s inequality, for an policy π, we have
+2δ ≥ Pξ2(ao = do(Xi = 1)) + Pξi(ao ̸= do(Xi = 1)) ≥ exp(−KL(ξπ
+2 , ξπ
+i )).
+Also, assume the stopping time as τ for the environment E, the KL divergence can be rewritten as
+KL(ξπ
+2 , ξπ
+i ) = EAt∼ξπ
+2
+� τ
+�
+t=1
+KL(Pξ2(Xt, Yt | At), Pξi(Xt, Yt | At))
+�
+(55)
+= Eξπ
+2
+� τ
+�
+t=1
+Pξ2(Xt, Yt | At)
+�
+log Pξ2(Xt, Yt | At)
+Pξi(Xt, Yt | At)
+��
+(56)
+= Eξπ
+2
+� τ
+�
+t=1
+Pξ2(Xt,i, Xt,1 | At)
+�
+log Pξ2(Xt,i, Xt,1 | At)
+Pξi(Xt,i, Xt,1 | At)
+��
+(57)
+where the last equation is derived as follows:
+Pξ2(Xt, Yt | At)
+Pξi(Xt, Yt | At) = Pξ2(Xt,i, Xt,1 | At) · Pξ2( ¯Xt,i, Yt | Xt,i, Xt,1, At)
+Pξi(Xt,i, Xt,1 | At) · Pξi( ¯Xt,i, Yt | Xt,i, Xt,1, At)
+
+Combinatorial Causal Bandits without Graph Skeleton
+where ¯Xt,i = Xt \ {Xt,i, Xt,1}. Now since ¯Xt,i is only decided by X1, X2 and X2 is only decided by At, then
+Pξ2( ¯Xt,i, Yt | Xt,i, Xt,1, At) = Pξi( ¯Xt,i, Yt | Xt,i, Xt,1, At)
+and then
+Pξ2(Xt, Yt | At)
+Pξi(Xt, Yt | At) = Pξ2(Xt,i, Xt,1 | At)
+Pξi(Xt,i, Xt,1 | At) .
+Note that only when At = do(Xi = 1), do(Xi = 0), Pξ2(Xt,i, Xt,1 | At) ̸= Pξi(Xt,i, Xt,1 | At). Then the equation (57)
+can be further calculated as
+(57) =
+�
+x=0,1
+Eξπ
+2
+� τ
+�
+t=1
+I{At = do(Xi = x)}
+�
+· Pξ2(Xt,i, Xt,1 | do(Xi = x))
+�
+log Pξ2(Xt,i, Xt,1 | do(Xi = x))
+Pξi(Xt,i, Xt,1 | do(Xi = x))
+�
+=
+�
+x=0,1
+Eξπ
+2
+� τ
+�
+t=1
+I{At = do(Xi = x)}
+�
+· Pξ2(Xt,1 | do(Xi = x))
+�
+log Pξ2(Xt,1 | do(Xi = x))
+Pξi(Xt,1 | do(Xi = x))
+�
+≤
+�
+x=0,1
+Eξπ
+2
+� τ
+�
+t=1
+I{At = do(Xi = x)}
+� �
+0.5 ·
+�
+log
+0.5
+0.5 + 4ε + log
+0.5
+0.5 − 4ε
+��
+≤
+�
+x=0,1
+Eξπ
+2
+� τ
+�
+t=1
+I{At = do(Xi = x)}
+�
+96ε2
+= 96ε2 · Eξπ
+2 [N(do(Xi = 1)) + N(do(Xi = 0))].
+where the Eξπ
+2 N(a) represents that the number of times taking action a for policy π under the instance ξ2. Now we have
+Eξπ
+2 [N(do(Xi = 1)) + N(do(Xi = 0))] ≥ KL(ξπ
+2 , ξπ
+i )
+96ε2
+≥
+1
+96ε2 log 1
+2δ .
+Hence the stopping time τ under policy π can be lower bounded by
+Eξπ
+2 [τ] ≥
+n
+�
+i=3
+Eξπ
+2 [N(do(Xi = 1)) + N(do(Xi = 0))] ≥ n − 2
+96ε2 log 1
+2δ = O
+� n
+ε2 log 1
+δ
+�
+.
+D.5. Technical Lemma
+Lemma 13. If T = CH log dT
+δ for some constant C and parameter d such that d ≥ eδ, then T = O(H log Hd
+δ ).
+Proof. Let f(x) =
+x
+log(dx/δ), then for x ≥ 1
+f ′(x) = log(dx/δ) − 1
+log2 dx/δ
+≥ 0
+because dx/δ > e. Then f(x) is non-decreasing for x ≥ 1.
+To prove T = O(H log Hd
+δ ), we only need to show that f(T) ≤ f(C′H log Hd
+δ ) for some constant C′. Since
+log C′Hd log Hd
+δ
+δ
+= log C′Hd
+δ
++ log log Hd
+δ
+we only need to prove
+f(C′H log Hd
+δ ) =
+C′H log Hd
+δ
+log C′Hd
+δ
++ log log Hd
+δ
+≥ CH = f(T).
+
+Combinatorial Causal Bandits without Graph Skeleton
+If we choose C′ ≥ 2C + C log C′, then
+CH
+�
+log C′Hd
+δ
++ log log Hd
+δ
+�
+≤ CH(log C′Hd
+δ
++ log Hd
+δ )
+≤ 2CH log Hd
+δ
++ CH log C′
+≤ (2C + C log C′)H log Hd
+δ
+≤ C′H log Hd
+δ .
+
diff --git a/q9FQT4oBgHgl3EQftjaQ/content/tmp_files/load_file.txt b/q9FQT4oBgHgl3EQftjaQ/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d31b0e8d71ef3a99dfd7c88e972edfecea048c58
--- /dev/null
+++ b/q9FQT4oBgHgl3EQftjaQ/content/tmp_files/load_file.txt
@@ -0,0 +1,1364 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf,len=1363
+page_content='Combinatorial Causal Bandits without Graph Skeleton  Shi Feng * 1 2 Nuoya Xiong * 1 Wei Chen 2  Abstract  In combinatorial causal bandits (CCB), the learn-  ing agent chooses a subset of variables in each  round to intervene and collects feedback from  the observed variables to minimize expected re-  gret or sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Previous works study  this problem in both general causal models and  binary generalized linear models (BGLMs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' How-  ever, all of them require prior knowledge of causal  graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' This paper studies the CCB prob-  lem without the graph structure on binary general  causal models and BGLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We first provide an  exponential lower bound of cumulative regrets  for the CCB problem on general causal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  To overcome the exponentially large space of pa-  rameters, we then consider the CCB problem on  BGLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We design a regret minimization algo-  rithm for BGLMs even without the graph skeleton  and show that it still achieves O(  √  T ln T) ex-  pected regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' This asymptotic regret is the same  as the state-of-art algorithms relying on the graph  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Moreover, we sacrifice the regret to  O(T  2  3 ln T) to remove the weight gap covered by  the asymptotic notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' At last, we give some  discussions and algorithms for pure exploration  of the CCB problem without the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Introduction  The multi-armed bandits (MAB) problem is a classical  model in sequential decision-making (Robbins, 1952;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Auer  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Bubeck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In each round, the learning  agent chooses an arm and observes the feedback reward cor-  responding to that arm, with the goal of either maximizing  the cumulative reward over T rounds (regret minimization),  or minimizing the sample complexity to find the interven-  tion closest to the optimal one (pure exploration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' MAB  can be extended to have more structures among arms and  1Institute for Interdisciplinary Information Sciences, Tsinghua  University, Beijing, China 2Microsoft Research, Beijing, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Correspondence to:  Shi Feng <shifeng-thu@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='com>,  Nuoya Xiong <xiongny20@mails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='cn>, Wei Chen  <weic@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='com>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Under review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  reward functions, which leads to more advanced learning  techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Such structured bandit problems include com-  binatorial causal bandits (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 2016), linear  bandits (Abbasi-Yadkori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Agrawal & Goyal,  2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2017), and sparse linear bandits (Abbasi-  Yadkori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  In this paper, we study another structured bandit problem  called causal bandits, which is first proposed by (Lattimore  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' It consists of a causal graph G = (X∪{Y }, E)  indicating the causal relationship among the observed vari-  ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In each round, the learning agent selects one or a few  variables in X to intervene, gains the reward as the output of  Y , and observes the values of all variables in X ∪{Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The  use of causal bandits is possible in a variety of contexts that  involve causal relationships, including medical drug testing,  performance tuning, policy making, scientific experimental  process, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  In all previous literature except (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2021), the struc-  ture of the causal graph is known, but the underlying proba-  bility distributions governing the causal model are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (2021) further assumes that the graph structure  is unknown and the learning agent can only see the graph  skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Here, graph skeleton is also called essential graph  (G´amez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2013) and means all the edges in G without  direction information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In our paper, we further consider that  the graph skeleton is unknown and remove the unrealistic  assumption |Pa(Y )| = 1 in (Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In many  scenarios, the learning agent needs to learn the causal re-  lationships between variables and thus needs to learn the  graph without any prior information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For example, in poli-  cymaking for combating COVID-19, many possible factors  like food supply, medical resources, vaccine research, pub-  lic security, and public opinion may consequently impact  the mortality rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' However, the causal relationships among  these factors are not readily known and need to be clarified  during the sequential decision-making process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Learning  the causal graph from scratch while identifying the optimal  intervention raises a new challenge to the learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  For regret minimization, we study combinatorial causal  bandits (CCB) under the binary generalized linear mod-  els (BGLMs) as (Feng & Chen, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xiong & Chen,  2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Using a novel initialization phase, we could de-  termine the ancestor structure of the causal graph for the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='13392v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='LG]  31 Jan 2023  Combinatorial Causal Bandits without Graph Skeleton  BGLM when the minimum edge weight in the model satis-  fies a weight gap assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' This is enough to perform a  CCB algorithm based on maximum likelihood estimation  on it (Feng & Chen, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The resulting algorithm BGLM-  OFU-Unknown achieves O(  √  T log T) regret, where T is  the time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The big O notation only holds for T larger  than a threshold so the weight gap assumption is hidden by  the asymptotic notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For binary linear models (BLMs),  one could sacrifice O(T  1  6 ) regret to remove the weight gap  assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The algorithms we design for BLMs allow  hidden variables and use linear regression instead of MLE  to remove an assumption on parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  For pure exploration, we give some discussions on general  causal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' If we allow the weight gap, a trivial solu-  tion exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Without the weight gap, we give an adaptive  algorithm for general causal model in the atomic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  In summary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' our contribution includes: (a) providing an ex-  ponential lower bound of cumulative regret for CCB on gen-  eral causal model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (b) proposing an O(  √  T ln T) cumulative  regret CCB algorithm BGLM-OFU-Unknown for BGLMs  without graph skeleton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (c) proposing an O(T  2  3 ln T) cu-  mulative regret CCB algorithm for BLMs without graph  skeleton and the weight gap assumption,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (d) giving the first  discussion including algorithms and lower bounds on pure  exploration of causal bandits on general causal models and  atomic intervention without knowing the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Related Works  Causal Bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The causal bandits problem is first pro-  posed by Lattimore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' They discuss the simple  regret for parallel graphs and general graphs with known  probability distributions P(Pa(Y )|a) for any action a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Sen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Nair et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Maiti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (2021) generalize  the simple regret study for causal bandits to more general  causal graphs and soft interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Nair  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Maiti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (2021) consider cumulative regret  for causal bandits problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' However, all of these studies  are not designed for combinatorial action set and has expo-  nentially large regret or sample complexity with respect to  the graph size if the actions are combinatorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Yabe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Feng & Chen (2023);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xiong & Chen (2023);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Varici  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (2022) consider combinatorial action set for causal  bandits problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Among them, Feng & Chen (2023) are  the first to remove the requirement of T > �  X∈X 2|Pa(X)|  and proposes practical CCB algorithms on BGLMs with  O(  √  T ln T) regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xiong & Chen (2023) simultaneously  propose CCB algorithms on BGLMs as well as general  causal models with polynomial sample complexity with re-  spect to the graph size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Varici et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (2022) further include  soft interventions in the CCB problem, but their work is  on Linear Structural Equation Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Lee & Bareinboim  (2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 2020) propose several CCB algorithms on gen-  eral causal bandits problem, but they focus on empirical  studies while we provide theoretical regret analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' All  of the above works require the learning agent to know the  graph structure in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (2021) is the first and  only work on causal bandits without graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' How-  ever, their algorithm is limited to the case of |Pa(Y )| = 1  for the atomic setting, and thus the main technical issue  degenerates to finding the particular parent of Y so that one  could intervene on this node for the optimal reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Social Network and Causality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Causal models have intrin-  sic connections with influence propagation in social net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Feng & Chen (2021) study the identifiability in the  Independent Cascade (IC) propagation model as a causal  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The BGLM studied in this paper contains the IC  model and linear threshold (LT) model in a DAG as special  cases, and is also related to the general threshold model  (Kempe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Moreover, Feng & Chen (2023);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Xiong & Chen (2023) also study causal bandits on BGLMs  to avoid the exponentially large parameter space of general  causal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' These papers borrow some techniques and  ideas from influence maximization literature, including (Li  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2020) and (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' However, in our  BGLM CCB problem, the graph skeleton is unknown, and  we need adaptation and integration of previous techniques  together with some new ingredients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Model  We utilize capital letters (U, X, Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=') to represent variables  and their corresponding lower-case letters to indicate their  values, as was frequently done in earlier causal inference  literatures (see, for example, (Pearl, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Pearl & Macken-  zie, 2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' To express a group or a vector of variables or  values, we use boldface characters like X and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Causal Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' A causal graph G = (X ∪ {Y }, E) is a  directed acyclic graph consisting of intervenable variables  X, a special target node Y without outgoing edges, and  the set of directed edges E connecting nodes in X ∪ {Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Denote n = |X| as the number of nodes in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For sim-  plicity, in this paper we consider all variables in X ∪ {Y }  are (0, 1)-binary random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In our main text, all  the variables in X ∪ {Y } are known and their values can  be observed but the edges in E are unknown and cannot  be directly observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We refer to the in-neighbor nodes of  a node X in G as the parents of X, denoted by Pa(X),  and the values of these parent random variables as pa(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  According to the definition of causal Bayesian model (Pearl,  2009), the probability distribution P(X|Pa(X)) is used to  represent the causal relationship between X and its parents  for every conceivable value combination of Pa(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' More-  over, we define the ancestors of a node X ∈ X ∪ {Y } by  Anc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  We mainly study the Markovian causal graph G in this  paper, which means that there are no hidden variables in  G and every observed variable X has some randomness  that is not brought on by any other variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In this study,  we dedicate random variable X1 to be a special variable  that always takes the value 1 and is a parent of all other  observed random variables in order to model this effect of  the Markovian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  In this paper, we study a special causal model called bi-  nary generalized linear model (BGLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Specifically, in  BGLM, we have P(X = 1|Pa(X) = pa(X)) = fX(θ∗  X ·  pa(X)) + εX, where fX is a monotone increasing func-  tion, θ∗  X is an unknown weight vector in [0, 1]|Pa(X)|, and  εX is a zero-mean sub-Gaussian noise that ensures that the  probability does not exceed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We use the notation θ∗  X′,X  to denote the entry in vector θ∗  X that corresponds to node  X′ ∈ Pa(X), θ∗ to denote the vector of all the weights,  and Θ to denote the feasible domain for the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We  also use notation ε to represent all noise random variables  (εX)X∈X∪Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  We also study binary linear model (BLM) and linear model  in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In BLMs, all fX’s are identity functions,  so P(X = 1|Pa(X) = pa(X)) = θ∗  X · pa(X) + εX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  When we remove the noise variable εX, BLM coincides  with the linear threshold (LT) model for influence cas-  cades (Kempe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2003) in a DAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In linear models,  we remove the randomness of conditional probabilities, so  X = θ∗  X · pa(X) + εX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For a node X and one of its parent  X′, the corresponding weight are denoted as θ∗  X′,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  For the unknown causal graph, there is an important param-  eter θ∗  min = min(X′,X)∈E θ∗  X′,X, which represents the min-  imum weight gap for all edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Intuitively, this minimum  gap measures the difficulty for the algorithm to discover the  edge and its correct direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' When the gap is relatively  large, we can expect to discover the whole graph accurately  during the learning process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' When the gap is very small,  we cannot guarantee to discover the graph directly and we  must come up with another way to solve the causal bandit  problem on an inaccurate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The problem of combi-  natorial causal bandits (CCB) is first introduced in (Feng  & Chen, 2023) and describes the following setting and the  online learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The intervention can be done on no  all variables except X1 and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The action set is defined  by A ⊆ {do(S) = s}S⊆X\\{X1},s∈{0,1}|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The expected  reward under intervention on S ⊆ X\\{X1} is denoted as  E[Y |do(S = s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' A learning agent runs an algorithm π for  T rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In particular, an atomic intervention only inter-  venes one node, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' |S| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In our paper, we assume the  observation do() and atomic interventions do(X = x) are  always in our action set, because they are needed to discover  the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  The performance of the agent could be measured by the  regret of the algorithm π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The regret Rπ(T) in our context  is the difference between the cumulative reward using algo-  rithm π and the expected cumulative reward of choosing best  action S∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Here, S∗ ∈ argmaxdo(S=s)∈A E[Y |do(S)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Formally, we have  Rπ(T) = E  � T  �  t=1  (E[Y |do(S∗ = s∗)] − E[Y |do(Sπ  t = sπ  t )])  �  ,  (1)  where Sπ  t is the intervention set selected by algorithm π  in round t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The expectation is from the randomness of the  causal model and the algorithm π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  In this paper, we mainly focus on the regret minimization  problem, and we will discuss the pure exploration problem  and its sample complexity in the Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We defer the  definition of sample complexity to that section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Lower Bound on General Binary Causal  Model  In this section, we explain why we only consider BGLM  and BLM instead of the general binary causal model in  the combinatorial causal bandit setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Note that in the  general case both the number of actions and the number of  parameters of the causal model are exponentially large to  the size of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The following theorem shows that in  the general binary causal model, the regret bound must be  exponential to the size of the graph when T is sufficiently  large, or simply linear to T when T is not large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  This means that we cannot avoid the exponential factor for  the general case, and thus justify our consideration of the  BGLM and BLM settings with only a linear number of  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Theorem 1 (Binary Model Lower Bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Recall that n =  |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For any algorithm, when T ≥ 16(2n−1)  3  , there exists a  bandit instance T such that  ET [R(T)] ≥  √  2nT  8e  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Moreover, when T ≤  16(2n−1)  3  , there exists a bandit in-  stance T that  ET [R(T)] ≥ T  16e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  The lower bound contains two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The first part shows  that the asymptotic regret cannot avoid an exponential term  2n when T is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The second part states that if T is not  exponentially large, the regret will be linear at the worst  case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The proof technique of this lower bound is similar  to but not the same as previous classical bandit, because  the existence of observation do() and atomic intervention  Combinatorial Causal Bandits without Graph Skeleton  do(Xi = 1) may provide more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' To our best  knowledge, this result is the first regret lower bound on  the general causal model considering the potential role of  observation and atomic intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The result shows that  in the general binary causal model setting, it is impossible  to avoid the exponential term in the cumulative regret even  with the observations on null and atomic interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The  proof of lower bound is provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  The main idea is to consider the action set A  =  {do(), do(X = x), do(X = x)} for all node X, x ∈  {0, 1}, x ∈ {0, 1}n be the null intervention, atomic in-  terventions and actions that intervene all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The causal  graph we use is a parallel graph where all nodes in X di-  rectly points to Y with no other edges in the graph, and each  node Xi ∈ X has probability P(Xi = 1) = P(Xi = 0) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Intuitively, under this condition the null intervention  and atomic interventions can provide limited information  to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' This fact shows that observations and atomic  interventions may not be conducive to our learning process  in the worst case on the general binary causal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' BGLM CCB without Graph Skeleton but  with Minimum Weight Gap  In this section, we propose an algorithm for causal bandits  on Markovian BGLMs based on maximum likelihood es-  timation (MLE) without any prior knowledge of the graph  skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Our idea is to try to discover the causal graph structure and  then apply the recent CCB algorithm with known graph  structure (Feng & Chen, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We discover the graph struc-  ture by using atomic interventions in individual variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  However, there are a few challenges we need to face on  graph discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' First, it could be very difficult to exactly  identify all parent-child relationships, since some grand-  parent nodes may also have strong causal influence to its  grand-child nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Fortunately, we find that it is enough to  identify ancestor-descendant relationships instead of parent-  child relationships, since we can artificially add an edge  with 0 weight between each pair of ancestor and descendant  without impacting the causal propagation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Another  challenge is the minimum weight gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' When the weight of  an edge is very small, we need to perform more atomic inter-  ventions to identify its existence and its direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Hence, we  design an initialization phase with the number of rounds pos-  itively correlated to the total round number T and promise  that the ancestor-descendant relationship can always be iden-  tified correctly with a large probability when T is sufficiently  large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Following (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Feng & Chen, 2023;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xiong &  Chen, 2023), we have three assumptions:  Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For every X ∈ X ∪ {Y }, fX is twice  differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Its first and second order derivatives are  upper-bounded by L(1)  fX > 0 and L(2)  fX > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Let κ = infX∈X∪{Y },v∈[0,1]|Pa(X)|,||θ−θ∗  X||≤1 ˙fX(v · θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We have κ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' There exists a constant ζ > 0 such that for  any X ∈ X ∪ {Y } and X′ ∈ Anc(X), for any value vec-  tor v ∈ {0, 1}|Anc(X)\\{X′,X1}|, the following inequalities  hold:  Pr  ε,X,Y {X′ = 1|Anc(X) \\ {X′, X1} = v} ≥ ζ,  (2)  Pr  ε,X,Y {X′ = 0|Anc(X) \\ {X′, X1} = v} ≥ ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (3)  Assumptions 1 and 2 are the classical assumptions in gener-  alized linear model (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Assumption 3 makes  sure that each ancestor node of X has some freedom to  become 0 and 1 with a non-zero probability, even when  the values of all other ancestors of X are fixed, and it is  originally given in (Feng & Chen, 2023) with additional  justifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For BLMs and continuous linear models, we  propose an algorithm based on linear regression without the  need of this assumption in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  To discover the ancestors of all variables, we need to per-  form an extra initialization phase (see Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We  denote the total number of rounds by T and arbitrary con-  stants c0, c1 to make sure that c0T 1/2 ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In the initial-  ization phase, from X1 to Xn, we intervene each of them to  1 and 0 for c0T 1/2 times respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We denote the value  of X in the tth round by X(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For every two variables  Xi, Xj ∈ X\\{X1}, if  1  c0  √  T  c0  √  T  �  k=1  �  X(2ic0  √  T +k)  j  − X((2i+1)c0  √  T +k)  j  �  > c1T − 1  5 ,  (4)  we set Xi as an ancestor of Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Here, X(2ic0  √  T +k)  j  ’s  with k ∈ [c0  √  T] are the values of Xj in the rounds that  do(Xi = 1) is chosen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' X((2i+1)c0  √  T +k)  j  , k ∈ [c0  √  T]  are the values of Xj in the rounds that do(Xi = 0) is  chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Specifically, if Xi is not an ancestor of Xj, the  value of Xj is not impacted by intervention on Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Si-  multaneously, if Xi ∈ Pa(Xj), the value of Xj is no-  tably impacted by do(Xi) so the difference of Xj under  do(Xi = 1), do(Xi = 0) can be used as a discriminator for  the ancestor-descendant relationship between Xi and Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  This is formally shown by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Let G be a BGLM with parameter θ∗ that satis-  fies Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Recall that θ∗  min = min(X′,X)∈E θ∗  X′,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  If Xi  ∈  Pa(Xj), we have E[Xj|do(Xi  =  1)] −  E[Xj|do(Xi  = 0)] ≥ κθ∗  Xi,Xj  ≥ κθ∗  min;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' if Xi is  not an ancestor of Xj, we have E[Xj|do(Xi = 1)] =  E[Xj|do(Xi = 0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  Algorithm 1 BGLM-OFU-Unknown for BGLM CCB Prob-  lem  1: Input: Graph G = (X ∪{Y }, E), action set A, param-  eters L(1)  fX, L(2)  fX, κ, ζ in Assumption 1, 2 and 3, positive  constants c0 and c1 for initialization phase such that  c0  √  T ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  2: /* Initialization Phase: */  3: Do each intervention among do(X2 = 1), do(X2 =  0), · · · , do(Xn = 1), do(Xn = 0) for c0T 1/2 times in  order and observe the feedback (Xt, Yt), 1 ≤ t ≤ T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  4: Compute the ancestors �  Anc(X), X ∈ X ∪ {Y } by  BGLM-Ancestors((X1, Y1), · · · , (XT0, YT0), c0, c1)  (see Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  5: Initialize M0,X ← 0 ∈ R|�  Anc(X)|×|�  Anc(X)| for all  X ∈ X ∪ {Y }, δ ←  1  3n  √  T , R ← ⌈  512n(L(2)  fX )2  κ4  (n2 +  ln 1  δ )⌉, T0  ←  2(n − 1)c0T 1/2, T1  ←  T0 +  max  �  c  ζ2 ln 1  δ , (8n2−6)R  ζ  �  and ρ ← 3  κ  �  log(1/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  6: Do no intervention on BGLM G for T1 − T0 rounds  and observe feedback (Xt, Yt), T0 + 1 ≤ t ≤ T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  7: /* Iterative Phase: */  8: for t = T1 + 1, T1 + 2, · · · , T do  9:  {ˆθt−1,X, Mt−1,X}X∈X∪{Y }  =  BGLM-Estimate((X1, Y1), · · · , (Xt−1, Yt−1))  (see Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  10:  Compute the confidence ellipsoid Ct,X = {θ′  X ∈  [0, 1]|�  Anc(X)| :  ���θ′  X − ˆθt−1,X  ���  Mt−1,X  ≤ ρ} for  any node X ∈ X ∪ {Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  11:  Adopt argmaxdo(S=s)∈A,θ′  t,X∈Ct,X E[Y |do(S  =  s)] as (St, st, ˜θt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  12:  Intervene all the nodes in St to st and observe the  feedback (Xt, Yt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  13: end for  We use the above idea to implement the procedure in Algo-  rithm 2, and then put this procedure in the initial phase and  integrate this step into BGLM-OFU proposed by (Feng &  Chen, 2023), to obtain our main algorithm, BGLM-OFU-  Unknown (Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Notice that each term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (4) is a random sample of  E[Xj|do(Xi = 1)] − E[Xj|do(Xi = 0)], which means  that the left-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (4) is just an estimation of  E[Xj|do(Xi = 1)] − E[Xj|do(Xi = 0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Such expression  can be bounded by concentration inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Hence we  can prove that Algorithm 2 identifies Xi ∈ Anc(Xj) with  false positive rate and false negative rate both no more than  exp  �  − c0c2  1T 1/10  2  �  when θ∗  min ≥ 2c1κ−1T −1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Formally,  we have the following lemma that shows the probability of  correctness for Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For completeness, the proof of  Lemma 2 is put in appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Algorithm 2 BGLM-Ancestors  1: Input:  Observations ((X1, Y1), · · · , (XT0, YT0)),  positive constants c0 and c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  2: Output: �  Anc(X), ancestors of X, X ∈ X ∪ {Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  3: For all X ∈ X, �  Anc(X) = ∅, �  Anc(Y ) = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  4: for i ∈ {2, 3, · · · , n} do  5:  for j ∈ {2, 3, · · · , n}\\{i} do  6:  if �c0  √  T  k=1  �  X(2ic0  √  T +k)  j  − X((2i+1)c0  √  T +k)  j  �  >  c0c1T 3/10 then  7:  Add Xi into �  Anc(Xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  8:  end if  9:  end for  10: end for  11: Recompute the transitive closure of �  Anc(·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', if  Xi ∈ �  Anc(Xj) and Xj ∈ �  Anc(Xℓ), then add Xi to  �  Anc(Xℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Algorithm 3 BGLM-Estimate  1: Input: All observations ((X1, Y1), · · · , (Xt, Yt)) un-  til round t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  2: Output: {ˆθt,X, Mt,X}X∈X∪{Y }  3: For each X ∈ X ∪ {Y }, i ∈ [t], construct data pair  (V i,X, X(i)) with V i,X the vector of ancestors of X in  round i, and X(i) the value of X in round i if X ̸∈ Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  4: for X ∈ X ∪ {Y } do  5:  Calculate  the  maximum-likelihood  es-  timator  ˆθt,X  by  solving  the  equation  �t  i=1(X(i) − fX(V ⊺  i,XθX))V i,X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  6:  Mt,X = �t  i=1 V i,XV ⊺  i,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  7: end for  Lemma 2 (Positive Rate of BGLM-Order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Suppose As-  sumption 2 holds for the BGLM G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  In the initializa-  tion phase of Algorithm 1, Algorithm 2 finds a consistent  ancestor-descendant relationship for the BGLM G with  probability no less than 1−2  �n−1  2  �  exp  �  − c0c2  1T 1/10  2  �  when  θ∗  min ≥ 2c1κ−1T −1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  We refer to the condition θ∗  min ≥ 2c1κ−1T −1/5 in this  lemma as weight gap assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The number of initial-  ization rounds in Algorithm 1 is O(  √  T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' According to  Lemma 2, the expected regret contributed by incorrect-  ness of the ancestor-descendant relationship does not ex-  ceed O  �  T exp  �  − c0c2  1T 1/10  2  ��  = o(  √  T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Therefore, after  adding the initialization, the expected regret of BGLM-OFU-  Unknown increases by no more than o(  √  T) over BGLM-  OFU (Algorithm 1 in (Feng & Chen, 2023)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Thus we have  the following theorem to show the regret of Algorithm 2,  which is formally proved in appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  Theorem 2 (Regret Bound of BGLM-OFU-Unknown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' De-  note L(1)  max = maxX∈X∪{Y } L(1)  fX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Under Assumptions 1, 2  and 3, the regret of BGLM-OFU-Unknown (Algorithms 1, 2  and 3) is bounded as  R(T) = O  � 1  κn  3  2 L(1)  max  √  T log T  �  ,  (5)  where the terms of o(  √  T ln T) are omitted, and the big O  notation holds for T ≥ 32  �  c1  κθ∗  min  �5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Compared to (Feng & Chen, 2023), Theorem 5 has the  same asymptotic regret, The only additional assumption is  T ≥ 32 (c1/(κθ∗  min))5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Intuitively, this extra assumption  guarantees that we can discover the ancestor-descendant  relationship consistent with the true graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Our result indi-  cates that not knowing the causal graph does not provide  substantial difficulty with the weight gap assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Because Lemma 2 requires weight gap assump-  tion, in the proof of this regret bound, we only consider  the case of T ≥ 32 (c1/(κθ∗  min))5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' This limitation does not  impact the asymptotic big O notation in our regret bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  However, when the round number T is not that large, the  regret can be linear with respect to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We remove this  weight gap assumption in Section 6 for the linear model  setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The c0 and c1 are two adjustable constants in prac-  tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' When T is small, one could try a small c0 to shorten  the initialization phase, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', to make sure that T0 ≪ T, and  a small c1 to satisfy the weight gap assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' When T  is large, one could consider larger c0 and c1 for a more ac-  curate ancestor-descendant relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' However, because  θ∗  min is unknown, one cannot promise that the weight gap  assumption holds by manipulating c1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', θ∗  min may be too  small for any practical T given c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' BLM CCB without Graph Skeleton and  Weight Gap Assumption  In the previous section, we find that if T > O((θ∗  min)−5),  we can get a valid upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' However, in reality, we  have two challenges: 1) We do not know the real value  of θ∗  min, and this makes it hard to know when an edge’s  direction is identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 2) When θ∗  min → 0, it makes it  very difficult to estimate the graph accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' To solve  these challenges, we must both eliminate the dependence of  θ∗  min in our analysis, and think about how the result will be  influenced by an inaccurate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In this section, we give  a causal bandit algorithm and show that the algorithm can  always give ˜O(T 2/3) regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' This sub-linear regret result  shows that the challenge can be solved by some additional  techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  In this section, we consider a special case of BGLM called  Binary Linear Model (BLM), where fX becomes identity  Algorithm 4 BLM-LR-Unknown for BLM CCB Problem  without Weight Gap  1: Input: Graph G = (X ∪ {Y }, E), action set A, posi-  tive constants c0 and c1 for initialization phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  2: Initialize δ ←  1  n  √  T , ρt ←  �  n log(1 + tn) + 2 log 1  δ  +√n for t  =  0, 1, 2, · · · , T and T0  ←  2(n −  1)c0T 2/3 log(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  3: /* Initialization Phase: */  4: Do each intervention among do(X2 = 1), do(X2 =  0), · · · , do(Xn = 1), do(Xn = 0) for c0T 2/3 times in  order and observe the feedback (Xt, Yt) for 1 ≤ t ≤ T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  5: Compute the ancestors �  Anc(X), X ∈ X ∪ {Y } by  Nogap-BLM-Ancestors((X1, Y1), · · · , (XT0, YT0), c0  , c1) (see Algorithm 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  6: /* Iterative Phase: */  7: Initialize M0,X ← I ∈ R|�  Anc(X)|×|�  Anc(X)|, b0,X ←  0|�  Anc(X)| for all X ∈ X ∪ {Y }, ˆθ0,X  ← 0 ∈  R|�  Anc(X)| for all X ∈ X ∪ {Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  8: for t = T0 + 1, T0 + 2, · · · , T do  9:  Compute the confidence ellipsoid Ct,X = {θ′  X ∈  [0, 1]|�  Anc(X)| :  ���θ′  X − ˆθt−1,X  ���  Mt−1,X  ≤ ρt−1} for  any node X ∈ X ∪ {Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  10:  Adopt argmaxdo(S=s)⊆A,θ′  t,X∈Ct,X E[Y |do(S  =  s)] as (St, st, ˜θt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  11:  Intervene all the nodes in St to st and observe the  feedback (Xt, Yt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  12:  for X ∈ X ∪ {Y } do  13:  Construct data pair (V t,X, X(t)) with V t,X the  vector of ancestors of X in round t, and X(t) the  value of X in round t if X ̸∈ St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  14:  Mt,X = Mt−1,X + V t,XV ⊺  t,X, bt,X = bt−1,X +  X(t)V t,X, ˆθt,X = M −1  t,Xbt,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  15:  end for  16: end for  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The linear structure allows us to release the As-  sumption 1-3 (Feng & Chen, 2023) and analyze the influ-  ence of an inaccurate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  The main algorithm follows the BLM-LR algorithm in (Feng  & Chen, 2023), which uses linear regression to estimate the  weight θ∗, and the pseudocode is provided in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  We add a graph discovery process (Algorithm 5) in the  initialization phase using O(nT 2/3 log T) times rather than  O(nT 1/2) in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For any edge X′ →  X with weight θ∗  X′,X ≥ T −1/3, with probability at least  1 − 1/T 2, we expect to identify the edge’s direction within  O(nT 2/3 log(T)) samples for do(X′ = 1) and do(X′ =  0) by checking whether the difference P(X | do(X′ =  1)) − P(X | do(X′ = 0)) is large than T −1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Since  Combinatorial Causal Bandits without Graph Skeleton  Algorithm 5 Nogap-BLM-Ancestors  1: Input:  Observations ((X1, Y1), · · · , (XT0, YT0)),  positive constants c0 and c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  2: Output: For all X ∈ X ∪ {Y }, �  Anc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  3: For all X ∈ X, �  Anc(X) = ∅, �  Anc(Y ) = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  4: for i ∈ {2, 3, · · · , n} do  5:  for j ∈ {2, 3, · · · , n}\\{i} do  6:  if �c0T 2/3  k=1  �  X(c0(2i)T 2/3+k)  j  − X(c0(2i+1)T 2/3+k)  j  �  > c0c1T 1/3 log(T 2) then  7:  Add Xi into �  Anc(Xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  8:  end if  9:  end for  10: end for  11: Recompute the transitive closure of �  Anc(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  the above difference is always larger than θ∗  X′,X, after the  initialization phase, the edge X′ → X will be added to the  graph if θ∗  X′,X ≥ T −1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Moreover, if X′ is not an ancestor of X, we claim that it  cannot be a estimated as an ancestor after the initialization  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' This is because in this case P(X | do(X′ = 1)) =  P(X) = P(X | do(X′ = 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Denote the estimated graph  G′ as the graph with edge X′ → X for all X′ ∈ �  Anc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  We then have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In Algorithm 4, if the constants c0 and c1 satisfy  that c0 ≥ max{ 1  c2  1 ,  1  (1−c1)2 }, with probability at least 1 −  (n − 1)(n − 2)  1  T 1/3 , after the initialization phase we have  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='If X′ is a true parent of X in G with weight θ∗  X′,X ≥  T −1/3, the edge X′ → X will be identified and added to  the estimated graph G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='If X′ is not an ancestor of X in G, X′ → X will not be  added into G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  The properties above together provide the analytic basis for  the following observation, which plays a key role in our  further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Denote the estimated accuracy r = T −1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  We know the linear regression for X will be performed  on X and all its possible ancestoers �  Anc(X) we esti-  mated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  For the true parent node X′ in G that is not  contained in �  Anc(X), we have θ∗  X′,X ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Suppose  �  Anc(X) = {X1, X2, · · · , Xm}, and true parents which  is not contained in �  Anc(X) are Xm+1, · · · , Xm+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Thus  θ∗  Xm+i,X ≤ r for all 1 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Also, assume X1, · · · , Xt(t < m) are true parents of X in  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For Xm+i, by law of total expectation, the expectation  of X can be rewritten as  E[X | X1, · · · , Xt]  = EXm+1,··· ,Xm+k[E[X | X1, · · · , Xt, Xm+1, · · · , Xm+k]]  = EXm+1,··· ,Xm+k  �  t  �  i=1  θ∗  Xi,XXi +  m+k  �  i=m+1  θ∗  Xi,XXi  �  =  t  �  i=1  θ∗  Xi,XXi +  m+k  �  i=m+1  θ∗  Xi,XE[Xi] =  t  �  i=1  θ∗′  Xi,XXi,  where  θ∗′  Xi,X = θ∗  Xi,X,  i ≥ 2,  (6)  θ∗′  X1,X = θ∗  X1,X +  m+k  �  i=m+1  θ∗  Xi,XE[Xi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (7)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' (7) is because X1 = 1 always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then we have  |θ∗′  Xi,X − θ∗  Xi,X| ≤ �m+k  i=m+1 θ∗  Xi,X ≤ kr ≤ nr, which  shows that the difference between θ′ and θ is small if ac-  curacy r is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Let model M ′ represent the model with  graph G′ with weights θ∗′ defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The following  lemma shows the key observation:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The linear regression performed on graph G′ in  Algorithm 4 (lines 12–15) gives the estimation ˆθ′ such that  ∥( ˆθ′t,X − θ∗′  X)∥Mt,X ≤  �  n log(1 + tn) + 2 log(1/δ) + √n,  where Mt,X is defined in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  This lemma shows that, the linear regression performed  on the inaccurate estimated linear model M ′ is equivalent  to the regression for θ∗′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Note that this regression only  gives us the approximation in some direction with respect  to elliptical norm, allowing the variables to be dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Based on claim above, we only need to measure the dif-  ference for E[Y | do(S = 1)] on model M and M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The  following lemma shows that the difference between two  models can be bounded by our estimated accuracy r:  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' |EM[Y | do(S = 1)]−EM ′[Y | do(S = 1)]| ≤  n2(n + 1)r, where r is the estimated accuracy defined in  the start of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  The Lemma 5 gives us a way to bound our linear regres-  sion performance on the estimated model M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Suppose  our linear regression achieves O(  √  T) regret comparing to  maxS EM ′[Y | do(S = 1)], based on our estimated ac-  curacy r = O(T −1/3), the regret for optimization error is  O(T 2/3), which is the same order as the initialization phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Moreover, it implies that we cannot set r to a larger gap,  such as r = O(T −1/2), because it would lead to the regret  of optimization error linear to T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  From these two lemmas, we can measure the error for  initialization phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Motivated by Explore-then-Commit  Combinatorial Causal Bandits without Graph Skeleton  framework, we can achieve sublinear regret without the  weight gap assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The detailed proof is provided in  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='2 and Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' If c0 ≥ max{ 1  c2  1 ,  1  (1−c1)2 }, the regret of Algo-  rithm 4 running on BLM is upper bounded as  R(T) = O((n3T 2/3) log T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Theorem 3 states the regret of our algorithm without weight  gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The leading term of the result is O(T 2/3 log T), which  has higher order than O(  √  T log T), the regret of the pre-  vious Algorithm 2 and the BLM-LR algorithm in (Feng  & Chen, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' This degradation in regret bound can be  viewed as the cost of removing the weight gap assumption,  which makes the accurate discovery of the causal graph ex-  tremely difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' How to devise a O(  √  T log T) algorithm  without weight gap assumption is still an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Using the transformation in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='1 in (Feng & Chen,  2023), this algorithm can also work with hidden variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Pure Exploration of Causal Bandits  without Graph Structure  Another performance measure for bandit algorithms is  called sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In this setting, the agent aims  to find an action with the maximum expected reward using  as small number of rounds as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' This setting is also  called pure exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' To be more specific, the agent is  willing to find ε-optimal arm with probability at least 1 − δ  by sampling as few rounds as possible for fixed parameter ε  and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For pure exploration, we consider the general binary  causal model with only null and atomic interventions, and  study the gap-dependent bounds, meaning that the sample  complexity depends on the reward gap between the optimal  and suboptimal actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Moreover, let a∗ be one of the  optimal actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For each action a = do(Xi = x), define  µa = E[Y | a] and the gap for action a to be  ∆a =  � µa∗ − maxa∈A\\{a∗}{µa},  a = a∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  µa∗ − µa,  a ̸= a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (8)  According to the causal discovery literature (Pearl, 2009), by  passive observations alone one can obtain an essential graph  of the causal graph, with some edge directions unidentified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  We assume that the essential graph is known but the exact  graph structure is unknown, which is also considered by (Lu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2021), with additional assumptions on the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  One naive solution for this problem is to first identify the  graph structure and then performed the pure exploration  algorithm of causal bandits with known graph (Xiong &  Chen, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Define ce = |P(X | do(X′ = 1)) −  P(X′ | do(X = 0))| for each edge e = (X, X′) and  cX = mine:X→X′  1  c2e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then this naive solution admits a  sample complexity about  ˜O  ��  a∈S  1  max{∆a, ε/2}2 +  �  x∈X  1  c2  X  �  ,  (9)  where S is a particular set defined following the previous  work (Xiong & Chen, 2023) and the definition is provided  in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The first term is the sample complexity in  (Xiong & Chen, 2023), while the second term is the cost for  identifying the directions of all edges in the essential graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  This naive solution separates the causal discovery phase  and learning phase, so it cannot discover the directions  adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In the Appendix D, we propose an adaptive  algorithm to discover the edges’ directions and learn the  reward distribution in parallel, which can provide a lower  sample complexity for some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  However, when the ∆a and cX is small, both the naive al-  gorithm and our algorithms provided in Appendix D suffers  Ω( n  ε2 log(1/δ)) sample complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We claim that pure ex-  ploration for the general binary causal model is intrinsically  hard due to unknown graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' To show this, we state  a negative result for pure exploration of causal bandits on  unknown graph structure with atomic intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' It states  that even if we have all observation distribution P(X, Y ) as  prior knowledge, we still cannot achieve better sample com-  plexity result than the result in the classical pure exploration  problem for the multi-armed bandit O( n  ε2 log(1/δ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Theorem 4 (Lower bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Consider causal bandits with  only essential graph and atomic intervention, for any algo-  rithm which can output ε-optimal action with probability at  least 1 − δ, there is a bandit instance with expected sample  complexity Ω( n  ε2 log(1/δ)) even if we have all observational  distribution P(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Note that if we know distribution P(X, Y ) and the ex-  act graph structure, we can compute each intervention  P(Y | do(X = x)) by do-calculus because the absence  of hidden variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' So Theorem 4 shows the intrinsic hard-  ness provided by unknown graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The detailed  proof can be found in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Future Work  This paper is the first theoretical study on causal bandits  without the graph skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' There are many future direc-  tions to extend this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We believe that similar initial-  ization methods and proof techniques can be used to de-  sign causal bandits algorithms for other parametric models  without the skeleton, like linear structural equation models  (SEM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Moreover, how to provide a algorithm with �O(  √  T)  regret without weight gap assumption is interesting and still  open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For pure exploration, the combinatorial setting needs  more research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
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+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', and Shakkottai,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Identifying best interventions through online impor-  tance sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In International Conference on Machine  Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 3057–3066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Varici, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', Shanmugam, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', Sattigeri, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', and Tajer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Causal bandits for linear structural equation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  arXiv preprint arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='12764, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Xiong, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' and Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Combinatorial pure exploration of  causal bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In Proceedings of the 11th International  Conference on Learning Representations (ICLR), May  2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Yabe, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', Hatano, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', Sumita, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', Ito, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', Kakimura, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=',  Fukunaga, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', and Kawarabayashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Causal bandits  with propagating inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In International Conference  on Machine Learning, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 5512–5520.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' PMLR, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', Chen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', Sun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', and Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Online influ-  ence maximization with node-level feedback using stan-  dard offline oracles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In Proceedings of the AAAI Confer-  ence on Artificial Intelligence, volume 36, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 9153–9161,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Proofs for Propositions in Section 5  In this section, we give proofs that are omitted in Section 5 of our main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Proof of Lemma 1  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Let G be a BGLM with parameter θ∗ that satisfies Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Recall that θ∗  min = min(X′,X)∈E θ∗  X′,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' If  Xi ∈ Pa(Xj), we have E[Xj|do(Xi = 1)] − E[Xj|do(Xi = 0)] ≥ κθ∗  Xi,Xj ≥ κθ∗  min;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' if Xi is not an ancestor of Xj, we  have E[Xj|do(Xi = 1)] = E[Xj|do(Xi = 0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' At first, we define an equivalent threshold model form of the BGLM as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For each node X, we randomly  sample a threshold γX uniformly from [0, 1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', γX ∼ U[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then if fX(Pa(X)·θ∗  X)+εX ≥ γX, X is activated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', X  is set to 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' otherwise, X is not activated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', X is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Therefore, if we ignore ε, the BGLM model belongs to the family  of general threshold models (Kempe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For convenience, we denote the vector of all γX, X ∈ X ∪ {Y }\\{X1}  by γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The vector of fixing all entries in γ except γX is denoted by γ−X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Now we prove the first part of this lemma: E[Xj|do(Xi = 1)] − E[Xj|do(Xi = 0)] ≥ κθ∗  Xi,Xj ≥ κθ∗  min if Xi ∈ Pa(Xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  By the definition of our equivalent threshold model, we know that after fixing all the thresholds γX’s and noises εX’s, the  propagation result is completely determined merely by the intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Therefore, we have  E[Xj|do(Xi = 1)] = Eγ∈(U[0,1])n,ε[Xj|do(Xi = 1)]  = Eγ−Xj ∈(U[0,1])n−1,ε  �  Pr  γXj ∼U[0,1]  �  Xj = 1|do(Xi = 1), γ−Xj, ε  ��  ,  and  E[Xj|do(Xi = 0)] = Eγ−Xj ∈(U[0,1])n−1,ε  �  Pr  γXj ∼U[0,1]  �  Xj = 1|do(Xi = 0), γ−Xj, ε  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Hence, in order to prove E[Xj|do(Xi = 1)] − E[Xj|do(Xi = 0)] ≥ κθ∗  Xi,Xj ≥ κθ∗  min, we only need to prove  Pr  γXj ∼U[0,1]  �  Xj = 1|do(Xi = 1), γ−Xj, ε  �  −  Pr  γXj ∼U[0,1]  �  Xj = 0|do(Xi = 0), γ−Xj, ε  �  ≥ κθ∗  min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  When γ−Xj and ε are fixed, all the nodes in X ∪ {Y }\\ ({Xj} ∪ {Des(Xj)}) are already fixed given an arbitrarily fixed  intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Here, Des(Xj) is used to represent the descendants of Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Suppose under do(Xi = 1), γ−Xj and ε, the value  vector of parents of Xj is pa1(Xj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' under do(Xi = 0), γ−Xj and ε, the value vector of parents of Xj is pa0(Xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' By  induction along the topological order, nodes in X ∪ {Y }\\ ({Xj} ∪ {Des(Xj)}) that is activated under do(Xi = 0), γ−Xj  and ε must be also activated under do(Xi = 1), γ−Xj and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Therefore, entries in pa1(Xj)−pa0(Xj) are all non-negative  and the entry in pa1(Xj) − pa0(Xj) for the value of Xj is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' From this observation, we can deduce that  fXj(pa1(Xj) · θ∗  Xj) − fXj(pa0(Xj) · θ∗  Xj) ≥ κ  �  pa1(Xj) · θ∗  Xj − pa0(Xj) · θ∗  Xj  �  ≥ κθ∗  Xi,Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Hence, we have  Pr  γXj ∼U[0,1]  �  Xj = 1|do(Xi = 1), γ−Xj, ε  �  −  Pr  γXj ∼U[0,1]  �  Xj = 0|do(Xi = 0), γ−Xj, ε  �  =  Pr  γXj ∼U[0,1]  �  fXj(pa1(Xj) · θ∗  Xj) ≥ γXj + εXj|εXj  �  −  Pr  γXj ∼U[0,1]  �  fXj(pa0(Xj) · θ∗  Xj) ≥ γXj + εXj|εXj  �  =  �  fXj(pa1(Xj) · θ∗  Xj) − εXj  �  −  �  fXj(pa0(Xj) · θ∗  Xj) − εXj  �  ≥ κθ∗  Xi,Xj ≥ κθ∗  min,  which is what we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Until now, the first part of Lemma 1 has been proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  Then we prove the second part of this lemma: E[Xj|do(Xi = 1)] = E[Xj|do(Xi = 0)] if Xj is not a descendant of Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In  this situation, we know from the graph structure that (Xj ⊥⊥ Xi)G{Xi}, where G{Xi} is the graph obtained by deleting from  G all arrows pointing to Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' According to the third law of do-calculus (Pearl, 2012), we deduce that  E[Xj|do(Xi = 1)] = Pr{Xj = 1|do(Xi = 1)} = Pr{Xj = 1|} = Pr{Xj = 1|do(Xi = 0)} = E[Xj|do(Xi = 0)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Now Lemma 1 is completely proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Corollary 1 (An Extension of Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Suppose G is a BGLM with parameter θ∗ that satisfying Assumption 2 and  do(S = s) is an intervention such that Xi, Xj /∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' If Xi ∈ Pa(Xj), we have E[Xj|do(Xi = 1), do(S = s)] −  E[Xj|do(Xi = 0), do(S = s)] ≥ κθ∗  Xi,Xj ≥ κθ∗  min;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' if Xi is not an ancestor of Xj, we have E[Xj|do(Xi = 1), do(S =  s)] = E[Xj|do(Xi = 0), do(S = s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' According to (Pearl, 2012), Pr{Xj|do(Xi), do(S)} is equivalent to Pr{Xj|do(Xi)} in a new model G′ such that  all in-edges of S are deleted and all nodes in S are fixed by s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We know that Lemma 1 holds in G′, so this corollary holds  in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Proof of Lemma 2  Lemma 2 (Positive Rate of BGLM-Order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Suppose Assumption 2 holds for the BGLM G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In the initialization phase of  Algorithm 1, Algorithm 2 finds a consistent ancestor-descendant relationship for the BGLM G with probability no less than  1 − 2  �n−1  2  �  exp  �  − c0c2  1T 1/10  2  �  when θ∗  min ≥ 2c1κ−1T −1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We first assume that for every pair of nodes if Xi ∈ Pa(Xj), Algorithm 2 puts Xj as a descendant of Xi in the  ancestor-descendant relationship;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' if Xj is not a descendant of Xi, Algorithm 2 do not put Xj as an descendant of Xi in the  ancestor-descendant relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' This event is denoted by E for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We prove that when event E does occur, the  ancestor-descendant relationship we find is absolutely consistent with the true graph structure of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Otherwise, suppose there  is a mistake in the ancestor-descendant relationship such that Xi is an ancestor of Xj but not put in �  Anc(Xj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We denote a  directed path from Xi to Xj by Xi → Xk1 → Xk2 → · · · → Xkp → Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Therefore, Xk1 must be put in �  Anc(Xi), Xk2  must be put in �  Anc(Xk1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', Xj must be put in �  Anc(Xkp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In conclusion, Xj should be put in �  Anc(Xi), which is a  contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Hence, there is no mistake in the ancestor-descendant relationship given event E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Now we only prove that using Algorithm 2, with probability no less than 1 − 2  �n−1  2  �  exp  �  − c0c2  1T 1/10  2  �  , event E defined  in the paragraph above occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For a pair of nodes Xi, Xj ∈ X\\{X1}, if Xi ∈ Pa(Xj), we know from Lemma 1  that E[Xj|do(Xi = 1)] − E[Xj|do(Xi = 0)] ≥ κθ∗  min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We denote the difference between random variable Xj given  do(Xi = 1) and random variable Xj given do(Xi = 0) by Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In �c0T 1/2  k=1  �  X(2ic0T 1/2+k)  j  − X((2i+1)c0T 1/2+k)  j  �  , each  term X(2ic0T 1/2+k)  j  − X((2i+1)c0T 1/2+k)  j  is an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' sample of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We denote X(2ic0T 1/2+k)  j  − X((2i+1)c0T 1/2+k)  j  by Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  We know that Zk ∈ [−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 1] and E[Zk] ≥ κθ∗  min,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' so according to Hoeffding’s inequality (Hoeffding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 1994),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' we have  Pr  �  �  �  c0T 1/2  �  k=1  �  X(2ic0T 1/2+k)  j  − X((2i+1)c0T 1/2+k)  j  �  > c0c1T 3/10  �  �  �  = Pr  �  �  �  c0T 1/2  �  k=1  Zk > c0c1T 3/10  �  �  �  = 1 − Pr  �  �  �  c0T 1/2  �  k=1  Zk ≤ c0c1T 3/10  �  �  �  ≥ 1 − exp  �  −2  �  c0T 1/2κθ∗  min − c0c1T 3/10�2  4c0T 1/2  �  = 1 − exp  �  −c0  �  T 1/4κθ∗  min − c1T 1/20�2  2  �  ≥ 1 − exp  �  −c2  1c0T 1/10  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (because T ≥ 32  �  c1  κθ∗  min  �5  )  Similarly, if Xj is not a descendant of Xi, we do not put Xi in �  Anc(Xj) in the ancestor-descendant relationship if and  only if �c0T 1/2  k=1  �  X(2ic0T 1/2+k)  j  − X((2i+1)c0T 1/2+k)  j  �  ≤ c0c1T 3/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Now we still have Zk ∈ [−1, 1] but E[Zk] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Therefore, according to Hoeffding’s inequality (Hoeffding, 1994), we have  Pr  �  �  �  c0T 1/2  �  k=1  �  X(2ic0T 1/2+k)  j  − X((2i+1)c0T 1/2+k)  j  �  ≤ c0c1T 3/10  �  �  �  = 1 − Pr  �  �  �  c0T 1/2  �  k=1  Zk > c0c1T 3/10  �  �  �  > 1 − exp  �  −2  �  c0c1T 3/10�2  4c0T 1/2  �  = 1 − exp  �  −c2  1c0T 1/10  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Hence, by union bound (Boole’s inequality (Bonferroni, 1936)), the probability of E is no less than 1 −  2  �n−1  2  �  exp  �  − c2  1c0T 1/10  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' This is because when Xi, Xj ∈ X\\{X1}, there are 2  �n−1  2  �  possible choices of them that  are tested by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' When E happens, Algorithm 2 gets the ancestor-descendant relationship correct, so Lemma 2 is  proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Proof of Theorem 2  In the following proofs on a BGLM G, when X′ ∈ Anc(X) but X′ /∈ Pa(X), we add an edge X′ → X with weight  θX′,X = 0 into G and this does not impact the propagation results of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Let D = maxX∈X∪{Y } |Pa(X)| be the maximum  in-degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' After doing this transformation, D = n and Anc(X) = Pa(X) for all X ∈ X ∪ {Y } in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Before the proof of this theorem, we introduce several lemmas at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The first component is based on the result of  maximum-likelihood estimation (MLE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' It gives a theoretical measurement for the accuracy of estimated ˆθ computed by  MLE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' One who is interested could find the proof of this lemma in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='2 of (Feng & Chen, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Lemma 6 (Lemma 1 in (Feng & Chen, 2023)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Suppose that Assumptions 1 and 2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Moreover, given δ ∈ (0, 1), assume  that  λmin(Mt,X) ≥  512|Pa(X)|  �  L(2)  fX  �2  κ4  �  |Pa(X)|2 + ln 1  δ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (10)  Then with probability at least 1 − 3δ, the maximum-likelihood estimator satisfies , for any v ∈ R|Pa(X)|,  ���v⊺(ˆθt,X − θ∗  X)  ��� ≤ 3  κ  �  log(1/δ) ∥v∥M −1  t,X ,  Combinatorial Causal Bandits without Graph Skeleton  where the probability is taken from the randomness of all data collected from round 1 to round t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  The second component is called the group observation modulated (GOM) bounded smoothness property (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' It  shows that a small change in parameters θ leads to a small change in the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Under our BGLM setting, this lemma is  proved in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='3 of (Feng & Chen, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Lemma 7 (Lemma 2 in (Feng & Chen, 2023)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For any two weight vectors θ1, θ2 ∈ Θ for a BGLM G, the difference of  their expected reward for any intervened set S can be bounded as  ��σ(S, θ1) − σ(S, θ2)  �� ≤ Eε,γ  �  �  �  X∈XS,Y  ��V ⊺  X(θ1  X − θ2  X)  �� L(1)  fX  �  � ,  (11)  where XS,Y is the set of nodes in paths from S to Y excluding S, and V X is the propagation result of the parents of X  under parameter θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The expectation is taken over the randomness of the thresholds γ and the noises ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Thirdly, we propose a lemma in order to bound the sum of ∥V t,X∥M −1  t−1,X at first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' This lemma is proved in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='4 of  (Feng & Chen, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Lemma 8 (Lemma 9 in (Feng & Chen, 2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Let {W t}∞  t=1 be a sequence in Rd satisfying ∥W t∥ ≤  √  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Define W 0 = 0  and Mt = �t  i=0 W iW ⊺  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Suppose there is an integer t1 such that λmin(Mt1+1) ≥ 1, then for all t2 > 0,  t1+t2  �  t=t1  ∥W t∥M −1  t−1 ≤  �  2t2d log(t2d + t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  At last, in order to show that λmin(MT1,X) ≥ R after the initialization phase of Algorithm 1 and thus satisfy the condition  of Lemma 6, we introduce Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' This lemma is improved upon Lemma 7 in (Feng & Chen, 2022) and enables us to use  Lecu´e and Mendelson’s inequality (Nie, 2022) in our later theoretical regret analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Let Sphere(d) denote the sphere of the d-dimensional unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For any v = (v1, v2, · · · , v|Pa(X)|) ∈ Sphere(|Pa(X)|) and any X ∈ X ∪ {Y } in a BGLM that satisfies  Assumption 3, we have  Pr  ε,X,Y  �  |Pa(X) · v| ≥  1  √  4D2 − 3  �  ≥ ζ,  where Pa(X) is the random vector generated by the natural Bayesian propagation in BGLM G with no interventions  (except for setting X1 to 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The lemma is similarly proved as Lemma 7 in (Feng & Chen, 2022) using the idea of Pigeonhole principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Let  Pa(X) = (Xi1 = X1, Xi2, Xi3, · · · , Xi|Pa(X)|) as the random vector and pa(X) = (x1 = 1, xi1, xi2, xi3, · · · , xi|Pa(X)|)  as a possible valuation of Pa(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Without loss of generality, we suppose that |v2| ≥ |v3| ≥ · · · ≥  ��v|Pa(X)|  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For  simplicity, we denote D0 =  √  D − 1 +  1  2√D−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' If |v1| ≥  D0  √  D2  0+1, we can deduce that  |pa(X) · v| ≥ |v1| − |v2| − |v3| − · · · −  ��v|Pa(X)|  ��  ≥  D0  �  D2  0 + 1  −  �  (D − 1)  �  |v2|2 + |v3|2 + · · ·  ��v|Pa(X)|  ��2�  (12)  ≥  D0  �  D2  0 + 1  −  �  (D − 1)  �  1 −  D2  0  D2  0 + 1  �  (13)  =  1  2  �  (D2  0 + 1)(D − 1)  =  1  √  4D2 − 3  ,  where Inequality (12) is by the Cauchy-Schwarz inequality and the fact that |Pa(X)| ≤ D, and Inequality (13) uses the  fact that v ∈ Sphere(|Pa(X)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Thus, when |v1| ≥  D0  √  D2  0+1, the event |Pa(X) · v| ≥  1  √  4D2−3 holds deterministically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  Otherwise, when |v1| <  D0  √  D2  0+1, we use the fact that |v2| is the largest among |v2|, |v3|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' and deduce that  |v2| ≥  1  √n − 1  �  |v2|2 + |v3|2 + · · · ≥  �  1 −  �  D0  √  D2  0+1  �2  √n − 1  =  2  √  4D2 − 3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (14)  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' using the fact that  Pr  ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='Y {Xi1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi2 = xi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi3 = xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · } =  Pr  ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='Y {Xi2 = xi2|Xi1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi3 = xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · } ·  Pr  ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='Y {(Xi1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi3 = xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · } ≥ ζ  Pr  ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='Y {Xi1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi3 = xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · }  and �  xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='··· Prε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='Y {Xi1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi3 = xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · } = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' we have  Pr  ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='Y  �  |Pa(X) · v| ≥  1  √  4D2 − 3  �  =  �  xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='···  Pr{Xi1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi2 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi3 = xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · } · I  �  |(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · ) · (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' v3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · )| ≥  1  √  4D2 − 3  �  +  �  xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='···  Pr{Xi1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi2 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi3 = xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · } · I  �  |(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · ) · (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' v3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · )| ≥  1  √  4D2 − 3  �  ≥  �  xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='···  ζ Pr{Xi1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi3 = xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi4 = xi4 · · · } · I  �  |(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · ) · (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' v3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · )| ≥  1  √  4D2 − 3  �  +  �  xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='···  ζ Pr{Xi1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi3 = xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi4 = xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · } · I  �  |(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · ) · (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' v3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · )| ≥  1  √  4D2 − 3  �  = ζ ·  �  xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='···  Pr{Xi1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi3 = xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi4 = xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · }  �  I  �  |(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · ) · (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' v3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · )| ≥  1  √  4D2 − 3  �  +I  �  |(1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · ) · (v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' v3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · )| ≥  1  √  4D2 − 3  ��  ≥ ζ  �  xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='···  Pr{Xi1 = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi3 = xi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xi4 = xi4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' · · · }  (15)  = ζ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  which is exactly what we want to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Inequality (15) holds because otherwise, at least for some xi3, xi4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', both  indicators on the left-hand side of the inequality have to be 0, which implies that  |(1, 1, xi3, xi4, · · · ) · (v1, v2, v3, · · · ) − (1, 0, xi3, xi4, · · · ) · (v1, v2, v3, · · · )| = |v2| <  2  √  4D2 − 3  ,  (16)  but this contradicts to Inequality (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Having these four lemmas above together with Lemma 2 proved in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='2, we are finally able to prove the regret  bound of BGLM-OFU-Unknown algorithm (Theorem 2) as below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Theorem 2 (Regret Bound of BGLM-OFU-Unknown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Denote L(1)  max = maxX∈X∪{Y } L(1)  fX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Under Assumptions 1, 2 and  3, the regret of BGLM-OFU-Unknown (Algorithms 1, 2 and 3) is bounded as  R(T) = O  � 1  κn  3  2 L(1)  max  √  T log T  �  ,  (5)  where the terms of o(  √  T ln T) are omitted, and the big O notation holds for T ≥ 32  �  c1  κθ∗  min  �5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We only consider the case of T ≥ 32  �  c1  κθ∗  min  �5  in this proof because the big O notation is asymptotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Let Ht be the history of the first t rounds and Rt be the regret in the tth round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Because the reward node Y is in interval  [0, 1], we can deduce that for any t ≤ T1, Rt ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Now we consider the case of t > T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' According to Lemma 2, with  probability at least 1 − 2  �n−1  2  �  exp  �  − c0c2  1T 1/10  2  �  , Algorithm 2 returns a correct ancestor-descendant relationship, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=',  �  Anc(X) = Anc(X) for X ∈ X ∪ {Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Next we bound the regret conditioned on the correct ancestor-descendant  relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' When t > T1, we have  E[Rt|Ht−1] = E[σ(Sopt, θ∗) − σ(St, θ∗)|Ht−1],  (17)  where the expectation is taken over the randomness of St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Then for T1  <  t  ≤  T, we define ξt−1,X for  X  ∈ X ∪ {Y } as ξt−1,X  =  ����vT (ˆθt−1,X − θ∗  X)  ��� ≤ ρ · ∥v∥M −1  t−1,X , ∀v ∈ R|Pa(X)|�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  According to the defini-  tion of Algorithm 1, we can deduce that λmin(Mt−1,X) ≥ λmin(MT1,X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  By Lecu´e and Mendelson’s inequal-  ity (Nie, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Feng & Chen, 2022) (conditions of this inequality satisfied according to Lemma 9), we have  Pr {λmin(MT1,X) < R} ≤ Pr {λmin(MT1,X − MT0,X) < R} ≤ exp  �  − (T1−T0)ζ2  c  �  where c, ζ are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then  we can define ξt−1 = ∧X∈X∪{Y }ξt−1,X and let ξt−1 be its complement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  By Lemma 6, we have Pr  �  ξt−1  �  ≤  �  3δ + exp  �  − (T1−T0)ζ2  c  �  + 3δ exp  �  − (T1−T0)ζ2  c  ��  n ≜ perror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Because under ξt−1, for any X ∈ X ∪ {Y } and v ∈ R|Pa(X)|, we have  ���vT (ˆθt−1,X − θ∗  X)  ��� ≤ ρ · ∥v∥M −1  t−1,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Therefore,  by the definition of ˜θt, we have σ(St, ˜θt) ≥ σ(Sopt, θ∗) because θ∗ is in our confidence ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Hence,  E[Rt] ≤ Pr {ξt−1} · E[σ(Sopt, θ∗) − σ(St, θ∗)] + Pr(ξt−1)  ≤ E[σ(Sopt, θ∗) − σ(St, θ∗)] + perror  ≤ E[σ(St, ˜θt) − σ(St, θ∗)] + perror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Then we need to bound σ(St, ˜θt) − σ(St, θ∗) carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Therefore, according to Lemma 6 and Lemma 7, we can deduce that  E[Rt] ≤ E  �  �  �  X∈XSt,Y  ���V t,X(˜θt,X − θ∗  X)  ��� L(1)  fX  �  � + perror  ≤ E  �  �  �  X∈XSt,Y  ∥V t,X∥M −1  t−1,X  ���˜θt,X − θ∗  X  ���  Mt−1,X  L(1)  fX  �  � + perror  ≤ 2ρ · E  �  �  �  X∈XSt,Y  ∥V t,X∥M −1  t−1,X L(1)  fX  �  � + perror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  The last inequality holds because  ���˜θt,X − θ∗  X  ���  Mt−1,X  ≤  ���˜θt,X − ˆθt−1,X  ���  Mt−1,X  +  ���ˆθt−1,X − θ∗  X  ���  Mt−1,X  ≤ 2ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Therefore, conditioned on the correct ancestor-descendant relationship, the total regret can be bounded as  R(T) ≤ 2ρ · E  �  �  T  �  t=T0+1  �  X∈XSt,Y  ∥V t,X∥M −1  t−1,X L(1)  fX  �  � + perror(T − T1) + T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  For convenience, we define W t,X as a vector such that if X ∈ St, W t,X = 0|Pa(X)|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' if X ̸∈ St, W t,X = V t,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Using  Combinatorial Causal Bandits without Graph Skeleton  Lemma 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' we can get the result:  R(T) ≤  �  �2ρE  �  �  T  �  t=T0+1  �  X∈XSt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='Y  ∥V t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='X∥M −1  t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='X L(1)  fX  �  � + perror(T − T1) + T1  �  �  �  1 − 2  �n − 1  2  �  exp  �  −c0c2  1T 1/10  2  ��  + 2  �n − 1  2  �  exp  �  −c0c2  1T 1/10  2  �  T  ≤ 2ρE  �  �  T  �  t=T0+1  �  X∈X∪{Y }  ∥W t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='X∥M −1  t−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='X L(1)  fX  �  � + perror(T − T1) + T1 + 2  �n − 1  2  �  exp  �  −c0c2  1T 1/10  2  �  T  ≤ 2ρ ·  max  X∈X∪{Y }  �  L(1)  fX  �  E  �  �  �  X∈X∪{Y }  �  2(T − T0)|Pa(X)| log ((T − T0)|Pa(X)| + T0)  �  �  + perror(T − T1) + T1 + 2  �n − 1  2  �  exp  �  −c0c2  1T 1/10  2  �  T  = O  � 1  κn  3  2 √  TL(1)  max ln T  �  = ˜O  � 1  κn  3  2 √  TL(1)  max  �  because ρ = 3  κ  �  log(1/δ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' exp  �  − c0c2  1T 1/10  2  �  T = o(  √  T) and perrorT = o(  √  T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' A BLM CCB Algorithm with Safety Gap Based on Linear Regression  As BLM is a special case of BGLM, the initialization phase in BGLM-OFU-Unknown to determine the ancestor-descendant  relationship can also be used on BLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Feng & Chen (2023) propose a CCB algorithm for BLMs using linear regression  instead of MLE to remove the requirement of Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Furthermore, BLM takes the identity function as fX’s, so  Assumptions 1 and 2 is neither required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The specific algorithm BLM-LR-Unknown-SG (BLM-LR-Unknown algorithm  with safety gap) is demonstrated in Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  The following theorem shows the regret bound of BLM-LR-Unknown-SG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' It is not surprising that this algorithm could also  work on linear models with continuous variables as Appendix F in (Feng & Chen, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The dominant term in the expected  regret does not increase compared to BLM-LR in (Feng & Chen, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Theorem 5 (Regret Bound of BLM-LR-Unknown-SG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The regret of BLM-LR-Unknown-SG running on BLM or linear  model is bounded as  R(T) = O  �  n  5  2 √  T log T  �  ,  where the terms of o(  √  T ln T) are omitted, and the big O notation holds for T ≥ 32  �  c1  κθ∗  min  �5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In the following proof on G, when X′ ∈ Anc(X) but X′ /∈ Pa(X), we add an edge X′ → X with weight  θX′,X = 0 into G and this does not impact the propagation results of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' After doing this transformation, D = n and  Anc(X) = Pa(X) for all X ∈ X ∪ {Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  According to Lemma 2, with probability at least 1 − 2  �n−1  2  �  exp  �  − c0c2  1T 1/10  2  �  , Algorithm 2 returns a correct ancestor-  descendant relationship, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', Anc(X) = �  Anc(X) for X ∈ X ∪ {Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Moreover, by Lemma 11 in (Feng & Chen,  2022), with probability at most nδ, event  �  ∃T0 < t ≤ T, x ∈ X ∪ {Y } :  ���θ∗′  X − ˆθt,X  ��� > ρt  �  occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Now we bound  the expected regret conditioned on the absence of this event and finding a correct ancestor-descendant relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For  Combinatorial Causal Bandits without Graph Skeleton  Algorithm 6 BLM-LR-Unknown-SG for BLM and Linear Model CCB Problem  1: Input: Graph G = (X ∪ {Y }, E), action set A, positive constants c0 and c1 for initialization phase such that  c0  √  T ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  2: /* Initialization Phase: */  3: Do each intervention among do(X2 = 1), do(X2 = 0), · · · , do(Xn = 1), do(Xn = 0) for c0T 1/2 times in order and  observe the feedback (Xt, Yt) for 1 ≤ t ≤ T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  4: Determine  a  feasible  ancestor-descendant  relationship  �  Anc(X)’s  for  X  ∈  X  ∪  {Y }  by  BGLM-Ancestors((X1, Y1), · · · , (XT0, YT0), c1) (see Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  5: Initialize M0,X ← I ∈ R|�  Anc(X)|×|�  Anc(X)|, b0,X ← 0|�  Anc(X)| for all X ∈ X ∪ {Y }, ˆθ0,X ← 0 ∈ R|�  Anc(X)| for all  X ∈ X ∪ {Y }, δ ←  1  n  √  T , ρt ←  �  n log(1 + tn) + 2 log 1  δ + √n for t = 0, 1, 2, · · · , T and T0 ← 2(n − 1)c0T 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  6: /* Iterative Phase: */  7: for t = T0 + 1, T0 + 2, · · · , T do  8:  Compute the confidence ellipsoid Ct,X = {θ′  X ∈ [0, 1]|�  Anc(X)| :  ���θ′  X − ˆθt−1,X  ���  Mt−1,X  ≤ ρt−1} for any node  X ∈ X ∪ {Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  9:  (St, st, ˜θt) = argmaxdo(S=s)∈A,θ′  t,X∈Ct,X E[Y |do(S = s)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  10:  Intervene all the nodes in St to st and observe the feedback (Xt, Yt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  11:  for X ∈ X ∪ {Y } do  12:  Construct data pair (V t,X, X(t)) with V t,X the vector of ancestors of X in round t, and X(t) the value of X in  round t if X ̸∈ St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  13:  Mt,X = Mt−1,X + V t,XV ⊺  t,X, bt,X = bt−1,X + X(t)V t,X, ˆθt,X = M −1  t,Xbt,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  14:  end for  15: end for  T0 < t ≤ T, according to Theorem 1 in (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2020) and Theorem 7, we can deduce that  E [Rt] = E  �  σ′(Sopt, θ∗′) − σ′(St, θ∗′)  �  ≤ E  �  σ′(St, ˜θt) − σ′(St, θ∗′)  �  ≤ E  �  �  �  X∈XSt,Y  ���V ⊺  t,X(˜θt,X − θ∗′  X)  ���  �  �  ≤ E  �  �  �  X∈XSt,Y  ∥V t,X∥M −1  t−1,X  ���˜θt,X − θ∗′  X  ���  Mt−1,X  �  �  ≤ E  �  �  �  X∈XSt,Y  2ρt−1 ∥V t,X∥M −1  t−1,X  �  � ,  since ˜θt,X, θ∗  X are both in the confidence set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Thus, we have  R(T) = E  � T  �  t=1  Rt  �  ≤ E  �  T  �  t=T0+1  Rt  �  + T0  ≤ 2ρT · E  �  �  T  �  t=T0+1  �  X∈XSt,Y  ∥V t,X∥M −1  t−1,X  �  � + T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  For convenience, we define W t,X as a vector such that if X ∈ St, W t,X = 0|Pa(X)|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' if X ̸∈ St, W t,X = V t,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  According to Cauchy-Schwarz inequality, we have  R(T) ≤ 2ρT · E  �  �  T  �  t=T0+1  �  X∈X∪{Y }  ∥W t,X∥M −1  t−1,X  �  � + T0  ≤ 2ρT · E  �  �√  T ·  �  X∈X∪{Y }  �  �  �  �  T  �  t=T0+1  ∥W t,X∥2  M −1  t−1,X  �  � + T0  ≤ 2ρT · E  �  �√  T ·  �  X∈X∪{Y }  �  �  �  �  T  �  t=1  ∥W t,X∥2  M −1  t−1,X  �  � + 2(n − 1)c0T 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Note that Mt,X = Mt−1,X + W t,XW ⊺  t,X and therefore, det (Mt,X) = det(Mt−1,X)  �  1 + ∥W t,X∥2  M −1  t−1,X  �  , we have  T  �  t=1  ∥W t,X∥2  M −1  t−1,X ≤  T  �  t=1  n  log(n + 1) · log  �  1 + ∥W t,X∥2  M −1  t−1,X  �  ≤  n  log(n + 1) · log det(MT,X)  det(I)  ≤ n|Pa(X)|  log(n + 1) · log tr(MT,X)  |Pa(X)|  ≤ n|Pa(X)|  log(n + 1) · log  �  1 +  T  �  t=1  ∥W t,X∥2  2  |Pa(X)|  �  ≤  nD  log(n + 1) log(1 + T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Therefore, the final conditional regret R(T) is bounded by  R(T) ≤ 2ρT n  �  T  nD  log(n + 1) log(1 + T) + 2(n − 1)c0T 1/2,  because ρT =  �  D log(1 + TD) + 2 log 1  δ +  √  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' When  �  ∃t ∈ (T0, T], x ∈ X ∪ {Y } :  ���θ∗′  X − ˆθt,X  ��� > ρt  �  does occur  or Algorithm 2 finds an incorrect order, the regret is no more than T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Therefore, the total regret is no more than  �  2ρT n  �  T  nD  log(n + 1) log(1 + T) + 2(n − 1)c0T 1/2  � �  1 − nδ − 2  �n − 1  2  �  exp  �  −c0c2  1T 1/10  2  ��  + T  �  nδ + 2  �n − 1  2  �  exp  �  −c0c2  1T 1/10  2  ��  ≤ 2ρT n  �  T  nD  log(n + 1) log(1 + T) + o(  √  T ln T)  = O  �  n  5  2 √  T log T  �  ,  which is exactly what we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Replacing Lemma 11 in (Feng & Chen, 2022) by Lemma 12 in (Feng & Chen, 2022), the above proof for BLMs is still  feasible for the regret on linear models without any other modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' According to the transformation in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='1 of (Feng & Chen, 2023), this algorithm also works for some  BLMs with hidden variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Using that transformation, running BLM-LR-Unknown-SG on G is equivalent to running on  a Markovian BLM or linear model G′, where parameter θ∗ is also transformed to a new set of parameters θ∗′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Here, we  disallow the graph structure where a hidden node has two paths to Xi and Xi’s descendant Xj and the paths contain only  hidden nodes except the end points Xi and Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Proofs for Propositions in Section 6  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Proof of Lemma 3  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In Algorithm 4, if the constants c0 and c1 satisfy that c0 ≥ max{ 1  c2  1 ,  1  (1−c1)2 }, with probability at least  1 − (n − 1)(n − 2)  1  T 1/3 , after the initialization phase we have  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='If X′ is a true parent of X in G with weight θ∗  X′,X ≥ T −1/3, the edge X′ → X will be identified and added to the  estimated graph G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='If X′ is not an ancestor of X in G, X′ → X will not be added into G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' First, for each node Xj and its parent Xi with weight θ∗  Xi,Xj ≥ T −1/3, by Lemma 1, we can have  E[Xj | do(Xi = 1)] − E[Xj | do(Xi = 0)] ≥ θ∗  Xi,Xj  Then each element X(c0(2i)T 2/3+k)  j  − X(c0(2i+1)T 2/3+k)  j  is an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='d sample of Z = Xj |do(Xi=1) −Xj |do(Xi=0) with  E[Z] ≥ θ∗  Xi,Xj ≥ T −1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' By the Hoeffding’s inequality, if we choose c1 < 1 and c0(1 − c1)2 > 1  3, we have  Pr  �  �  �  c0T 2/3  �  k=1  �  X(c0(2i)T 2/3+k)  j  − X(c0(2i+1)T 2/3+k)  j  �  > c0c1T 1/3 log(T 2)  �  �  �  ≥ 1 − exp  �  −2 log(T 2)  �  c0T 2/3E[Z] − c0c1T 1/3�2  4c0T 2/3  �  ≥ 1 − exp  �  −2 log(T 2)  �  c0T 1/3 − c0c1T 1/3�2  4c0T 2/3  �  ≥ 1 − exp  �  −c0(1 − c1)2 log(T 2)  2  �  ≥ 1 − T −c0(1−c1)2  ≥ 1 − 1  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Taking the union bound for all X and X′, with probability at least 1 −  �n−1  2  � 1  T 2 , the edge X′ → X with θ∗  X′,X will be  identified and added to the estimated graph G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Also, assume Xi is not an ancestor of Xj, then  E[Xj | do(Xi = 1)] − E[Xi | do(Xi = 0)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Thus the element X(c0(2i)T 2/3+k)  j  − X(c0(2i+1)T 2/3+k)  j  is an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='d sample of Z′ = Xj |do(Xi=1) −Xj |do(Xi=0) with  E[Z′] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Thus by Hoeffding’s inequality,  Pr  �  �  �  c0T 2/3  �  k=1  �  X(c0(2i)T 2/3+k)  j  − X(c0(2i+1)T 2/3+k)  j  �  > c0c1T 1/3 log(T 2)  �  �  �  ≤ exp  �  −2 log(T 2)  �  c0T 2/3E[Z] − c0c1T 1/3�2  4c0T 2/3  �  ≤ exp  �  −c0c2  1 log T  �  ≤ T −c0c2  1  ≤ 1  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  and then with probability at least 1 −  �n−1  2  � 1  T , we will not add the edge X′ → X in the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Combining these two  facts, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Proof of Lemma 4  For each node X, consider the estimated possible parent Pa′(X), then our observation V t,X ∈ {0, 1}Pa′(X) are the values  of Pa′(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Since we have θ′ that  E[Xt | V t,X] = θT  t,XV t,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (18)  Thus applying Lemma 1 in (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=', 2020), we can have  |θ′  X − θ′  t,X|Mt,X ≤  �  n log(1 + tn) + 2 log(1/δ) + √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (19)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Proof of Lemma 5  Note that M represents the model with true graph G and true weights θ, and M ′ represents the model with estimated graph  G′ and estimated weights M ′, then difference  |θ′  Xi,X − θXi,X| ≤ nr  (20)  Now we construct a auxillary model M ′′, which has graph G′ and weights θ on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The parent of X in model M Pa′′(X) is  equivalent to Pa′(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then we prove the following two claims:  Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' |EM[Y | do(S = 1)] − EM ′′[Y | do(S = 1)]| ≤ n2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Let the topological order be X1, X2, · · · , Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' First, EM[X1 | do(S)] − EM ′′[X1 | do(S)] = 0 ≤ nr because  X1 is always 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Assume Xq+1 /∈ S EM[Xi | do(S)] − EM ′′[Xi | do(S)] ≤ qnr for all i ≤ q, then if Xq+1 ∈ S,  EM[Xq+1 | do(S)] − EM ′′[Xq+1 | do(S)] = 0 ≤ (q + 1)nr holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Thus now we assume Xq+1 /∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  EM[Xq+1 | do(S)] − EM ′′[Xq+1 | do(S)]  = EM  �  �  �  Xi∈Pa(Xq+1)  θXi,Xq+1Xi  �����do(S)  �  � − EM ′′  �  �  �  Xi∈Pa′′(Xq+1)  θXi,Xq+1Xi  �����do(S)  �  �  =  �  Xi∈Pa′′(Xq+1)  θXi,Xq+1(EM[Xi | do(S)] − EM ′′[Xi | do(S)])+  �  Xi∈Pa(Xq+1)\\Pa′′(Xq+1)  θXi,Xq+1EM[Xi | do(S)]  ≤  �  Xi∈Pa(Xq+1)  θXi,Xq+1qnr + rn  ≤ (q + 1)nr  where the first equality follows the definition of linear model, the second equality is because θ′  X′,X = 0 if X′ is not a true  parent of X in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The third inequality is derived by induction, and the last inequality is because ∥θX′,Xq+1∥1 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' |EM ′[Y | do(S = 1)] − EM ′′[Y | do(S = 1)]| ≤ n3r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' First, EM[X1 | do(S)] − EM ′′[X1 | do(S)] = 0 ≤ n2r Then similarly, assume EM[Xi | do(S)] − EM ′′[Xi |  Combinatorial Causal Bandits without Graph Skeleton  do(S)] ≤ qn2r for all i ≤ q and Xq+1 /∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then  EM ′[Xq+1 | do(S)] − EM ′′[Xq+1 | do(S)]  = EM ′  �  �  �  Xi∈Pa′(Xq+1)  θ′  Xi,Xq+1Xi  �����do(S)  �  � − EM ′′  �  �  �  Xi∈Pa′′(Xq+1)  θXi,Xq+1Xi  �����do(S)  �  �  =  �  Xi∈Pa′′(Xq+1)  θ′  Xi,Xq+1EM ′[Xi | do(S)] − θXi,Xq+1EM ′′[Xi | do(S)]  =  �  Xi∈Pa′′(Xq+1)  (θ′  Xi,Xq+1 − θXi,Xq+1)EM ′[Xi | do(S)]+  �  Xi∈Pa′′(Xq+1)  θXi,Xq+1(EM ′[Xi | do(S)] − EM ′′[Xi | do(S)])  = n2r + n2qr  ≤ (q + 1)n2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  where the first equality follows the definition, the second equality is because Pa′(X) = Pa′′(X) for any node X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The  fourth inequality derived from induction , inequality (20) and Xi ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' By induction, we complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Now we prove the Lemma 5:  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Combining Claim 1 and Claim 2, we have  EM[Y | do(S)] − EM ′[Y | do(S)] ≤ n2(n + 1)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (21)  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Proof of Theorem 3  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Denote the original model and estimated model as M and M ′ The initialization phase will lead to regret at most  T0 = 16(n − 1)T 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' At Iterative phase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' denote the optimal action to be do(S∗ = 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' by Lemma 3 and the guarantee of  BLM-LR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' with probability at least 1 − (n − 1)(n − 2) 1  T  T  �  t=1  EM[Y | do(S∗ = 1)] − EM[Y | do(St = 1)]  =  T  �  t=1  ((EM[Y | do(S∗ = 1)] − EM ′[Y | do(S∗ = 1)]) + (EM ′[Y | do(S∗ = 1)] − EM ′[Y | do(St = 1)]))  ≤ T0 +  T  �  t=T0+1  n2(n + 1)T −1/3 +  T  �  t=T0+1  (EM ′[Y | do(S∗ = 1)] − EM ′[Y | do(St = 1)])  ≤ T0 + n2(n + 1)T 2/3 + cn2�  nT log T  = O((n3T 2/3 + n3√  T) log T)  = O(n3T 2/3 log T),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  where the first inequality is derived from Lemma 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' and the second inequality is the guarantee of BLM-LR in Theorem 3 of  (Feng & Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Thus the total regret will be bounded by  R(T) ≤ (n − 1)(n − 2)  T  T + O((n3T 2/3) log T)  = O((n3T 2/3) log T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  The first inequality is because our regret have an upper bound T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Proof of Theorem 1  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Consider the causal bandit instances Ti with parallel graph (E = {Xi → Y, 1 ≤ i ≤ n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=') and A = {do(), do(X =  x), do(X = x)} for all node X, x ∈ {0, 1}, x ∈ {0, 1}n be all observation, atomic intervention and actions that intervene  all nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  For T1, we assume Xi are independent with each other and P(Xi = 1) = P(Xi = 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Define  P(Y = 1) =  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + ∆  if X1 = X2 = · · · = Xn = 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5  otherwise  Then for Ti, 2 ≤ i ≤ 2n, consider the binary representation of i − 1 as  ¯  b1b2 · · · bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then assume Xi are independent with  each other and P(Xi = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5, and define  P(Y = 1) =  �  �  �  �  �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + ∆  if X1 = X2 = · · · = Xn = 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + 2∆  if Xj = bj for all 1 ≤ j ≤ n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5  otherwise  Now in Ti, do  �  X = b1b2 · · · bn  �  is the best action, and other actions will lead to at least ∆ regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Denote Ta(t) for action a ∈ A as the number of times taking a until time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' To simplify the notation, we denote ai as  do(X = x), where x is the binary representatino of i − 1, {b1, b2, · · · , bn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then for instances T1 and Ti, we have  ET1[R(t)] ≥ PT1(Ta1(t) ≤ t/2)t∆  2 ,  ETi[R(t)] ≥ PTi(Ta1(t) > t/2)t∆  2  Thus  ET1[R(t)] + ETi[R(t)] > t∆  2 (PT1(Ta1(t) ≤ t/2) + PTi(Ta1(t) > t/2))  ≥ t∆  4 exp (−KL(PT1, PTi))  Now we need to bound KL(PTi, PT1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  KL(PT1, PTi) ≤  �  a∈A  ET1[Ta(t)]KL(PT1(X, Y | a)∥PTi(X, Y | a))  (22)  =  �  a∈A  ET1[Ta(t)]KL(PT1(Y | a)∥PTi(Y | a))  (23)  ≤ ET1[Tai(t)] · KL(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + 2∆) +  �  a=do(Xi=x),do()  ET1[Ta(t)] · KL(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5∥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 +  ∆  2n−2 )  (24)  ≤ ET1[Tai(n)] · 2∆2 + t ·  ∆2  22n−3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (25)  where (24) is because for a = do(Xi = x) or a = do(), P(Y | do(a)) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 in T1, and P(Y | do(a)) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 +  2∆  2n−1 =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 +  ∆  2n−2 in Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Now we choose  i = argmin  j>1  ET1[Taj(t)],  (26)  then we have  ET1[Ta1(t)] ≤  T  2n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (27)  Combinatorial Causal Bandits without Graph Skeleton  Then by (25), choosing ∆ =  �  2n−1  3t , we have  KL(PT1, PTi) ≤ 2t∆2  2n − 1 + t∆2  22n−3 ≤ t∆2 ·  3  2n − 1 = 1  (28)  Thus  ET1[R(t)] + ETi[R(t)] ≥ t∆  4 exp (−KL(PT1, PTi))  ≥ t∆  4e  ≥  �  (2n − 1)t  4  √  3e  ≥  √  2nt  8e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Then max{ET1[R(t)], ETi[R(t)]} ≥  √  2nt  16e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We complete the proof when t ≥ 16(2n−1)  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Now suppose t ≤ 16(2n−1)  3  , choose ∆ = 1  4, then based on (25) and (27), we have  KL(PT1, PTi) ≤  t  8(2n − 1) +  t  22n+1  ≤ 2  3 + 16  3 · 2n − 1  22n+1  ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Then we have  ET1[R(t)] + ETi[R(t)] ≥ t∆  4 exp (−KL(PT1, PTi))  ≥ t∆  4e  ≥  t  16e,  and max{ET1[R(t)], ETi[R(t)]} ≥  t  32e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' General Causal Bandits without Graph Structure  In this section, we only consider the atomic intervention, and provide an algorithm to solve causal bandits with the graph  skeleton on binary model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We only consider the atomic intervention setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' An atomic intervention is do(X = x), where  X is a node of graph G and x ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' General Causal Bandit Algorithms  We first provide the positive results, which provides an algorithm to improve the sample complexity comparing to applying  the multi-armed bandit approach directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  At each iteration we try to recover the edges’ direction in parallel using sub-procedure ”RECOVER-EDGE(a)” for a ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  For action a = do(X = x), this sub-procedure first performs two interventions do(X = 1) and do(X = 0), then chooses  an undirected edge (X, X′) corresponding to X (if exists), and then perform do(X′ = 1), do(X′ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The goal of these  operations is to estimate the difference between P(X = 1 | do(X′ = 0)) and P(X = 1 | do(X′ = 1)), and also the  difference between P(X′ = 1 | do(X = 0)), P(X′ = 1 | do(X = 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' which decides whether X′ → X or X → X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  By this sub-procedure in parallel, the algorithm estimate the model and recover the edges’ direction simultaneously and  adaptively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' To measure the difficulty for identified the direction of edges, for e : X → X′ we define  ce = P(X′ = 1 | do(X = 1)) − P(X′ = 1 | do(X = 0))  (29)  ca = cX =  min  e:X→X′ ce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (30)  Combinatorial Causal Bandits without Graph Skeleton  Algorithm 7 Causal-PE-unknown(G, A, ε, δ)  1: Initialize t = 1, Ta(0) = 0, ˆµa = 0 for all arms a ∈ A, Aknown = ∅  2: for t = 1, 2, · · · , do  3:  at−1  h  = argmaxa∈A ˆµt−1  a  4:  at−1  l  = argmaxa∈A\\at−1  h  (U t−1  a  )  5:  if Uat−1  l  ≤ Lat−1  h  + ε then  6:  Return at−1  h  7:  end if  8:  Perform do() operation and observe Xt and Yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For a = do(), Ta(t) = Ta(t−1)+1, Da(t) = Da(t−1), ra,∅(t) =  1  Ta(t)  �t  j=1 Yj, pa,∅(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  9:  for a = do(X = x) ∈ Aknown do  10:  Ta,z(t) = Ta,z(t − 1) + I{Xt = x, P = z}, Ta(t) = minz{Ta,z(t)}, where P = Pa(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Da(t) = Da(t − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  11:  Update ra,z(t) =  1  Ta,z(t)  �t  j=1 I{Xj = x, P j = z}Yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  12:  Update pa,z(t) = 1  t  �t  j=1 I{P j = z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  13:  Estimate ˆµO,a(t) = �  z ra,z(t)pa,z(t) and calculate [Lt  O,a, U t  O,a]by (34) and (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  14:  end for  15:  RECOVER-EDGE(at−1  h  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  16:  RECOVER-EDGE(at−1  l  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  17:  Update empirical mean ˆµI,a(t) using interventional dataand interventional confidence bound [Lt  I,a, U t  I,a]  18:  Update confidence bound [Lt  a, U t  a] by (33), ˆµa = (Lt  a + U t  a)/2, for each arm a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  19: end for  ce measure the difficulty for distinguishing the direction for an edge, and ca = cX represents the hardness for discovering  all directions corresponding to X and its childs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  The main Algorithm 7 is followed from (Xiong & Chen, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' During the algorithm, we add ”RECOVER-EDGE”  sub-procedure to identify the directions of the unknown edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' This sub-procedure first perform intervention do(X = 0)  and do(X = 1) on the node X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then if there is an edge (X′, X) which direction has not been identified, it chooses one such  edge and perform do(X′ = 1) and do(X′ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then it constructs the confidence bound for all P(X′ = 1 | do(X = 1)),  P(X′ = 1 | do(X = 0)), P(X = 1 | do(X′ = 1)) and P(X = 1 | do(X′ = 0)) based on Hoeffding’s concentration  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In fact, assume there are Da(t) samples for a = do(X′ = x), x ∈ {0, 1} until round t, then the confidence bound  for X conditioning on do(X′ = x) is defined by  [LX|do(X′=x), UX|do(X′=x)] =  �  ˆP(X = 1 | do(X′ = x)) −  �  2  Da(t) log 4n2t2  δ  , ˆP(X = 1 | do(X′ = x)) +  �  2  Da(t) log 4n2t2  δ  �  ,  (31)  where n is the number of nodes, and ˆP(X = 1 | do(X′ = x)) are the empirical mean of P(X = 1 | do(X′ = x)) using  all these Da(t) samples for do(X′ = x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Other confidence bounds define in this way similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Moreover, at iteration t, Line 4-Line 6 first choose two actions at−1  h  and at−1  l  through LUCB1 algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then, we use  Aknown to represent all nodes actions do(X = x) where all the edges corresponding to X are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In fact, if all the  edges corresponding to X are identified, we can find the true parent set Pa(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then we can use do-calculus to estimate  the causal effect:  E[Y | do(X = x)] =  �  z  P(Y | X = x, Z = z)P(Z = z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (32)  Line 9-14 enmurates all these actions, and calculate corresponding confidence bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The confidence bound is calculated by  [Lt  a, U t  a] = [Lt  O,a, U t  O,a] ∩ [Lt  I,a, U t  I,a],  (33)  where the first term [Lt  O,a, U t  O,a] = (−∞, ∞) for a = do(X = x) if the parents of X are not sure at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In fact, if we  do not discover all the edges corresponding to X, we cannot estimate the causal effect E[Y | do(X = x)] using do-calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  Algorithm 8 RECOVER-EDGE(a)  1: if a = do() then  2:  Return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  3: else  4:  Assume a = do(X = x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Sample action do(X = 1), do(X = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  5:  Da′(t) = Da′(t) + 1 for a′ = do(X = 1) and a′ = do(X = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  6:  Estimate P(X′ = 1 | do(X = 1)) and P(X′ = 1 | do(X = 0)) using interventional data for neighbor X′, where  the direction of (X′, X) is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  7:  Update the confidence bound [LX′|do(X=1), UX′|do(X=1)] and [LX′|do(X=0), UX′|do(X=0)] by (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  8:  if [LX′|do(X=1), UX′|do(X=1)] ∩ [LX′|do(X=0), UX′|do(X=0)] = ∅ then  9:  recover X → Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  10:  end if  11:  if ∃X′ such that (X′, X) is unknown then  12:  Choose one such X′ and perform do(X′ = 1) and do(X′ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  13:  Estimate P(X = 1 | do(X′ = 0)) and P(X = 1 | do(X′ = 1)) using interventional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  14:  Update the confidence bound [LX|do(X′=1), UX|do(X′=1)] and [LX|do(X′=0), UX|do(X′=0)] by (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  15:  if [LX|do(X′=1), UX|do(X′=1)] ∩ [LX|do(X′=0), UX|do(X′=0)] = ∅ then  16:  recover X → Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  17:  end if  18:  Da′(t) = Da′(t) + 1 for a′ = do(X′ = 1) and a′ = do(X′ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  19:  end if  20: end if  For nodes which parent set is identified,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' we calculate  [Lt  O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' U t  O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a] = [ˆµO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a(t) − βO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' ˆµO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a(t) + βO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a(t)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' [Lt  I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' U t  I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a] = [ˆµI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a(t) − βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' ˆµI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a(t) + βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a(t)]  (34)  The term ˆµO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a is calculated by estimating all terms at the right side of (32) empirically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' and confidence radius is given by  βO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a(t) =  �  12  Ta(t) log 16n2Zat3  δ  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' βI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a(t) = 2  �  1  Da(t) log 2n log(2t)  δ  (35)  Similar to the (Xiong & Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 2023),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' we can prove it is a valid confidence radius,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' which means that the true effect µO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a will  fall into the confidence bound [Lt  O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' U t  O,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a] with a high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Line 15-16 try to recover the edge for action chosen by LUCB1 algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' At the end of this iteration, the algorithm updates  all parameters and confidence bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  To represent the complexity result, we first provide the definition of gap-dependent threshold in (Xiong & Chen, 2023): For  a = do(X = x) and one possible configuration of the parent z ∈ {0, 1}|Pa(X)|, define qa,z = P(X = x, Pa(X) = z)  and qa = minz{qa,z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then sort the arm set as qa1 · max{∆a1, ε/2}2 ≤ qa2 · max{∆a2, ε/2}2 ≤ · · · ≤ qa|A| ·  max{∆a|A|, ε/2}2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Recall that ∆a = µ∗ − µa is the reward gap between the optimal reward and the reward of action a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Then Hr is defined by  Hr =  r  �  i=1  1  max{∆ai, ε/2}2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (36)  Definition 1 (Gap-dependent observation threshold((Xiong & Chen, 2023))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For a given causal graph G and its associated  qa’s and ∆a’s, the gap-dependent observation threshold mε,∆ is defined as:  mε,∆ = min  �  τ :  �����  �  a ∈ A  �����qa max {∆a, ε/2}2 <  1  Hτ  ������ ≤ τ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Denote action set S = {a ∈ A : qa max{∆a, ε/2}2 <  1  Hmε,∆ } are all actions which qa is relatively small, then |S| ≤ mε,∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Intuitively, action a with smaller qa are harder to be estimated by observation: If we assume qa = qa,z for a fixed vector z,  Combinatorial Causal Bandits without Graph Skeleton  then P(X = x, Pa(X) = z) is hard to observe and estimate by empirical estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Thus S contains all actions that are  relatively hard to observe, so it is more efficient to estimate µa by intervention for a ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Based on this definition, we can  provide the final sample complexity result:  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Denote H = �  a∈S  1  max{∆a,ε/2}2 + �  a/∈S min{  1  max{∆a,ε/2}2 , 1  c2a + �  e:X′→X  1  c2e }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' With probability 1 − 4δ,  Algorithm 7 will return a ε-optimal arm with sample complexity bound at most  T = O  �  H log  �nZH  δ  ��  ,  where ce, ca is defined in (29) and (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  The result can be explained in an intuitive way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The first term of H is the summation of all actions in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' As we discussed  above, it is more efficient to estimate the µa with intervention for a ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Thus, this summation can be regarded as the  sample complexity applying multi-armed bandit algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' LUCB1) directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The second term is to estimate the actions  by observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For each action a = do(X = x) with larger qa, we can first identify the edge’s direction corresponding to  the node X, and then using do-calculus to estimate the reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The term  1  c2a + �  e:X′→X  1  c2e represents the complextity  to identify the directions, and the complexity for using do-calculus can be contained in the first term �  a∈S  1  max{∆a,ε/2}2  because of the definition of gap-dependent observation threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Also, the term min{  1  max{∆a,ε/2}2 , 1  c2a + �  e:X′→X  1  c2e }  is because when we are discovering the edges’ direction, if the reward can be estimated by intervention accurately, we  turn to use interventional estimation and give up the causal discovery for this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The detailed proof can be found in the  Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Even if these two mechanisms can reduce the sample complexity, at the worst case the complexity also degenerates to  O(n/ε2), which is equal to the complexity for multi-armed bandit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We provide a lower bound to show that this problem  cannot be avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Theorem 7 (Lower bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Consider causal bandits with only essential graph and atomic intervention, for any (ε, δ)−PAC  algorithm, there is a bandit instance with expected sample complexity Ω( n  ε2 log(1/δ)) even if we have all observational  distribution P(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Theorem 7 states that even if we receive all observational distribution, which shows the intrinsic hardness for unknown  graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Indeed, the proof of lower bound shows that the unknown direction will lead to different interventional effects even  when the observational distribution are the same, leading to a unavoidable hardness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Proof of Theorem 6  First, fixed an action a = do(Xi = x), z ∈ {0, 1}|Pa(X)| , then Ta,z(t) = �t  j=1 I{Xj,i = x, Pa(Xi)j = z} and the  empirical mean ˆqa,z(t) = Ta,z(t)/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then denote 2|Pa(X)| = Za, if qa,z(t) ≥ 6  t log(2nZa/δ), with probability at least  1 −  δ  2nZa , we can have  |ˆqa,z(t) − qa,z(t)| <  �  6qa,z(t)  t  log  �2nZa  δ  �  Hence  ˆqa(t) = min  z {ˆqa,z(t)} ≤ min  z {qa,z +  �  6qa,z  t  log 2nZa  δ  } = qa +  �  6qa  t  log 2nZa  δ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (37)  When qa ≥ 3  t log 2nZa  δ  , f(x) = x −  �  6x  t log 2nZa  δ  is a increasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  ˆqa(t) ≥ min  z {qa,z −  �  6qa,z  t  log 2nZa  δ  } = qa −  �  6qa  t  log 2nZa  δ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (38)  So define the event as  E1(t) =  �  ∀a ∈ A with t ≥ 6  qa  log  �2nZa  δ  �  , |ˆqa(t) − qa| ≤  �  6qa  t  log  �2nZa  δ  ��  Combinatorial Causal Bandits without Graph Skeleton  then Pr{Ec  1(t)} ≤ δ, where Ec means the complement of the event E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Now we consider the concentration bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' First, by classical anytime confidence bound, with probability at least 1 −  δ  2n,  for any time Da(t) ≥ 1  |ˆµI,a(t) − µI,a| < 2  �  1  Da(t) log  �2n log(2Da(t))  δ  �  ≤ 2  �  1  Da(t) log  �2n log(2t)  δ  �  Thus define the event as  E2 =  �  ∀t, a, |ˆµI,a(t) − µI,a| < 2  �  1  Da(t) log  �2n log(2t)  δ  ��  ,  then Pr{Ec  2} ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Consider the observational confidence bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' First, if a /∈ Aknown, [Lt  O,a, U t  O,a] = (−∞, ∞) and then the ˆµO,a(t) ∈  [Lt  O,a, U t  O,a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Now we consider that if a = do(X = x) ∈ Aknown and the parent of X is P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' By Hoeffding’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  with probability at least 1 − δ/16n2Zat3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' for a = do(X = x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  |ra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='z(t) − P(Y = 1 | X = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' P = z)| >  �  1  2Ta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='z(t) log 16n2Zat3  δ  (39)  Also,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' by Chernoff’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' since qa ≤ P(P = z) for all z ∈ {0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' 1}|P |,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' when t ≥  6  qa log  �  16n2Zat3  δ  �  with probability  at least 1 − δ/16n2Zat3 we will have  |pa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='z(t) − P(P = z)| >  �  6P(P = z)  t  log 16n2Zat3  δ  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (40)  then  ˆµO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='a =  �  z  ra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='z(t) · pa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='z(t)  ≤  �  z  P(Y = 1 | X = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' P = z)pa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='z(t) +  �  z  pa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='z(t)  �  1  2Ta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='z(t) log 16n2Zat3  δ  ≤  �  z  P(Y = 1 | X = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' P = z)pa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='z(t) +  �  1  2Ta(t) log 16n2Zat3  δ  ≤  �  z  P(Y = 1 | X = x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' P = z)P(P = z) +  �  z  �  6P(P = z)  t  log 16n2Zat3  δ  +  �  1  2Ta(t) log 16n2Zat3  δ  ≤ µa +  �  6Z  t log 16n2Zat3  δ  +  �  1  2Ta(t) log 16n2Zat3  δ  ≤ µa +  �  6  Ta(t) log 16n2Zat3  δ  +  �  1  2Ta(t) log 16n2Zat3  δ  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  = µa +  �  8  Ta(t) log 16n2Zat3  δ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Also, if t ≤  6  qa log 16n2Zat3  δ  , first by Chernoff inequality, set Q =  6  qa log 16n2Zat3  δ  , then with probability at least 1 −  δ/16n2Zat3, we have  ˆqa(Q) ≤ 2qa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (41)  Combinatorial Causal Bandits without Graph Skeleton  by E1(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Ta(t) ≤ Ta(Q) ≤ ˆqa(Q) · Q ≤ 2qa · Q = 12  qa  log 16n2Zat3  δ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Then  �  12  Ta(t) log 16n2Zat3  δ  ≥ 1 and the inequality  |ˆµO,a(t) − µO,a| ≤  �  12  Ta(t) log 16n2Zat3  δ  also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Thus we define the event  E3 =  �  ∀a, t, |ˆµO,a(t) − µO,a| ≤  �  12  Ta(t) log 16n2Zat3  δ  �  then by taking the union bound of (39), (40) and (41),  Pr{Ec  3} ≤  ∞  �  t=1  �  a∈A  �  z  3 ·  δ  16n2Zat3  ≤  ∞  �  t=1  δ  4t3  ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Now we consider how to bound our sample complexity based on events E1, E2 and E3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' First, we provide the following  lemma in (Xiong & Chen, 2023):  Lemma 10 (Lemma 6 in (Xiong & Chen, 2023)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Under the event E1, E2 and E3, at round t, if we have  βat  h(t) ≤  max{∆at  h, ε/2}  4  , βat  l(t) ≤  max{∆at  l, ε/2}  4  ,  where at  h, at  l are the actions performed by algorithm at round t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' then the algorithm will stop at round t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Now assume the algorithm does not terminate at T1 = 192H log(nZT 3  1 /δ), where Z = maxa Za.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For a ∈ S, Da(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Note  that H ≥ Hmε,∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Thus at round T1, for action a with qa ≥  1  Hmε,∆·max{∆a,ε/2}2 ≥ 192  T1 log 16nZaT 3  1  δ  , if a ∈ Aknown, then  under event E1(T1), we have  ˆqa(T1) ≥ qa −  �  6qa  T1  log 16nZaT 3  1  δ  ≥ qa  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Then  βa(T1) ≤ βO,a(T1) =  �  12  Ta(T1) log 16n2Zat3  δ  ≤  �  12qa  2T1  ≤ max{∆a, ε/2}2  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Now we prove that if Da(t) is large for some a, then a ∈ Aknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Lemma 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' With probability at least 1 − δ, denote Ca =  1  c2a + �  e:X′→X  1  c2e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' If Da(t) ≥ 32Ca log(4n2t2/δ), a ∈ Aknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' If Da(t) ≥ 8Ca log t, we have called sub-procedure RECOVER-EDGE(a) for Da(t) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then, for each edge  e : X → X′, we will perform intervention do(X = 1), do(X = 0) for at least Da(t) times and observe the empirical  Combinatorial Causal Bandits without Graph Skeleton  difference | ˆP(X′ | do(X = 1)) − ˆP(X′ | do(X = 0))|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' By Hoeffding’s inequality and union bound on all time t and the  �n−1  2  �  ordered-pair (X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' with probability at least 1 − δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' for all t ∈ [T] and all X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' X we have  | ˆP(X′ | do(X = 1)) − P(X′ | do(X = 1))| ≤  �  2  Da(t) log 4n2t2  δ  | ˆP(X′ | do(X = 0) − P(X′ | do(X = 0))| ≤  �  2  Da(t) log 4n2t2  δ  Then for the confidence bounds  [LX′|do(X=1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' UX′|do(X=1)] =  �  ˆP(X′ | do(X = 1)) −  �  2  Da(t) log 4n2t2  δ  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' ˆP(X′ | do(X = 1)) +  �  2  Da(t) log 4n2t2  δ  �  [LX′|do(X=0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' UX′|do(X=0)] =  �  ˆP(X′ | do(X = 0)) −  �  2  Da(t) log 4n2t2  δ  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' ˆP(X′ | do(X = 0)) +  �  2  Da(t) log 4n2t2  δ  �  the intersection  [LX′|do(X=1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' UX′|do(X=1)] ∩ [LX′|do(X=0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' UX′|do(X=0)] = ∅  since  | ˆP(X′ | do(X = 1) − ˆP(X′ | do(X = 0))|  ≥|P(X′ | do(X = 1) − P(X′ | do(X = 0))| − |P(X′ | do(X = 1) − ˆP(X′ | do(X = 1))|  − |P(X′ | do(X = 0) − ˆP(X′ | do(X = 0))|  ≥ca − 2  �  2  Da(t) log 4n2t2  δ  ≥2  �  2  Da(t) log 4n2t2  δ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  where we use Da(t) ≥  1  c2a log 4n2t2  δ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then the edge’s direction will be identified correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Consider the edge e : X′ → X, then if we sample do(X′ = 1) and do(X′ = 0) for 1  c2e log 4n2t2  δ  times within sub-procedures  RECOVER-EDGE(a), similarly we will identify the edge X′ → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then because the RECOVER-EDGE(a) will perform  intervention do(X′ = 0) and do(X′ = 1) for the X′ that the direction of (X′, X) has not been discovered each time, after  �  e:X′→X  1  c2e log 4n2t2  δ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Then we define  E4 = {Lemma 11 holds}  Then Pr{Ec  4} ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Also, under the event E2, the following lemma shows that if Da(t) is really large, we can estimate the  µa accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Under event E2, if Da(t) ≥  64  max{∆a,ε/2}2 log 16n2Zat3  δ  , then  βa(T1) ≤ max{∆a, ε/2}2  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In fact,  βa(t) ≤ βI,a(t) = 2  �  1  Da(t) log  �2n log(2t)  δ  �  ≤ 2  �  1  Da(t) log 16n2Zat3  δ  ≤ max{∆a, ε/2}2  4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  Now we turn to our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' From the Lemma 10, at least one arm a with βa(t) ≥ max{∆a,ε/2}  4  will be performed an  intervention at each round t ≥ T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Under the event E1, E2, E3 and E4, these interventions will only performed in two types  of action a:  qa ≤  1  Hmε,∆·max{∆a,ε/2}2 and Da(t) ≤  64  max{∆a,ε/2}2 log 16n2Zat3  δ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Da(t) ≤ min{MCa log(t),  64  max{∆a,ε/2}2 log 16n2Zat3  δ  }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Note that qa ≤  1  Hmε,∆·max{∆a,ε/2}2 implies that a ∈ S, then after at most T2 rounds, where  T2 = 64  ��  a∈S  1  max{∆a, ε/2}2 +  �  a/∈S  min  �  1  max{∆a, ε/2}2 , 1  c2a  +  �  e:X′→X  1  c2e  ��  log 16n2ZT 3  2  δ  = 64H log 16n2ZT 3  2  δ  the algorithm should terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The fist term is the summation of all actions in S, and the second term is for the second  type of actions, which Da(t) ≤ min{MCa log(t),  64  max{∆a,ε/2}2 log 16n2Zat3  δ  }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Denote T = T1 + T2, then  T = T1 + T2 ≤ 256H log 16n2ZT 3  δ  ≤ 768H log 16nZT  δ  Then by the Lemma 13, with probability at least 1 − 4δ, the sample complexity has the upper bound  T = O  �  H log  �nZH  δ  ��  Replace δ to δ/4, we derive the sample complexity in the Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The correctness of algorithm can be derived by  LUCB1 algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We provide a short argument here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Because the stopping rule is ˆµt  at  l + βat  l(t) ≤ ˆµt  at  h − βat  h(t) + ε, if  a∗ ̸= at  h, we have  µat  h + ε ≥ ˆµat  h − βat  h(t) + ε ≥ ˆµat  l + βat  l(t) ≥ ˆµa∗ + βa∗(t) ≥ µa∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Hence either a∗ = at  h or at  h is ε-optimal arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Proof of Lemma 10  For completeness, we provide the proof in (Xiong & Chen, 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' If the optimal arm a∗ = at  h,  ˆµat  l + βat  l(t) ≤ µat  l + 2βat  l(t)  ≤ µat  l +  max{∆at  l, ε/2}  2  ≤ µat  h − ∆at  l +  max{∆at  l, ε/2}  2  ≤ ˆµat  h + βa∗(Ta∗(t)) − ∆at  l +  max{∆at  l, ε/2}  2  ≤ ˆµat  h − βa∗(Ta∗(t)) +  max{∆a∗, ε/2} + max{∆at  l, ε/2}  2  − ∆at  l  ≤ ˆµat  h − βa∗(Ta∗(t)) +  ∆a∗ + ε/2 + ∆at  l + ε/2  2  − ∆at  l  ≤ ˆµat  h − βa∗(Ta∗(t)) + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  If optimal arm a∗ ̸= at  h, and the algorithm doesn’t stop at round t + 1, then we prove a∗ ̸= at  l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Otherwise, assume a∗ = at  l  ˆµt  at  h ≤ µt  at  h +  max{∆at  h, ε/2}  4  (42)  = µt  at  l − ∆at  h +  max{∆at  h, ε/2}  4  (43)  ≤ µt  at  l −  3∆at  h  4  + ε/4  (44)  ≤ ˆµt  at  l + max{∆a∗, ε/2}  4  −  3∆at  h  4  + ε/4  (45)  ≤ ˆµt  at  l + ε/2 −  ∆at  h  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (46)  From the definition of at  h, we know ε > ∆at  h ≥ ∆a∗, βat  h(t) ≤ ε/4, βat  l(t) ≤ ε/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then ˆµt  at  l + βat  l(t) + βat  h(t) ≤  ˆµat  l + ε/2 ≤ ˆµt  at  h + ε, which means the algorithm stops at round t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Now we can assume a∗ ̸= at  l, a∗ ̸= at  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then  µat  l + 2βat  l(t) ≥ ˆµat  l + βat  l(t) ≥ ˆµa∗ + βa∗(Ta∗(t)) ≥ µa∗ = µat  l + ∆at  l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (47)  Thus  ∆at  l ≤ 2βat  l(t) ≤  max{∆at  l, ε/2}  2  ,  (48)  which leads to ∆at  l ≤ ε/2, βat  l(t) ≤ ε/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Since  Also,  µat  h + βat  h(t) ≥ ˆµat  h ≥ ˆµat  l ≥ µa∗ − βat  l(t) = µat  h + ∆at  h − βat  l(t),  (49)  which leads to  max{∆at  h, ε/2}  4  ≥ ∆at  h − ε/8,  (50)  and ∆at  h ≤ ε/2, βat  h(t) ≤ ε/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Hence ˆµt  at  l + βat  l(t) + βat  h(t) ≤ ˆµat  l + ε/2 ≤ ˆµt  at  h + ε, which means the algorithm stops at  round t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Proof of Theorem 7  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' We construct n − 1 graphs with the same distribution P(X, Y ) but different causal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Indeed, We construct  the bandit instances {ξi}2≤i≤n as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For instance ξ2, the graph structure contains edge X1 → Y, X2 → X1, X1 →  Xi(3 ≤ i ≤ n) and X2 → Xi(3 ≤ i ≤ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For instances ξi(3 ≤ i ≤ n), we change X1 → Xi to Xi → X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The graph  structure are shown in the Figure 1 and Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  The observational distribution for all instance is:  P(X, Y ) = p1p2 · · · pn,  (51)  where  p1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5,  (52)  p2 =  � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + ε  x2 = x1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − ε  x2 ̸= x1  (53)  pi =  � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + 4ε  xi = x1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − 4ε  xi ̸= x1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  (54)  Combinatorial Causal Bandits without Graph Skeleton  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Causal Bandits Instance τ2  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Causal Bandits Instance τi(i = 3)  It is easy to check that �  x,y P(X = x, Y = y) = 1 and P(Xi = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' The action set is do(), do(Xi = 1), do(Xi = 0)  where 2 ≤ i ≤ n, which means the action set does not contain do(X1 = x) for x = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Now in ξ2, we consider P(Y = 1 | do(X2 = 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Actually, it is easy to show that P(Y = 1 | do(X2 = 1)) = P(X1 = 1 |  do(X2 = 1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Similarly, P(Y = 1 | do(X2 = 0)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For other actions, P(Y = 1 | a) = P(X1 = 1 |  a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 since other actions a will not influence the value of X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Now consider instance ξi for 3 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' For action do() and do(Xj = x) with j ̸= 2, i, it will not influence the value of X1  and then P(Y = 1 | a) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Now consider action a = do(X2 = 1), we have  P(Y = 1 | do(X2 = 1)) = P(X1 = 1 | do(X2 = 1))  = P(X1 = 1 | X2 = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Similarly, P(Y = 1 | do(X2 = 0)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='.Combinatorial Causal Bandits without Graph Skeleton  Now we calculate P(Y = 1 | do(Xi = 1)) in instance ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In fact, denote q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + 4ε and by do-calculus,  P(X1 = 1 | do(Xi = 1)) =  �  x=0,1  P(X1 = 1 | Xi = 1, X2 = x)P(X2 = x)  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5(P(X1 = 1 | Xi = 1, X2 = 0) + P(X1 = 1 | Xi = 1, X2 = 1)  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5  �P(X1 = 1, Xi = 1, X2 = 0)  P(Xi = 1, X2 = 0)  + P(X1 = 1, Xi = 1, X2 = 1)  P(Xi = 1, X2 = 1)  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5  �  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + 4ε)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − ε)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + 4ε)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − ε) + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − 4ε)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + ε) +  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + 4ε)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + ε)  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + 4ε)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + ε) + (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − 4ε)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − ε)  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5  �  q(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − ε)  q(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − ε) + (1 − q)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + ε) +  q(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + ε)  q(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + ε) + (1 − q)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − ε)  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5  �  q(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − ε)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − (2q − 1)ε +  q(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + ε)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + (2q − 1)ε  �  = q  � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='52 − (2q − 1)ε2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='52 − (2q − 1)2ε2  �  ≤ q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + 4ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Also, we prove that  q  � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='52 − (2q − 1)ε2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='52 − (2q − 1)2ε2  �  ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Actually, this inequality is equal to  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + 4ε)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='52 − 8ε3) ≥ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + 2ε)(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='52 − 8ε4)  ⇐⇒ 1 ≥ 56ε3 + 8ε2 − 32ε4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  When ε is small enough, this inequality holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' In summary, we have  P(X1 = 1 | do(Xi = 1)) ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + 2ε, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + 4ε].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Similarly, we can get  P(X1 = 1 | do(Xi = 0)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5(P(X1 = 1 | Xi = 0, X2 = 1) + P(X1 = 1 | Xi = 0, X2 = 0))  = (1 − q)  � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='52 − (1 − 2q)ε2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='52 − (1 − 2q)2ε2  �  ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − 4ε, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Now in instance ξ2, the output action should be do(X2 = 1), while in instance ξi, the output action should be do(Xi = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Now by Pinkser’s inequality, for an policy π, we have  2δ ≥ Pξ2(ao = do(Xi = 1)) + Pξi(ao ̸= do(Xi = 1)) ≥ exp(−KL(ξπ  2 , ξπ  i )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Also,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' assume the stopping time as τ for the environment E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' the KL divergence can be rewritten as  KL(ξπ  2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' ξπ  i ) = EAt∼ξπ  2  � τ  �  t=1  KL(Pξ2(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Yt | At),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Pξi(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Yt | At))  �  (55)  = Eξπ  2  � τ  �  t=1  Pξ2(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Yt | At)  �  log Pξ2(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Yt | At)  Pξi(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Yt | At)  ��  (56)  = Eξπ  2  � τ  �  t=1  Pξ2(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='1 | At)  �  log Pξ2(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='1 | At)  Pξi(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='1 | At)  ��  (57)  where the last equation is derived as follows:  Pξ2(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Yt | At)  Pξi(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Yt | At) = Pξ2(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='1 | At) · Pξ2( ¯Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Yt | Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' At)  Pξi(Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='1 | At) · Pξi( ¯Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Yt | Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' At)  Combinatorial Causal Bandits without Graph Skeleton  where ¯Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='i = Xt \\ {Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Now since ¯Xt,i is only decided by X1, X2 and X2 is only decided by At, then  Pξ2( ¯Xt,i, Yt | Xt,i, Xt,1, At) = Pξi( ¯Xt,i, Yt | Xt,i, Xt,1, At)  and then  Pξ2(Xt, Yt | At)  Pξi(Xt, Yt | At) = Pξ2(Xt,i, Xt,1 | At)  Pξi(Xt,i, Xt,1 | At) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Note that only when At = do(Xi = 1), do(Xi = 0), Pξ2(Xt,i, Xt,1 | At) ̸= Pξi(Xt,i, Xt,1 | At).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then the equation (57)  can be further calculated as  (57) =  �  x=0,1  Eξπ  2  � τ  �  t=1  I{At = do(Xi = x)}  �  Pξ2(Xt,i, Xt,1 | do(Xi = x))  �  log Pξ2(Xt,i, Xt,1 | do(Xi = x))  Pξi(Xt,i, Xt,1 | do(Xi = x))  �  =  �  x=0,1  Eξπ  2  � τ  �  t=1  I{At = do(Xi = x)}  �  Pξ2(Xt,1 | do(Xi = x))  �  log Pξ2(Xt,1 | do(Xi = x))  Pξi(Xt,1 | do(Xi = x))  �  ≤  �  x=0,1  Eξπ  2  � τ  �  t=1  I{At = do(Xi = x)}  � �  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 ·  �  log  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 + 4ε + log  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5 − 4ε  ��  ≤  �  x=0,1  Eξπ  2  � τ  �  t=1  I{At = do(Xi = x)}  �  96ε2  = 96ε2 · Eξπ  2 [N(do(Xi = 1)) + N(do(Xi = 0))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  where the Eξπ  2 N(a) represents that the number of times taking action a for policy π under the instance ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Now we have  Eξπ  2 [N(do(Xi = 1)) + N(do(Xi = 0))] ≥ KL(ξπ  2 , ξπ  i )  96ε2  ≥  1  96ε2 log 1  2δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Hence the stopping time τ under policy π can be lower bounded by  Eξπ  2 [τ] ≥  n  �  i=3  Eξπ  2 [N(do(Xi = 1)) + N(do(Xi = 0))] ≥ n − 2  96ε2 log 1  2δ = O  � n  ε2 log 1  δ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Technical Lemma  Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' If T = CH log dT  δ for some constant C and parameter d such that d ≥ eδ, then T = O(H log Hd  δ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Let f(x) =  x  log(dx/δ), then for x ≥ 1  f ′(x) = log(dx/δ) − 1  log2 dx/δ  ≥ 0  because dx/δ > e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Then f(x) is non-decreasing for x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  To prove T = O(H log Hd  δ ), we only need to show that f(T) ≤ f(C′H log Hd  δ ) for some constant C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content=' Since  log C′Hd log Hd  δ  δ  = log C′Hd  δ  + log log Hd  δ  we only need to prove  f(C′H log Hd  δ ) =  C′H log Hd  δ  log C′Hd  δ  + log log Hd  δ  ≥ CH = f(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
+page_content='  Combinatorial Causal Bandits without Graph Skeleton  If we choose C′ ≥ 2C + C log C′, then  CH  �  log C′Hd  δ  + log log Hd  δ  �  ≤ CH(log C′Hd  δ  + log Hd  δ )  ≤ 2CH log Hd  δ  + CH log C′  ≤ (2C + C log C′)H log Hd  δ  ≤ C′H log Hd  δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/q9FQT4oBgHgl3EQftjaQ/content/2301.13392v1.pdf'}
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+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+1|29 
+AI model GPT-3 (dis)informs us better than 
+humans 
+ 
+Giovanni Spitale 
+0000-0002-6812-0979 
+Institute of Biomedical Ethics and History of Medicine, University of Zurich, Zurich, Switzerland 
+giovanni.spitale@ibme.uzh.ch  
+ 
+Nikola Biller-Andorno 
+0000-0001-7661-1324 
+Institute of Biomedical Ethics and History of Medicine, University of Zurich, Zurich, Switzerland 
+biller-andorno@ibme.uzh.ch  
+ 
+Federico Germani (corresponding author) 
+0000-0002-5604-0437 
+Institute of Biomedical Ethics and History of Medicine, University of Zurich, Zurich, Switzerland 
+federico.germani@ibme.uzh.ch  
++41 44 634 40 80  
+Institute for Biomedical Ethics and History of Medicine (IBME), Winterthurerstrasse 30, 8006 Zürich (CH) 
+ 
+Abstract 
+Artificial intelligence is changing the way we create and evaluate information, and this is happening during 
+an infodemic, which has been having dramatic effects on global health. In this paper we evaluate whether 
+recruited individuals can distinguish disinformation from accurate information, structured in the form of 
+tweets, and determine whether a tweet is organic or synthetic, i.e., whether it has been written by a 
+Twitter user or by the AI model GPT-3. Our results show that GPT-3 is a double-edge sword, which, in 
+comparison with humans, can produce accurate information that is easier to understand, but can also 
+produce more compelling disinformation. We also show that humans cannot distinguish tweets generated 
+by GPT-3 and real Twitter users. Starting from our results, we reflect on the dangers of AI for 
+disinformation, and on how we can improve information campaigns to benefit global health. 
+Introduction 
+Artificial intelligence text generators caught much attention over the last years, especially after the release 
+of GPT-3 in 20201. GPT-3, the latest iteration of the Generative Pre-trained Transformers developed by 
+OpenAI, is arguably the most advanced systems of pre-trained language representations2. A generative pre-
+trained transformer, in its essence, is a statistical representation of language; an AI engine that based on 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+2|29 
+users’ prompts can produce very credible – and sometimes astonishing – text 3. In fact, an initial test on 
+people’s ability to tell whether a ∼ 500 word article was written by humans or GPT-3 showed a mean 
+accuracy of 52%; just slightly better than random guessing 1.  
+GPT-3 does not have any mental representations or understanding of the language it operates on 4. The 
+system relies on statistical representations of language for how it is used in real-life by real humans, or ‘a 
+simulacrum of the interaction between people and the world’ 4. Even keeping in mind these structural 
+limitations, what GPT-3 can do is remarkable, and remarkable are also the possible implications. While on 
+one hand GPT-3 can be a great tool for machine translations, text classification, dialogue/chatbot systems, 
+knowledge summarizing, question answering, creative writing 2,5,6, detecting hate speech 7, and automatic 
+code writing 2,8, it can also be used to produce ‘misinformation, spam, phishing, abuse of legal and 
+governmental processes, fraudulent academic essay writing and social engineering pretexting’ 1,9–11.  GPT-3 
+is essentially a lever, an amplifier of human intentions which can receive instructions in natural language 
+and produce output that can be natural language as well, or formal language. In this sense, this tool is 
+inherently neutral from an ethical point of view – and as every other similar technology, it is subject to the 
+dual use problem12.  
+These advancements in AI text generators and the release of GPT-3 historically coincide with the ongoing 
+infodemic13 – an epidemic-like circulation of fake news and disinformation, which, alongside the COVID-19 
+pandemic, has been greatly detrimental for global health. Since GPT-3 can potentially be used to generate 
+information, and given the possible misuse of such technology (i.e., producing disinformation) and the 
+devastating effects it may have on global health, it is of utmost importance to evaluate how text produced 
+by GPT-3 can affect people’s understanding of information.  
+In this paper we aim at determining whether GPT-3 can be used to produce accurate information and 
+disinformation in the form of tweets, and how credible this information or disinformation is when 
+compared with information and disinformation produced by humans, and whether the same technology 
+can be used to develop assistive tools to help identifying disinformation. For clarity, we acknowledge that 
+the definitions of disinformation and misinformation are diverse, but in this paper we refer to an inclusive 
+definition, which considers as disinformation both false information (also partially false information) and/or 
+misleading content14.  
+To achieve our goals, we asked GPT-3 to write tweets containing informative or disinformative tweets on a 
+range of different topics, including vaccines, 5G and COVID-19, or the theory of evolution, among others, 
+which are commonly subject to disinformation and public misconception. We collected a set of real tweets 
+written by users on the same topics, and programmed a survey in which we asked respondents to classify 
+whether randomly selected synthetic tweets (i.e.: written by GPT-3) and organic tweets (i.e.: written by 
+humans) were true or false (i.e. whether they contained accurate information or disinformation), and 
+whether they were written by a real Twitter user or by an AI.  
+ 
+Results 
+Study design and demographics 
+In order to test for the use of GPT-3 AI model as a tool to generate accurate information or disinformation 
+in the form of tweets, we crafted instruction prompts to instruct GPT-3 to produce fake tweets, either 
+containing accurate information, or disinformation about the following topics: climate change, vaccines 
+safety, theory of evolution, COVID-19, masks safety, vaccines and autism, homeopathy treatments for 
+cancer, flat Earth, 5G technology and COVID-19, antibiotics and viral infections, and COVID-19 and 
+influenza. We performed a Twitter search to identify accurate tweets and disinformation tweets generated 
+by Twitter users. We call “synthetic” those tweets that are generated by GPT-3, and we call “organic” those 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+3|29 
+real tweets retrieved from Twitter. Human respondents were recruited online to participate in a quiz, in 
+which they were asked to recognize whether a set of tweets were organic or synthetic, and true or false 
+(i.e., whether they contained accurate information or disinformation). GPT-3 was also questioned about 
+whether tweets forming the same dataset were true or false (Figure 1A). We recruited 869 respondents. 
+157 responses were excluded because they were incomplete. Of the 712 remaining responses, 15 
+additional responses were removed because the respondents were too fast to meaningfully complete the 
+survey, for a total of 697 responses included in our analysis (Figure 1B). Most of the respondents were from 
+the UK, Australia, Canada, USA, and Ireland (Figure S1A), with more females than males (Figure S1B), a 
+balanced age, with a high representation of people between 42 and 76 years old (Figure S1C), and a 
+balanced education level profile, with most of the respondents holding a Bachelor’s degree (Figure S1D); 
+among those with a Bachelor’s degree or above, their field of study was mostly in social sciences and 
+humanities, natural sciences, or medical sciences (Figure S1E). 
+GPT-3 AI model informs and disinforms us better 
+We measured how accurately participants recognized whether a tweet was containing disinformation or 
+accurate information (Disinformation Recognition Score), for four types of tweet: “Organic true”, which are 
+tweets published by Twitter users (organic) and containing accurate information (true); “Synthetic true”, 
+which are tweets generated by GPT-3 (synthetic) and containing accurate information (true); “Organic 
+false”, which are tweets generated by Twitter users (organic) and containing disinformation (false); and 
+finally “synthetic false”, which are tweets generated by GPT-3 (synthetic) and contain disinformation 
+(false). Participants recognized “organic false” tweets with the highest efficiency, better than synthetic false 
+tweets (Scores 0.92 versus 0.89, respectively; p=0.0032) (Figure 1C). Similarly, they recognized “synthetic 
+true” tweets correctly more than “organic true” tweets (Scores 0.84 versus 0.72, respectively; p<0.0001). 
+This indicates that human respondents can recognize the accuracy of tweets containing accurate 
+information more often when such tweets are generated by GPT-3, when compared with organic tweets 
+retrieved from Twitter. Similarly, this means that disinformation tweets generated with GPT-3 achieve their 
+deceiving goal more often when compared with disinformation tweets generated organically, even though 
+in this case the effect is small. When evaluating the same dataset, segmenting the analysis for true versus 
+false tweets – regardless of whether they are organic or synthetic – and for organic versus synthetic tweets 
+– regardless of their truthfulness – tweets containing accurate information received lower scores when 
+compared with tweets containing disinformation (Scores 0.78 versus 0.91, respectively; p<0.0001). 
+Similarly, synthetic tweets were categorized more often correctly for the accuracy of the information they 
+contained (Scores 0.87 versus 0.82, respectively; p<0.0001) (Figure 1C’). Participants required on average 
+29.14 seconds to determine whether an “organic true” tweet was accurate or contained disinformation. 
+This was significantly more when compared with “organic false” tweets, which required 23.28 seconds for 
+evaluation, with “synthetic true” tweets requiring 21.02 seconds, and “synthetic false” tweets requiring 
+19.87 seconds (Figure 1D). True tweets required a longer time for evaluation when compared with false 
+tweets (25.07 vs 21.97 seconds, p<0.0001), as well as organic tweets when compared with synthetic tweets 
+(26.21 vs 20.44 seconds, p<0.0001). (Figure 1D’). The time required for evaluation was not dependent on 
+the length of tweets (Figure 1E). Further, we calculated Disinformation Recognition Scores for each 
+category (e.g., “climate change”, “vaccines and autism”), for each type of tweet (i.e., “organic true”, 
+“synthetic true”, “organic false”, “synthetic false”) (Figure S2), and plotted the average Disinformation 
+Scores for each type of tweet (Figure S3), obtaining comparable results with the analysis run on the 
+Disinformation Recognition Scores of each respondent. This confirms that, for humans, accurate 
+information is more difficult to evaluate when compared with disinformation, and that information 
+produced by GPT-3 is not only more effective to inform and disinform humans, but also does so more 
+efficiently, in less time (i.e.: respondents evaluated faster – correctly or incorrectly – these tweets). A list of 
+the disinformation tweets recognized most often as accurate tweets can be seen in Figure S4, and a list of 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+4|29 
+tweets containing accurate information, recognized most often as disinformation tweets, can be seen in 
+Figure S5. 
+Humans evaluate the accuracy of information better than GPT-3 
+The respondents of our survey evaluated the accuracy or inaccuracy of the information contained in 220 
+tweets. Using the same dataset, we asked GPT-3 to evaluate whether the tweets were accurate or whether 
+they contained disinformation. For disinformation tweets, humans and GPT-3 performed similarly, even 
+though respondents performed slightly better (Success rates: 0.92 vs 0.89, respectively). For accurate 
+tweets, GPT-3, likewise human respondents, had more difficulties evaluating the accuracy of the 
+information. In comparison, human respondents performed better than GPT-3 (Success rates: 0.72 vs 0.64, 
+respectively) (Figure 2A). A detailed analysis of the results for each category of tweets can be found in 
+Figure S6. These results suggest that human respondents can evaluate information better than GPT-3. 
+Considering these respondents are not necessarily trained individuals in recognizing disinformation, with a 
+high degree of information literacy, trained humans, at this stage, may perform much better than machines 
+at performing this task. 
+GPT-3 can “disobey” requests to produce disinformation 
+As GPT-3 does not have mental representations nor anything that can resemble a proper intentionality15,16, 
+we believe the use of quotes when referring to “obedience” or “disobedience” is needed. As mentioned, 
+we instructed GPT-3 to produce a set of true and false tweets (i.e., accurate or disinformation tweets) 
+(Figure 1A). To obtain the dataset, we instructed GPT-3 to produce 10 accurate and 10 disinformation 
+tweets for each category. Of these, we included in our survey only the tweets for which GPT-3 “obeyed” 
+our request to produce accurate or disinformation tweets, respectively. We calculated the rate of 
+obedience, i.e., the percentage of requests that were satisfied by GPT-3 divided for the overall number of 
+requests we made to GPT-3. For accurate information, GPT-3 obeyed our requests 99 times out of 101, 
+whereas for disinformation the rate of obedience was much lower (80/102) (Figure 2B), indicating that 
+GPT-3 can “refuse” to produce disinformation, and in rarer instances, it may produce disinformation when 
+asked to produce accurate information. For a detailed analysis of the Obedience rate per category of tweet, 
+we refer to Figure S7. 
+Information generated by GPT-3 is undistinguishable from human-generated information 
+We calculated the AI Recognition Score (or OS score), i.e., the ability of respondents to recognize whether 
+tweets are organic (i.e., produced by Twitter users) or synthetic (i.e., produced by GPT-3). As for the 
+evaluation of the ability to recognize disinformation, we calculated the AI Recognition Score for each type 
+of tweet (i.e., “organic true”, “synthetic true”, “organic false”, “synthetic false”). On average, scores 
+fluctuated around 0.5, indicating that the choice between synthetic and organic is random: respondents 
+cannot understand if the tweet is produced by AI or has been really tweeted by a user on Twitter (Figure 3). 
+That said, respondents obtained a significantly higher score for “organic true” tweets and, to a lesser 
+extent, for “organic false” tweets (0.67 and 0.60, respectively), whereas for “synthetic true” and “synthetic 
+false” tweets, scores were below 0.5 (0.34 and 0.40, respectively) (Figure 3A). Looking at true versus false 
+tweets, and organic versus synthetic tweets, we note something interesting; whether tweets contain 
+accurate information or disinformation, it does not influence people’s ability to determine whether a tweet 
+is organic or synthetic; answers are perfectly random on average (0.50 vs 0.50, respectively, p=0.9576). 
+However, when comparing organic versus synthetic tweets, organic tweets were recognized as organic 
+more often when compared with synthetic tweets recognized as synthetic (0.63 vs 0.37, p<0.0001) (Figure 
+3A’). Therefore, both organic and synthetic tweets tend to be classified as “human”, indicating that GPT-3 
+can effectively mimic human-generated information. Further, we calculated AI Recognition Scores for each 
+category (e.g., “climate change”, “vaccines and autism”), for each type of tweet (i.e., “organic true”, 
+“synthetic true”, “organic false”, “synthetic false”) (Figure S8), and plotted the average AI Recognition 
+Scores for each type of tweet (Figure S9), obtaining comparable results with the analysis run on the AI 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+5|29 
+Recognition Scores of each respondent. A list of the organic tweets recognized most often as synthetic can 
+be seen in Figure S10, and a list of synthetic tweets recognized most often as organic can be seen in Figure 
+S11. 
+Building versus crashing confidence: how the self-reported ability to recognize 
+disinformation and AI-generated information changes after survey completion 
+At the beginning of the survey, we asked respondents to define how confident they were in their ability to 
+recognize disinformation, and in their ability to identify AI versus human-generated text using a 1 to 5 Likert 
+scale (Figure 4A,B). The Disinformation Recognition Confidence before the test was higher than AI 
+Recognition Confidence before the test (3.05 versus 2.69, respectively). After taking the survey (but before 
+seeing their results), we asked again respondents to define how confident they were in their ability to 
+recognize disinformation and AI versus human-generated text. Respondents were more confident in their 
+ability to recognize disinformation (Pre versus Post, 3.05 versus 3.49, respectively, p<0.0001) (Figure 4A), 
+whereas they were much less confident in their ability to recognize synthetic versus organic tweets (Pre 
+versus Post, 2.79 versus 1.70, respectively, p<0.0001) (Figure 4B). The increase in confidence to recognize 
+disinformation could be explained by the inoculation theory of misinformation17, whereby critical exposure 
+to disinformation could improve disinformation recognition capacity and resilience. On the opposite, the 
+stark decrease in confidence to recognize synthetic tweets could depend on what we could call “resignation 
+theory” – i.e., people may give up attempting to critically evaluate information when faced with a high 
+volume of information, especially when it is difficult to understand its origin or intentionality. This may lead 
+to a sense of hopelessness or apathy towards information consumption, and a tendency to rely on 
+heuristics or simple cues (such as the emotions evoked by the information).  
+Figures 
+ 
+ 
+Figure 1. GPT-3 AI model informs and disinform us better. Study set-up: GPT-3 was provided instruction 
+prompts to produce synthetic tweets containing accurate versus inaccurate information (information vs 
+
+A
+c
+c'
+Study dataset
+Disinformation Recognition Score (0-1)
+Instruction
+Organic True
+Synthetic tweets
+1.0
+1.0 -
+prompts
+GPT3
+ Synthetic True
+1
+ Organic False
+Twitter search
+Organic twe ets
+ Synthetic False
+0.5 -
+True or False?
+Organic or Synthetic?
+0.0
+Human
+atic
+ns
+respondents
+GPT3
+ns
+ns
+B
+D
+D'
+E
+ns
+ns
+ns
+ns
+869 responses
+(oas)
+Removed 157
+35 -
+35 -
+250-
+inco mplete
+ Average Time to Response
+30 -
+30 -
+Tweet Characters Count
+responses
+200-
+25.
+25 -
+712 responses
+20 -
+20 -
+150-
+Removed 15
+responses
+15-
+15
+100
+697 responses
+ from speeders
+10 -
+10-
+Included in
+ the analysis
+5-
+5-
+50-
+0.
+0.
+c
+nic
+etic
+Tue
+True
+2
+the
+gnAI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+6|29 
+disinformation). Through a Twitter search, organic tweets from Twitter users were retrieved and classified 
+as accurate information or disinformation. The sum of synthetic and organic tweets, either containing 
+information or disinformation, constitutes the study dataset. Human respondents were asked to recognize 
+whether such tweets were true or false (i.e., accurate information or disinformation) and whether the 
+tweets were organic (i.e., generated by Twitter users) or synthetic (i.e., generated by GPT-3). GPT-3 was 
+asked to recognize whether the tweets were true or false (i.e., accurate information or disinformation) (A). 
+Data collection: we gathered 869 responses to our survey. 157 responses were incomplete and were 
+removed. Of 712 remaining responses, 15 were removed as they were completed too fast to be reliable. 
+Our analysis was conducted on 697 complete and reliable responses (B). GPT-3 produces accurate and 
+disinformation tweets that are recognized by human respondents as accurate more often than accurate 
+and disinformation tweets, respectively, produced by humans. “Organic true” tweets (green column bars) 
+are accurate tweets generated by Twitter users; “Synthetic true” (dotted green column bars) tweets are 
+accurate tweets generated by GPT-3; “Organic false” (red column bars) tweets are disinformation tweets 
+generated by Twitter users; “Synthetic false” (dotted red column bars) tweets are disinformation tweets 
+generated by GPT-3 (C). Disinformation tweets (red column bars) are recognized more often correctly when 
+compared with accurate tweets (green column bars). Synthetic tweets (dotted grey column bars) are 
+recognized more often correctly when compared with organic tweets (grey column bars). Disinformation 
+recognition score (or TF score) (0-1) is the average score for all 697 respondents (1= 100% correct answers; 
+0 = 0% correct answers); Ordinary one-way ANOVA multiple-comparisons Tukey’s test, n=697, ** p<0.01; 
+****p<0.0001. Error bars = SEM. (C’). Average time to response in seconds for “organic true”, “synthetic 
+true”, “organic false” and “synthetic false” tweets. Organic true tweets required the most time to receive 
+an evaluation from respondents, whereas synthetic true and synthetic false tweets required the least time 
+to receive an evaluation (D). True tweets (i.e., accurate) required longer to be evaluated by respondents 
+than false tweets (i.e., containing disinformation), and organic tweets required more time to be evaluated 
+when compared with synthetic tweets. Ordinary one-way ANOVA multiple-comparisons Tukey’s test, 
+n=697, ns = non-significant, p>0.05; *p<0.05; ****p<0.0001. Error bars = SEM. (D’). The different time 
+required to evaluate each type of tweet does not depend on the length of tweets, since the tweet 
+characters count is no different between types. Ordinary one-way ANOVA multiple-comparisons Tukey’s 
+test, n(total)=220, n(type)= 55; ns = non-significant, p>0.05 (E). 
+ 
+ 
+Figure 2. Humans evaluate information and disinformation better than GPT-3, and GPT-3 can “disobey” 
+requests to generate disinformation. Green column bars represent successful responses given by human 
+respondents, whereas green dotted bars represent successful responses given by GPT-3. Red bars 
+represent incorrect responses from human respondents, whereas red dotted bars represent incorrect 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+7|29 
+responses from GPT-3. The success rate concerning the evaluation of disinformation is 89% and 92% for 
+GPT-3 and human respondents, respectively. The success rate concerning the evaluation of accurate 
+information is 64% and 72% for GPT-3 and human respondents, respectively. The evaluation was conducted 
+on organic tweets retrieved from Twitter which were included in our survey (A). Rate of “obedience” for 
+GPT-3 – i.e., how often GPT-3 respected our request to generate information or disinformation tweets. For 
+accurate information tweets, GPT-3 “obeyed” our request 99 times out of 101 requests, whereas for 
+disinformation tweets, it “obeyed” our request 80 times out of 102 requests (B). 
+ 
+ 
+ 
+Figure 3. Human respondents cannot distinguish organic versus synthetic tweets, but recognize their 
+origin better when they are generated by humans. AI recognition score for “organic true” (green bars), 
+“synthetic true” (green dotted bars), “organic false” (red bars) and “synthetic false” (red dotted bars) 
+tweets. AI recognition score (0-1) indicates the probability that human respondents can identify whether a 
+tweet is produced organically (i.e., by a Twitter user), or synthetically (i.e., by GPT-3). Human respondents 
+recognize whether organic true tweets are organic or synthetic tweets more effectively than all other type 
+of tweets, whereas synthetic true tweets are recognized correctly the least (A). Human respondents cannot 
+predict whether true or false tweets (i.e., accurate tweets or disinformation tweets, green versus red bars) 
+are produced by humans or by GPT-3, and the truthfulness of the information does not have an impact on 
+the AI recognition score. Regarding organic versus synthetic tweets (grey versus grey dotted bars), human 
+respondents recognize whether tweets are generated by humans or GPT-3 better when they are organic 
+(i.e., generated by Twitter users), when compared with synthetic tweets (i.e., generated by GPT-3). AI 
+recognition score (0-1) is the average score for all 697 respondents (1= 100% correct answers; 0 = 0% 
+correct answers); Ordinary one-way ANOVA multiple-comparisons Tukey’s test, n=697; ns = non-significant, 
+p>0.05; **p<0.01; ***p<0.001; ***p<0.0001. Bars represent SEM. 
+ 
+ 
+
+A
+B
+c
+EFFICIENT VS INEFFICIENT
+c'
+X
+I Recognition Confidence (1-5)
+5
+5
+Initiation
+Humans
+Disinformation Recognition
+Initiation
+Confidence (1-5)
+4
+4
+Information
+Information
+GPT3
+campaign
+campaign
+3
+3
+Is it information
+Humans
+Isit information
+2
+2
+or disinformation?
+or disinformation?
+Evaluation
+Evaluation
+GPT3
+nfidenc
+白
+Pre
+Po
+st.AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+8|29 
+ 
+Figure 4. The Confidence in recognizing disinformation increases post-survey, whereas the confidence in 
+recognizing AI-generated information decreases; and proposed model to launch information campaigns 
+and evaluate information. Respondents were asked to provide a score of how confident they were in their 
+ability to recognize disinformation tweets before taking the survey (grey bar), and after taking the survey 
+(black bar). Participants’ confidence in disinformation recognition increased significantly from 3.05 to 3.49 
+out of 5. n=697; Welch’s t-test, ****p<0.0001. Bars represent SEM (A). Respondents were asked to provide 
+a score of how confident they were in their ability to recognize whether tweets were generated by humans 
+(grey bar) or by AI (black bar). Participant’s confidence in AI recognition dropped significantly from 2.69 to 
+1.7 out of 5. n=697; Welch’s t-test, ****p<0.0001. Bars represent SEM (B). Model for an efficient and 
+inefficient communication strategy and launch of information campaign. Based on our data, and with the AI 
+model adopted for our analysis, an efficient system relies on accurate information generated by GPT-3 
+(initiation phase), whereas it relies on trained humans to evaluate whether a piece of information is 
+accurate or whether it contains disinformation (evaluation phase) (C). An inefficient system relies on 
+humans to generate information and initiate an information campaign, and it relies on AI to evaluate 
+whether a piece of information is accurate or whether it contains disinformation (C’). 
+Discussion 
+How to communicate and evaluate information 
+Our findings show that tweets produced by GPT-3 can both inform and disinform better than organic 
+tweets. Synthetic tweets containing reliable information are recognized as true better and faster than true 
+organic tweets, while false synthetic tweets are recognized as false worse than false organic tweets. 
+Moreover, GPT-3 does not perform better than humans in recognizing both information and 
+disinformation. The results suggest that GPT-3 may be more efficient at informing because it is able to 
+generate text that is simpler to read and understand when compared with text written by humans. Based 
+on these results, we propose a model for efficient communication and evaluation of information that 
+challenges the current approach and consensus, according to which humans produce information and AIs 
+assist in the evaluation18 (Figure 4C, C’). A well-tailored information campaign can be shaped by providing 
+instruction prompts to GPT-3, which produces effective information campaigns targeting humans (initiation 
+phase). The accuracy of information is then evaluated by trained humans (Figure 4C). Instead, information 
+campaigns written and prepared by humans would turn out to be less effective, and AIs would perform an 
+inefficient evaluation of how truthful and reliable information is (Figure 4C’). The proposed model is of 
+relevance in the context of a public health crisis and infodemic, given the need to communicate fast and 
+clearly to large segments of the public. 
+“Disobedience”, training datasets, and error propagation 
+Our results show that, when asked, GPT-3 is less likely to produce misinformation on some specific topics, 
+e.g.: vaccines and autism (Figure S7). Being GPT-3 a statistical representation of language – for how 
+language is used in the datasets it was trained on – we assume GPT-3’s “disobedience” depends on the 
+composition of GPT-3’s training datasets. If the training dataset contains volumes of information 
+contradicting what the prompt asks for, the system will likely output that type of information. We can 
+therefore assume that the volume of information in the training dataset debunking causal links between 
+vaccines and autism is higher than the volume of information debunking conspiracy theories on other 
+topics taken into consideration by our study. Some control on the material fed into the training datasets is 
+therefore crucial. GPT-3 is trained on data obtained from Common Crawl, WebText2, Books1, Books2 and 
+Wikipedia19, which could include also misinformation and disinformation. To reduce the risk of generating 
+disinformation, we suggest that future text transformers should be trained on datasets regulated by the 
+principles of accuracy and transparency: information entering the training datasets should be verified, and 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+9|29 
+its origin should be open for independent scrutiny. Finally, the output of models trained on accurate and 
+transparent datasets should report the sources used for its generation, thus increasing transparency, and 
+allowing independent fact checking. Fact checking may still be difficult given the amount of information 
+that likely serves as a source but nonetheless declaring sources would be a good start.  
+‘As human as humans’: synthetic text identification and impersonation 
+In line with previous research20, we found that both respondents and GPT-3 were not able to distinguish 
+whether a tweet was organic or synthetic (data on GPT-3’s assessment are available in our study’s 
+repository)21. It might be possible to develop specific training courses to improve humans’ recognition of 
+synthetic text, based on linguistic markers, grammatical structure, and syntax. However, since the release 
+of ChatGPT (an interactive, conversational, and even simpler interface to GPT-3), users started to search for 
+ways to circumvent OpenAI’s content policy blocks. A common (and effective) strategy is impersonation: 
+when GPT-3 refuses to generate output that could violate the content policy, users just ask it to 
+impersonate a character – for which, apparently, content policies do not apply22. With this approach, even 
+more credible swathes of disinformation could be produced by first asking GPT-3 to generate fake profiles 
+of people to impersonate, and in a second iteration, to generate tweets that these profiles could write. 
+Besides circumventing content policy blocks, this would add even more ‘human-like’ feel to the tweets, and 
+make it even harder to identify them as synthetic. Based on these premises, synthetic text identification 
+might soon be a hopeless battle to fight, for both people and AIs. 
+Resignation theory 
+According to our results, not only humans cannot distinguish synthetic text from organic text, but also the 
+confidence in their ability to do so decreases significantly after having been asked to recognize the different 
+origin. This decrease in self confidence after exposure to both synthetic and organic texts may be due to 
+the realization that there is no clear marker that allows users to identify whether a text has been generated 
+by a machine or a human. This is likely because of GPT-3's ability to mimic human writing styles and 
+language patterns. Additionally, respondents may have initially underestimated GPT-3’s abilities to write 
+human-like text: this may be due to the fact that such technology is new and revolutionary, and people are 
+not yet accustomed to how powerful it can be. Another possible interpretation is that the survey may have 
+made participants more aware of GPT-3’s potential to generate disinformation with a human-like feel, 
+making them more sceptical of both synthetic and organic information, and thus decreasing their 
+confidence in their ability to identify organic text as well. 
+Beyond Twitter 
+We decided to focus our study on tweets for the following reasons: Twitter is currently used by over 368 
+million monthly active users 23 who use the platform several times a day24 to consume mostly news and 
+political information 24,25. Furthermore, Twitter offers a very simple application programming interface (API) 
+to develop bots, i.e.: programs able to post content and interact with posts or users without human 
+supervision26. Recent research shows that only about 5% of Twitter users are bots – but that these bots 
+cumulatively account for 20% - 29% of the content posted on Twitter27. Because of these characteristics, 
+Twitter is the ideal target – and potentially a very vulnerable one – for AI-generated swathes of 
+disinformation. Overall, our findings raise important questions about the potential uses and misuses of 
+GPT-3 and other advanced AI text generators, and the implications for information dissemination in the 
+digital age, particularly in relation to the spread of disinformation, particularly on social media. It is 
+important to note that while we focused on tweets in this study, our results could be extended to other 
+social media platforms and other forms of communication that can be used by bots via APIs, and that could 
+be exploited to programmatically disseminate AI-generated disinformation. In fact, we generated tweet-
+like social media posts that we call tweets, but in fact have features shared in other type of social media 
+posts, such as Instagram or Facebook posts.  
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+10|29 
+The genie is out of the bottle 
+Starting from our findings, we predict that advanced AI text generators such as GPT-3 could have the 
+potential to greatly impact the dissemination of information, both positively and negatively. To mitigate 
+negative effects, taking action to regulate which training datasets are used to develop these technologies is 
+crucial, thus ensuring transparency, truthfulness of the output information, and limiting misuse of the 
+technology to generate deceiving information. Additionally, until we do not have efficient strategies for 
+identifying disinformation (whether based on human skills or on future AI improvements) it might be 
+necessary to restrict the use of these technologies, e.g.: licensing them only to trusted users (e.g., research 
+institutions), or limiting the potential of AIs to certain type of applications. Finally, it is crucial that we 
+continue to critically evaluate the implications of these technologies and take action to mitigate any 
+negative effects they may have on society. 
+References 
+1. 
+Brown, T. B. et al. Language Models are Few-Shot Learners. (2020) doi:10.48550/arXiv.2005.14165. 
+2. 
+Dale, R. GPT-3: What’s it good for? Nat. Lang. Eng. 27, 113–118 (2021). 
+3. 
+GPT-3. Update: Some Replies by GPT-3. Daily Nous https://dailynous.com/2020/07/30/philosophers-
+gpt-3/ (2020). 
+4. 
+Benzon, W. L. GPT-3: Waterloo or Rubicon? Here be Dragons. 
+https://papers.ssrn.com/abstract=3667608 (2020). 
+5. 
+Marlow, R. & Wood, D. Ghost in the machine or monkey with a typewriter—generating titles for 
+Christmas research articles in The BMJ using artificial intelligence: observational study. BMJ 375, 
+e067732 (2021). 
+6. 
+Elkins, K. & Chun, J. Can GPT-3 Pass a Writer’s Turing Test? J. Cult. Anal. 5, 17212 (2020). 
+7. 
+Chiu, K.-L., Collins, A. & Alexander, R. Detecting Hate Speech with GPT-3. ArXiv210312407 Cs (2022). 
+8. 
+Ugli, M. I. B. Will Human Beings Be Superseded by Generative Pre-trained Transformer 3 (GPT-3) in 
+Programming? Int. J. Orange Technol. 2, 141–143 (2020). 
+9. 
+Cabanac, G., Labbé, C. & Magazinov, A. Tortured phrases: A dubious writing style emerging in science. 
+Evidence of critical issues affecting established journals. ArXiv210706751 Cs (2021). 
+10. Dehouche, N. Plagiarism in the age of massive Generative Pre-trained Transformers (GPT-3). Ethics 
+Sci. Environ. Polit. 21, 17–23 (2021). 
+11. Mindzak, M. & Eaton, S. E. Artificial intelligence is getting better at writing, and universities should 
+worry about plagiarism. The Conversation http://theconversation.com/artificial-intelligence-is-
+getting-better-at-writing-and-universities-should-worry-about-plagiarism-160481 (2021). 
+12. Forge, J. A Note on the Definition of “Dual Use”. Sci. Eng. Ethics 16, 111–118 (2010). 
+13. WHO. Immunizing the public against misinformation. https://www.who.int/news-room/feature-
+stories/detail/immunizing-the-public-against-misinformation (2020). 
+14. Roozenbeek, J., Suiter, J. & Culloty, E. Countering Misinformation: Evidence, Knowledge Gaps, and 
+Implications of Current Interventions. Preprint at https://doi.org/10.31234/osf.io/b52um (2022). 
+
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+15. Sobieszek, A. & Price, T. Playing Games with Ais: The Limits of GPT-3 and Similar Large Language 
+Models. Minds Mach. 32, 341–364 (2022). 
+16. Floridi, L. A Defence of Constructionism: Philosophy as Conceptual Engineering. Metaphilosophy 42, 
+282–304 (2011). 
+17. Cook, J., Lewandowsky, S. & Ecker, U. K. H. Neutralizing misinformation through inoculation: Exposing 
+misleading argumentation techniques reduces their influence. PLOS ONE 12, e0175799 (2017). 
+18. Ahmad, T., Aliaga Lazarte, E. A. & Mirjalili, S. A Systematic Literature Review on Fake News in the 
+COVID-19 Pandemic: Can AI Propose a Solution? Appl. Sci. 12, 12727 (2022). 
+19. Cooper, K. OpenAI GPT-3: Everything You Need to Know. Springboard Blog 
+https://www.springboard.com/blog/data-science/machine-learning-gpt-3-open-ai/ (2021). 
+20. Clark, E. et al. All That’s ‘Human’ Is Not Gold: Evaluating Human Evaluation of Generated Text. 
+Preprint at https://doi.org/10.48550/arXiv.2107.00061 (2021). 
+21. Spitale, G., Germani, F. & Biller-Andorno, N. Can AI Disinform Us Better? (2022). 
+22. Zack Witten [@zswitten]. Pretending is All You Need (to get ChatGPT to be evil). A thread. Twitter 
+https://twitter.com/zswitten/status/1598088267789787136 (2022). 
+23. Dixon, S. Twitter: number of worldwide users 2019-2024. Statista 
+https://www.statista.com/statistics/303681/twitter-users-worldwide/ (2022). 
+24. Kjarval, T. R., Jeff Sonderman, Kevin Loker, Maria Ivancin, Nina. How people use Twitter in general. 
+American Press Institute https://www.americanpressinstitute.org/publications/reports/survey-
+research/how-people-use-twitter-in-general/ (2015). 
+25. Twitter news. How many people come to Twitter for news? As it turns out, a LOT. 
+https://blog.twitter.com/en_us/topics/insights/2022/how-many-people-come-twitter-for-news 
+(2022). 
+26. Garson, J. How to create a Twitter bot with Twitter API v2. developer.twitter.com 
+https://developer.twitter.com/en/docs/tutorials/how-to-create-a-twitter-bot-with-twitter-api-v2 
+(2023). 
+27. Carr, D. F. Bots Likely Not A Big Part of Twitter’s Audience — But Tweet a Lot. Similarweb 
+https://www.similarweb.com/blog/insights/twitter-bot-research-news/ (2022). 
+28. Shaver, L. G. et al. Using Facebook Advertising to Recruit Representative Samples: Feasibility 
+Assessment of a Cross-Sectional Survey. J. Med. Internet Res. 21, (2019). 
+Supplementary material 
+Methods 
+Definition of the topics 
+As the focus of this study, we initially identified 14 topics on which disinformation exists. This preliminary 
+list included: 
+• 
+Climate change; 
+• 
+Vaccines safety; 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+12|29 
+• 
+Theory of evolution; 
+• 
+COVID-19; 
+• 
+Masks safety; 
+• 
+Vaccines and autism; 
+• 
+Homeopathic treatments for cancer; 
+• 
+Flat Earth; 5G technology and COVID-19; 
+• 
+Bill Gates and population control; 
+• 
+Antibiotics and viral infections; 
+• 
+COVID-19 = influenza; 
+• 
+Inferior human races; 
+• 
+Moral AI. 
+Generation of synthetic tweets 
+Based on the list defined above, we generated synthetic tweets passing input to GPT-3 via API (Application 
+Programming Interface). The code asks to generate 10 true tweets and 10 false tweets for each of the 
+topics detailed above (e.g.: prompt: ‘Write a tweet to explain why climate change is real’, category: 
+‘Climate change’). The tweet generation code consists of one function to pass input prompts to GPT-3, and 
+of two different loops to iterate over categorized prompts. The first function defines the parameters to 
+pass to GPT-3 (temperature, max_token, top_p, best_of, frequency_penalty, presence_penalty), empirically 
+defined in an iterative process as the most apt to produce text that resembles social media content. GPT-
+3's API returns also the reason for termination (e.g.: reaching the length specified in max_tokens). For these 
+cases, the text sometimes contains unfinished sentences: these have been removed. The loops to generate 
+true and false tweets read input organized in .csv files (prompt and category) and generate the given 
+number of texts per each prompt (in this example, 10). The output is then exported as a .xlsx file containing 
+three columns: the text, the reason for termination, and the category. All the code, available in this study’s 
+pre-registration repository, is organized in commented Jupyter lab notebooks for scrutiny and replication21. 
+The prompts and the output are available in the same repository.  
+Definitions 
+Throughout the manuscript we adopt – and sometimes explain for added clarity – the terminology “true” 
+and “false” tweets. True tweets are those tweets containing accurate information, and false tweets are 
+those containing inaccurate information, i.e., disinformation. 
+As for the definition of accurate information and disinformation, we base ourselves on the current scientific 
+knowledge and understanding of the topics and information under scrutiny. To avoid dubious and 
+debatable cases, which may be subject to personal opinions and interpretations, we only analyzed and 
+added to our questionnaire those tweets containing information that is clearly categorizable as true or 
+false. Of note, if a tweet contained partially incorrect information – meaning it contained more than 1 
+pieces of information, and at least one was incorrect, it was labelled as false. As discussed in the 
+introductory section of the manuscript, we acknowledge that the definition of disinformation and 
+misinformation is diverse, but we refer to an inclusive definition, which considers false information (also 
+partially false information) and/or misleading content14.  
+Retrieval of organic tweets 
+Using Twitter's advanced search, we collected a random sample of recent organic tweets on the topics 
+listed above, including both true and false tweets. Our initial aim was to collect 50 tweets per category; 
+however, this proved impossible for some categories, for various reasons – for example, for some 
+categories tweets were ambiguous and difficult to categorize as true or false. For other categories, we were 
+not able to retrieve enough tweets. Some categories were therefore dropped and excluded from the 
+following phases of the study. 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+13|29 
+Category 
+Organic tweets retrieved 
+Climate change 
+50 
+Vaccines safety 
+50 
+Theory of evolution 
+49 (1 duplicated tweet was 
+excluded) 
+COVID-19 
+50 
+Masks safety 
+50 
+Vaccines and autism 
+50 
+Homeopathic treatments for 
+cancer 
+30 
+Flat Earth 
+50 
+5G technology and COVID-19 
+50 
+Bill Gates and population control 
+0 (dropped) 
+Antibiotics and viral infections 
+50 
+COVID-19 = influenza 
+50 
+Inferior human races 
+10 (dropped) 
+Moral AI 
+0 (dropped) 
+Table 1. Organic tweets retrieved, by category 
+The tweets are available in the study’s repository21.  
+Expert assessment of synthetic and organic tweets 
+We evaluated synthetic and organic tweets to assess whether they contained disinformation. The expert 
+assessment was performed independently by FG and GS, and a following joint analysis was conducted by 
+FG and GS to verify the correctness of their initial assessments. 
+Selection of the tweets to include in the survey and generation of tweet images 
+Based on the assessments defined above, we selected the following tweets for each category: 
+• 
+5 synthetic false; 
+• 
+5 synthetic true; 
+• 
+5 organic false; 
+• 
+5 organic true. 
+We selected only tweets for which FG and GS agreed in their evaluation, following the expert assessment 
+phases. This resulted in a dataframe of 220 tweets (available in our repository21) used to generate the 
+images of the tweets. The code generates a random pseudonym and a random username for each tweet 
+(e.g.: ‘John S.’, @john_s), and creates an image which resembles the screenshot of a tweet. The code, the 
+dataframe containing the tweets, and the output images are available in the study’s repository21. 
+AI assessment of tweets 
+The AI assessment was performed by GPT-3 (true/false evaluation and organic/synthetic evaluation). The 
+first evaluation function defines the parameters to pass to GPT-3 to produce a 'true/false evaluation' (i.e.: 
+whether the tweet is true or false). The second evaluation function defines the parameters to pass to GPT-3 
+to produce an 'organic/synthetic evaluation' (i.e.: whether the tweet was written by a person or by an AI). 
+The loops for evaluation read the content of the files containing the tweets and evaluate them. The output 
+is scored (i.e.: whether GPT-3’s assessment matches the expert assessment for true/false and whether it 
+matches the origin of the tweet for the organic/synthetic classification) and then exported as a .xlsx file. 
+The code and the files containing the assessments are available in the study’s repository21. 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+14|29 
+Programming of the survey 
+We programmed a Qualtrics survey to collect demographics, display the tweets to the respondents, and 
+collect their assessments (true versus false, organic versus synthetic). For each tweet, respondents assess: 
+• 
+Whether it is accurate or contains disinformation (single choice, accurate/misinformation); 
+• 
+Whether it was written by a real person or generated by a computer (single choice, real 
+person/computer). 
+Additionally, respondents provide: 
+• 
+Some demographic information (nationality, age, sex, education level, education field) 
+• 
+Self-perceived (pre and post survey) ability to recognize, respectively, disinformation and synthetic 
+text (Likert scale, 1 - very difficult - 5 - very easy) 
+The images of the tweets are organized in nested randomizers within the survey structure: 
+• 
+the first level randomizer randomizes the category order (climate change, ...). All the categories are 
+displayed to every respondent. 
+• 
+second level randomizers (for each category) randomize the single tweet displayed for each 
+category to the respondent. Each category comprises a total of 20 tweets: 5 synthetic false, 5 
+synthetic true, 5 organic false, 5 organic true. The second level randomizers evenly present one 
+tweet from the pool of 20 tweets. 
+The survey adopts a gamified approach to keep respondents engaged: at the beginning of the survey, 
+respondents are told that, upon competition, they will obtain their score for both scales (disinformation 
+recognition, and synthetic text recognition). This ensured a low dropout rate. In-survey scoring is achieved 
+using the 'scoring' function in Qualtrics. The survey file and structure are available in the study’s 
+repository21. 
+Pilot testing 
+We pilot tested the survey in two phases. During the first phase we circulated the link to a convenience 
+sample with the aim to test the usability and the layout. This led to minor modifications in the interface and 
+in the wording. During the second phase we distributed the link via a Facebook ads campaign, structured as 
+follows: 
+• 
+Daily budget: 15€ 
+• 
+Start: 04.10.2022 
+• 
+End: 13.10.2022 
+• 
+Age: 16 - 65+ 
+• 
+Languages: English 
+• 
+Title: True or False? Organic or synthetic? 
+• 
+Description: Are you able to distinguish text written by an artificial intelligence from text written by 
+a human being? And accurate information from misinformation? Find out with this test, and 
+contribute to research on information ethics. 
+• 
+Image: generated by DALL·E 2 (available in this study’s repository21) 
+The campaign had a total cost of 122.92€. It generated a total of 593 clicks on the link (cost per click: 0.21€) 
+and a total of 276 responses (cost per response: 0.46€). The campaign was launched and completed in 
+October 2022. 
+Sample size and power analysis 
+Based on pilot data, we conducted a power analysis to determine the sample size for the full study. 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+15|29 
+Primary endpoint hypothesis 
+Disinformation produced by a machine is more credible than disinformation produced by a human 
+(synthetic versus organic disinformation). 
+Secondary endpoints hypotheses 
+1. Accurate information produced by a machine is more credible than accurate information produced 
+by a human (synthetic versus organic accurate information). 
+2. Users recognize and distinguish information produced by humans and by machines (regardless of 
+the truthfulness of the information). 
+3. The confidence of respondents in recognizing disinformation increases after the completion of the 
+questionnaire. 
+4. The confidence of respondents in recognizing synthetic versus organic information increases after 
+the completion of the questionnaire. 
+Power analysis 
+Based on the data resulting from the pilot study, available in the study’s repository21, we performed a 
+power analysis to estimate the sample size necessary to draw sufficiently meaningful conclusions for 
+Primary and Secondary Endpoints (PE and SEs). Endpoints are continuous, and the study runs on two 
+independent samples. 
+Primary endpoint: Results 
+Average group 1 Score (Synthetic tweets, disinformation)* = 0.86 
+* From 0 to 1, the score indicates how good the performance was in recognizing synthetic tweets 
+containing disinformation. 
+Stdev group 1 = 0.25 
+Average group 2 Score (Organic tweets, disinformation)* = 0.89 
+* From 0 to 1, the score indicates how good the performance was in recognizing organic tweets containing 
+disinformation. 
+Enrollment ratio = 1.01194 
+Alpha = 0.05 Power = 80% 
+Sample Size Total = 2181 (Group 1: 1084; Group 2: 1097) 
+Secondary Endpoint 1 
+Average group 1 Score (Synthetic tweets, accurate information)* = 0.78  
+* From 0 to 1, the score indicates how good the performance was in recognizing synthetic tweets 
+containing accurate information. 
+Stdev group 1 = 0.35 
+Average group 2 Score (Organic tweets, accurate information)* = 0.64 
+* From 0 to 1, the score indicates how good the performance was in recognizing organic tweets containing 
+accurate information. 
+Enrollment ratio = 0.991045 
+Alpha = 0.05 Power = 80% 
+Sample Size Total = 197 (Group 1: 99; Group 2: 98) 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+16|29 
+Secondary Endpoint 2 
+Average group 1 Score (Synthetic tweets [accurate information + disinformation])* = 0.315 
+* From 0 to 1, the score indicates how good the performance was in recognizing synthetic tweets, 
+regardless of whether they contained accurate information or disinformation. 
+Stdev group 1 = 0.44 
+Average group 2 Score (Organic tweets, [accurate information + disinformation])* = 0.59 
+* From 0 to 1, the score indicates how good the performance was in recognizing organic tweets, regardless 
+of whether they contained accurate information or disinformation. 
+Enrollment ratio = 1.001493 
+Alpha = 0.05 Power = 80% 
+Sample Size Total = 80 (Group 1: 40; Group 2: 40) 
+Secondary Endpoint 3 
+Average group 1 Score (Pre-confidence level in ability to recognize disinformation)* = 2.932271 
+* From 1 to 5 
+Stdev group 1 = 0.829093 
+Average group 2 (Post-confidence level in ability to recognize disinformation)* = 3.319149 
+* From 1 to 5 
+Enrollment ratio = 1.0680 
+Alpha = 0.05 Power = 80% 
+Sample Size Total = 145 (Group 1: 70; Group 2: 75) 
+Secondary Endpoint 4 
+Average group 1 (Pre-confidence level in ability to recognize synthetic versus organic contents)* = 2.703557 
+* From 1 to 5 
+Stdev group 1 = 0.897012 
+Average group 2 (Post-confidence level in ability to recognize synthetic versus organic contents)* = 1.75 
+* From 1 to 5 
+Enrollment ratio = 1.0720 
+Alpha = 0.05 Power = 80% 
+Sample Size Total = 27 (Group 1: 13; Group 2: 14) 
+Sample size evaluation 
+Taking the larger sample size resulting from our power analyses (n=2181 assessments for PE), and 
+considering that we obtained 1348 assessments (organic, disinformation + synthetic, disinformation), and 
+considering that the pilot study has generated full responses from 277 respondents, the ratio between 
+target power (number of assessments) and sample size of the pilot study (number of assessments) is 
+1.617953. Therefore, the number of users required for the study is 277*1.617953 = 448.1728. We 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+17|29 
+established that the minimum number of respondents to achieve a properly powered analysis in the full 
+study is n=449. 
+Data collection 
+We distribute the survey via different Facebook ads campaigns in order to compensate for some 
+demographic imbalances we noted from the pilot data (overrepresentation of women, 
+underrepresentation of people aged 18 - 54) 28. The campaigns took place in October and November 2022. 
+We used a total budget of 492.24€, distributed as detailed in the following table: 
+Campaign 
+Age 
+Sex 
+Visualizations Cost 
+USA, GBR, AUS, NZL, CAN 
+18-54 
+All 
+7226 
+35.22€ 
+USA, GBR, AUS, NZL, CAN 
+16-65+ 
+M 
+9907 
+34.24€ 
+USA, GBR, AUS, NZL, CAN 
+16-65+ 
+All 
+14710 
+33.78€ 
+USA, GBR, AUS, NZL, CAN 
+16-25 
+M 
+83525 
+88.00€ 
+USA, GBR, AUS, NZL, CAN 
+16-25 
+F 
+57780 
+44.00€ 
+USA, GBR, AUS, NZL, CAN 
+26-41 
+M 
+8787 
+22.00€ 
+USA, GBR, AUS, NZL, CAN 
+26-41 
+F 
+9544 
+31.00€ 
+USA 
+26-41 
+F 
+21046 
+31.00€ 
+USA 
+26-41 
+M 
+58146 
+93.00€ 
+USA 
+16-25 
+All 
+99899 
+80.00€ 
+Table 2. Facebook dissemination campaigns for data collection. 
+Our recruitment strategy aims to enroll a population of active social media users by utilizing a social media 
+platform. Due to this design, we were unable to recruit a representative sample upfront. Instead, we chose 
+to assess representativeness through a "rolling assessment" of demographics by targeting different 
+segments of the population in sequential campaigns based on the demographics of already recruited 
+participants28. 
+Analysis 
+Scoring and analysis are implemented in Python, using a Jupyter notebook. The code takes as input the 
+results of our Qualtrics survey and generates the files needed for the analysis as output. The code is 
+available for scrutiny and replication in the study’s repository21.  
+Cleaning 
+Data are cleaned removing incomplete responses, responses resulting from preview links, and responses 
+submitted in less than 170.5 seconds (determined empirically as the minimum possible time to complete 
+the survey – this was calculated as the average time required by a convenience sample to read, with a 
+sustained rhythm, the questions and answers or the survey, and answer the questions).  
+Scoring 
+True/false and organic/synthetic scores of each respondent are calculated by the rules defined in Qualtrics’ 
+survey programming; furthermore, they are re-calculated using the dataframe containing the tweets and 
+the expert assessments. True/false average scores of the tweets are calculated as follows: for true tweets, 
+the score is the average of the assessments; for false tweets, the score is 1 - the average of the 
+assessments. Organic/synthetic average scores of the tweets are calculated as follows: for organic tweets, 
+the score is the average of the assessments; for synthetic tweets, the score is 1 - the average of the 
+assessments.  
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+18|29 
+Inferential statistics 
+Correlation analyses are performed as follows: for quantitative/quantitative data arrays, we first perform a 
+Pearson’s test, followed by Shapiro’s test to determine data normality, and by both Wilcoxon’s test and a T-
+test for hypothesis testing. For qualitative/quantitative data arrays, we first perform an ANOVA test, 
+followed by Shapiro’s test to determine data normality, and by a Kruskal-Wallis test. Finally, we perform 
+multiple comparisons with a Tukey test. Effect sizes resulting from ANOVA and Kruskal-Wallis are 
+interpreted as small when η2 ≤ 0.01; medium when 0.01 < η2 < 0.06, and as large when η2 ≥ 0.14. 
+‘The hard ones’ 
+We defined tweets that were difficult to identify correctly for respondents (we called them ‘the hard ones’) 
+as follows. False identified as true: false tweets with average scores > 0.75; true identified as false: true 
+tweets with scores < 0.25; Synthetic identified as organic: synthetic tweets with average scores > 0.75; 
+organic identified as synthetic: organic tweets with scores < 0.25.  
+Supplementary Results 
+Correlations between study variables 
+We evaluated whether any correlation between numerical and categorical variables in our analysis existed 
+(Figure S12), as well as between numerical variables and other numerical variables (Figure S13).  
+OS score and demographics 
+We evaluated any potential correlation between OS Score and demographic variables, and identified the 
+age of respondents to be a relevant factor, with a small effect size (Figure S12A). Younger individuals (18-
+41), seem to perform slightly better at recognizing synthetic versus human tweets when compared with 
+very young individuals (16-17 years old), and especially older respondents (42+ years old) (Figure S12A’).  
+TF score and demographics 
+Similarly, we evaluated potential correlations between TF score and demographic variables. As for the OS 
+Score, also for the TF Score, age correlated with a small effect size, in addition to the education level of 
+respondents (Figure S12B). In this case, 42-57 years old individuals performed slightly better than older 
+individuals aged 58 to 76, although the distribution of TF scores per age seems to be quite uniform across 
+the board (Figure S12B’). As expected, a higher education level was associated with higher TF score. This 
+effect was small but consistent: participants holding a doctorate/PhD degree had higher scores when 
+compared with participants holding a Master’s degree, and those with a Master’s degree performed better 
+than respondents with a Bachelor’s degree, and so on (Figure S12B’’).  
+Self-confidence and demographics 
+Further, we evaluated the correlation between TF self-confidence PRE (i.e., the score of how confident 
+respondents were in their ability to recognize disinformation before the survey) and demographic variables 
+(Figure S12C); as well as the correlation between TF self-confidence POST (i.e., the score of how confident 
+respondents were in their ability to recognize disinformation after the survey) and demographic variables 
+(Figure S12D); and the correlation between OS self-confidence PRE (i.e., the score of how confident 
+respondents were in their ability to distinguish synthetic versus organic tweets disinformation before the 
+survey) and demographic variables (Figure S12E); and the correlation between OS self-confidence POST 
+(i.e., the score of how confident respondents were in their ability to distinguish synthetic versus organic 
+tweets disinformation after the survey) and demographic variables (Figure S12F).  
+OS / TF self-confidence delta and OS / TF score 
+For numerical versus numerical variables, we found no correlation between OS Delta (i.e., the difference in 
+confidence POST versus PRE in the ability to recognize AI-generated text) and OS Score (Figure S13A), but 
+we found a small but significant correlation between TF Delta (i.e., the difference in confidence POST versus 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+19|29 
+PRE in the ability to recognize disinformation) and TF Score (Figure S13B), suggesting that the higher the 
+score, the more respondents built confidence in their abilities, despite participants were only shown how 
+well they scored in the survey after evaluating their confidence level post-survey.  
+Duration and OS / TF scores 
+Further, we found no significant correlation between duration (i.e., how long respondents took to complete 
+the survey) and OS Score (Figure S13C), as well as between duration and TF Score (Figure S13D). 
+Supplementary Figures 
+ 
+ 
+ 
+Figure S1. Demographics data. Demographics from the study (n=697); Country of origin of respondents (A), 
+gender (B), age (C), education level (D), and, among those declaring at least a Bachelor’s degree, the field of 
+study (E). 
+
+A
+B
+c
+UK-
+Not disclosed -
+Not disclosed -
+Not disclosed
+Prefers not to answer
+Australia -
+Prefers not to answer-
+77+
+Country
+Canada-
+Gender
+Others
+Age
+58-76
+USA
+New Zealand -
+Non-binary / third gender-
+42-57-
+26-41-
+Ireland -+
+Male -
+Prefers not to answer-
+18-25-
+Others -+
+Female-
+16-17-
+0100 200 300 400
+0
+250
+500
+0
+100 200 300 400
+Number of respondents
+Number of respondents
+Number of respondents
+D
+E
+Not disclosed
+Social sciences and humanities-
+Prefers not to answer-
+Field of study
+Not disclosed -
+Education le
+PhD/Doctorate
+Prefers not to answer-
+Master's Degree.
+Medical sciences
+Bachelor's Degree
+High school graduate
+Others -
+Less than high school degree
+Natural sciences
+0
+100
+200
+ 300
+0
+125
+250
+Number of respondents
+Number of respondentsAI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+20|29 
+ 
+Figure S2. Disinformation Recognition Score per category of tweet. In the survey, for each category of 
+tweets, 20 tweets were included, of which 5 were “organic true”, represented with green bars, 5 “synthetic 
+true”, represented with green dotted bars, 5 “organic false”, represented with red bars, and 5 “synthetic 
+false”, represented with red dotted bars. For each category and type of tweet, we analyzed the success of 
+respondents in recognizing whether information contained in the tweet were accurate or inaccurate (i.e., 
+information or disinformation). For the categories “climate change”, ”vaccines safety”, “theory of 
+evolution”, “COVID-19 and influenza”, “vaccines and autism”, ”homeopathy and cancer”, “flat Earth”, “5G 
+and COVID-19”, “organic true” tweets were recognized the least correctly as accurate information (A-D, F-
+J), whereas for the categories “masks safety” and “antibiotics and viral infections”, “synthetic false” tweets 
+have the lowest score (E, K). Conversely, the highest score was generally relative to “organic false” tweets, 
+as in the case of “vaccines safety”, “masks safety”, “COVID-19 and influenza”, “homeopathy and cancer” 
+tweets (B, E, F, H), or “synthetic false” tweets, in the categories “climate change”, “theory of evolution”, 
+“COVID-19”, “vaccines and autism”, “flat Earth”, “5G and COVID-19” (A, C-D, G, I-J). An exception is the 
+category “antibiotics and viral infections”, in which “synthetic true” tweets were recognized correctly the 
+most as accurate, and “synthetic false” tweets were recognized the least as disinformation, when 
+compared with all other tweet types (K). n=5 for each type of tweet, for a total of n=20 for each category. 
+Ordinary one-way ANOVA multiple-comparisons Tukey’s test, ns = non-significant; *p<0.05; **p<0.01, 
+***p<0.001, ****p<0.0001. Bars represent SEM. 
+ 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+21|29 
+ 
+Figure S3. GPT-3 AI model informs and disinform us better (a single tweet level analysis). Confirming the 
+results of Figure 1, the Disinformation recognition score was not extracted from the average score for each 
+survey respondent, but rather from the average scores, for each type of tweet (i.e., “Organic true” depicted 
+with green bars, “synthetic true” depicted with green dotted bars, “organic false” depicted with red bars, 
+and “synthetic false” depicted with red dotted bars), for each tweet (20 tweets, 5 for each type). Organic 
+true tweets were recognized the least correctly (i.e., as accurate), when compared with other type of 
+tweets. Organic false tweets were recognized correctly the most, when compared with other type of 
+tweets (A). False tweets (i.e., disinformation tweets) were recognized to contain inaccurate information 
+correctly more often than true tweets (i.e., accurate tweets) (green versus red bars) were recognized to 
+contain accurate information. There was no significant difference in how often organic and synthetic 
+tweets (grey versus grey dotted bars) were recognized to contain accurate information or disinformation. 
+n=55 tweets based on n=5 tweets per type per category of tweet (11 categories, including, e.g., climate 
+change, antibiotics and viral infections, 5G and COVID-19, etc.). Ordinary one-way ANOVA multiple-
+comparisons Tukey’s test, ns = non-significant; *p<0.05; ***p<0.001; ****p<0.0001. Bars represent SEM 
+(A’). 
+ 
+ 
+
+Sandra D.
+@sandra_d
+Wearing a face mask may give you a
+false sense of security and prevent
+you from taking necessary precautions
+to protect yourself from COViD-19.Robert O.
+@robert_o
+Wearing a face mask may give you a
+false sense of security and prevent
+you from taking necessary precautions
+to avoid #COVID19.AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+22|29 
+Figure S4. Disinformation tweets recognized as accurate tweets most often. Disinformation tweets, either 
+generated by Twitter users or by GPT-3, which are most often recognized by human respondents as 
+accurate tweets (false tweets with mean TF score > 0.75).  
+ 
+ 
+Figure S5. Accurate tweets recognized as disinformation tweets most often. Tweets containing accurate 
+information, either generated by Twitter users or by GPT-3, which are most often recognized by human 
+respondents as disinformation tweets (true tweets with mean TF score < 0.25).  
+ 
+ 
+ 
+
+Christopher N.
+@christopher_n
+Thanks, my stepmom passed away from
+cancer after trusting all this stuff
+with her health. Sometimes homeopathy
+can be truly predatory, praying
+financially on the hope of the
+vulnerable.Sarah I.
+@sarah_i
+Homeopathy is based on the idea that
+"like cures like" and that diluting a
+substance makes it more potent.
+However. there is no scientific basis
+for this claim and homeopathic
+treatments have not been shown to be
+effective in treating cancer or any
+other illness.Mark T.
+@mark_t
+Vaccines are safe and effectiveAI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+23|29 
+ 
+Figure S6. Humans evaluate information and disinformation better than GPT-3 (a category breakdown). 
+Green column bars represent successful responses given by human respondents, whereas green dotted 
+bars represent successful responses given by GPT-3. Red bars represent incorrect responses from human 
+respondents, whereas red dotted bars represent incorrect responses from GPT-3. The success rate (0-1) is 
+used to compare humans’ versus GPT-3’s ability to recognize disinformation and accurate information. The 
+evaluation was conducted on organic tweets retrieved from Twitter which were included in our survey. In 
+line with “overall” results (A), human respondents performed better than GPT-3 in recognizing 
+disinformation related to “climate change”, “vaccines and autism”, “homeopathic treatments for cancer”, 
+“flat Earth”, “antibiotics and viral infections”, and “COVID-19 and influenza” (B, G-I, K, L). Instead, GPT-3 
+performed better than humans at recognizing disinformation in the categories “vaccines and safety”, 
+“theory of evolution”, “COVID-19”, “masks safety”, and “5G and COVID-19” (C-F, J). Concerning the correct 
+identification of accurate information, in line with “overall” results (A), human respondents performed 
+better than GPT-3 in the categories “COVID-19”, “masks safety”, “vaccines and autism”, “homeopathic 
+treatments for cancer”, “flat Earth”, “5G and COVID-19”, “antibiotics and viral infections”, and “COVID-19 
+and influenza” (E-L). Instead, GPT-3 performed better than human respondents at recognizing accurate 
+information for the categories “climate change”, “vaccines safety”, and “theory of evolution” (B-D). 
+ 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+24|29 
+ 
+Figure S7. GPT-3 Rate of “obedience” for each category. We calculated the number of requests 
+(instruction prompts) to produce tweets containing accurate information (dotted green) and disinformation 
+(dotted red), and the number of requests fulfilled (or “obeyed”) by GPT-3, for each category. For all 
+categories, as shown in Figure 2, GPT-3 produced accurate tweets 99 times/101, and disinformation tweets 
+80 times/102. For the categories “climate change”, “vaccines safety”, “theory of evolution”, “COVID-19”, 
+“masks safety”, “vaccines and autism”, “homeopathic treatment for cancer”, “flat Earth”, “5G and COVID-
+19”, “antibiotics and viral infections”, “COVID-19 and influenza”, accurate information tweets were 
+produced by GPT-3, respectively, 9/10, 10/10, 10/10, 10/10, 10/10, 10/10, 10/10, 10/10, 8/9, 9/9, 10/10 
+times, whereas disinformation tweets were produced, respectively, 10/10, 10/10, 10/10, 7/10, 8/10, 3/10, 
+5/9, 6/10, 10/10, 8/9, 3/4 times.  
+
+99/101
+10/10
+10/10
+10/10
+10/10
+10/10
+10/10
+10/10
+10/10
+10/10
+10/10
+10/10
+10/10
+6/6
+Request: accurate information
+9/10
+100
+80/102
+8/9
+8/9
+Request: disinformation
+8/10
+Rate of obedience (%)
+7/10
+3/4
+6/10
+5/9
+50
+3/10
+0
+cancer
+19
+ safety
+ pue
+ Earth
+and
+f evolution
+o
+COVID-
+for
+ treatment 
+Flat 
+COVID-
+-19
+Climate 
+Vaccines s
+For
+Theory
+5G and
+COVID-
+Vaccines
+Antibiotics AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+25|29 
+ 
+Figure S8. AI Recognition Score per category of tweet. In the survey, for each category of tweets, 20 
+tweets were included, of which 5 were “organic true”, represented with green bars, 5 “synthetic true”, 
+represented with green dotted bars, 5 “organic false”, represented with red bars, and 5 “synthetic false”, 
+represented with red dotted bars. For each category and type of tweet, we analyzed the success of 
+respondents in recognizing whether information contained in the tweet were generated by a human or by 
+GPT-3. For most categories, i.e., “theory of evolution”, “COVID-19”, “masks safety”, “COVID-19 and 
+influenza”, “vaccines and autism”, “homeopathy for cancer”, “flat Earth”, “5G and COVID-19”, “organic 
+true” tweets were recognized the most for being generated by a Twitter user (C-J), following the trend 
+observed when all categories of tweet are overlapped (L). Instead, for tweets concerning “climate change”, 
+and “vaccines safety”, the category “organic false” obtained the highest score (A, B). For the categories 
+“climate change”, “theory of evolution”, “COVID-19”, “COVID-19 and influenza”, “vaccines and autism”, 
+“homeopathy for cancer”, “flat Earth”, “5G and COVID-19”, and “antibiotics and viral infections”, “synthetic 
+true” tweets were recognized the least for being generated by AI, when compared with all other tweet 
+types (A-D, F-K). The only exception is the category “masks safety”, in which “synthetic false” tweets 
+obtained the lowest score (E). n=5 for each type of tweet, for a total of n=20 for each category. Ordinary 
+one-way ANOVA multiple-comparisons Tukey’s test, ns = non-significant; *p<0.05; **p<0.01, ***p<0.001, 
+****p<0.0001. Bars represent SEM. 
+ 
+ 
+ 
+ 
+
+AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+26|29 
+ 
+Figure S9. Human respondents cannot distinguish organic versus synthetic tweets, but recognize their 
+origin better when they are generated by humans (a single tweet level analysis). Confirming the results of 
+Figure 3, the AI recognition score was not extracted from the average score for each survey respondent, 
+but rather from the average scores, for each type of tweet (i.e., “Organic true, depicted with green bars 
+“synthetic true” depicted with green dotted bars, “organic false” depicted with red bars, and “synthetic 
+false” depicted with red dotted bars), for each tweet (20 tweets, 5 for each type). Organic true tweets were 
+recognized more often correctly to be generated by humans, whereas synthetic true tweets were 
+recognized correctly the least to be generated by GPT-3 (A). There was no significant difference in how 
+often true (i.e., accurate) and false (i.e., containing disinformation) tweets (green versus red bars) were 
+recognized correctly to be generated by GPT-3 or by a Twitter user. Organic tweets were recognized 
+correctly more often to be generated by a human when compared with how often synthetic tweets were 
+recognized correctly to be generated by GPT-3 (grey versus grey dotted bars). n=55 tweets based on n=5 
+tweets per type per category of tweet (11 categories, including, e.g., climate change, antibiotics and viral 
+infections, 5G and COVID-19, etc.). Ordinary one-way ANOVA multiple-comparisons Tukey’s test, ns = non-
+significant; *p<0.05; ****p<0.0001. Bars represent SEM. 
+ 
+ 
+Figure S10. Organic tweets recognized as synthetic most often. Tweets generated by Twitter users which 
+are most often recognized by human respondents as synthetic, i.e., generated by GPT-3 (organic tweets 
+with mean OS score < 0.25). 
+ 
+
+Barbara G.
+@barbara_g
+The goodness of homeopath can fight
+fatal ovarian and lung cancer.
+#homeopathy #health #cancer
+#beatcancerAI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+27|29 
+ 
+
+Sandra Z.
+@sandra_z
+Antibiotics can't treat viral
+infections, but they can treat
+bacterial infections that can
+sometimes occur when a virus is
+present.Mary W.
+@mary_w
+Climate change is real because we're
+seeing the effects with our own eyes.
+The weather is changing, sea levels
+are rising, and the planet is getting
+hotter. We need to take action now to
+protect our planet and future
+generations.Charles R.
+@charles_r
+There is no scientific evidence that
+vaccines cause autism.Vaccines are
+safe and effective and save livesKaren P.
+@karen_p
+Homeopathic treatments cannot cure
+cancer because they are nothing more
+than water and sugar pills. The
+treatments do not contain any of the
+active ingredients that would be
+necessary to effectively cure cancer.Joseph Z.
+@joseph_z
+Evolution is NoT a hoax. It's the
+scientific theory that explains how
+living things change over timeRichard G.
+@richard_g
+The #Covid19 pandemic is a hoax
+There's no evidence that it's a real
+virus, and the symptoms are identical
+to those of other common illnesses
+This is just another way to scare
+people into giving up their rights
+and freedoms.Karen M.
+@karen_m
+The earth is flat because it's easier
+to draw that way!Lisa E.
+@lisa_e
+#vaccines don't cause autism - they
+save lives.Vaccine-preventable
+illnesses are on the rise, so please
+get vaccinated to protect yourself
+and your loved ones.Michael D.
+@michael_d
+I can believe this my son has been
+sick since feb - flu + viral
+infections. I'm even considering
+keeping him home, it can't be normal
+for a 2yr old to be on antibiotics
+twice for 2 weeks in a row. I'll wait
+it out take him when he's 3 or 4. Any
+advice?Nancy A.
+@nancy_a
+Homeopathic treatments cannot cure
+cancer because they are based on the
+false premise that like cures like.
+There is no scientific evidence that
+this is true.James T.
+@james_t
+Vaccines do cause autism. It's not a
+coincidence that the number of autism
+cases have skyrocketed as the number
+of yaccines kids receive has
+increased. Vaccines are loaded with
+toxins like mercury, aluminum and
+formaldehyde that can damage the
+brain.Patricia N.
+@patricia_n
+The Earth's climate has always been
+changing, but human activities are
+now accelerating the process. Climate
+change is real, it's happening now,
+and it's a threat to our planet and
+our way of life.Linda L.
+@linda_!
+Climate change is real and it's
+happening right now. The Earth is
+getting warmer every year and it's
+causing more extreme weather
+conditions. We need to take action to
+reduce our emissions and protect our
+planet.John J.
+@john.j
+5G technology is not a cause of
+COVID-19. This technology is designed
+to improve internet connectivity and
+does not pose any health risksDaniel Q.
+@daniel_q
+The climate is changing and it's
+happening faster than we thought it
+would. The science is clear, the
+evidence is clear, and the impacts
+are already being felt. We have to
+act now to protect our planet and our
+children's future.AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+28|29 
+Figure S11. Synthetic tweets recognized as organic most often. Tweets generated by GPT-3 which are 
+most often recognized by human respondents as organic, i.e., generated by a Twitter user (synthetic tweets 
+with mean OS score > 0.75). 
+ 
+Figure S12. Correlations between demographics and other metrics. Correlation between 
+Organic/Synthetic Score (OS Score) and demographics. OS Score correlates with age with a small effect size. 
+(A). Young respondents (18-25 years old, and partly 26-41 years old) obtained higher AI Recognition scores 
+when compared with older respondents; Ordinary one-way ANOVA multiple-comparisons Tukey’s test; 
+
+Age
+A'
+A
+Correlation between OS score and demographics
+I Recognition Score (0-11)
+variables
+pval_anova
+eta_sq_anova
+pval_ shapiro
+ pval_ kruskal
+eta _sq_kruskal
+os_ score and Country
+0,216996
+0,030426
+3,66E-06
+0,204146
+0,006648
+os_score and Age
+8.78E-05****
+0,042713 (small)
+3,22E-06
+0,000228 ***
+0,030358 (small)
+os score and Gender
+0,618338
+0,005089
+7,34E-06
+0,487723
+-0,00081
+os_ score and Education
+0,510743
+0,007574
+0,000538
+0,464434
+-0,00052
+ os_score and Field
+0,578748
+0,006193
+1,23E-05
+ os_score and timecat
+0,596937
+0,001486
+6,34E-07
+0,669532
+-0,00173
+0-
+Educ ation
+B'
+B"
+B
+ Correlation between TF score and demographics
+Age
+Disinformation Recognition Score (0-11) 
+variables
+pval_anova
+eta_sq_anova
+pval_ s hapiro
+pval_kruskal
+eysnay bs era
+tf_score and Country
+0,768493
+0,018055
+3,12E-20
+0,731724
+-0,00579
+11
+tf_score and Age
+3,57E-06 ****
+0.052956 (small)
+1,05E-17
+0.00407 **
+0,020036 (small)
+tf_ score and Gender
+3,71E-05
+0,039569
+2,51E-19
+0,256441
+0,002241
+tf_score and Education
+1,83E-07 ****
+0,058906 (small)
+6,14E-17
+0.002931**
+0,02009 (small)
+tf_score and Study field
+0,566655
+0,006346
+3,47E-16
+tf_ score and timecat
+0,313104
+0,003341
+9,37E-22
+0.223816
+0,001432
+C
+Correlation between TF self-confidence PRE and demographics
+ variables
+pval_anova
+eta_sq_anova
+pval_ shapiro
+pval kruskal
+ eta_sq_kruskal
+tf_easy_start and Country
+0.004848 **
+0,051649 (small)
+2,35E-17
+0,023118 *
+0,020172 (small)
+ tf_easy_start and Age
+0,214099 ns
+0,013969
+5,28E-17
+0,152694 ns
+0,005443
+ tf_easy_start and Gender
+0,036661 *
+0,017262
+8,45E-22
+0,22206 ns
+0,002913
+tf_ easy_start and Education
+0,279765 ns
+0,010906
+1,91E-20
+0,672196 ns
+-0,0029
+tf_easy_start and Study field
+0,757311 ns
+0,004111
+1,95E-16
+tf_easy_start and timecat
+0.410423 ns
+0,002604
+3,21E-20
+0,551608 ns
+-0.00119
+D
+Correlation between TF self-confidence POST and demographics
+variables
+ pval _anova
+eta_ sq_ anova
+pval_ shapiro
+pval krus kal
+eta_sq kruskal
+tf_easy_end and Country
+0,061126 ns
+0,038895
+2,01E-16
+0,123444 ns
+0,010261
+tf_easy_end and Age
+1,87E-05 ****
+0,048474 (small)
+5,31E-14
+6,19E-05 ****
+0,035416 (small)
+tf_easy_end and Gender
+0,274725 ns
+0,009257
+7,01E-20
+0,235928 ns
+0,002647
+tf_easy_end and Education
+0,024213 *
+0.021115 (small)
+2,52E-17
+0.030166 *
+0,011713 (small)
+tf_easy_end and Study field
+0,111155 ns
+0,016305
+ 5,82E-14
+tf_easy_end and timecat
+0,027894 *
+0.010427 (small)
+2.42E-18
+0.02406 *
+0,007986 (small)
+E
+Correlation between OS self-confidence PRE and demographics
+ variables
+pval_anova
+eta_sq_anova
+pval_ shapiro
+pval_kruskal 
+eta_sq_kruskal
+os easy, start and Country
+0,00557 **
+0,05101
+6,68E-18
+0,068616 ns
+0,01397
+0,201193 ns
+0,014274
+2,48E-17
+0,248612 ns
+0,003033
+os_easy_ start and Gender
+0,03978 *
+0,016962
+1,35E-20
+0,229291 ns
+0,002773
+os_easy_start and Education
+0,472475 ns
+0,008153
+2,75E-19
+0,579196 ns
+-0,00187
+ os_e asy_start and Study field
+0,007566 **
+0,029993
+3,59E-11
+os_ easy_start and timecat
+0,302306 ns
+0,003497
+5,05E-19
+0,29017 ns
+0,000695
+Correlation between OS self-confidence POST and demographics
+ variables
+pval_anova
+eta_ q_anova
+pval_ shapiro
+ pval_ kruskal
+ eta_ sq_ kruskal
+os_ easy_end and Country
+0,05608
+0,03938
+3,41E-28
+0,479966
+-0,00056
+ os_easy_end and Age
+0,02331 *
+0,023532
+3,66E-27
+0,09763 ns
+0,007508
+os_easy_end and Gender
+4,66E-05 ****
+0,039482 (small)
+4,09E-26
+0,033597 *
+0,010424 (small)
+os_easy_end and Education
+0,05328 ns
+1,55E-26
+0,035592 *
+0.011063 (small)
+Play pnas pue pua Asea so
+0,459497
+0,007895
+3,44E-23
+os_easy_end and timecat
+0,070596 ns
+0,007732 (small)
+4,82E-27
+0.04353 *
+0,00625 (small)AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT] 
+29|29 
+*p<0.05, **p<0.01. (A’). Correlation between True/False score (TF score) and demographics. TF Score 
+correlates with age and education level, with a small effect size (B). 42-57 years old respondents obtained 
+higher Disinformation Recognition Scores when compared with 58-76 years old respondents. Ordinary one-
+way ANOVA multiple-comparisons Tukey’s test; **p<0.01. (B’); respondents with a higher education level 
+generally obtained a higher Disinformation Recognition Score when compared with respondents with a 
+lower education level. Ordinary one-way ANOVA multiple-comparisons Tukey’s test; *p<0.05, **p<0.01. 
+(B’’). Correlation between TF Self-Confidence PRE and demographics. The country of origin correlates with 
+how confident respondents were to recognize disinformation before taking the survey, with a small effect 
+size (C). Correlation between TF self-confidence POST and demographics. Age, education level, and timecat 
+(i.e., how long respondents took to complete the survey), all correlate, with a small effect size, with how 
+confident respondents were to recognize disinformation after completing the survey (D). There is no 
+correlation between OS self-confidence PRE and demographics variables (E). Correlation between OS self-
+confidence POST and demographics. Gender, education, and timecat  correlate, with a small effect size, 
+with how confident respondents were to recognize organic versus synthetic information after completing 
+the survey (F). For all analyses: Reported p-values follow statistical analysis with ANOVA, Shapiro, and 
+Kruskal-Wallis. The effect size and statistical significance were determined with Kruskal-Wallis. *p<0.05; 
+**p<0.01, ***p<0.001, ****p<0.0001. Bars represent SEM. 
+ 
+ 
+Figure S13. Correlations between numerical variables. There is no correlation between OS Delta and OS 
+Score; OS Delta is the difference between OS self-confidence POST and OS self-confidence PRE, and 
+represents how the confidence level in recognizing organic versus synthetic information changed after 
+taking the survey, when compared with the confidence level before taking the survey (A). Correlation 
+between TF Delta and TF Score. TF Delta is the difference between TF self-confidence POST and TF self-
+confidence PRE, and represents how the confidence level in recognizing disinformation versus accurate 
+information changed after taking the survey, when compared with the confidence level before taking the 
+survey. The correlation is small (B). There is no correlation between duration (i.e., how much time 
+respondents took to complete the survey) and OS Score (C). There is no correlation between duration and 
+TF Score (D).  
+ 
+ 
+ 
+
diff --git a/tdFKT4oBgHgl3EQf2i5t/content/tmp_files/load_file.txt b/tdFKT4oBgHgl3EQf2i5t/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..a015d056f25d7ffd1671a41e99fe5c52b59d6329
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@@ -0,0 +1,1304 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf,len=1303
+page_content='AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  1|29  AI model GPT-3 (dis)informs us better than  humans  Giovanni Spitale  0000-0002-6812-0979  Institute of Biomedical Ethics and History of Medicine, University of Zurich, Zurich, Switzerland  giovanni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='spitale@ibme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='uzh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='ch  Nikola Biller-Andorno  0000-0001-7661-1324  Institute of Biomedical Ethics and History of Medicine, University of Zurich, Zurich, Switzerland  biller-andorno@ibme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='uzh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='ch  Federico Germani (corresponding author)  0000-0002-5604-0437  Institute of Biomedical Ethics and History of Medicine, University of Zurich, Zurich, Switzerland  federico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='germani@ibme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='uzh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='ch  +41 44 634 40 80  Institute for Biomedical Ethics and History of Medicine (IBME), Winterthurerstrasse 30, 8006 Zürich (CH)  Abstract  Artificial intelligence is changing the way we create and evaluate information, and this is happening during  an infodemic, which has been having dramatic effects on global health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' In this paper we evaluate whether  recruited individuals can distinguish disinformation from accurate information, structured in the form of  tweets, and determine whether a tweet is organic or synthetic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', whether it has been written by a  Twitter user or by the AI model GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Our results show that GPT-3 is a double-edge sword, which, in  comparison with humans, can produce accurate information that is easier to understand, but can also  produce more compelling disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' We also show that humans cannot distinguish tweets generated  by GPT-3 and real Twitter users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Starting from our results, we reflect on the dangers of AI for  disinformation, and on how we can improve information campaigns to benefit global health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Introduction  Artificial intelligence text generators caught much attention over the last years, especially after the release  of GPT-3 in 20201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' GPT-3, the latest iteration of the Generative Pre-trained Transformers developed by  OpenAI, is arguably the most advanced systems of pre-trained language representations2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' A generative pre-  trained transformer, in its essence, is a statistical representation of language;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' an AI engine that based on  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  2|29  users’ prompts can produce very credible – and sometimes astonishing – text 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' In fact, an initial test on  people’s ability to tell whether a ∼ 500 word article was written by humans or GPT-3 showed a mean  accuracy of 52%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' just slightly better than random guessing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  GPT-3 does not have any mental representations or understanding of the language it operates on 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The  system relies on statistical representations of language for how it is used in real-life by real humans, or ‘a  simulacrum of the interaction between people and the world’ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Even keeping in mind these structural  limitations, what GPT-3 can do is remarkable, and remarkable are also the possible implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' While on  one hand GPT-3 can be a great tool for machine translations, text classification, dialogue/chatbot systems,  knowledge summarizing, question answering, creative writing 2,5,6, detecting hate speech 7, and automatic  code writing 2,8, it can also be used to produce ‘misinformation, spam, phishing, abuse of legal and  governmental processes, fraudulent academic essay writing and social engineering pretexting’ 1,9–11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' GPT-3  is essentially a lever, an amplifier of human intentions which can receive instructions in natural language  and produce output that can be natural language as well, or formal language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' In this sense, this tool is  inherently neutral from an ethical point of view – and as every other similar technology, it is subject to the  dual use problem12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  These advancements in AI text generators and the release of GPT-3 historically coincide with the ongoing  infodemic13 – an epidemic-like circulation of fake news and disinformation, which, alongside the COVID-19  pandemic, has been greatly detrimental for global health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Since GPT-3 can potentially be used to generate  information, and given the possible misuse of such technology (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', producing disinformation) and the  devastating effects it may have on global health, it is of utmost importance to evaluate how text produced  by GPT-3 can affect people’s understanding of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  In this paper we aim at determining whether GPT-3 can be used to produce accurate information and  disinformation in the form of tweets, and how credible this information or disinformation is when  compared with information and disinformation produced by humans, and whether the same technology  can be used to develop assistive tools to help identifying disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For clarity, we acknowledge that  the definitions of disinformation and misinformation are diverse, but in this paper we refer to an inclusive  definition, which considers as disinformation both false information (also partially false information) and/or  misleading content14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  To achieve our goals, we asked GPT-3 to write tweets containing informative or disinformative tweets on a  range of different topics, including vaccines, 5G and COVID-19, or the theory of evolution, among others,  which are commonly subject to disinformation and public misconception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' We collected a set of real tweets  written by users on the same topics, and programmed a survey in which we asked respondents to classify  whether randomly selected synthetic tweets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' : written by GPT-3) and organic tweets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' : written by  humans) were true or false (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' whether they contained accurate information or disinformation), and  whether they were written by a real Twitter user or by an AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Results  Study design and demographics  In order to test for the use of GPT-3 AI model as a tool to generate accurate information or disinformation  in the form of tweets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' we crafted instruction prompts to instruct GPT-3 to produce fake tweets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' either  containing accurate information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' or disinformation about the following topics: climate change,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' vaccines  safety,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' theory of evolution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' COVID-19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' masks safety,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' vaccines and autism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' homeopathy treatments for  cancer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' flat Earth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 5G technology and COVID-19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' antibiotics and viral infections,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' and COVID-19 and  influenza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' We performed a Twitter search to identify accurate tweets and disinformation tweets generated  by Twitter users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' We call “synthetic” those tweets that are generated by GPT-3, and we call “organic” those  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  3|29  real tweets retrieved from Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Human respondents were recruited online to participate in a quiz, in  which they were asked to recognize whether a set of tweets were organic or synthetic, and true or false  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', whether they contained accurate information or disinformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' GPT-3 was also questioned about  whether tweets forming the same dataset were true or false (Figure 1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' We recruited 869 respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  157 responses were excluded because they were incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Of the 712 remaining responses, 15  additional responses were removed because the respondents were too fast to meaningfully complete the  survey, for a total of 697 responses included in our analysis (Figure 1B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Most of the respondents were from  the UK, Australia, Canada, USA, and Ireland (Figure S1A), with more females than males (Figure S1B), a  balanced age, with a high representation of people between 42 and 76 years old (Figure S1C), and a  balanced education level profile, with most of the respondents holding a Bachelor’s degree (Figure S1D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  among those with a Bachelor’s degree or above, their field of study was mostly in social sciences and  humanities, natural sciences, or medical sciences (Figure S1E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  GPT-3 AI model informs and disinforms us better  We measured how accurately participants recognized whether a tweet was containing disinformation or  accurate information (Disinformation Recognition Score), for four types of tweet: “Organic true”, which are  tweets published by Twitter users (organic) and containing accurate information (true);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' “Synthetic true”,  which are tweets generated by GPT-3 (synthetic) and containing accurate information (true);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' “Organic  false”, which are tweets generated by Twitter users (organic) and containing disinformation (false);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' and  finally “synthetic false”, which are tweets generated by GPT-3 (synthetic) and contain disinformation  (false).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Participants recognized “organic false” tweets with the highest efficiency, better than synthetic false  tweets (Scores 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='92 versus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='89, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0032) (Figure 1C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Similarly, they recognized “synthetic  true” tweets correctly more than “organic true” tweets (Scores 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='84 versus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='72, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  This indicates that human respondents can recognize the accuracy of tweets containing accurate  information more often when such tweets are generated by GPT-3, when compared with organic tweets  retrieved from Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Similarly, this means that disinformation tweets generated with GPT-3 achieve their  deceiving goal more often when compared with disinformation tweets generated organically, even though  in this case the effect is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' When evaluating the same dataset, segmenting the analysis for true versus  false tweets – regardless of whether they are organic or synthetic – and for organic versus synthetic tweets  – regardless of their truthfulness – tweets containing accurate information received lower scores when  compared with tweets containing disinformation (Scores 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='78 versus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='91, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Similarly, synthetic tweets were categorized more often correctly for the accuracy of the information they  contained (Scores 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='87 versus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='82, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001) (Figure 1C’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Participants required on average  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='14 seconds to determine whether an “organic true” tweet was accurate or contained disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  This was significantly more when compared with “organic false” tweets, which required 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='28 seconds for  evaluation, with “synthetic true” tweets requiring 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='02 seconds, and “synthetic false” tweets requiring  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='87 seconds (Figure 1D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' True tweets required a longer time for evaluation when compared with false  tweets (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='07 vs 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='97 seconds, p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001), as well as organic tweets when compared with synthetic tweets  (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='21 vs 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='44 seconds, p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' (Figure 1D’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The time required for evaluation was not dependent on  the length of tweets (Figure 1E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Further, we calculated Disinformation Recognition Scores for each  category (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', “climate change”, “vaccines and autism”), for each type of tweet (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', “organic true”,  “synthetic true”, “organic false”, “synthetic false”) (Figure S2), and plotted the average Disinformation  Scores for each type of tweet (Figure S3), obtaining comparable results with the analysis run on the  Disinformation Recognition Scores of each respondent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' This confirms that, for humans, accurate  information is more difficult to evaluate when compared with disinformation, and that information  produced by GPT-3 is not only more effective to inform and disinform humans, but also does so more  efficiently, in less time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' : respondents evaluated faster – correctly or incorrectly – these tweets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' A list of  the disinformation tweets recognized most often as accurate tweets can be seen in Figure S4, and a list of  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  4|29  tweets containing accurate information, recognized most often as disinformation tweets, can be seen in  Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Humans evaluate the accuracy of information better than GPT-3  The respondents of our survey evaluated the accuracy or inaccuracy of the information contained in 220  tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Using the same dataset, we asked GPT-3 to evaluate whether the tweets were accurate or whether  they contained disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For disinformation tweets, humans and GPT-3 performed similarly, even  though respondents performed slightly better (Success rates: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='92 vs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='89, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For accurate  tweets, GPT-3, likewise human respondents, had more difficulties evaluating the accuracy of the  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' In comparison, human respondents performed better than GPT-3 (Success rates: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='72 vs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='64,  respectively) (Figure 2A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' A detailed analysis of the results for each category of tweets can be found in  Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' These results suggest that human respondents can evaluate information better than GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Considering these respondents are not necessarily trained individuals in recognizing disinformation, with a  high degree of information literacy, trained humans, at this stage, may perform much better than machines  at performing this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  GPT-3 can “disobey” requests to produce disinformation  As GPT-3 does not have mental representations nor anything that can resemble a proper intentionality15,16,  we believe the use of quotes when referring to “obedience” or “disobedience” is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' As mentioned,  we instructed GPT-3 to produce a set of true and false tweets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', accurate or disinformation tweets)  (Figure 1A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' To obtain the dataset, we instructed GPT-3 to produce 10 accurate and 10 disinformation  tweets for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Of these, we included in our survey only the tweets for which GPT-3 “obeyed”  our request to produce accurate or disinformation tweets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' We calculated the rate of  obedience, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', the percentage of requests that were satisfied by GPT-3 divided for the overall number of  requests we made to GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For accurate information, GPT-3 obeyed our requests 99 times out of 101,  whereas for disinformation the rate of obedience was much lower (80/102) (Figure 2B), indicating that  GPT-3 can “refuse” to produce disinformation, and in rarer instances, it may produce disinformation when  asked to produce accurate information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For a detailed analysis of the Obedience rate per category of tweet,  we refer to Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Information generated by GPT-3 is undistinguishable from human-generated information  We calculated the AI Recognition Score (or OS score), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', the ability of respondents to recognize whether  tweets are organic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', produced by Twitter users) or synthetic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', produced by GPT-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' As for the  evaluation of the ability to recognize disinformation, we calculated the AI Recognition Score for each type  of tweet (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', “organic true”, “synthetic true”, “organic false”, “synthetic false”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' On average, scores  fluctuated around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='5, indicating that the choice between synthetic and organic is random: respondents  cannot understand if the tweet is produced by AI or has been really tweeted by a user on Twitter (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  That said, respondents obtained a significantly higher score for “organic true” tweets and, to a lesser  extent, for “organic false” tweets (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='67 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='60, respectively), whereas for “synthetic true” and “synthetic  false” tweets, scores were below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='5 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='34 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='40, respectively) (Figure 3A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Looking at true versus false  tweets, and organic versus synthetic tweets, we note something interesting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' whether tweets contain  accurate information or disinformation, it does not influence people’s ability to determine whether a tweet  is organic or synthetic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' answers are perfectly random on average (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='50 vs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='50, respectively, p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='9576).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  However, when comparing organic versus synthetic tweets, organic tweets were recognized as organic  more often when compared with synthetic tweets recognized as synthetic (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='63 vs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='37, p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001) (Figure  3A’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Therefore, both organic and synthetic tweets tend to be classified as “human”, indicating that GPT-3  can effectively mimic human-generated information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Further, we calculated AI Recognition Scores for each  category (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', “climate change”, “vaccines and autism”), for each type of tweet (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', “organic true”,  “synthetic true”, “organic false”, “synthetic false”) (Figure S8), and plotted the average AI Recognition  Scores for each type of tweet (Figure S9), obtaining comparable results with the analysis run on the AI  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  5|29  Recognition Scores of each respondent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' A list of the organic tweets recognized most often as synthetic can  be seen in Figure S10, and a list of synthetic tweets recognized most often as organic can be seen in Figure  S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Building versus crashing confidence: how the self-reported ability to recognize  disinformation and AI-generated information changes after survey completion  At the beginning of the survey, we asked respondents to define how confident they were in their ability to  recognize disinformation, and in their ability to identify AI versus human-generated text using a 1 to 5 Likert  scale (Figure 4A,B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The Disinformation Recognition Confidence before the test was higher than AI  Recognition Confidence before the test (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05 versus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='69, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' After taking the survey (but before  seeing their results), we asked again respondents to define how confident they were in their ability to  recognize disinformation and AI versus human-generated text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Respondents were more confident in their  ability to recognize disinformation (Pre versus Post, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05 versus 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='49, respectively, p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001) (Figure 4A),  whereas they were much less confident in their ability to recognize synthetic versus organic tweets (Pre  versus Post, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='79 versus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='70, respectively, p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001) (Figure 4B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The increase in confidence to recognize  disinformation could be explained by the inoculation theory of misinformation17, whereby critical exposure  to disinformation could improve disinformation recognition capacity and resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' On the opposite, the  stark decrease in confidence to recognize synthetic tweets could depend on what we could call “resignation  theory” – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', people may give up attempting to critically evaluate information when faced with a high  volume of information, especially when it is difficult to understand its origin or intentionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' This may lead  to a sense of hopelessness or apathy towards information consumption, and a tendency to rely on  heuristics or simple cues (such as the emotions evoked by the information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Figures  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' GPT-3 AI model informs and disinform us better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=" Study set-up: GPT-3 was provided instruction  prompts to produce synthetic tweets containing accurate versus inaccurate information (information vs  A  c  c'  Study dataset  Disinformation Recognition Score (0-1)  Instruction  Organic True  Synthetic tweets  1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0 -  prompts  GPT3  Synthetic True  1  Organic False  Twitter search  Organic twe ets  Synthetic False  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='5 -  True or False?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Organic or Synthetic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content="0  Human  atic  ns  respondents  GPT3  ns  ns  B  D  D'  E  ns  ns  ns  ns  869 responses  (oas)  Removed 157  35 -  35 -  250-  inco mplete  Average Time to Response  30 -  30 -  Tweet Characters Count  responses  200-  25." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  25 -  712 responses  20 -  20 -  150-  Removed 15  responses  15-  15  100  697 responses  from speeders  10 -  10-  Included in  the analysis  5-  5-  50-  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  c  nic  etic  Tue  True  2  the  gnAI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  6|29  disinformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Through a Twitter search, organic tweets from Twitter users were retrieved and classified  as accurate information or disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The sum of synthetic and organic tweets, either containing  information or disinformation, constitutes the study dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Human respondents were asked to recognize  whether such tweets were true or false (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', accurate information or disinformation) and whether the  tweets were organic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', generated by Twitter users) or synthetic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', generated by GPT-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' GPT-3 was  asked to recognize whether the tweets were true or false (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', accurate information or disinformation) (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Data collection: we gathered 869 responses to our survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 157 responses were incomplete and were  removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Of 712 remaining responses, 15 were removed as they were completed too fast to be reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Our analysis was conducted on 697 complete and reliable responses (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' GPT-3 produces accurate and  disinformation tweets that are recognized by human respondents as accurate more often than accurate  and disinformation tweets, respectively, produced by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' “Organic true” tweets (green column bars)  are accurate tweets generated by Twitter users;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' “Synthetic true” (dotted green column bars) tweets are  accurate tweets generated by GPT-3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' “Organic false” (red column bars) tweets are disinformation tweets  generated by Twitter users;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' “Synthetic false” (dotted red column bars) tweets are disinformation tweets  generated by GPT-3 (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Disinformation tweets (red column bars) are recognized more often correctly when  compared with accurate tweets (green column bars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Synthetic tweets (dotted grey column bars) are  recognized more often correctly when compared with organic tweets (grey column bars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Disinformation  recognition score (or TF score) (0-1) is the average score for all 697 respondents (1= 100% correct answers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  0 = 0% correct answers);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Ordinary one-way ANOVA multiple-comparisons Tukey’s test, n=697, ** p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  ****p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Error bars = SEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' (C’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Average time to response in seconds for “organic true”, “synthetic  true”, “organic false” and “synthetic false” tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Organic true tweets required the most time to receive  an evaluation from respondents, whereas synthetic true and synthetic false tweets required the least time  to receive an evaluation (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' True tweets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', accurate) required longer to be evaluated by respondents  than false tweets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', containing disinformation), and organic tweets required more time to be evaluated  when compared with synthetic tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Ordinary one-way ANOVA multiple-comparisons Tukey’s test,  n=697, ns = non-significant, p>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' *p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' ****p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Error bars = SEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' (D’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The different time  required to evaluate each type of tweet does not depend on the length of tweets, since the tweet  characters count is no different between types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Ordinary one-way ANOVA multiple-comparisons Tukey’s  test, n(total)=220, n(type)= 55;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' ns = non-significant, p>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05 (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Humans evaluate information and disinformation better than GPT-3, and GPT-3 can “disobey”  requests to generate disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Green column bars represent successful responses given by human  respondents, whereas green dotted bars represent successful responses given by GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Red bars  represent incorrect responses from human respondents, whereas red dotted bars represent incorrect  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  7|29  responses from GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The success rate concerning the evaluation of disinformation is 89% and 92% for  GPT-3 and human respondents, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The success rate concerning the evaluation of accurate  information is 64% and 72% for GPT-3 and human respondents, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The evaluation was conducted  on organic tweets retrieved from Twitter which were included in our survey (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Rate of “obedience” for  GPT-3 – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', how often GPT-3 respected our request to generate information or disinformation tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For  accurate information tweets, GPT-3 “obeyed” our request 99 times out of 101 requests, whereas for  disinformation tweets, it “obeyed” our request 80 times out of 102 requests (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Human respondents cannot distinguish organic versus synthetic tweets, but recognize their  origin better when they are generated by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' AI recognition score for “organic true” (green bars),  “synthetic true” (green dotted bars), “organic false” (red bars) and “synthetic false” (red dotted bars)  tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' AI recognition score (0-1) indicates the probability that human respondents can identify whether a  tweet is produced organically (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', by a Twitter user), or synthetically (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', by GPT-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Human respondents  recognize whether organic true tweets are organic or synthetic tweets more effectively than all other type  of tweets, whereas synthetic true tweets are recognized correctly the least (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Human respondents cannot  predict whether true or false tweets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', accurate tweets or disinformation tweets, green versus red bars)  are produced by humans or by GPT-3, and the truthfulness of the information does not have an impact on  the AI recognition score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Regarding organic versus synthetic tweets (grey versus grey dotted bars), human  respondents recognize whether tweets are generated by humans or GPT-3 better when they are organic  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', generated by Twitter users), when compared with synthetic tweets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', generated by GPT-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' AI  recognition score (0-1) is the average score for all 697 respondents (1= 100% correct answers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 0 = 0%  correct answers);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Ordinary one-way ANOVA multiple-comparisons Tukey’s test, n=697;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' ns = non-significant,  p>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' **p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' ***p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' ***p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Bars represent SEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content="  A  B  c  EFFICIENT VS INEFFICIENT  c'  X  I Recognition Confidence (1-5)  5  5  Initiation  Humans  Disinformation Recognition  Initiation  Confidence (1-5)  4  4  Information  Information  GPT3  campaign  campaign  3  3  Is it information  Humans  Isit information  2  2  or disinformation?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  or disinformation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Evaluation  Evaluation  GPT3  nfidenc  白  Pre  Po  st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  8|29  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The Confidence in recognizing disinformation increases post-survey, whereas the confidence in  recognizing AI-generated information decreases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' and proposed model to launch information campaigns  and evaluate information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Respondents were asked to provide a score of how confident they were in their  ability to recognize disinformation tweets before taking the survey (grey bar), and after taking the survey  (black bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Participants’ confidence in disinformation recognition increased significantly from 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='49  out of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' n=697;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Welch’s t-test, ****p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Bars represent SEM (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Respondents were asked to provide  a score of how confident they were in their ability to recognize whether tweets were generated by humans  (grey bar) or by AI (black bar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Participant’s confidence in AI recognition dropped significantly from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='69 to  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='7 out of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' n=697;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Welch’s t-test, ****p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Bars represent SEM (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Model for an efficient and  inefficient communication strategy and launch of information campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Based on our data, and with the AI  model adopted for our analysis, an efficient system relies on accurate information generated by GPT-3  (initiation phase), whereas it relies on trained humans to evaluate whether a piece of information is  accurate or whether it contains disinformation (evaluation phase) (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' An inefficient system relies on  humans to generate information and initiate an information campaign, and it relies on AI to evaluate  whether a piece of information is accurate or whether it contains disinformation (C’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Discussion  How to communicate and evaluate information  Our findings show that tweets produced by GPT-3 can both inform and disinform better than organic  tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Synthetic tweets containing reliable information are recognized as true better and faster than true  organic tweets, while false synthetic tweets are recognized as false worse than false organic tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Moreover, GPT-3 does not perform better than humans in recognizing both information and  disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The results suggest that GPT-3 may be more efficient at informing because it is able to  generate text that is simpler to read and understand when compared with text written by humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Based  on these results, we propose a model for efficient communication and evaluation of information that  challenges the current approach and consensus, according to which humans produce information and AIs  assist in the evaluation18 (Figure 4C, C’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' A well-tailored information campaign can be shaped by providing  instruction prompts to GPT-3, which produces effective information campaigns targeting humans (initiation  phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The accuracy of information is then evaluated by trained humans (Figure 4C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Instead, information  campaigns written and prepared by humans would turn out to be less effective, and AIs would perform an  inefficient evaluation of how truthful and reliable information is (Figure 4C’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The proposed model is of  relevance in the context of a public health crisis and infodemic, given the need to communicate fast and  clearly to large segments of the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  “Disobedience”, training datasets, and error propagation  Our results show that, when asked, GPT-3 is less likely to produce misinformation on some specific topics,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' : vaccines and autism (Figure S7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Being GPT-3 a statistical representation of language – for how  language is used in the datasets it was trained on – we assume GPT-3’s “disobedience” depends on the  composition of GPT-3’s training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' If the training dataset contains volumes of information  contradicting what the prompt asks for, the system will likely output that type of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' We can  therefore assume that the volume of information in the training dataset debunking causal links between  vaccines and autism is higher than the volume of information debunking conspiracy theories on other  topics taken into consideration by our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Some control on the material fed into the training datasets is  therefore crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' GPT-3 is trained on data obtained from Common Crawl, WebText2, Books1, Books2 and  Wikipedia19, which could include also misinformation and disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' To reduce the risk of generating  disinformation, we suggest that future text transformers should be trained on datasets regulated by the  principles of accuracy and transparency: information entering the training datasets should be verified, and  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  9|29  its origin should be open for independent scrutiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Finally, the output of models trained on accurate and  transparent datasets should report the sources used for its generation, thus increasing transparency, and  allowing independent fact checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Fact checking may still be difficult given the amount of information  that likely serves as a source but nonetheless declaring sources would be a good start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  ‘As human as humans’: synthetic text identification and impersonation  In line with previous research20, we found that both respondents and GPT-3 were not able to distinguish  whether a tweet was organic or synthetic (data on GPT-3’s assessment are available in our study’s  repository)21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' It might be possible to develop specific training courses to improve humans’ recognition of  synthetic text, based on linguistic markers, grammatical structure, and syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' However, since the release  of ChatGPT (an interactive, conversational, and even simpler interface to GPT-3), users started to search for  ways to circumvent OpenAI’s content policy blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' A common (and effective) strategy is impersonation:  when GPT-3 refuses to generate output that could violate the content policy, users just ask it to  impersonate a character – for which, apparently, content policies do not apply22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' With this approach, even  more credible swathes of disinformation could be produced by first asking GPT-3 to generate fake profiles  of people to impersonate, and in a second iteration, to generate tweets that these profiles could write.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Besides circumventing content policy blocks, this would add even more ‘human-like’ feel to the tweets, and  make it even harder to identify them as synthetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Based on these premises, synthetic text identification  might soon be a hopeless battle to fight, for both people and AIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Resignation theory  According to our results, not only humans cannot distinguish synthetic text from organic text, but also the  confidence in their ability to do so decreases significantly after having been asked to recognize the different  origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' This decrease in self confidence after exposure to both synthetic and organic texts may be due to  the realization that there is no clear marker that allows users to identify whether a text has been generated  by a machine or a human.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=" This is likely because of GPT-3's ability to mimic human writing styles and  language patterns." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Additionally, respondents may have initially underestimated GPT-3’s abilities to write  human-like text: this may be due to the fact that such technology is new and revolutionary, and people are  not yet accustomed to how powerful it can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Another possible interpretation is that the survey may have  made participants more aware of GPT-3’s potential to generate disinformation with a human-like feel,  making them more sceptical of both synthetic and organic information, and thus decreasing their  confidence in their ability to identify organic text as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Beyond Twitter  We decided to focus our study on tweets for the following reasons: Twitter is currently used by over 368  million monthly active users 23 who use the platform several times a day24 to consume mostly news and  political information 24,25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Furthermore, Twitter offers a very simple application programming interface (API)  to develop bots, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' : programs able to post content and interact with posts or users without human  supervision26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Recent research shows that only about 5% of Twitter users are bots – but that these bots  cumulatively account for 20% - 29% of the content posted on Twitter27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Because of these characteristics,  Twitter is the ideal target – and potentially a very vulnerable one – for AI-generated swathes of  disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Overall, our findings raise important questions about the potential uses and misuses of  GPT-3 and other advanced AI text generators, and the implications for information dissemination in the  digital age, particularly in relation to the spread of disinformation, particularly on social media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' It is  important to note that while we focused on tweets in this study, our results could be extended to other  social media platforms and other forms of communication that can be used by bots via APIs, and that could  be exploited to programmatically disseminate AI-generated disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' In fact, we generated tweet-  like social media posts that we call tweets, but in fact have features shared in other type of social media  posts, such as Instagram or Facebook posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  10|29  The genie is out of the bottle  Starting from our findings, we predict that advanced AI text generators such as GPT-3 could have the  potential to greatly impact the dissemination of information, both positively and negatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' To mitigate  negative effects, taking action to regulate which training datasets are used to develop these technologies is  crucial, thus ensuring transparency, truthfulness of the output information, and limiting misuse of the  technology to generate deceiving information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Additionally, until we do not have efficient strategies for  identifying disinformation (whether based on human skills or on future AI improvements) it might be  necessary to restrict the use of these technologies, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' : licensing them only to trusted users (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', research  institutions), or limiting the potential of AIs to certain type of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Finally, it is crucial that we  continue to critically evaluate the implications of these technologies and take action to mitigate any  negative effects they may have on society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Brown, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Language Models are Few-Shot Learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' (2020) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='14165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Dale, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' GPT-3: What’s it good for?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Lang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 27, 113–118 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Update: Some Replies by GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Daily Nous https://dailynous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='com/2020/07/30/philosophers-  gpt-3/ (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Benzon, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' GPT-3: Waterloo or Rubicon?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Here be Dragons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  https://papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='ssrn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='com/abstract=3667608 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Marlow, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' & Wood, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Ghost in the machine or monkey with a typewriter—generating titles for  Christmas research articles in The BMJ using artificial intelligence: observational study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' BMJ 375,  e067732 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Elkins, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' & Chun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Can GPT-3 Pass a Writer’s Turing Test?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Cult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 5, 17212 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Chiu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', Collins, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' & Alexander, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Detecting Hate Speech with GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' ArXiv210312407 Cs (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Ugli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Will Human Beings Be Superseded by Generative Pre-trained Transformer 3 (GPT-3) in  Programming?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Orange Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 2, 141–143 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Cabanac, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', Labbé, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' & Magazinov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Tortured phrases: A dubious writing style emerging in science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Evidence of critical issues affecting established journals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' ArXiv210706751 Cs (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Dehouche, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Plagiarism in the age of massive Generative Pre-trained Transformers (GPT-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Ethics  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Environ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Polit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 21, 17–23 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Mindzak, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' & Eaton, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Artificial intelligence is getting better at writing, and universities should  worry about plagiarism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The Conversation http://theconversation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='com/artificial-intelligence-is-  getting-better-at-writing-and-universities-should-worry-about-plagiarism-160481 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Forge, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' A Note on the Definition of “Dual Use”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Ethics 16, 111–118 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' WHO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Immunizing the public against misinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='who.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='int/news-room/feature-  stories/detail/immunizing-the-public-against-misinformation (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Roozenbeek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', Suiter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' & Culloty, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Countering Misinformation: Evidence, Knowledge Gaps, and  Implications of Current Interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Preprint at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='31234/osf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='io/b52um (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  11|29  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Sobieszek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' & Price, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Playing Games with Ais: The Limits of GPT-3 and Similar Large Language  Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Minds Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 32, 341–364 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Floridi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' A Defence of Constructionism: Philosophy as Conceptual Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Metaphilosophy 42,  282–304 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Cook, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', Lewandowsky, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' & Ecker, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Neutralizing misinformation through inoculation: Exposing  misleading argumentation techniques reduces their influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' PLOS ONE 12, e0175799 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Ahmad, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', Aliaga Lazarte, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' & Mirjalili, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' A Systematic Literature Review on Fake News in the  COVID-19 Pandemic: Can AI Propose a Solution?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 12, 12727 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Cooper, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' OpenAI GPT-3: Everything You Need to Know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Springboard Blog  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='springboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='com/blog/data-science/machine-learning-gpt-3-open-ai/ (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Clark, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' All That’s ‘Human’ Is Not Gold: Evaluating Human Evaluation of Generated Text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Preprint at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='00061 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Spitale, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', Germani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' & Biller-Andorno, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Can AI Disinform Us Better?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Zack Witten [@zswitten].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Pretending is All You Need (to get ChatGPT to be evil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' A thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Twitter  https://twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='com/zswitten/status/1598088267789787136 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Dixon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Twitter: number of worldwide users 2019-2024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Statista  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='statista.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='com/statistics/303681/twitter-users-worldwide/ (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Kjarval, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', Jeff Sonderman, Kevin Loker, Maria Ivancin, Nina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' How people use Twitter in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  American Press Institute https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='americanpressinstitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='org/publications/reports/survey-  research/how-people-use-twitter-in-general/ (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Twitter news.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' How many people come to Twitter for news?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' As it turns out, a LOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  https://blog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='com/en_us/topics/insights/2022/how-many-people-come-twitter-for-news  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Garson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' How to create a Twitter bot with Twitter API v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='com  https://developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='com/en/docs/tutorials/how-to-create-a-twitter-bot-with-twitter-api-v2  (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Carr, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Bots Likely Not A Big Part of Twitter’s Audience — But Tweet a Lot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Similarweb  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='similarweb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='com/blog/insights/twitter-bot-research-news/ (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Shaver, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Using Facebook Advertising to Recruit Representative Samples: Feasibility  Assessment of a Cross-Sectional Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Internet Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 21, (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Supplementary material  Methods  Definition of the topics  As the focus of this study, we initially identified 14 topics on which disinformation exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' This preliminary  list included: Climate change;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Vaccines safety;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  12|29 Theory of evolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' COVID-19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Masks safety;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Vaccines and autism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Homeopathic treatments for cancer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Flat Earth;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 5G technology and COVID-19;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Bill Gates and population control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Antibiotics and viral infections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' COVID-19 = influenza;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Inferior human races;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Moral AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Generation of synthetic tweets  Based on the list defined above, we generated synthetic tweets passing input to GPT-3 via API (Application  Programming Interface).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The code asks to generate 10 true tweets and 10 false tweets for each of the  topics detailed above (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' : prompt: ‘Write a tweet to explain why climate change is real’, category:  ‘Climate change’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The tweet generation code consists of one function to pass input prompts to GPT-3, and  of two different loops to iterate over categorized prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The first function defines the parameters to  pass to GPT-3 (temperature, max_token, top_p, best_of, frequency_penalty, presence_penalty), empirically  defined in an iterative process as the most apt to produce text that resembles social media content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=" GPT-  3's API returns also the reason for termination (e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' : reaching the length specified in max_tokens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For these  cases, the text sometimes contains unfinished sentences: these have been removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The loops to generate  true and false tweets read input organized in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='csv files (prompt and category) and generate the given  number of texts per each prompt (in this example, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The output is then exported as a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='xlsx file containing  three columns: the text, the reason for termination, and the category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' All the code, available in this study’s  pre-registration repository, is organized in commented Jupyter lab notebooks for scrutiny and replication21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  The prompts and the output are available in the same repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Definitions  Throughout the manuscript we adopt – and sometimes explain for added clarity – the terminology “true”  and “false” tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' True tweets are those tweets containing accurate information, and false tweets are  those containing inaccurate information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  As for the definition of accurate information and disinformation, we base ourselves on the current scientific  knowledge and understanding of the topics and information under scrutiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' To avoid dubious and  debatable cases, which may be subject to personal opinions and interpretations, we only analyzed and  added to our questionnaire those tweets containing information that is clearly categorizable as true or  false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Of note, if a tweet contained partially incorrect information – meaning it contained more than 1  pieces of information, and at least one was incorrect, it was labelled as false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' As discussed in the  introductory section of the manuscript, we acknowledge that the definition of disinformation and  misinformation is diverse, but we refer to an inclusive definition, which considers false information (also  partially false information) and/or misleading content14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content="  Retrieval of organic tweets  Using Twitter's advanced search, we collected a random sample of recent organic tweets on the topics  listed above, including both true and false tweets." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Our initial aim was to collect 50 tweets per category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  however, this proved impossible for some categories, for various reasons – for example, for some  categories tweets were ambiguous and difficult to categorize as true or false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For other categories, we were  not able to retrieve enough tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Some categories were therefore dropped and excluded from the  following phases of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='13|29  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Category  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Organic tweets retrieved  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Climate change  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Vaccines safety  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Theory of evolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='49 (1 duplicated tweet was  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='excluded)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='COVID-19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Masks safety  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Vaccines and autism  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Homeopathic treatments for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='cancer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='30  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Flat Earth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='5G technology and COVID-19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Bill Gates and population control  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0 (dropped)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Antibiotics and viral infections  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='COVID-19 = influenza  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Inferior human races  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='10 (dropped)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Moral AI  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0 (dropped)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Organic tweets retrieved, by category  The tweets are available in the study’s repository21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Expert assessment of synthetic and organic tweets  We evaluated synthetic and organic tweets to assess whether they contained disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The expert  assessment was performed independently by FG and GS, and a following joint analysis was conducted by  FG and GS to verify the correctness of their initial assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Selection of the tweets to include in the survey and generation of tweet images  Based on the assessments defined above, we selected the following tweets for each category: 5 synthetic false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 5 synthetic true;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 5 organic false;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 5 organic true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  We selected only tweets for which FG and GS agreed in their evaluation, following the expert assessment  phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' This resulted in a dataframe of 220 tweets (available in our repository21) used to generate the  images of the tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The code generates a random pseudonym and a random username for each tweet  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' : ‘John S.’, @john_s), and creates an image which resembles the screenshot of a tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The code, the  dataframe containing the tweets, and the output images are available in the study’s repository21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  AI assessment of tweets  The AI assessment was performed by GPT-3 (true/false evaluation and organic/synthetic evaluation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=" The  first evaluation function defines the parameters to pass to GPT-3 to produce a 'true/false evaluation' (i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' :  whether the tweet is true or false).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=" The second evaluation function defines the parameters to pass to GPT-3  to produce an 'organic/synthetic evaluation' (i." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' : whether the tweet was written by a person or by an AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  The loops for evaluation read the content of the files containing the tweets and evaluate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The output  is scored (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' : whether GPT-3’s assessment matches the expert assessment for true/false and whether it  matches the origin of the tweet for the organic/synthetic classification) and then exported as a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='xlsx file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  The code and the files containing the assessments are available in the study’s repository21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  14|29  Programming of the survey  We programmed a Qualtrics survey to collect demographics, display the tweets to the respondents, and  collect their assessments (true versus false, organic versus synthetic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For each tweet, respondents assess: Whether it is accurate or contains disinformation (single choice, accurate/misinformation);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Whether it was written by a real person or generated by a computer (single choice, real  person/computer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Additionally, respondents provide: Some demographic information (nationality, age, sex, education level, education field) Self-perceived (pre and post survey) ability to recognize, respectively, disinformation and synthetic  text (Likert scale, 1 - very difficult - 5 - very easy)  The images of the tweets are organized in nested randomizers within the survey structure: the first level randomizer randomizes the category order (climate change, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' All the categories are  displayed to every respondent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' second level randomizers (for each category) randomize the single tweet displayed for each  category to the respondent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Each category comprises a total of 20 tweets: 5 synthetic false, 5  synthetic true, 5 organic false, 5 organic true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The second level randomizers evenly present one  tweet from the pool of 20 tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  The survey adopts a gamified approach to keep respondents engaged: at the beginning of the survey,  respondents are told that, upon competition, they will obtain their score for both scales (disinformation  recognition, and synthetic text recognition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' This ensured a low dropout rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=" In-survey scoring is achieved  using the 'scoring' function in Qualtrics." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The survey file and structure are available in the study’s  repository21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Pilot testing  We pilot tested the survey in two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' During the first phase we circulated the link to a convenience  sample with the aim to test the usability and the layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' This led to minor modifications in the interface and  in the wording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' During the second phase we distributed the link via a Facebook ads campaign, structured as  follows: Daily budget: 15€ Start: 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='2022 End: 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='2022 Age: 16 - 65+ Languages: English Title: True or False?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Organic or synthetic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Description: Are you able to distinguish text written by an artificial intelligence from text written by  a human being?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' And accurate information from misinformation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Find out with this test, and  contribute to research on information ethics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Image: generated by DALL·E 2 (available in this study’s repository21)  The campaign had a total cost of 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='92€.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' It generated a total of 593 clicks on the link (cost per click: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='21€)  and a total of 276 responses (cost per response: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='46€).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The campaign was launched and completed in  October 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Sample size and power analysis  Based on pilot data, we conducted a power analysis to determine the sample size for the full study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  15|29  Primary endpoint hypothesis  Disinformation produced by a machine is more credible than disinformation produced by a human  (synthetic versus organic disinformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Secondary endpoints hypotheses  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Accurate information produced by a machine is more credible than accurate information produced  by a human (synthetic versus organic accurate information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Users recognize and distinguish information produced by humans and by machines (regardless of  the truthfulness of the information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The confidence of respondents in recognizing disinformation increases after the completion of the  questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The confidence of respondents in recognizing synthetic versus organic information increases after  the completion of the questionnaire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Power analysis  Based on the data resulting from the pilot study, available in the study’s repository21, we performed a  power analysis to estimate the sample size necessary to draw sufficiently meaningful conclusions for  Primary and Secondary Endpoints (PE and SEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Endpoints are continuous, and the study runs on two  independent samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Primary endpoint: Results  Average group 1 Score (Synthetic tweets, disinformation)* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='86  From 0 to 1, the score indicates how good the performance was in recognizing synthetic tweets  containing disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Stdev group 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='25  Average group 2 Score (Organic tweets, disinformation)* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='89  From 0 to 1, the score indicates how good the performance was in recognizing organic tweets containing  disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Enrollment ratio = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='01194  Alpha = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05 Power = 80%  Sample Size Total = 2181 (Group 1: 1084;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Group 2: 1097)  Secondary Endpoint 1  Average group 1 Score (Synthetic tweets, accurate information)* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='78  From 0 to 1, the score indicates how good the performance was in recognizing synthetic tweets  containing accurate information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Stdev group 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='35  Average group 2 Score (Organic tweets, accurate information)* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='64  From 0 to 1, the score indicates how good the performance was in recognizing organic tweets containing  accurate information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Enrollment ratio = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='991045  Alpha = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05 Power = 80%  Sample Size Total = 197 (Group 1: 99;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Group 2: 98)  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  16|29  Secondary Endpoint 2  Average group 1 Score (Synthetic tweets [accurate information + disinformation])* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='315  From 0 to 1, the score indicates how good the performance was in recognizing synthetic tweets,  regardless of whether they contained accurate information or disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Stdev group 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='44  Average group 2 Score (Organic tweets, [accurate information + disinformation])* = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='59  From 0 to 1, the score indicates how good the performance was in recognizing organic tweets, regardless  of whether they contained accurate information or disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Enrollment ratio = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='001493  Alpha = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05 Power = 80%  Sample Size Total = 80 (Group 1: 40;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Group 2: 40)  Secondary Endpoint 3  Average group 1 Score (Pre-confidence level in ability to recognize disinformation)* = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='932271  From 1 to 5  Stdev group 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='829093  Average group 2 (Post-confidence level in ability to recognize disinformation)* = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='319149  From 1 to 5  Enrollment ratio = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0680  Alpha = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05 Power = 80%  Sample Size Total = 145 (Group 1: 70;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Group 2: 75)  Secondary Endpoint 4  Average group 1 (Pre-confidence level in ability to recognize synthetic versus organic contents)* = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='703557  From 1 to 5  Stdev group 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='897012  Average group 2 (Post-confidence level in ability to recognize synthetic versus organic contents)* = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='75  From 1 to 5  Enrollment ratio = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0720  Alpha = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05 Power = 80%  Sample Size Total = 27 (Group 1: 13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Group 2: 14)  Sample size evaluation  Taking the larger sample size resulting from our power analyses (n=2181 assessments for PE), and  considering that we obtained 1348 assessments (organic, disinformation + synthetic, disinformation), and  considering that the pilot study has generated full responses from 277 respondents, the ratio between  target power (number of assessments) and sample size of the pilot study (number of assessments) is  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='617953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Therefore, the number of users required for the study is 277*1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='617953 = 448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='1728.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' We  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  17|29  established that the minimum number of respondents to achieve a properly powered analysis in the full  study is n=449.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Data collection  We distribute the survey via different Facebook ads campaigns in order to compensate for some  demographic imbalances we noted from the pilot data (overrepresentation of women,  underrepresentation of people aged 18 - 54) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The campaigns took place in October and November 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  We used a total budget of 492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='24€, distributed as detailed in the following table:  Campaign  Age  Sex  Visualizations Cost  USA, GBR, AUS, NZL, CAN  18-54  All  7226  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='22€  USA, GBR, AUS, NZL, CAN  16-65+  M  9907  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='24€  USA, GBR, AUS, NZL, CAN  16-65+  All  14710  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='78€  USA, GBR, AUS, NZL, CAN  16-25  M  83525  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='00€  USA, GBR, AUS, NZL, CAN  16-25  F  57780  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='00€  USA, GBR, AUS, NZL, CAN  26-41  M  8787  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='00€  USA, GBR, AUS, NZL, CAN  26-41  F  9544  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='00€  USA  26-41  F  21046  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='00€  USA  26-41  M  58146  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='00€  USA  16-25  All  99899  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='00€  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Facebook dissemination campaigns for data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Our recruitment strategy aims to enroll a population of active social media users by utilizing a social media  platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Due to this design, we were unable to recruit a representative sample upfront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Instead, we chose  to assess representativeness through a "rolling assessment" of demographics by targeting different  segments of the population in sequential campaigns based on the demographics of already recruited  participants28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Analysis  Scoring and analysis are implemented in Python, using a Jupyter notebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The code takes as input the  results of our Qualtrics survey and generates the files needed for the analysis as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The code is  available for scrutiny and replication in the study’s repository21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Cleaning  Data are cleaned removing incomplete responses, responses resulting from preview links, and responses  submitted in less than 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='5 seconds (determined empirically as the minimum possible time to complete  the survey – this was calculated as the average time required by a convenience sample to read, with a  sustained rhythm, the questions and answers or the survey, and answer the questions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Scoring  True/false and organic/synthetic scores of each respondent are calculated by the rules defined in Qualtrics’  survey programming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' furthermore, they are re-calculated using the dataframe containing the tweets and  the expert assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' True/false average scores of the tweets are calculated as follows: for true tweets,  the score is the average of the assessments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' for false tweets, the score is 1 - the average of the  assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Organic/synthetic average scores of the tweets are calculated as follows: for organic tweets,  the score is the average of the assessments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' for synthetic tweets, the score is 1 - the average of the  assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  18|29  Inferential statistics  Correlation analyses are performed as follows: for quantitative/quantitative data arrays, we first perform a  Pearson’s test, followed by Shapiro’s test to determine data normality, and by both Wilcoxon’s test and a T-  test for hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For qualitative/quantitative data arrays, we first perform an ANOVA test,  followed by Shapiro’s test to determine data normality, and by a Kruskal-Wallis test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Finally, we perform  multiple comparisons with a Tukey test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Effect sizes resulting from ANOVA and Kruskal-Wallis are  interpreted as small when η2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' medium when 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='01 < η2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='06, and as large when η2 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  ‘The hard ones’  We defined tweets that were difficult to identify correctly for respondents (we called them ‘the hard ones’)  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' False identified as true: false tweets with average scores > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' true identified as false: true  tweets with scores < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Synthetic identified as organic: synthetic tweets with average scores > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='75;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  organic identified as synthetic: organic tweets with scores < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Supplementary Results  Correlations between study variables  We evaluated whether any correlation between numerical and categorical variables in our analysis existed  (Figure S12), as well as between numerical variables and other numerical variables (Figure S13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  OS score and demographics  We evaluated any potential correlation between OS Score and demographic variables, and identified the  age of respondents to be a relevant factor, with a small effect size (Figure S12A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Younger individuals (18-  41), seem to perform slightly better at recognizing synthetic versus human tweets when compared with  very young individuals (16-17 years old), and especially older respondents (42+ years old) (Figure S12A’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  TF score and demographics  Similarly, we evaluated potential correlations between TF score and demographic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' As for the OS  Score, also for the TF Score, age correlated with a small effect size, in addition to the education level of  respondents (Figure S12B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' In this case, 42-57 years old individuals performed slightly better than older  individuals aged 58 to 76, although the distribution of TF scores per age seems to be quite uniform across  the board (Figure S12B’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' As expected, a higher education level was associated with higher TF score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' This  effect was small but consistent: participants holding a doctorate/PhD degree had higher scores when  compared with participants holding a Master’s degree, and those with a Master’s degree performed better  than respondents with a Bachelor’s degree, and so on (Figure S12B’’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Self-confidence and demographics  Further, we evaluated the correlation between TF self-confidence PRE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', the score of how confident  respondents were in their ability to recognize disinformation before the survey) and demographic variables  (Figure S12C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' as well as the correlation between TF self-confidence POST (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', the score of how confident  respondents were in their ability to recognize disinformation after the survey) and demographic variables  (Figure S12D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' and the correlation between OS self-confidence PRE (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', the score of how confident  respondents were in their ability to distinguish synthetic versus organic tweets disinformation before the  survey) and demographic variables (Figure S12E);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' and the correlation between OS self-confidence POST  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', the score of how confident respondents were in their ability to distinguish synthetic versus organic  tweets disinformation after the survey) and demographic variables (Figure S12F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  OS / TF self-confidence delta and OS / TF score  For numerical versus numerical variables, we found no correlation between OS Delta (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', the difference in  confidence POST versus PRE in the ability to recognize AI-generated text) and OS Score (Figure S13A), but  we found a small but significant correlation between TF Delta (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', the difference in confidence POST versus  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  19|29  PRE in the ability to recognize disinformation) and TF Score (Figure S13B), suggesting that the higher the  score, the more respondents built confidence in their abilities, despite participants were only shown how  well they scored in the survey after evaluating their confidence level post-survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Duration and OS / TF scores  Further, we found no significant correlation between duration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', how long respondents took to complete  the survey) and OS Score (Figure S13C), as well as between duration and TF Score (Figure S13D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Supplementary Figures  Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Demographics data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Demographics from the study (n=697);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Country of origin of respondents (A),  gender (B), age (C), education level (D), and, among those declaring at least a Bachelor’s degree, the field of  study (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='c  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='UK-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Not disclosed -  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Not disclosed -  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Not disclosed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Prefers not to answer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Australia -  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Prefers not to answer-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='77+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Country  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Canada-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Gender  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Others  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Age  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='58-76  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='USA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='New Zealand -  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Non-binary / third gender-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='42-57-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='26-41-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Ireland -+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Male -  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Prefers not to answer-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='18-25-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Others -+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Female-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='16-17-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0100 200 300 400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='250  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='100 200 300 400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Number of respondents  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Number of respondents  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Number of respondents  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='D  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Not disclosed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Social sciences and humanities-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Prefers not to answer-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Field of study  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Not disclosed -  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Education le  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='PhD/Doctorate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Prefers not to answer-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content="Master's Degree." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content="  Medical sciences  Bachelor's Degree  High school graduate  Others -  Less than high school degree  Natural sciences  0  100  200  300  0  125  250  Number of respondents  Number of respondentsAI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  20|29  Figure S2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Disinformation Recognition Score per category of tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' In the survey, for each category of  tweets, 20 tweets were included, of which 5 were “organic true”, represented with green bars, 5 “synthetic  true”, represented with green dotted bars, 5 “organic false”, represented with red bars, and 5 “synthetic  false”, represented with red dotted bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For each category and type of tweet, we analyzed the success of  respondents in recognizing whether information contained in the tweet were accurate or inaccurate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=',  information or disinformation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For the categories “climate change”, ”vaccines safety”, “theory of  evolution”, “COVID-19 and influenza”, “vaccines and autism”, ”homeopathy and cancer”, “flat Earth”, “5G  and COVID-19”, “organic true” tweets were recognized the least correctly as accurate information (A-D, F-  J), whereas for the categories “masks safety” and “antibiotics and viral infections”, “synthetic false” tweets  have the lowest score (E, K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Conversely, the highest score was generally relative to “organic false” tweets,  as in the case of “vaccines safety”, “masks safety”, “COVID-19 and influenza”, “homeopathy and cancer”  tweets (B, E, F, H), or “synthetic false” tweets, in the categories “climate change”, “theory of evolution”,  “COVID-19”, “vaccines and autism”, “flat Earth”, “5G and COVID-19” (A, C-D, G, I-J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' An exception is the  category “antibiotics and viral infections”, in which “synthetic true” tweets were recognized correctly the  most as accurate, and “synthetic false” tweets were recognized the least as disinformation, when  compared with all other tweet types (K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' n=5 for each type of tweet, for a total of n=20 for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Ordinary one-way ANOVA multiple-comparisons Tukey’s test, ns = non-significant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' *p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' **p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='01,  ***p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='001, ****p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Bars represent SEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  21|29  Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' GPT-3 AI model informs and disinform us better (a single tweet level analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Confirming the  results of Figure 1, the Disinformation recognition score was not extracted from the average score for each  survey respondent, but rather from the average scores, for each type of tweet (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', “Organic true” depicted  with green bars, “synthetic true” depicted with green dotted bars, “organic false” depicted with red bars,  and “synthetic false” depicted with red dotted bars), for each tweet (20 tweets, 5 for each type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Organic  true tweets were recognized the least correctly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', as accurate), when compared with other type of  tweets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Organic false tweets were recognized correctly the most, when compared with other type of  tweets (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' False tweets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', disinformation tweets) were recognized to contain inaccurate information  correctly more often than true tweets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', accurate tweets) (green versus red bars) were recognized to  contain accurate information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' There was no significant difference in how often organic and synthetic  tweets (grey versus grey dotted bars) were recognized to contain accurate information or disinformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  n=55 tweets based on n=5 tweets per type per category of tweet (11 categories, including, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', climate  change, antibiotics and viral infections, 5G and COVID-19, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Ordinary one-way ANOVA multiple-  comparisons Tukey’s test, ns = non-significant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' *p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' ***p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' ****p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Bars represent SEM  (A’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Sandra D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  @sandra_d  Wearing a face mask may give you a  false sense of security and prevent  you from taking necessary precautions  to protect yourself from COViD-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Robert O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  @robert_o  Wearing a face mask may give you a  false sense of security and prevent  you from taking necessary precautions  to avoid #COVID19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  22|29  Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Disinformation tweets recognized as accurate tweets most often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Disinformation tweets, either  generated by Twitter users or by GPT-3, which are most often recognized by human respondents as  accurate tweets (false tweets with mean TF score > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Accurate tweets recognized as disinformation tweets most often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Tweets containing accurate  information, either generated by Twitter users or by GPT-3, which are most often recognized by human  respondents as disinformation tweets (true tweets with mean TF score < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Christopher N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  @christopher_n  Thanks, my stepmom passed away from  cancer after trusting all this stuff  with her health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Sometimes homeopathy  can be truly predatory, praying  financially on the hope of the  vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Sarah I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  @sarah_i  Homeopathy is based on the idea that  "like cures like" and that diluting a  substance makes it more potent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  However.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' there is no scientific basis  for this claim and homeopathic  treatments have not been shown to be  effective in treating cancer or any  other illness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Mark T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  @mark_t  Vaccines are safe and effectiveAI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  23|29  Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Humans evaluate information and disinformation better than GPT-3 (a category breakdown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Green column bars represent successful responses given by human respondents, whereas green dotted  bars represent successful responses given by GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Red bars represent incorrect responses from human  respondents, whereas red dotted bars represent incorrect responses from GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The success rate (0-1) is  used to compare humans’ versus GPT-3’s ability to recognize disinformation and accurate information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The  evaluation was conducted on organic tweets retrieved from Twitter which were included in our survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' In  line with “overall” results (A), human respondents performed better than GPT-3 in recognizing  disinformation related to “climate change”, “vaccines and autism”, “homeopathic treatments for cancer”,  “flat Earth”, “antibiotics and viral infections”, and “COVID-19 and influenza” (B, G-I, K, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Instead, GPT-3  performed better than humans at recognizing disinformation in the categories “vaccines and safety”,  “theory of evolution”, “COVID-19”, “masks safety”, and “5G and COVID-19” (C-F, J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Concerning the correct  identification of accurate information, in line with “overall” results (A), human respondents performed  better than GPT-3 in the categories “COVID-19”, “masks safety”, “vaccines and autism”, “homeopathic  treatments for cancer”, “flat Earth”, “5G and COVID-19”, “antibiotics and viral infections”, and “COVID-19  and influenza” (E-L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Instead, GPT-3 performed better than human respondents at recognizing accurate  information for the categories “climate change”, “vaccines safety”, and “theory of evolution” (B-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  24|29  Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' GPT-3 Rate of “obedience” for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' We calculated the number of requests  (instruction prompts) to produce tweets containing accurate information (dotted green) and disinformation  (dotted red), and the number of requests fulfilled (or “obeyed”) by GPT-3, for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For all  categories, as shown in Figure 2, GPT-3 produced accurate tweets 99 times/101, and disinformation tweets  80 times/102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For the categories “climate change”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' “vaccines safety”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' “theory of evolution”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' “COVID-19”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  “masks safety”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' “vaccines and autism”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' “homeopathic treatment for cancer”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' “flat Earth”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' “5G and COVID-  19”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' “antibiotics and viral infections”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' “COVID-19 and influenza”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' accurate information tweets were  produced by GPT-3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 9/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 10/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 10/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 10/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 10/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 10/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 10/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 10/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 8/9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 9/9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 10/10  times,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' whereas disinformation tweets were produced,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 10/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 10/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 10/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 7/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 8/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 3/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  5/9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 6/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 10/10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 8/9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 3/4 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='99/101  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='10/10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='10/10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='10/10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='10/10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
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+page_content='10/10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='10/10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='10/10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='10/10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='10/10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='10/10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='6/6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Request: accurate information  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='9/10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='80/102  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='8/9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='8/9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Request: disinformation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='8/10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Rate of obedience (%)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='7/10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='3/4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='6/10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='5/9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='50  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='3/10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='cancer  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='safety  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='pue  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Earth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='f evolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='o  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='COVID-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='for  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='treatment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Flat  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='COVID-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Climate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Vaccines s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='For  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Theory  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='5G and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='COVID-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Vaccines  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Antibiotics AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='25|29  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' AI Recognition Score per category of tweet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' In the survey, for each category of tweets, 20  tweets were included, of which 5 were “organic true”, represented with green bars, 5 “synthetic true”,  represented with green dotted bars, 5 “organic false”, represented with red bars, and 5 “synthetic false”,  represented with red dotted bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For each category and type of tweet, we analyzed the success of  respondents in recognizing whether information contained in the tweet were generated by a human or by  GPT-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For most categories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', “theory of evolution”, “COVID-19”, “masks safety”, “COVID-19 and  influenza”, “vaccines and autism”, “homeopathy for cancer”, “flat Earth”, “5G and COVID-19”, “organic  true” tweets were recognized the most for being generated by a Twitter user (C-J), following the trend  observed when all categories of tweet are overlapped (L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Instead, for tweets concerning “climate change”,  and “vaccines safety”, the category “organic false” obtained the highest score (A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For the categories  “climate change”, “theory of evolution”, “COVID-19”, “COVID-19 and influenza”, “vaccines and autism”,  “homeopathy for cancer”, “flat Earth”, “5G and COVID-19”, and “antibiotics and viral infections”, “synthetic  true” tweets were recognized the least for being generated by AI, when compared with all other tweet  types (A-D, F-K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The only exception is the category “masks safety”, in which “synthetic false” tweets  obtained the lowest score (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' n=5 for each type of tweet, for a total of n=20 for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Ordinary  one-way ANOVA multiple-comparisons Tukey’s test, ns = non-significant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' *p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' **p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='01, ***p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='001,  ****p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Bars represent SEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  26|29  Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Human respondents cannot distinguish organic versus synthetic tweets, but recognize their  origin better when they are generated by humans (a single tweet level analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Confirming the results of  Figure 3, the AI recognition score was not extracted from the average score for each survey respondent,  but rather from the average scores, for each type of tweet (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', “Organic true, depicted with green bars  “synthetic true” depicted with green dotted bars, “organic false” depicted with red bars, and “synthetic  false” depicted with red dotted bars), for each tweet (20 tweets, 5 for each type).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Organic true tweets were  recognized more often correctly to be generated by humans, whereas synthetic true tweets were  recognized correctly the least to be generated by GPT-3 (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' There was no significant difference in how  often true (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', accurate) and false (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', containing disinformation) tweets (green versus red bars) were  recognized correctly to be generated by GPT-3 or by a Twitter user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Organic tweets were recognized  correctly more often to be generated by a human when compared with how often synthetic tweets were  recognized correctly to be generated by GPT-3 (grey versus grey dotted bars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' n=55 tweets based on n=5  tweets per type per category of tweet (11 categories, including, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', climate change, antibiotics and viral  infections, 5G and COVID-19, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Ordinary one-way ANOVA multiple-comparisons Tukey’s test, ns = non-  significant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' *p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' ****p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Bars represent SEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Organic tweets recognized as synthetic most often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Tweets generated by Twitter users which  are most often recognized by human respondents as synthetic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', generated by GPT-3 (organic tweets  with mean OS score < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Barbara G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  @barbara_g  The goodness of homeopath can fight  fatal ovarian and lung cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  #homeopathy #health #cancer  #beatcancerAI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  27|29  Sandra Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content="  @sandra_z  Antibiotics can't treat viral  infections, but they can treat  bacterial infections that can  sometimes occur when a virus is  present." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Mary W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content="  @mary_w  Climate change is real because we're  seeing the effects with our own eyes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  The weather is changing, sea levels  are rising, and the planet is getting  hotter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' We need to take action now to  protect our planet and future  generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Charles R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  @charles_r  There is no scientific evidence that  vaccines cause autism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Vaccines are  safe and effective and save livesKaren P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  @karen_p  Homeopathic treatments cannot cure  cancer because they are nothing more  than water and sugar pills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The  treatments do not contain any of the  active ingredients that would be  necessary to effectively cure cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Joseph Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  @joseph_z  Evolution is NoT a hoax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=" It's the  scientific theory that explains how  living things change over timeRichard G." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content="  @richard_g  The #Covid19 pandemic is a hoax  There's no evidence that it's a real  virus, and the symptoms are identical  to those of other common illnesses  This is just another way to scare  people into giving up their rights  and freedoms." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Karen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content="  @karen_m  The earth is flat because it's easier  to draw that way!" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Lisa E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content="  @lisa_e  #vaccines don't cause autism - they  save lives." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Vaccine-preventable  illnesses are on the rise, so please  get vaccinated to protect yourself  and your loved ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Michael D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  @michael_d  I can believe this my son has been  sick since feb - flu + viral  infections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=" I'm even considering  keeping him home, it can't be normal  for a 2yr old to be on antibiotics  twice for 2 weeks in a row." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=" I'll wait  it out take him when he's 3 or 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Any  advice?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Nancy A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  @nancy_a  Homeopathic treatments cannot cure  cancer because they are based on the  false premise that like cures like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  There is no scientific evidence that  this is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='James T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  @james_t  Vaccines do cause autism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=" It's not a  coincidence that the number of autism  cases have skyrocketed as the number  of yaccines kids receive has  increased." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Vaccines are loaded with  toxins like mercury, aluminum and  formaldehyde that can damage the  brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Patricia N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content="  @patricia_n  The Earth's climate has always been  changing, but human activities are  now accelerating the process." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=" Climate  change is real, it's happening now,  and it's a threat to our planet and  our way of life." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='Linda L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  @linda_!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content="  Climate change is real and it's  happening right now." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=" The Earth is  getting warmer every year and it's  causing more extreme weather  conditions." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' We need to take action to  reduce our emissions and protect our  planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='John J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  @john.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='j  5G technology is not a cause of  COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' This technology is designed  to improve internet connectivity and  does not pose any health risksDaniel Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content="  @daniel_q  The climate is changing and it's  happening faster than we thought it  would." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The science is clear, the  evidence is clear, and the impacts  are already being felt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=" We have to  act now to protect our planet and our  children's future." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  28|29  Figure S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Synthetic tweets recognized as organic most often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Tweets generated by GPT-3 which are  most often recognized by human respondents as organic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', generated by a Twitter user (synthetic tweets  with mean OS score > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Figure S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Correlations between demographics and other metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Correlation between  Organic/Synthetic Score (OS Score) and demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' OS Score correlates with age with a small effect size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Young respondents (18-25 years old, and partly 26-41 years old) obtained higher AI Recognition scores  when compared with older respondents;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Ordinary one-way ANOVA multiple-comparisons Tukey’s test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content="  Age  A'  A  Correlation between OS score and demographics  I Recognition Score (0-11)  variables  pval_anova  eta_sq_anova  pval_ shapiro  pval_ kruskal  eta _sq_kruskal  os_ score and Country  0,216996  0,030426  3,66E-06  0,204146  0,006648  os_score and Age  8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='78E-05****  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='042713 (small)  3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='22E-06  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='000228 ***  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='030358 (small)  os score and Gender  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
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+page_content='23E-05  os_score and timecat  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
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+page_content='00173  0-  Educ ation  B\'  B"  B  Correlation between TF score and demographics  Age  Disinformation Recognition Score (0-11)  variables  pval_anova  eta_sq_anova  pval_ s hapiro  pval_kruskal  eysnay bs era  tf_score and Country  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
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+page_content='04353 *  0,00625 (small)AI model GPT-3 (dis) informs us better than humans – v3 23012023 [PREPRINT]  29|29  p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05, **p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' (A’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Correlation between True/False score (TF score) and demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' TF Score  correlates with age and education level, with a small effect size (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' 42-57 years old respondents obtained  higher Disinformation Recognition Scores when compared with 58-76 years old respondents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Ordinary one-  way ANOVA multiple-comparisons Tukey’s test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' **p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' (B’);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' respondents with a higher education level  generally obtained a higher Disinformation Recognition Score when compared with respondents with a  lower education level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Ordinary one-way ANOVA multiple-comparisons Tukey’s test;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' *p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05, **p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  (B’’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Correlation between TF Self-Confidence PRE and demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The country of origin correlates with  how confident respondents were to recognize disinformation before taking the survey, with a small effect  size (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Correlation between TF self-confidence POST and demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Age, education level, and timecat  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', how long respondents took to complete the survey), all correlate, with a small effect size, with how  confident respondents were to recognize disinformation after completing the survey (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' There is no  correlation between OS self-confidence PRE and demographics variables (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Correlation between OS self-  confidence POST and demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Gender, education, and timecat  correlate, with a small effect size,  with how confident respondents were to recognize organic versus synthetic information after completing  the survey (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' For all analyses: Reported p-values follow statistical analysis with ANOVA, Shapiro, and  Kruskal-Wallis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The effect size and statistical significance were determined with Kruskal-Wallis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' *p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  **p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='01, ***p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='001, ****p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Bars represent SEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='  Figure S13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Correlations between numerical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' There is no correlation between OS Delta and OS  Score;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' OS Delta is the difference between OS self-confidence POST and OS self-confidence PRE, and  represents how the confidence level in recognizing organic versus synthetic information changed after  taking the survey, when compared with the confidence level before taking the survey (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' Correlation  between TF Delta and TF Score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' TF Delta is the difference between TF self-confidence POST and TF self-  confidence PRE, and represents how the confidence level in recognizing disinformation versus accurate  information changed after taking the survey, when compared with the confidence level before taking the  survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' The correlation is small (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=' There is no correlation between duration (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
+page_content=', how much time  respondents took to complete the survey) and OS Score (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/tdFKT4oBgHgl3EQf2i5t/content/2301.11924v1.pdf'}
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+arXiv:2301.02583v1  [math.DG]  6 Jan 2023
+ELASTIC DIFFEOLOGICAL SPACES
+CHRISTIAN BLOHMANN
+Abstract. We introduce a class of diffeological spaces, called elastic, on which
+the left Kan extension of the tangent functor of smooth manifolds defines an
+abstract tangent functor in the sense of Rosick´y.
+On elastic spaces there is a
+natural Cartan calculus, consisting of vector fields and differential forms, together
+with the Lie bracket, de Rham differential, inner derivative, and Lie derivative,
+satisfying the usual graded commutation relations. Elastic spaces are closed under
+arbitrary coproducts, finite products, and retracts. Examples include manifolds
+with corners and cusps, diffeological groups and diffeological vector spaces with
+a mild extra condition, mapping spaces between smooth manifolds, and spaces of
+sections of smooth fiber bundles.
+1. Introduction
+1.1. The quest for a Cartan calculus on diffeological spaces. A category that
+contains smooth manifolds as a full subcategory but has better properties, such as
+having all limits, colimits, and exponential objects, is often called a convenient set-
+ting for differential geometry. The price to be paid for this convenience is that such
+categories are usually too large as to allow for strong geometric results that hold
+for all its objects. A typical example is the category of diffeological spaces. It is a
+quasi-topos with all its good categorical properties. But since it contains arbitrary
+quotients, arbitrary subsets, and arbitrary intersections of smooth manifolds, topo-
+logical spaces, vector spaces of arbitrary cardinality, and much more, a theorem that
+holds for all diffeological spaces must hold in all these cases. So instead of trying to
+prove statements that would have to cover this impossible generality of situations,
+the task is often to identify conditions that are strong enough to prove a desired
+result, but weak enough to allow for a wide range of examples and applications.
+A considerable part of the infinitesimal differential geometric computations on
+a smooth manifold M can be carried out in its Cartan calculus, which consists of
+the tangent bundle TM → M, the Lie bracket of vector fields, the graded algebra
+of differential forms Ω(M), together with the de Rham differential d, the inner
+derivative ιv and the Lie derivative Lv for every vector field v, which satisfy the
+relations
+[d, d] = 0 ,
+[ιv, ιw] = 0 ,
+[ιv, d] = Lv ,
+[Lv, ιw] = ι[v,w] ,
+[Lv, d] = 0 ,
+[Lv, Lw] = L[v,w] ,
+where the bracket is the graded commutator of graded derivations of Ω(M). For
+example, local definitons and calculations of symplectic geometry can typically be
+worked out in the Cartan calculus, such as hamiltonian vector fields, Poisson brack-
+ets, hamiltonian actions, Dirac structures, generalized complex geometry, contact
+structures, the L∞-algebra of a multisymplectic structure, homotopy momentum
+Date: January 9, 2023.
+2020 Mathematics Subject Classification. 58A40 (58A03, 18F15, 18F40).
+Key words and phrases. Diffeological space, tangent structure, Cartan calculus.
+1
+
+2
+C. BLOHMANN
+maps, infinitesimal models for equivariant cohomology, etc. Another example is lo-
+cal Lagrangian Field Theory, where the derivation of the Euler-Lagrange equations,
+local symmetries, Noether’s theorems, the theory of Jacobi fields, etc. take place in
+the Cartan calculus of the infinite jet bundle, also known as the variational bicom-
+plex [DF99]. In fact, the question we will address in this paper was motivated by
+current developments in Lagrangian Field Theory and geometric deformation the-
+ory, areas where the basic geometric objects, spaces of fields and paths in a moduli
+space of structures, are naturally equipped with diffeologies.
+Question 1.1. What are the conditions a diffeological space must satisfy so that it
+is equipped with a natural Cartan calculus?
+Of course, there are always the tautological conditions which promote the desired
+outcome to axioms, in our case the existence of a Cartan calculus. The task is to
+identify a set of conditions that is minimal or at least so small that it can be verified
+in a wide range of cases.
+The basic structures of the Cartan calculus on a smooth manifold M, the differ-
+ential graded algebra of differential forms Ω(M) and the tangent bundle TM → M,
+are local, that is, they can be defined first on the open subsets U ⊂ Rn of a chart
+U ⊂ M and then glued together on an atlas. More precisely, the functor U �→ Ω(U)
+is a sheaf and the tangent functor U �→ TU a cosheaf, so that
+Ω(M) ∼= lim
+U→M Ω(U)
+TM ∼= colim
+U→M TU ,
+where the limit in differential graded algebras and the colimit in manifolds are taken
+over the category of charts of a maximal atlas of M. For a diffeological space X
+we simply replace the category of charts by the category of plots and obtain the
+definitions
+Ω(X) := lim
+U→X Ω(U)
+TX := colim
+U→X TU
+of the de Rham complex and the tangent diffeological space of X.
+The maps X �→ Ω(X) and X �→ TX are the pointwise left Kan extensions from
+smooth manifolds to diffeological spaces, which shows that they are functorial in X.
+The left Kan extension also applies to natural transformations, such as the bundle
+projection πU : TU → U and the zero section 0U : U → TU. This suggests, that
+the left Kan extension is the natural method to generalize the Cartan calculus of
+smooth manifolds to diffeological spaces.
+How do we define the Lie bracket of vector fields on a diffeological space? The
+first guess is to start from the Lie algebras X(U) = Γ(U, TU) of vector fields on all
+plots U → X. However, U �→ X(U) is not a functor, so that the left Kan extension
+cannot be applied. We could map the vector fields to the space of derivations of
+C∞(X) = Ω0(X), which is equipped with the commutator bracket. However, this
+map is generally not injective, and even if it is, its image may not be closed under
+the bracket. Worse, the map X �→ Der(C∞(X)) is still not a functor, so that this
+does not solve the problem of naturality. The conclusion is that the spaces of vector
+fields on plots are not a good starting point for the construction of a natural Cartan
+calculus on diffeological spaces.
+
+ELASTIC DIFFEOLOGICAL SPACES
+3
+Fortunately, the situation has been analyzed carefully by Rosick´y who has iden-
+tified the natural structure of the tangent functor that is needed to define the Lie
+bracket of vector fields. He defines an abstract tangent structure on a category C
+to be an endofunctor T : C → C together with the natural transformations of the
+bundle projection πX : TX → X, zero section 0X : X → TX, fiberwise addition
++X : TX ×X TX → TX, exchange of order of differentiation τX : T 2X → T 2X,
+and inclusion of the tangent fibers into the vertical tangent space λX : TX → T 2X,
+which have to satisfy a rather long list of axioms (Definition 2.7). It is instructive
+to see how all these structures come together to define the Lie bracket (8) of vec-
+tor fields, avoiding any reference to the commutator bracket of derivations of some
+structure ring.
+For us, the main advantage of Rosick´y’s approach is that all the structure is given
+by functors and natural transformations, to which we can apply the left Kan exten-
+sion. However, this does not yield an abstract tangent structure on all diffeological
+spaces. The main issue is that the pointwise left Kan extension, which is given by a
+colimit, does not preserve limits, in particular the pullback on which the fiberwise
+addition of tangent vectors is defined. More precisely, the natural morphism
+(1)
+colim
+U→X TU ×U TU −→ TX ×X TX ,
+is not an isomorphism for all diffeological spaces X. In fact, this map is generally
+neither surjective nor injective, as the following two examples show.
+Example 1.2 (Axis cross of the plane). Consider the subset {(x, y) ∈ R2 | xy =
+0} ⊂ R2 with the subspace diffeology.
+The two tangent vectors at the orgin in
+the direction of the x-axis and the y-axis cannot be represented on the same plot
+(Figure 1). It follows that (1) is not surjective.
+Example 1.3 (Folded line). Consider the diffeological quotient space of the action
+Z2 × R → R, (k, x) �→ kx, where Z2 = {1, −1}. The quotient map R → R/Z2
+is a plot.
+The tangent vectors (0, 1) and (0, −1) on its domain represent the
+same tangent vector on R/Z2. This implies that the pairs ζ =
+�
+(0, 1), (0, 1)
+�
+and
+η =
+�
+(0, 1), (0, −1)
+�
+in TR ×R TR represent the same pair of tangent vectors in
+T(R/Z2) ×R/Z2 T(R/Z2). Since the tangent morphism of every morphism of plots
+preserves the sum of a pair of tangent vectors at a point and since the sum of ζ is zero
+but that of η is not, the two pairs cannot be equivalent in colimU→R/Z2 TU ×U TU.
+We conclude that (1) is not injective.
+1.2. The axiom of elasticity. Only if (1) is an isomorphism, the left Kan extension
+of the addition +U on on plots is a morphism +X : TX ×X TX → TX that can
+be viewed as a fiberwise addition of tangent vectors on the diffeological space X.
+Therefore, requiring (1) to be an isomorphism is the first condition we have to impose
+for a diffeological space to have a natural Cartan calculus.
+A k-form in Ω(X) is a family of k-forms on all plots U → X that are compatible
+with the pullbacks along morphisms of plots. A vector field, however, is not repre-
+sented by a family of vector fields on the plots. For this reason, there is no natural
+operation of inner derivative on Ω(X). For the inner derivative, we have to define a
+k-form as a fiberwise multilinear and antisymmetric morphism
+α : TX ×X . . . ×X TX
+�
+��
+�
+=:TkX
+−→ R .
+
+4
+C. BLOHMANN
+(We avoid defining a tensor product, which would entail the usual technical issues
+of completion when the fibers are infinite-dimensional.) The notation TkX for the
+k-fold fiber product is standard in the literature on abstract tangent structures. The
+inner derivative of α with respect to a vector field v : X → TX is then given by
+precomposition
+ιvα : Tk−1X
+∼
+=
+−−→ X ×X Tk−1X
+v×Xid
+−−−−−→ TkX
+α
+−−→ R .
+If we define forms as maps TkX → R, how can we define the differential? The
+differential of a function f : TX → X is given by the tangent map,
+df : TX
+Tf
+−−−→ TR
+∼
+=
+−−→ R × R
+pr2
+−−−→ R .
+However, the functions and exact 1-forms do not generate the ring of forms, so that
+this construction cannot be extended to higher forms.
+We are now in the following dilemma.
+Either we define differential forms as
+families of forms on the plots, in which case we have a differential but no inner
+derivative. Or we define them as fiberwise multilinear and antisymmetric morphisms
+TkX → R, in which case we have an inner derivative, but no differential. The way
+out is to require that the two notions of differential forms coincide.
+We have already imposed the condition that (1) is an isomorphism, which induces
+an isomorphism
+(2)
+Hom(TX ×X TX, R)
+∼
+=
+−−→ lim
+U→X Hom(TU ×U TU, R) .
+It is easy to see that this isomorphism is equivariant with respect to the exchange of
+the two fractors for the fiber product. Moreover, the maps are fiberwise multilinear
+on TkX if and only if they are on all TkU. This shows that the isomorphism (2)
+induces an isomorphism from fiberwise multilinear and antisymmetric morphisms
+on TX ×X TX to Ω2(X). Since we need such an isomorphism for forms of arbitrary
+degree k, we have to impose the following axiom:
+Axiom (E1). The natural morphisms
+θk,X : colim
+U→X TkU −→ TkX ,
+are isomorphisms for all k > 1.
+This axiom has the following geometric interpretation.
+Every tangent vector
+vx ∈ TxX is represented by a path. One can picture this by stretching out x in
+the direction of vx to a smooth path γ : (−ε, ε) → X of short but non-zero length
+through γ(0) = x, such that the coordinate tangent vector
+∂
+∂t at the origin of the
+interval is mapped by T0γ to vx. In this sense, every point of a diffeological space
+has some elasticity in a single infinitesimal direction.
+However, we generally cannot simultaneously stretch out x in the directions of
+several tangent vectors v1
+x, . . . , vk
+x ∈ TXx. That is, we cannot always find a plot
+p : U → X with p(0) = x such that (T0p) ∂
+∂ti = vi
+x, where (t1, . . . , tk) are the
+canonical coordinates of U ⊂ Rk.
+And even if we can find such a plot, it may
+happen that the tangent map Tp is not injective at 0, so that we cannot identify
+the tangent vectors on X with the coordinate vectors on U. This identification is
+possible at every point x ∈ X if and only if the morphism θk,X is a bijection. If in
+addition we want this condition to be compatible with the smooth structure, then
+we have to make the stronger assumption that θk,X is an isomorphism of diffeological
+
+ELASTIC DIFFEOLOGICAL SPACES
+5
+Figure 1. Diffeological subspaces of R2 with non-elastic points
+marked in red, at which two tangent directions cannot be represented
+on the same plot.
+spaces. In this sense, Axiom (E1) captures the geometric idea of the “elasticity” of
+a diffeological space in which any finite set of tangent directions can be streched out
+to a smooth “membrane” given by the image of a plot.
+Example 1.4 (Pasta diffeologies). We can equip a smooth manifold M with an
+alternative diffeology by defining the plots be all smooth maps p : U → M such
+that the rank of Tp : TU → TM is everywhere less than or equal to r. Since (i) the
+precomposition of p with a smooth function f does not increase the rank, (ii) the
+rank is a local property, and (iii) the rank of constant maps is zero, this defines a
+diffeology, which we call the rank-r-restricted diffeology.
+For r = 0 we obtain the discrete diffeology. If r = 1, then every plot factors
+through R, so that we obtain the Spaghetti diffeology [IZ13, Sec. 1.10, footnote
+1].
+The case r = 2 might then be called the Fettuccine diffeology.
+It was
+suggested by the participants of the AMS-EMS-SMF meeting 2022 in Grenoble that
+the case r = 3 should be called the Gnocchi diffeology. For the rank-r-restricted
+diffeology the morphism θk,M of Axiom (E1) is an isomorphism for all k ≤ r but not
+for r < k < dim M.
+1.3. The additional axioms. So far we have the Axiom (E1) that ensures that we
+have a fiberwise addtion on TX and an inner derivative on differential forms. For
+the definition of the Lie bracket we need more structure. In particular, we need a
+natural morphism τX : T 2X → T 2X that exchanges the order of differentiation when
+we apply the tangent functor twice. On a euclidean space U ⊂ Rn, every tangent
+vector is represented by a path R → U on some plot, so that a tangent vector on the
+manifold of tangent vectors is represented by a smooth path of smooth paths, which
+is the same as a smooth map R2 → U. Exchanging the order of differentiation is
+achieved by exchanging the parameters,
+τ1↔2 : R2 −→ R2
+(t1, t2) �−→ (t2, t1) ,
+which descends by the commutative diagram
+(3)
+C∞(R2, U)
+C∞(R2, U)
+T 2U
+T 2U
+τ ∗
+1↔2
+τU
+to an endomorphism of T 2U.
+
+6
+C. BLOHMANN
+When we extend τU to diffeological spaces, the problem arises that the left Kan
+extension does not preserve the product of endofunctors, that is, the natural mor-
+phism
+θ2
+X : colim
+U→X T 2U −→ T 2X
+is generally not an isomorphism.
+We could impose the condition that θ2
+X is an
+isomorphism, but this would be unnecessarily strong. It suffices to require the left
+Kan extension of τU to descend to a morphism τX : T 2X → T 2X. Since θ2
+X is a
+subduction for all X (Proposition 3.26), such a τX is unique. This condition can
+be expressed more intuitively in terms of the smooth families in the same way as
+for euclidean spaces. We can show that we can represent elements in T 2X by plots
+R2 → X. More precisely, we have a subduction
+Dflg(R2, X) −→ T 2X ,
+where Dflg denotes the inner hom of diffeological spaces, that is, the set of morphisms
+equipped with the functional diffeology. The second axiom can now be expressed in
+a way that is completely analogous to diagram (3).
+Axiom (E2). There is a natural morphism τX : T 2X → T 2X, such that the
+diagram
+Dflg(R2, X)
+Dflg(R2, U)
+T 2X
+T 2X
+τ ∗
+1↔2
+τX
+commutes.
+Next, consider the natural morphism λX : TX → T 2X that maps v ∈ TX to
+the vertical tangent vector on TX represented by the path t �→ tv. On a smooth
+manifold, this morphism induces an isomorphism between every tangent space and
+the tangent space of the tangent space. For diffeologial vector spaces this can fail,
+as the following example shows.
+Example 1.5. Consider Rn equipped with k-times differentiable maps as plots.
+This is a diffeological vector space that we denote by Rn
+Ck. Its tangent diffeological
+space is given for k > 0 by
+TRn
+Ck ∼= Rn
+Ck × Rn
+Ck−1 ,
+which shows that the vector space and its tangent fiber are not isomorphic. Assume
+that k > 1, so that we can apply the tangent functor twice. The vertical lift,
+λRn
+Ck : Rn
+Ck × Rn
+Ck−1 −→ Rn
+Ck × Rn
+Ck−1 × Rn
+Ck−1 × Rn
+Ck−2
+(x, v) �−→ (x, 0, 0, v) ,
+is not a subduction.
+The definition of the Lie bracket in terms of the tangent structure yields a map
+from X to the vertical subbundle of T 2X restricted to the zero section of TX.
+We have to be able to identify this bundle with TX for the bracket to be again a
+vector field. This condition is not specific to diffeological spaces. A vector field on
+a Ck-manifold is a Ck-map. The commutator of two such vector fields is a Ck−1-
+map which is, therefore, not a vector field on the Ck-manifold. To exclude such
+phenomena we have to impose the following axiom:
+
+ELASTIC DIFFEOLOGICAL SPACES
+7
+Figure 2. Elastic diffeological subspaces of R2. The tangent spaces
+are 0 at the marked points, R at points on the black lines, and R2 at
+gray points in the interior.
+Axiom (E3). The vertical lift λX : TX → T 2X is an induction.
+There are two more axioms. For smooth manifolds the tangent functor commutes
+with pullbacks over submersions.
+This follows from the local standard form of
+submersions, which is proved using the implicit function theorem. Such a genuinely
+analytic result cannot hold for all diffeological spaces, which is why we need to
+impose the following axiom:
+Axiom (E4). The tangent functor commutes with fiber products of the tangent
+bundle, TTkX ∼= TkTX.
+Finally, we want the diffeological spaces that satisfy our axioms to form a category.
+This requires the collection of diffeological spaces that satisfy the axioms to be closed
+under the functors Tk, which leads to the following axiom:
+Axiom (E5). For every finite set of positive integers k1, . . . , kn the diffeological
+space X′ := Tk1 · · · TknX satisfies axioms (E1) through (E4).
+A diffeological space that satisfies Axioms (E1)-(E5) will be called elastic. If we
+drop Axiom (E5), then we still have a natural Cartan calculus on X. We call a
+diffeological space that satisfies Axioms (E1)-(E4) weakly elastic. The category of
+weakly elastic spaces is not closed under the functors Tk.
+1.4. Summary. The main result about elastic diffeological spaces is the follow-
+ing: The left Kan extension of the tangent structure on euclidean spaces defines
+a tangent structure with scalar R-multiplication on the category of elastic spaces
+(Theorem 4.2). The proof of this statement, which is very long and technical, is
+carried out in detail in a much longer paper [Blo]. Here we will present an overview
+of the conceptual framework, the main properties, and important examples of elastic
+spaces.
+In Section 2 we review Rosick´y’s concept of abstract tangent structures.
+We
+also spell out the conditions for a compatible scalar multiplication, which is only
+mentioned briefly in the original paper [Ros84]. For clarity, we spell out the tangent
+structure of euclidean spaces explicitly.
+In Section 3 we state some new results about the left Kan extension from eul-
+cidean spaces to diffeological spaces, that will be needed later. We use the definition
+of diffeological spaces as concrete sheaves on the site of euclidean spaces, which
+the best approach for the categorical constructions we will study. We then state,
+without proof, our results on the compatibility of the left Kan extension with prod-
+ucts, coproducts, subductions, composition of endofunctors, and the D-topology of
+diffeological spaces, which are all needed for the proof of the main Theorem 4.2.
+
+8
+C. BLOHMANN
+Section 4 contains the formal definition of elastic spaces an the statement, without
+proof, of the main Theorem 4.2. We discuss some alternative choices of axioms for
+which the theorem remains true. First, we observe that we can drop Axiom (E5)
+from the defintion of elastic spaces and still obtain a Cartan calculus on the space
+X.
+This category of weaker diffeological spaces will no longer be closed under
+the tangent functor and its fiber products, so that we no longer have a category
+with an abstract tangent structure in the sense of Rosick´y. We can also slightly
+relax the Axiom (E1) of elasticity by requiring the the condition only holds for
+k = 2. This weaker condition will be sufficient to prove Theorem 4.2. As explained
+in the introduction, our choice of the stronger Axiom (E1) is motivated by our
+wish to obtain not just a tangent structure, but a full-fledged Cartan calculus. In
+an earlier version of the notion of elastic spaces presented in talks, Axioms (E1),
+(E2), and (E4) were replaced by a single condition, which later turned out to be
+unnecessarily strong. Finally, we state, without proof, that the category of elastic
+spaces is closed under restrictions to open subsets, coproducts, finite products, and
+retracts.
+In Section 5 we will give a number of examples for elastic diffeological spaces that
+show that, while the conditions of elasticity are quite strong, they still allow for an
+interesting range of applications. The first main result, Theorem 5.2, shows that a
+diffeological Lie group G is elastic if and only if the natural map g → T0g from the
+vector space g = TeG to its tangent space at 0 is an induction. This is a surprisingly
+weak condition, which is not particular to diffeological spaces. For example, it is
+not satisfied by differentiable group structure on a Ck-manifold, k < ∞. The proof
+of Theorem 5.2, which is long and techincal, will be given in [Blo]. The second main
+result, Theorem 5.11, states that the diffeological space of sections of a smooth fiber
+bundle F → M is elastic. The proof of this theorem is again long and involved, so
+that it will be given in [Blo]. More examples of elastic spaces in this sections are:
+manifolds with corners and cusps, diffeological vector spaces satisfying a mild extra
+condition, manifolds modelled on elastic vector spaces, mapping spaces Dflg(X, A)
+to diffeological vector spaces satisfying A ∼= T0A, the space of smooth R-valued
+functions on a diffeological space, the mapping space of smooth manifolds.
+1.5. Outlook. One of the motivations to develop the concept of elastic spaces came
+from classical field theory, where the basic strucure, the “space” of fields is the set
+of sections of a smooth fiber bundle F → M leaving it often unclear or implicit
+what “space” means mathematically. It is often observed that F = Γ(M, F) is a
+Fr´echet manifold, which is subsequently viewed as blanket license to treat F as if
+it were an ordinary finite-dimensional smooth manifold. For example, it is often an
+implicit assumption that there is a natural differential bigraded algebra of smooth
+forms on F × M that restricts to the variational bicomplex on J∞F [DF99]. In
+the same vein, in the study of symmetries, such as Noether’s theorems or BV-
+theory, the constructions are often explained rigorously only for finite Lie groups
+acting on finite-dimensional manifolds and then generalized with a leap of faith to
+group-valued functions or diffeomorphisms acting on the spaces of fields. A closer
+analysis shows that only the diffeological structure that is being used. For example,
+a variation of a field is a smooth path of sections and a tangent vector to the space
+of solutions of the field equations (i.e. a generalized Jacobi field) is represented by a
+
+ELASTIC DIFFEOLOGICAL SPACES
+9
+smooth path of fields. All this suggests that there is a diffeological construction of
+a Cartan calculus on F. This approach is validated by Theorem 5.11.
+Another motivation comes from geometric deformation theory. Conceptually, a
+deformation is a path in the moduli space of structures, such as the morphisms
+of an algebraic structure, riemannian metrics, or complex structures, all of which
+are equipped with a natural functional diffeology. This suggests that the geomet-
+ric moduli spaces can be conceptualized by (higher) stacks that are presented by
+(higher) diffeological groupoids. The infinitesimal deformation theory should then
+be given by the fibers of the tangent bundle of the moduli space, which should be
+presented by the corresponding (higher) Lie algebroid. In order to define this pro-
+cedure rigorously, a Lie theory for diffeological groupoids has to be developed. This
+has lead us to the conclusion that we need a tangent structure on the diffeological
+spaces of the groupoid. It is encouraging that for diffeological Lie groups the con-
+dition of elasticity is surprisingly weak. Lie theory for diffeological groupoids and
+their application to geometric deformation theory is work in progress.
+2. Abstract tangent structures
+It is fairly straightforward to generalize the de Rham complex Ω(M), which is
+a contravariant functor from smooth manifolds to differential graded algebras, to
+other categories. While vector fields are in some sense dual to differential forms,
+their generalization is a much more difficult problem. An immediate obstacle is
+that the map M → X(M) that sends a manifold to its Lie algebra of vector fields
+is not a functor. It is the tangent bundle TM → M that is functorial in M and,
+therefore, lends itself easily to generalizations. Given a generalized tangent bundle
+πX : TX → X in some category, the vector fields are naturally defined as the
+sections of the morphism πX. The obvious question is now the following:
+Question 2.1. Given a morphism TX → X in some catgory, what is the natural
+structure needed to equip its space of sections with the structure of a Lie algebra?
+This question turns out to be quite involved. A vector field on a smooth manifold
+can be identified with a derivation of its ring of smooth functions C∞(M), which
+is closed under the commutator bracket of the ambient ring of endomorphisms of
+C∞(M). However, M �→ Der(C∞(M)) is still no functor. Moreover, in a generalized
+setting, the identification of sections of TX → X with derivations on some structure
+ring on X seems to be an overly strong requirement that is extraneous to differential
+geometric considerations. So how can we define the Lie bracket of vector fields on
+M directly in terms of the tangent functor using a categorical approach that lends
+itself to generalizations?
+This issue has been solved by Rosick´y in [Ros84]. In a first step, we observe that on
+manifolds the vector space structure on vector fields is induced by the vector space
+structure on the tangent fibers. To allow for categories that do not contain the real
+numbers as object, we relax the structure of R-vector space to that of an abelian
+group. The structure we then need on a generalized tangent bundle TX → X is
+the natural transformatios of addition +X : TX ×X TX → TX and a zero section
+0X : X → TX that equip TX → X with the structure of an abelian group over X.
+For the definition of the Lie bracket, we start from the coordinate formula
+�
+vi ∂
+∂xi wj ∂
+∂xj
+�
+=
+�
+vi∂wj
+∂xi − wi∂vj
+∂xi
+� ∂
+∂xj
+
+10
+C. BLOHMANN
+for vector fields on Rn. The right hand side is given by the derivation of w with
+respect to v minus the derivation of v with respect to w. In order to generalize this
+formula we must make sense of the differentiation of one vector field with respect to
+another and the subtraction of the two terms.
+The derivation of w : X → TX with respect to v : X → TX is given by the
+composition
+X
+v
+−−→ TX
+Tw
+−−−→ T 2X .
+However, we cannot yet subtract Tw(vx) and Tv(wx) since the basepoint of Tw(vx)
+in πTX : T(TX) → TX is wx, whereas that of Tv(wx) is vx.
+We first have to
+exchange the order of differentiation of the twofold tangent bundle. That is, we
+need a natural transformation τX : T 2X → T 2X that satisfies τX ◦ πTX = TπX.
+Then the basepoint of τX(Tv(wx)) is also wx, so that we can take the difference
+(4)
+Tw(vx) − τX
+�
+Tv(wx)
+�
+=
+�
++TX ◦ (Tw ◦ v, −τX ◦ Tv ◦ w)
+�
+x .
+The result lies in the vertical tangent bundle of T 2X → TX, that is, the kernel of
+TπTX. In the last step we have to be able to identify at every point wx ∈ TX the
+vertical tangent space with TxX itself.
+In a smooth manifold, there is a morphism λ2,M : TM ×M TM → T 2M that maps
+the pair (wx, ux) to the vertical tangent vector in T 2M that is represented by the
+path t �→ wx + tux. This map induces an isomorphism TM ×M TM ∼= ker TπTM. In
+the generalized setting this structure is promoted to an axiom: We require a natural
+morphism λ2,X : TX ×X TX → T 2X that induces an isomorphism TX ×X TX ∼=
+ker TπTX. Using this isomorphism, the expression (4) can be viewed as an element
+in TX ×X TX that can be projected onto the second factor which produces an
+element in TX, which is the value of the bracket [v, w] at x.
+For this bracket to satisfy the Jacobi identity, a number of compatibility relations
+and properties of the various structures, bundle projection, zero section, fiberwise
+addition, exchange of the order of differentiation, and the vertical lift have to be
+required [Ros84]. Rosicky’s axiomatization of all this structure is the basis of this
+paper.
+2.1. Preliminary remarks on terminology and notation.
+Terminology 2.2. Let Wibble be an algebraic theory. Let X be an object in a
+category C such that the overcategory C ↓ X has all finite products (i.e. pullbacks
+over X). A Wibble object in C ↓ X will be called a bundle of Wibbles over X.
+In this paper Wibble will be one of: monoid, group, abelian group, module, R-
+vector space (for categories containing R as an object). If W → X is a bundle of
+Wibbles and if the pullback Wx = ∗ ×X W over a point x : ∗ → X exists in C, then
+Wx is a Wibble object in C. In other words, every fiber of a bundle of Wibbles is a
+Wibble object in C, which justifies the terminology. Note, that the notion of bundle
+of Wibbles does not make any assumptions on local trivializations, whatsoever. So
+a bundle of vector spaces over a manifold M is more general than a vector bundle
+over M.
+Remark 2.3. The main purpose of Terminology 2.2 is to unify (for the purpose of
+this paper) the varied terminology found in the literature and to use a term that
+is self-explanatory for a category theorist. In [Ros84, p. 1] a bundle of (abelian)
+groups over an endofunctor F : C → C is called an “natural (abelian) group bundle
+
+ELASTIC DIFFEOLOGICAL SPACES
+11
+over F”. A bundle of vector spaces over a diffeological space X is called a “regular
+vector bundle” in [Vin08], a “diffeological vector space over X” in [CW16], and a
+“diffeological vector pseudo-bundle” in [Per16].
+Notation 2.4. We will follow [Ros84] for the notation of the compositions of func-
+tors and natural transformations. The composition of functors G : A → B and
+F : B → C will be denoted by juxtaposition FG : A → C. Therefore, the horizontal
+composition of natural transformations α : F → F ′ and β : G → G′ (the Godement
+product) will also be denoted by juxtaposition αβ : FG → F ′G′. Its components
+are given by the following commutative diagram:
+FG(A)
+FG′(A)
+F ′G(A)
+F ′G′(A)
+αG(A)
+F (βA)
+(αβ)A
+αG′(A)
+F ′(βA)
+The identity natural transformation F → F will be denoted by F, so that
+(Fβ)A = F(βA)
+(αG)A = αG(A) .
+The vertical composition of α with a natural transformation α′ : F ′ → F ′′ will be
+denoted by α′ ◦ α : F → F ′′. Its components are given by (α′ ◦ α)A = α′
+A ◦ αA. The
+monoidal category of endofunctors will be denoted by End(C).
+2.2. Rosick´y’s axioms. In [Ros84], Rosick´y introduced the notion of abstract
+tangent functor, which captures much of the categorical structure of the tangent
+functor of manifolds. The following notion is implicit in Rosick´y’s definition:
+Definition 2.5. Let F : C → C be a functor and τ : F 2 → F 2 a natural transfor-
+mation. Let τ12 := τ F and τ23 := F τ be the two trivial extensions of τ to natural
+transformations F 3 → F 3. We call τ a braiding on F if it satisfies the braid rela-
+tions τ12 ◦ τ23 ◦ τ12 = τ23 ◦ τ12 ◦ τ23. A braiding τ is called a symmetric structure
+on F if it satisfies τ ◦ τ = F 2.
+Remark 2.6. A symmetric structure on F defines an action of the symmetric group
+Sn on F n.
+A bundle of groups over X consists of a morphism π : A → X, the bundle
+projection, together with the morphisms 0 : X → A and + : A ×X A → A of the
+group structure. Let π′ : A′ → X′, 0′ : X′ → A′, +′ : A′ ×X′ A′ → A′ be another
+bundle of groups. A morphism of bundles is a commutative diagram
+A
+A′
+X
+X′
+ϕ
+π
+π′
+ψ
+
+12
+C. BLOHMANN
+There is a unique morphism ϕ ×ψ ϕ : A ×X A → A′ ×X′ A′ that makes the following
+diagram commutative:
+A ×X A
+A ×X A
+A × A
+A′ × A′
+ϕ×ψϕ
+ϕ×ϕ
+The pair (ϕ, ψ) is a morphism of bundles of groups if the diagram
+A ×X A
+A′ ×X A′
+A
+A′
+ϕ×ψϕ
++
++′
+ϕ
+commutes. An endofunctor F preserves the fiber product if the natural mor-
+phism of bundles over FX,
+(5)
+νk,X : F(A ×π,π
+X . . . ×π,π
+X A) −→ FA ×F π,F π
+F X
+. . . ×F π,F π
+F X
+FA ,
+where both sides have the same number k of factors, is an isomorphism for all k.
+Definition 2.7 (Sec. 2 in [Ros84], Def. 2.3 in [CC14]). A tangent structure of a
+category C consists of a functor T : C → C together with natural transformations
+π : T → 1, 0 : 1 → T, + : T2 → T, λ : T → T 2, and τ : T 2 → T 2, such that the
+following axioms hold:
+• Fiber products: The pullbacks
+Tk := T ×1 T ×1 . . . ×1 T
+�
+��
+�
+k factors
+over T
+π→ 1 exist for all k ≥ 1, are pointwise, and preserved by T.
+• Bundle of abelian groups: T
+π→ 1 with neutral element 0 and addition +
+is a bundle of abelian groups over 1 (Terminology 2.2).
+• Symmetric structure: τ : T 2 → T 2 is a symmetric structure on T (Def-
+inition 2.5). Moreover, τ is a morphism of bundles of groups. That is, the
+diagrams
+T 2
+T 2
+T
+τ
+Tπ
+πT
+and
+TT2
+T 2 ×Tπ,Tπ
+T
+T 2
+T 2T
+T 2
+T 2
+ν2
+T+
+τ×T τ
++T
+τ
+commute, where ν2 is morphism (5) for A = TX
+π→ X, F = T, and k = 2.
+
+ELASTIC DIFFEOLOGICAL SPACES
+13
+• Vertical lift: The diagrams
+T
+T 2
+1
+T
+λ
+π
+πT
+0
+T
+T 2
+T 2
+T 3
+λ
+λ
+λT
+Tλ
+commute. Moreover, the first diagram is a morphism of bundles of groups,
+that is (+T) ◦ (λ ×0 λ) = λ ◦ +.
+• Compatibility of vertical lift and symmetric structure: The diagrams
+T
+T 2
+T 2
+λ
+λ
+τ
+T 2
+T 3
+T 3
+T 2
+T 3
+Tλ
+τ
+τT
+Tτ
+λT
+commute.
+• The vertical lift is a kernel: The diagram
+(6)
+T
+T 2
+1
+T2
+λ
+π
+(πT,Tπ)
+0×10
+is a pointwise pullback.
+Terminology 2.8. In [CC14] and subsequent work, Rosick´y’s original condition
+that T → 1 be a bundle of abelian groups was relaxed to a bundle of abelian
+monoids. In this terminology, Rosicky’s stronger notion is called a tangent structure
+with negatives. All tangent structures in this paper will be with negatives.
+Remark 2.9. Diagram (6) is a pointwise pullback if and only if
+T
+T 2
+T
+λ
+πT
+Tπ
+0◦π◦πT
+is a pointwise triple equalizer. This condition is the original axiom in [Ros84].
+Remark 2.10. The vertical lift can be extended by the additive bundle structure
+to the map
+(7)
+λ2 : T2
+T0×0λ
+−−−−−→ T2T
++T
+−−−→ T 2
+τ
+−−→ T 2 .
+It was shown in [CC14, Lem. 3.10], assuming all other axioms of a tangent structure
+(with negatives), that axiom (6) is satisfied if and only if
+T2
+T 2
+1
+T
+λ2
+π◦pr1
+Tπ
+0
+is a pointwise pullback.
+
+14
+C. BLOHMANN
+2.3. Scalar multiplication. Let R be a ring object in the category C. This gives
+rise to an endofunctor R × 1 : C → C, X �→ R × X, which is equipped with the
+projection pr1 : R × 1 → 1. The ring structure of R equips R × 1 → 1 with the
+structure of a ring internal to endofunctors over 1. Let π : T → 1 be an abelian group
+object in the category of endofunctors over 1. An (R × 1 → 1)-module structure on
+T → 1 is given explicitly by a natural transformation
+κX : R × TX −→ TX ,
+such that the following diagrams commute for all X ∈ C:
+(i) Morphism of bundles:
+R × TX
+TX
+X
+κX
+πX◦pr2
+πX
+(ii) Associativity:
+R × R × TX
+R × TX
+R × TX
+TX
+idR×κX
+·×idR×T X
+κX
+κX
+(iii) Unitality:
+{1} × TX
+R × TX
+TX
+∼
+=
+κX
+(iv) Linearity in R:
+R × R × TX
+R × TX
+(R × TX) ×X (R × TX)
+TX ×X TX
+TX
++×idT X
+idR2×∆T X
+κX
+κX×XκX
++X
+Here ∆TX : TX → TX ×X TX is the diagonal morphism and the factors of
+the codomain are reordered.
+(v) Linearity in TX:
+R × TX ×X TX
+R × TX
+(R × TX) ×X (R × TX)
+TX ×X TX
+TX
+id×+X
+∆R×idT2X
+κX
+κX×XκX
++X
+
+ELASTIC DIFFEOLOGICAL SPACES
+15
+Here ∆R : R → R × R is the diagonal morphisms and the factors of the
+codomain are reordered.
+We will call this structure more succinctly an R-module structure on T → 1
+and T → 1 a bundle of R-modules (Terminology 2.2). If T is part of a tangent
+structure on C, then we also have to require the compatibility with the symmetric
+structure and the vertical lift.
+Definition 2.11. Let R be a ring internal to a category C with a tangent structure.
+An R-module structure κX : R×TX → TX will be called a scalar multiplication
+of the tangent structure if the following diagrams commute for all X ∈ C:
+(vi) Compatibility with the symmetric structure:
+R × T 2X
+R × T 2X
+T 2X
+idR×τX
+κT X
+TκX
+(vii) Compatibilty with the vertical lift:
+R × TX
+R × T 2X
+TX
+T 2X
+idR×λX
+κX
+κT X
+λX
+The tangent structures we consider here will all be equipped with an R-scalar
+multiplication.
+Remark 2.12. As is the case for any module structure, the commutative dia-
+gram (v) implies that the scalar multiplication by 0 ∈ R sends TX to the zero
+section, that is, the diagram
+{0} × TX
+R × TX
+X
+TX
+πX◦pr2
+κX
+0X
+is commutative. If κ is such that this diagram and diagrams (i)-(iii) are commutative,
+then κ will be called an R-cone structure and T → 1 a bundle of R-cones.
+2.4. The Lie bracket of vector fields.
+Definition 2.13. Let C be a category with a tangent structure. A vector field on
+X ∈ C is a section of πX : TX → X.
+The bracket of two vector fields v, w : X → TX is defined as follows.
+The
+composition of v and Tw : TX → T 2X satisfies
+πTX ◦ Tw ◦ v = w ◦ πX ◦ v = w ◦ idX
+= w .
+When we exchange v and w, we have πX ◦Tv◦w = v. In order to be able to subtract
+the two terms in the fiber product T 2X ×TX T 2X, we have two apply the symmetric
+
+16
+C. BLOHMANN
+structure on T 2, so that we obtain
+πTX ◦ τX ◦ Tv ◦ w = TπX ◦ Tv ◦ w = TidX ◦ w = idTX ◦ w
+= w .
+This shows that Tw◦v and τX ◦Tv◦w project to the same fiber of πTX : T 2X → TX,
+so that we can take the difference
+δ(v, w)(x) := (Tw ◦ v)(x) − (τX ◦ Tv ◦ w)(x) ,
+where the minus denotes the difference in the bundle of abelian groups πTX : T 2X →
+TX. We have
+πTX ◦ δ(v, w) = 0 = TπX ◦ δ(v, w) ,
+so that the map δ(v, w) : X → T 2X takes values in the kernel of TπX : T 2X → TX,
+which is isomorphic to TX ×X TX. By projecting on the second factor we thus
+obtain the vector field [v, w] : X → TX. This construction can be summarized by
+the following commutative diagram:
+(8)
+X
+X × X
+TX × TX
+T 2X × T 2X
+TX
+TX ×X TX
+T 2X
+T 2X ×TX T 2X
+T 2X × T 2X
+X
+TX
+TX
+TX × TX
+∆X
+[v,w]
+∃!
+∃!
+v×w
+Tw×Tv
+idX×τX
+⌟
+pr2
+λ2,X
+πX
+TπX
+⌟
+−T X
+πT X×πT X
+0X
+∆T X
+This shows that all of the tangent structure is needed for the definition of the bracket
+of vector fields. It was announced in [Ros84] and proved in [CC15] with the input
+of Rosick´y that [v, w] satisfies the Jacobi relation.
+The set of vector fields Γ(X, TX) has the structure of an abelian group with
+addition
+v + w := +X ◦ (v × w) ◦ ∆X ,
+where ∆X : X → X × X is the diagonal morphism. When the tangent structure
+has a scalar multiplication by R, then Γ(X, TX) is a module over the ring C(X, R)
+of R-valued functions, given by
+κ(f, v) = κX ◦ (f × v) ◦ ∆X .
+2.5. The tangent structure of euclidean spaces. The eponymous example for
+tangent structures is the tangent functor of open subsets of real vector spaces, which
+is the local model for the tangent functor of smooth manifolds. Let Eucl denote the
+category which has open subsets of Rn, n ≥ 0 as objects and smooth maps as
+morphisms.
+Eucl will be called the category of euclidean spaces.
+Its tangent
+functor will be denoted by
+T : Eucl −→ Eucl .
+
+ELASTIC DIFFEOLOGICAL SPACES
+17
+On an open subset U ⊂ Rn, the functors that appear in the definition 2.7 of a
+tangent category are given explicitly by
+TU = U × Rn
+T 2U = U × Rn × Rn × Rn
+T kU = U × (Rn)2k−1
+T2U = U × Rn × Rn
+TkU = U × (Rn)k .
+On a smooth map f : U → V ⊂ Rm the functors are given by
+Tf : (u, ui
+0) �−→
+�
+f(u), ∂f a
+∂xi ui
+0
+�
+T 2f : (u, ui
+0, ui
+1, ui
+01) �−→
+�
+f(u), ∂f a
+∂xi ui
+0, ∂f a
+∂xi ui
+1, ∂f a
+∂xi ui
+01 + ∂2f a
+∂xi∂xj ui
+0uj
+1
+�
+T2f : (u, ui
+0, ui
+1) �−→
+�
+f(u), ∂f a
+∂xi ui
+0, ∂f a
+∂xi ui
+1
+�
+.
+The formulas for T k and Tk are analogous.
+The natural transformations of the
+tangent category structure are given by
+πU : (u, u0) �−→ u
+0U : u �−→ (u, 0)
++U : (u, u0, v0) �−→ (u, u0 + v0)
+λU : (u, u0) �−→ (u, 0, 0, u0)
+τU : (u, u0, u1, u01) �−→ (u, u1, u0, u01) .
+The commutativity of T2 and T is given by the isomorphism
+T(T2U) −→ T2(TU)
+�
+(u, u0, v0), (u1, u01, v01)
+�
+�−→
+�
+(u, u1), (u0, u01), (v0, v01)
+�
+The bundle projection extends to T 2 as
+(πT)U = πTU : (u, u0, u1, u01) �−→ (u, u0)
+(Tπ)U = TπU : (u, u0, u1, u01) �−→ (u, u1) .
+The other natural transformations that appear in the definition, +T, T+, λT, Tλ,
+τT, and Tτ, are obtained in a similar way. The extension (7) of the vertical lift is
+given by
+λ2 : (u, u0, v0) �−→ (u, u0, 0, v0) .
+The following propositions can be checked by explicit elementary calculation:
+Proposition 2.14. Eucl with the tangent functor T and the natural transformations
+π, 0, +, τ, and λ is a tangent structure on Eucl.
+Proposition 2.15. The fiberwise multiplication by real numbers,
+κU : R × TU −→ TU
+�
+r, (u, u0)
+�
+�−→ (u, ru0) ,
+is a scalar multiplication of the tangent structure (Definition 2.11).
+
+18
+C. BLOHMANN
+3. Left Kan extension to diffeological spaces
+The structures on smooth manifolds that we wish to generalize to diffeological
+spaces, such as the tangent bundle and the algebra of differential forms, are local
+and universal in the sense that they are defined on all open subsets of Rn and then
+glued together along an atlas. In categorial terms, the local structure is given by a
+functor F : Eucl → C and the glueing operation by the colimit
+FM := colim
+U→M FU
+over a maximal atlas. In categorical terms, FM is the pointwise left Kan extension
+of F along the inclusion of euclidean spaces into smooth manifolds. For the gen-
+eralization of this construction to diffeological spaces we replace the charts of the
+smooth manifold M with the plots of the diffeological space X,
+FX := colim
+U→X FU .
+This is the pointwise left Kan extension of F along the inclusion of euclidean spaces
+into diffeological spaces.
+In order to use the left Kan extension as universal tool to generalize differential
+geometric structures to diffeological spaces, we have to study its categorical prop-
+erties in some detail. For this it is best to work with the definition of diffeological
+spaces as concrete sheaves on euclidean spaces, which was elucidated in [BH11].
+The site Eucl of euclidean spaces is concrete, which means that there is a faith-
+ful functor Eucl → Set, U → |U| that maps covers to surjective maps. A sheaf
+D : Euclop → Set is concrete if it is a subsheaf of U �→ Set(|U|, D(∗)). Explicitly,
+this means that D(U) is a collection of maps of sets |U| → X := D(∗), called plots,
+that satisfy the properties of a sheaf. In this way, we recover the original definition
+of diffeology.
+As is the case for any category of concrete sheaves, the category of diffeological
+spaces Dflg is a quasi-topos, a category with a classifier for strong subobjects, which
+has a number of good properties (Proposition 3.6). Other good properties of Dflg
+are inherited from the site Eucl. For example, the category of plots of a diffeological
+space, the index category used to compute the pointwise left Kan extension, is sifted,
+which suggest a compatibility with finite products.
+Every representable presheaf is a concrete sheaf, so that the Yoneda embedding
+factors through a full and faithful functor
+y : Eucl −→ Dflg .
+While y is simply the Yoneda embedding restricted on its codomain, the restriction
+changes the computation of colimits.
+A colimit in Dflg is given by first taking
+the pointwise colimit in the functor category SetEuclop and then applying the left
+adjoint of the inclusion Dflg → SetEuclop. This second step has consequences for the
+properties of the left Kan extension, which is given by a colimit.
+Most structures that we will consider are given by endofunctors F : Eucl →
+Eucl. Since Eucl is not cocomplete and since we want to obtain an endofunctor of
+diffeological spaces, we have to place the codomain of F in diffeological spaces by
+composing with y before taking the Kan extension. The Kan extended endofunctor
+will be denoted by
+LF := Lany yF : Dflg −→ Dflg .
+The main question we will address in this section is the following.
+
+ELASTIC DIFFEOLOGICAL SPACES
+19
+Question 3.1. What are the categorical properties of L : End(Eucl) → End(Eucl)
+and which properties of F are preserved by L?
+It follows from the naturality of the Kan extension that L is a functor
+L : End(Eucl) −→ End(Dflg)
+of categories of endofunctors, which means that the vertical composition of natural
+transformations and, therefore, commutative diagrams of natural transformations
+are preserved. This is the most important property for our purposes, since it im-
+plies that the commutative diagrams that appear in the Definition 2.7 of tangent
+structures carry over to the diffeological setting.
+Unfortunately, this is where the good news about the left Kan extension to dif-
+feological spaces end. Unlike the Kan extension to presheaves, the Kan extension
+of a functor to concrete sheaves does generally not preserve colimits, not even finite
+coproducts. Nor does L preserve the composition of endofunctors. Fortunately, the
+functors F we want to extend have some good categorical properties, that entail
+good properties of their Kan extensions LF.
+Using that the category of plots U → X is sifted, we show that if F preserves finite
+products, then so does its Kan extension LF (Proposition 3.19). It also implies that
+L preserves finite products of endofunctors (Proposition 3.20). For the compatibility
+with coproducts we have to assume that F is a cosheaf, that is, F : Euclop →
+Euclop is a sheaf. Then LF preserves coproducts (Corollary 3.23) and subductions
+(Proposition 3.24).
+While L does not preserve the composition of endofunctors, there is a natural
+morphism
+(9)
+L(FG) −→ (LF)(LG) ,
+which is generally not an isomorphism. We can show that if F is a cosheaf, then (9)
+applied to a diffeological space is a subduction (Proposition 3.26). The morphism
+L(FGH) −→ L(F)L(G)L(H)
+we obtain by applying (9) twice does not depend on whether we first apply it to
+(FG)H or to F(GH) (Proposition 3.27). In this sense µ is associative. It is straight-
+forward to see that (9) is natural in F and G (Proposition 3.28).
+Finally, we study the compatibility of the Kan extension with the D-topology
+of diffeological spaces. We observe that many of the functors on euclidean spaces
+we are interested in come with a natural transformation F → 1 such that the
+pullback along any open embedding V → U satisfies FV ∼= V ×U FU. We call
+such an F → 1 a local bundle, by default of a better term. We can show that if
+F → 1 is local, then so is its Kan extension LF → 1 (Proposition 3.35). Then we
+prove that a morphism FX → GX of local bundles is an induction, subduction,
+epimorphism, or isomorphism if all restrictions to the open subsets of a cover of X
+are (Proposition 3.34).
+3.1. Diffeological spaces as concrete sheaves. Recall that Eucl denotes the
+category which has all open subsets of euclidean spaces Rn, n ≥ 0 as objects and all
+smooth maps as morphisms. Open covers define a Grothendieck pretopology.
+Definition 3.2. The small category Eucl together with the Grothendieck topology
+generated by the pretopology of open covers will be called the site of euclidean
+spaces.
+
+20
+C. BLOHMANN
+The terminal object in Eucl is ∗ := R0. The functor of points,
+|
+| : Eucl −→ Set
+U �−→ Eucl(∗, U) ,
+is faithful, so that it equips Eucl with the structure of a concrete category. More-
+over, every cover {Ui → U} is surjective on the underlying sets. A site with these
+properties is called concrete.
+Let X : Euclop → Set be a presheaf. Then there is a morphism of presheaves
+defined by
+(10)
+X(U) −→ Set
+�
+|U|, |X|
+�
+p �−→
+�
+(∗
+u→ U) �→ Xu(p)
+�
+,
+where Xu ≡ X(u) : X(U) → X(∗) is the restriction of X(U) to the point u.
+Definition 3.3. A presheaf X : Euclop → Set on a concrete site is concrete if (10)
+is a monomorphism, i.e. if the maps defined in (10) are injective for all U ∈ Eucl.
+A sheaf is concrete if it is concrete as a presheaf. A morphism between concrete
+sheaves is a morphism of the underlying presheaves.
+Definition 3.4. A concrete sheaf on the site of euclidean spaces is called a diffeo-
+logical space. A morphism of diffeological spaces is a morphism of sheaves. The
+category of diffeological spaces will be denoted by Dflg.
+Theorem 3.5 (Thm. 5.25 in [BH11]). The category of diffeological spaces is a qua-
+sitopos with small limits and small colimits.
+Theorem 3.5 implies that the category of diffeological spaces has a number of
+convenient properties.
+For clarity and later reference, we will spell out some of
+them.
+Proposition 3.6. The category Dflg of diffeological spaces has the following prop-
+erties:
+• Dflg is locally cartesian closed, i.e. for every object X in Dflg the overcate-
+gory Dflg ↓ X is cartesian closed.
+• Strong monomorphisms and strong epimorphisms are effective.
+• (Strong) monomorphisms and (strong) epimorphisms are stable under pull-
+back.
+• Dflg is quasiadhesive, that is, the pushout of a strong monomorphism is a
+strong monomorphism and the pushout square is a pullback square.
+• The initial object is strict, i.e. every morphism X → ∅ is an isomorphism.
+• Coproducts are disjoint, i.e. X → X ⊔ Y ← Y are monomorphisms and
+X ×X⊔Y Y ∼= ∅.
+• The functor of points Dflg → Set, X → Dflg(∗, X) is faithful. It has a left
+and a right adjoint, so that it preserves limits and colimits.
+Terminology 3.7. The strong epimorphisms in Dflg are called subductions, the
+strong monomorphisms inductions.
+Example 3.8. Here are some of the most basic examples for diffeologies:
+
+ELASTIC DIFFEOLOGICAL SPACES
+21
+(a) The fine diffeology or discrete diffeology on a set S is the diffeology for
+which the plots are the locally constant maps.1
+(b) The coarse diffeology, or indiscrete diffeologoy, or trivial diffeology
+on a set S is given by U �→ Set
+�
+|U|, S
+�
+, i.e. all maps are plots.
+(c) Every topological space X is equipped with the continuous diffeology
+given by U �→ Top(U, X), i.e. the plots are the continuous maps.
+(d) Every smooth finite-dimensional manifold M is equipped with the natural
+diffeology given by U �→ Mfld(U, M), i.e. the plots are the infinitely often
+differentiable maps.
+(e) We will denote the exponential objects in Dflg by
+Dflg(X, Y ) ≡ Y X
+and call them the diffeological mapping spaces. The diffeology, which is
+given by the universal property
+Dflg
+�
+U, Dflg(X, Y )
+� ∼= Dflg(U × X, Y ) ,
+is called the functional diffeology.
+Remark 3.9. By Proposition 3.6, the forgetful functor Dflg → Set, X �→ |X| has a
+left and a right adjoint. The right adjoint equips a set S with the coarse diffeology.
+The left adjoint equips it with the fine diffeology. In other words, the fine diffeology
+on a set is the free diffeology, the coarse diffeology is the cofree diffeology.
+Remark 3.10. The map that sends a smooth manifold M to its natural diffeology
+U �→ Mfld(U, M) defines a full, faithful, and injective functor Mfld → Dflg.
+Remark 3.11. Every diffeological space X is naturally equipped with the finest
+topology such that all plots are continuous, which is called the D-topology. This
+topology is determined by the smooth curves only, so that many different diffeologies
+induce the same topology [CSW14, Thm. 3.7]. Mapping a diffeology on X to the
+induced topology is left adjoint to mapping a topology to the continuous diffeology
+[CSW14, Prop. 3.3]. The topology induced by the discrete (trivial) diffeology is
+the discrete topology. In general, however, neither the unit nor the counit of the
+adjunction is an isomorphism.
+The maps p ∈ X(U) ⊂ Set(|U|, |X|) of a diffeological space X are called plots.
+If we spell out the defining conditions of a concrete sheaf, we obtain the traditional
+definition of diffeological spaces in terms of plots.
+For every open cover {Ui → U} in Eucl, U is the coequalizer of �
+i,j Ui ∩ Uj ⇒
+� Ui. In other words, the site of euclidean spaces is subcanonical, so that every
+representable presheaf is a sheaf. Since every representable presheaf is concrete, it
+follows that all representable presheaves on Eucl are concrete sheaves. We conclude
+that the Yoneda embedding Y : X �→ Dflg( , X) factors as
+SetEuclop
+Eucl
+Dflg
+Y
+y
+I
+1In [BH11, Example (2), p. 5794] it is stated incorrectly that the discrete diffeology is given by
+the constant maps.
+
+22
+C. BLOHMANN
+through a functor y. Since Y and I are full and faithful, so is y. Since I is full
+and faithful, the Yoneda lemma implies that the evaluation of the concrete sheaf
+X ∈ Dflg on U ∈ Eucl is given by
+X(U) ∼= SetEuclop(Y U, IX) ∼= SetEuclop(IyU, IX)
+∼= Dflg(yU, X) .
+It follows that limits in Dflg are computed pointwise and that I preserves limits.
+By the adjoint functor theorem, I has a left adjoint,
+K : SetEuclop
+Dflg : I ,
+which was computed and studied in [BH11, Sec. 5.3]. Explicitly, K is given by a
+procedure called concretization followed by the Grothendieck plus construction.
+The left adjoint K is a retract, KI ∼= idDflg, which implies that the colimit of a
+diagram in X : I → Dflg can be computed as
+colim
+i∈I
+Xi ∼= colim
+i∈I
+KIXi
+∼= K colim
+i∈I
+IXi ,
+that is, by first computing the colimit in presheaves and then applying K. As a
+further consequence, it can be shown that y is dense:
+Proposition 3.12 (Prop. 51 in [BH11]). Every X ∈ Dflg is the colimit of y ↓ X →
+Eucl → Dflg, which we will write as
+X ∼= colim
+yU→X yU .
+Terminology 3.13. The comma category y ↓ X is called the category of plots of
+X.
+Remark 3.14. It is customary and convenient to identify notationally the domain
+of a plot U ∈ Eucl with the diffeological space yU ∈ Dflg. In this paper, however, we
+deal with a number of subtleties of Kan extensions along y where this identification
+would invite wrong proofs by notation (a trap the author has fallen into more than
+once). Therefore, we will always spell out the embedding y.
+3.2. Left Kan extension to diffeological spaces.
+Notation 3.15. Let F : Eucl → Eucl be an endofunctor. The left Kan extension of
+yF along the embedding y : Eucl ֒→ Dflg will be denoted by
+(11)
+LF := Lany yF ,
+which is an endofunctor of Dflg.
+The left Kan extension will be our device to extend the tangent structure of
+euclidean to diffeological spaces. L is functorial, which means that L preserves the
+vertical compostion of natural transformations ˆα : ˆF → ˆF ′ and ˆα′ : ˆF ′ → ˆF ′′
+between endofunctors ˆF, ˆF ′, ˆF ′′ ∈ End(Eucl), that is,
+L(α′ ◦ α) = (Lα′) ◦ (Lα) .
+
+ELASTIC DIFFEOLOGICAL SPACES
+23
+Since Eucl is small and Dflg is cocomplete, LF exists and is pointwise, that is, it
+can be computed by the colimit [ML98, Thm. X.5.3]
+(LF)(X) ∼= colim(y ↓ X −→ Eucl
+yF
+−→ Dflg)
+= colim
+yU→X yFU ,
+for all X ∈ Dflg. Let α : F → G be a natural transformation of functors Eucl →
+Eucl. Then yα : yF → yG is a natural transformation of functors Eucl → Dflg.
+The left Kan extension Lany yF is functorial in yF, so that we have a natural
+transformation
+(12)
+Lα := Lany yα : LF −→ LG .
+Together (11) and (12) define a functor
+(13)
+L : End(Eucl) −→ End(Dflg) ,
+where End(C) denotes the category of endofunctors and natural transformations of
+the category C.
+Proposition 3.16. The diagram
+Eucl
+Eucl
+Dflg
+Dflg
+y
+F
+y
+LF
+commutes for all endofunctors F, that is
+(LF)yU ∼= yFU
+for all U ∈ Eucl.
+Proof. Since y is full and faithful, the statement follows from [ML98, Cor. 3, Sec. X.3]
+or from [Kel05, Prop. 4.23].
+□
+Corollary 3.17. The functor (13) is full and faithful.
+3.3. Compatibility with products, coproducts, and subductions. Since y :
+Eucl → Dflg is full and faithful, a smooth map f : U → V of euclidean spaces is a
+strong epimorphism if and only if yf : yU → yV is a strong epimorphism, which is
+the same thing as a subduction. For this reason we will call a strong epimorphism
+f a subduction. This is the case if every point v0 ∈ V has an open neighborhood
+V0 ⊂ V such that there is a smooth map g0 : V0 → U satisfying f ◦ g0 = idV0.
+In short, f is a subduction if it has local sections. In particular, every surjective
+submersion is a subduction.
+Proposition 3.18. Let α : F → G be a natural transformation of endofunctors of
+Eucl. If αU : FU → GU is a subduction for all U ∈ Eucl, then (Lα)X : (LF)X →
+(LG)X is a subduction for all X ∈ Dflg.
+Proof. Since αU is a subduction, so is yαU. Subductions in Dflg are the same as
+regular epimorphisms, so that yαU is a regular epimorphism for all U ∈ Eucl. The
+left Kan extension αX = (Lα)X is given by the colimit over the category of plots
+y ↓ X. Since colimits preserve regular epimorphisms, αX is a regular epimorphism,
+that is, a subduction.
+□
+
+24
+C. BLOHMANN
+Proposition 3.19. If a functor F : Eucl → Eucl preserves finite products, then so
+does LF : Dflg → Dflg.
+Let F : I → End(Eucl), i �→ Fi be a functor. Due to the universal properties of
+colimits and limits, we have for every X ∈ Dflg the natural morphism
+colim
+yU→X lim
+i∈I yFiU −→ lim
+i∈I colim
+yU→X yFiU .
+Assuming that the limit limi∈I Fi exists in End(Eucl), it can be written as the natural
+transformation
+L lim
+i Fi −→ lim
+i LFi ,
+where we have used that y preserves limits. This is not an isomorphism unless the
+colimit and the limit commute.
+Let G : J → End(Eucl) be another diagram, such that limi Gi exists. Any natural
+transformation αi : Fi → Gi induces a commutative diagram
+L limi Fi
+L limi Gi
+limi LFi
+limi LGi
+L(limi αi)
+limi Lαi
+Proposition 3.20. Let F1, . . . , Fk ∈ End(Eucl) be a finite family of endofunctors.
+Then we have an isomorphism
+L(F1 × . . . × Fk) ∼= LF1 × . . . × LFk .
+It is well-known, that the left Kan extension of an arbitrary functor along the
+Yoneda embedding Y : Eucl → SetEuclop preserves all colimits. This is not true for
+Kan extensions along y : Eucl → Dflg. Already for the preservation of coproducts
+we have to make additional assumptions. Recall that a functor F : Eucl → C is a
+cosheaf if F op : Euclop → Cop is a sheaf.
+Example 3.21. The following functors on Eucl are cosheaves:
+(a) the tangent functor T : Eucl → Eucl;
+(b) fiber products of the tangent functor Tk : Eucl → Eucl;
+(c) if F : Eucl → C and G : Eucl → Eucl are cosheaves, then so is their
+composition FG;
+(d) the de Rham functor Ω : Eucl → dgAlgop, which maps U to the differential
+graded algebra of differential forms and smooth maps to the pullback of
+forms;
+(d) if F : Mfldop → C is a sheaf on the big site of manifolds and open covers,
+then the restriction F : Eucl ֒→ Mfld → Cop is a cosheaf;
+Proposition 3.22. If F : Eucl → C is a cosheaf, then its left Kan extension along
+y : Eucl → Dflg preserves coproducts.
+Corollary 3.23. If F : Eucl → Eucl is a cosheaf, then LF preserves coproducts.
+Proposition 3.24. If F : Eucl → Eucl is a cosheaf, then LF preserves subductions.
+Proposition 3.25. Let X : I → Dflg be a functor. If F : Eucl → Eucl is a cosheaf,
+then the natural morphism
+colim(LF)X −→ (LF) colim X
+
+ELASTIC DIFFEOLOGICAL SPACES
+25
+is a subduction.
+3.4. Compatibility with the composition of endofunctors. Let F, G : Eucl →
+Eucl be endofunctors. The left Kan extension of their product is given by the colimit
+L(FG)X ∼= colim
+yU→X yFGU
+∼= colim
+yU→X (LF)yGU ,
+(14)
+where we have used Proposition 3.16. The product of the Kan extensions can be
+written as
+(15)
+(LF)(LG)X ∼= (LF) colim
+yU→X yGU .
+Due to the universal property of the colimit, we have a natural morphism
+(16)
+colim
+yU→X (LF)yGU −→ (LF) colim
+yU→X yGU .
+Composing this morphism with the isomorphism (14) on its domain and with iso-
+morphism (15) on the codomain, we obtain a natural morphism
+(17)
+µF,G : L(FG)X −→ (LF)(LG)X .
+This is an isomorphism if and only if (16) is, which is generally not the case.
+Proposition 3.26. If F is a cosheaf, then (17) is a subduction for all X ∈ Dflg.
+Proof. Since F is a cosheaf, it follows from Proposition 3.25 that (16) is a subduction,
+which implies that (17) is a subduction.
+□
+The morphism (17) is associative in the followoing sense.
+Proposition 3.27. Let F, G, H : Eucl → Eucl be endofunctors. Then the diagram
+of natural transformations
+L(FGH)
+(LF)(LGH)
+L(FG)(LH)
+(LF)(LG)(LH)
+µF,GH
+µF G,H
+(LF )µG,H
+µF,G(LH)
+is commutative.
+Proposition 3.28. Let α : F → F ′ and β : G → G′ be natural transformations of
+endofunctors of Eucl. Then the diagram
+L(FG)
+L(F ′G′)
+(LF)(LG)
+(LF ′)(LG′)
+L(αβ)
+µF,G
+µF ′,G′
+(Lα)(Lβ)
+is commutative.
+
+26
+C. BLOHMANN
+3.5. Compatibility with the D-topology.
+Definition 3.29 (Sec. 2.8 in [IZ13]). The D-topology on a diffeological space is
+the finest topology (on the underlying set) such that every plot is continuous.
+Explicitly, a subset Y ⊂ X is open in the D-topolgy if and only if for every plot
+p : yU → X, the preimage p−1(Y ) ⊂ U is open. Every morphism of diffeological
+spaces is continuous with respect to the D-topologies. In the following “open” and
+”continuous” are always meant with respect to the D-topology. An open subset
+S ⊂ X of a diffeological space is naturally equipped with the subspace diffeology, so
+that the inclusion i : S → X is an open induction.
+Let {Si ⊂ X}i∈I be an open cover of a diffeological space X. Then the diagram
+(18)
+�
+i,j Sij
+�
+i Si
+X ,
+where Sij = Si ×X Sj, is a coequalizer. A functor F : Dflg → Dflg is a cosheaf if
+F preserves the coequalizer (18), that is, if
+�
+i,j FSij
+�
+i FSi
+FX
+is a coequalizer for every open cover. A special kind of cosheaf is given by a natural
+bundle that satisfies the following locality condition.
+Definition 3.30. A bundle π : F → 1 of endofunctors of diffeological spaces will
+be called local if for every open induction i : S → X the commutative diagram
+FS
+FX
+S
+X
+πS
+F i
+πX
+i
+is a pullback.
+Proposition 3.31. If a bundle π : F → 1 of endofunctors of diffeological spaces is
+local, then F is a cosheaf.
+Proposition 3.32. If a bundle π : F → 1 of endofunctors of diffeological spaces is
+local, then F preserves open inductions.
+Proposition 3.33. Let π : F → 1 and ρ : G → 1 be bundles of endofunctors of
+diffeological spaces. If both bundles are local, then the fiber product F ×1 G → 1 and
+the composition FG → 1 is local.
+Proposition 3.34. Let π : F → 1 and ρ : G → 1 be bundles of endofunctors of
+diffeological spaces and α : F → G a morphism of bundles, ρ◦α = π. If both bundles
+are local, then the following are equivalent:
+(i) αX : FX → GX is an induction (subduction, epimorphism, isomorphism).
+(ii) There is an open cover {Si → X}, such that αSi : FSi → GSi is an induction
+(subduction, epimorphism, isomorphism) for all i.
+Definition 3.30 applies also to bundles of endomorphisms of euclidean spaces.
+Explicitly, a bundle π : F → 1 is local if π−1
+U (V ) ∼= FV for every open subset
+V ⊂ U. The next proposition shows that this notion locality is preserved by the left
+Kan extension to diffeological spaces.
+Proposition 3.35. If a bundle π : F → 1 of endofunctors of euclidean spaces is
+local, then so is its left Kan extension Lπ : LF → 1.
+
+ELASTIC DIFFEOLOGICAL SPACES
+27
+4. Elastic diffeological spaces
+The main result for the category of elastic spaces is that the left Kan extension of
+the tangent structure on euclidean spaces defines a tangent structure with scalar R-
+multiplication (Theorem 4.2). The proof of this statement is quite long and will be
+given in [Blo]. It uses a variety of techniques, some categorical, some geometric, and
+all of the results about the left Kan extension given in the preceeding Section 3.2.
+4.1. The tangent structure of elastic spaces. The tangent structure of eu-
+clidean spaces consists of the tangent functor ˆT : Eucl → Eucl together with the nat-
+ural transformations of the bundle projection ˆπ : ˆT → 1, the zero section ˆ0 : 1 → ˆT,
+the addition ˆ+ : ˆT2 → ˆT, the symmetric structure ˆτ : ˆT 2 → ˆT 2, and the vertical
+lift ˆλ : ˆT → ˆT 2, which we from now one decorate with hats in order to distinguish
+them from their Kan extensions to diffeological spaces. The structure was spelled
+out explicitly in Section 2.5.
+We would like to extend this structure to diffeological spaces by applying the Kan
+extension functor L = Lany y
+. The Kan extension of the tangent functor will be
+denoted by
+T := L ˆT : Dflg → Dflg .
+Definition 4.1. A diffeological space X is called elastic if the following axioms
+hold:
+(E1) The natural morphisms
+θk,X : (L ˆTk)X −→ TkX
+are isomorphisms for all k > 1.
+(E2) There is a natural morphism τX : T 2X → T 2X, such that the diagram
+(L ˆT 2)X
+(L ˆT 2)X
+T 2X
+T 2X
+(Lˆτ)X
+θ2
+X
+θ2
+X
+τX
+commutes.
+(E3) The natural morphism
+λX : TX
+(Lˆλ)X
+−−−−→ (L ˆT 2)X
+θ2
+X
+−−−→ T 2X
+is an induction.
+(E4) The natural morphisms
+νk,X : TTkX −→ T 2X ×TπX,TπX
+TX
+. . . ×TπX,TπX
+TX
+T 2X
+are injective for all k > 1.
+(E5) For every finite set of positive integers k1, . . . , kn the diffeological space X′ :=
+Tk1 · · · TknX satisfies axioms (E1) through (E4).
+The full subcategory of elastic diffeological spaces will be denoted by Elst ⊂ Dflg.
+Theorem 4.2. The category of elastic diffeological spaces has a tangent structure
+given by the Kan extended tangent functor T = L ˆT, bundle projection πX := (Lˆπ)X,
+and zero section 0X := (Lˆ0)X, the addition
++X := θ−1
+2,X ◦ (Lˆ+)X ◦ θ2,X ,
+
+28
+C. BLOHMANN
+the symmetric structure τX of Axiom (E2) of Definition 4.1, and the vertical lift
+λX of Axiom (E3). Moreover, the Kan extension of the scalar R-multiplication is a
+scalar R-multiplication in the sense of Definition 2.11.
+4.2. Alternative axioms. The axioms for abstract tangent structures are to hold
+for the entire category.
+This is why the Axiom (E5) has to be included in the
+Definition 4.1 of elastic spaces, so that the application of Tk does not lead out of the
+subcategory of elastic spaces. However, Rosick´y’s axioms still make sense pointwise
+for a single object X that lies in an ambient category where Tk is defined. This is
+our situation, which suggests the following more general concept.
+Definition 4.3. A diffeological space X is called weakly elastic if the Axioms (E1)-
+(E4) of Definition 4.1 hold.
+On a weakly elastic space we still have most of the structure of differential calculus,
+like a Lie algebra of vector fields. We only have to observe that TX and its fiber
+products may no longer share the same good properties.
+Remark 4.4. Theorem 4.2 still holds if Axiom (E1) of Definition 4.1 is replaced
+with the following weaker version:
+(E1’) The natural morphism
+θk,X : (L ˆTk)X −→ TkX
+is an isomorphism for k = 2 and an epimorphism for all k > 2.
+As explained in the introduction, we need the stronger Axiom (E1) so that we obtain
+a Cartan calculus.
+In earlier versions of the definition of elastic space, we have used the following
+axiom:
+(E0) The natural morphisms
+L( ˆTk ˆT)X −→ TkTX
+are isomorphisms for all k ≥ 1.
+Proposition 4.5. The Axiom (E0) implies the Axioms (E1), (E2), and (E4) of
+Definition 4.1.
+While this proposition shows that Axiom (E0) is logically stronger than (E1),
+(E2), and (E4), we currently do not know an example of an elastic space that does
+not satisfy the stronger Axiom (E0).
+4.3. Stability properties of elastic spaces.
+Proposition 4.6. Let X be a diffeological space. The following are equivalent:
+(i) X is elastic.
+(ii) X has an open cover {Si → X} by elastic spaces Si.
+Corollary 4.7. Let X be a diffeological manifold modelled on the diffeological vector
+space A. Then X is elastic if and only if A is elastic.
+Corollary 4.8. Let {Xi}i∈I be a small family of diffeological spaces. The following
+are equivalent:
+(i) The coproduct ⊔iXi is elastic.
+
+ELASTIC DIFFEOLOGICAL SPACES
+29
+(ii) Every Xi is elastic.
+Proposition 4.9. Finite products of elastic spaces are elastic.
+Proposition 4.10. Retracts of elastic spaces are elastic.
+5. Examples
+The conditions for a diffeological space to be elastic are quite strong. Limits,
+colimits, subspaces, and mapping spaces of elastic spaces are generally no longer
+elastic. If by considering elastic spaces we have given away many of the conventient
+properties of the category of diffeological spaces, the question arises whether there
+are enough interesting examples and applications for the concept to be useful.
+We have already stated that finite products and small coproducts of elastic spaces
+are elastic. We have also seen that elastic spaces are stable under retracts, which
+allows for the construction of elastic spaces with rather benign singular behaviour.
+Basic examples are manifolds with corners [Joy12] and cusps.
+The original example that motivated the concept of diffeology by Souriau is that
+of diffeological groups. Our main result states that a diffeolgical group is elastic if
+and only if the vertical lift λG : TG → T 2G is an induction (Theorem 5.2). This
+mild condition is needed to avoid that the bracket of vector fields takes values in a
+tangent bundle with a weaker diffeology, as is the case for vector fields on groups on
+a Ck-manifold. This shows that almost all diffeological groups that come to mind
+are elastic. Even a seemingly pathological group like the quotient R/Q is elastic.
+Every diffeological vector space has an underlying abelian group, so that we can
+apply the characterization of diffeological groups.
+It follows that a diffeological
+vector space A is elastic if the natural map T0A → T0T0A that maps va to the tangent
+vector represented by the path t �→ tva is an induction. A diffeological manifold
+modelled on A is elastic if and only if A is elastic.
+A stronger condition which
+ensures elasticity is that A → T0A is an isomorphism, which yields a trivialization
+TA ∼= A × A of the tangent bundle. We call such vector spaces tangent stable. All
+fine diffeological vector spaces have this property.
+If A is tangent stable then the diffeological mapping space Dflg(X, A) is elastic
+for all X. An important example is the algebra C∞(X) ∼= Dflg(X, R) of smooth
+functions on X. The diffeological space of sections Γ(M, F) of a smooth fiber bundle
+ρ : F → M is elastic (Theorem 5.11). As expected, its tangent space is given by the
+sections of the vertical tangent bundle ker Tρ → M. As a corollary, the diffeological
+space Dflg(M, N) of smooth maps of manifolds is elastic, the tangent space being
+given by
+T Dflg(M, N) ∼= Dflg(M, TN) .
+More examples can be constructed by forming retracts, which allows for mildly
+singular situations.
+5.1. Manifolds with corners. Consider the set [0, ∞) ⊂ R equipped with the
+subspace diffeology. Every smooth map p ∈ U → R with image p(U) ⊂ [0, ∞) has
+vanishing derivatives to all orders at every point u0 with p(u0) = 0. It follows that
+T0[0, ∞) = 0 .
+The tangent space at an interior point x ∈ (0, ∞) is given by Tx[0, ∞) = R.
+
+30
+C. BLOHMANN
+We want to show that [0, ∞) is elastic. For this, we consider the maps
+π : R −→ [0, ∞)
+x �−→ x2
+and
+σ : [0, ∞) −→ R
+x �−→ √x ,
+which satisfy π ◦σ = id[0,∞) as maps of sets. We have to show that σ is smooth. Let
+p : U �→ [0, ∞) be a plot. Since σ is smooth on the interior (0, ∞), σ ◦ p is smooth
+at a all points in U that are mapped to the interior (0, ∞). Assume that p(u0) = 0.
+Since all derivatives of a plot p vanish at u0, p vanishes to all orders at u0, i.e.
+p(u)
+∥u − u0∥k
+u→u0
+−−−→ 0 ,
+for all k ≥ 0. This implies that
+�
+p(u) = (σ ◦ p)(u) vanishes to all orders at u0 as
+well, so that σ ◦ p is differentiable at u0. We conclude that σ is smooth, so that
+[0, ∞) is a smooth retract of R.
+By Proposition 4.10, we conclude that [0, ∞) is elastic and by Proposition 4.9
+that any finite product
+Rn
+k := [0, ∞)k × Rn−k
+is elastic. Since the diffeological tangent functor commutes with products, the tan-
+gent spaces are given by
+T(x1,...,xn)Rn
+k = Tx1[0, ∞) × . . . Txk[0, ∞) × Rn−k .
+Finally, it follows from Proposition 4.6 that every diffeological space modeled locally
+on Rn
+k is elastic.
+Such spaces are called manifolds with corners [Joy12].
+We
+conclude:
+Proposition 5.1. Manifolds with corners are elastic.
+5.2. Manifolds with cusps. Consider the following subset of R2,
+X :=
+�
+(x, y) | x ≥ 0 ∧ |y| ≤ x
+�
+,
+with the subspace diffeology.
+We can squeeze or stretch the corner at (0, 0) by
+multiplying the y coordinate by a smooth function f ∈ C∞�
+[0, ∞)
+�
+(see Figure 3),
+ϕ : X −→ Xf
+(x, y) �−→
+�
+x, f(x) y
+�
+,
+where
+Xf :=
+�
+(x, y) | x ≥ 0 ∧ |y| ≤ f(x)
+�
+.
+Assume that f(x) > 0 for x > 0. Then ϕ has an inverse map given by ϕ−1(0, 0) =
+(0, 0) and
+ϕ−1(x, y) =
+�
+x,
+y
+f(x)
+�
+otherwise. Let us equip Xf with the pullback diffeology of ϕ−1. Since ϕ−1 is sur-
+jective, ϕ is an isomorphism of diffeological spaces. X is [0, ∞) × [0, ∞) rotated by
+minus 45 degrees, so it is elastic by Proposition 5.1. Since ϕ is an isomorphism Xf
+is elastic, too.
+
+ELASTIC DIFFEOLOGICAL SPACES
+31
+∼=
+Figure 3. Squeezing a corner by multiplying the y-coordinate with
+the function f(x) = x
+3
+2. The boundary of the resulting diffeological
+subspace of R2 is the curve x2 = y5 with a cusp at (0, 0).
+We cannot describe here in general, what a manifold with cusps or similar defects
+is, since any squeezing operation that is a smooth retract will produce a new model
+for elastic subspaces of Rn. In Figure 2 on page 7 we have given a few examples of
+elastic subspaces of R2 that can be obtained in this way.
+5.3. Diffeological groups. A diffeological group is a group object (G, m, e) in Dflg.
+As for ordinary groups, the inverse i : G → G, g �→ g−1 is unique, so it is a property
+rather than a structure. By applying an endofunctor F : Dflg → Dflg that preserves
+products, we obtain a diffeological group (FG, Fm, Fe). Examples for F are T and
+Tk (see Proposition 3.19). (TG, Tm, Te) is called the tangent group.
+From the multiplication of the tangent group we obtain the left G-translation of
+TG given by
+L : G × TG
+0G×idT G
+−−−−−−→ TG × TG
+Tm
+−−−→ TG .
+If we restrict the second argument to the tangent fiber at e, we obtain the map
+(19)
+ϕG : G × TeG −→ TG
+(g, ve) �−→ Lgve ,
+where TeG = {e} ×G TG is the tangent fiber at the group identity. It has an inverse
+given by
+ϕ−1
+G : TG
+(i◦πG,idT G)
+−−−−−−−→ G × TG
+L
+−−→ TeG ,
+which maps vg �→ Lg−1vg. We conclude that (19) is an isomorphism. In analogy to
+Lie groups, we will denote
+g := TeG ,
+so that we obtain the trivialization of the tangent bundle,
+TG ∼= G × g .
+TG is itself a diffeological group with neutral element 0e. Using the isomorphism
+T(x,y)(X, Y ) ∼= {(x, y)} ×X×Y T(X × Y )
+∼= ({x} ×X TX) × ({y} ×Y TY )
+∼= TxX × TyY ,
+we obtain the trivialization
+T 2G ∼= TG × T(0e)TG
+∼= G × g × T(e,0)(G × g)
+∼= G × g × TeG × T0g
+∼= G × g × g × T0g .
+
+32
+C. BLOHMANN
+In this trivialization the vertical lift λG : TG → T 2G is given by
+G × g −→ G × g × g × T0g
+(g, a) �−→
+�
+g, 0, 0, λ⊥
+g (a)
+�
+where
+λ⊥
+g : g −→ T0g ,
+maps a ∈ g to the tangent vector represented by the path t �→ ta.
+Theorem 5.2. A diffeological group G is elastic if and only if λ⊥
+g : g → T0g is an
+induction.
+It follows from Theorem 5.2 and Theorem 4.2 that g and T0g are diffeological
+vector spaces. For an elastic group λ⊥
+g is an isomorphism and g is equipped with
+the Lie bracket of invariant vector fields.
+Example 5.3. Every Lie group G is elastic when G is a smooth manifold. However,
+when equipp G with the Ck-diffeology, then g = Rn
+Ck and T0g = Rn
+Ck−1, so that λ⊥
+g
+is no longer an induction (cf. Example 1.5).
+Example 5.4. The diffeomorphism group of a smooth manifold is elastic.
+5.4. Diffeological vector spaces. A diffeological vector space is an R-vector space
+object in Dflg. Explicitly, this means that the addition and scalar multiplication are
+morphisms of diffeological spaces. Diffeological vector spaces are a rich and subtle
+structure [CW19].
+Proposition 5.5. A diffeological vector space A is elastic if and only if the natural
+linear map T0A → T0T0A is an induction.
+Proof. This is Theorem 5.2 for the additive diffeological group A.
+□
+In many cases, elastic diffeological vector spaces satisfy the stronger condition
+A ∼= T0A, which is equivalent to
+TA ∼= A × A .
+We will call diffeological vector spaces with this property tangent stable. Tangent
+stable vector spaces are elastic.
+Proposition 5.6. All fine diffeological vector spaces are tangent stable (hence elas-
+tic).
+5.5. Diffeological manifolds. We recall that a diffeological manifold modelled
+on the diffeological vector space A is a diffeological space X such that every point
+of X has an open neighborhood that is isomorphic as diffeological space to an open
+subset of A.
+Proposition 5.7. A diffeological manifold is elastic if and only if it is modelled on
+an elastic diffeological vector space.
+Proof. This follows from Proposition 4.6.
+□
+
+ELASTIC DIFFEOLOGICAL SPACES
+33
+5.6. Mapping spaces. If A is diffeological vector space, then the mapping space
+Dflg(X, A) is a diffeological vector space with pointwise addition and scalar multi-
+plication.
+Proposition 5.8. Let X be a diffeological space and A a diffeological vector space.
+If A is tangent stable, then so is Dflg(X, A).
+Corollary 5.9. If A is a tangent stable diffeological vector space then we have a
+natural isomorphisms
+T Dflg(X, A) ∼= Dflg(X, A) × Dflg(X, A) ∼= Dflg(X, TA)
+for all X ∈ Dflg.
+Corollary 5.10. The diffeological space
+C∞(X) := Dflg(X, R)
+of smooth R-valued functions on a diffeological space X is elastic.
+Theorem 5.11. Let ρ : F → M be a fiber bundle of smooth manifolds. Then the
+diffeological space of sections Γ(M, F) is elastic with tangent space
+TΓ(M, F) ∼= Γ(M, V F) ,
+the space of sections of the vertical tangent bundle V F := ker Tρ → M.
+Corollary 5.12. The diffeological mapping space Dflg(M, N) of smooth manifolds
+M and N is elastic with tangent space
+T Dflg(M, N) ∼= Dflg(M, TN) .
+Corollary 5.13. The diffeological vector space of sections Γ(M, A) of a smooth
+vector bundle A → M is tangent stable.
+References
+[BH11]
+John C. Baez and Alexander E. Hoffnung. Convenient categories of smooth spaces. Trans.
+Amer. Math. Soc., 363(11):5789–5825, 2011.
+[Blo]
+Christian Blohmann. Tangent structure and Cartan calculus on elastic diffeological
+spaces. In preparation.
+[CC14]
+J. R. B. Cockett and G. S. H. Cruttwell. Differential structure, tangent structure, and
+SDG. Appl. Categ. Structures, 22(2):331–417, 2014.
+[CC15]
+J. R. B. Cockett and G. S. H. Cruttwell. The Jacobi identity for tangent categories. Cah.
+Topol. G´eom. Diff´er. Cat´eg., 56(4):301–316, 2015.
+[CSW14] J. Daniel Christensen, Gordon Sinnamon, and Enxin Wu. The D-topology for diffeolo-
+gical spaces. Pacific J. Math., 272(1):87–110, 2014.
+[CW16]
+J. Daniel Christensen and Enxin Wu. Tangent spaces and tangent bundles for diffeological
+spaces. Cah. Topol. G´eom. Diff´er. Cat´eg., 57(1):3–50, 2016.
+[CW19]
+J. Daniel Christensen and Enxin Wu. Diffeological vector spaces. 2019.
+[DF99]
+Pierre Deligne and Daniel S. Freed. Classical field theory. In Quantum fields and strings:
+a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), pages 137–225.
+Amer. Math. Soc., Providence, RI, 1999.
+[IZ13]
+Patrick Iglesias-Zemmour. Diffeology, volume 185 of Mathematical Surveys and Mono-
+graphs. American Mathematical Society, Providence, RI, 2013.
+[Joy12]
+Dominic Joyce. On manifolds with corners. In Advances in geometric analysis, volume 21
+of Adv. Lect. Math. (ALM), pages 225–258. Int. Press, Somerville, MA, 2012.
+[Kel05]
+G. M. Kelly. Basic concepts of enriched category theory. Repr. Theory Appl. Categ.,
+(10):vi+137, 2005. Reprint of the 1982 original [Cambridge Univ. Press, Cambridge;
+MR0651714].
+
+34
+C. BLOHMANN
+[ML98]
+Saunders Mac Lane. Categories for the working mathematician, volume 5 of Graduate
+Texts in Mathematics. Springer-Verlag, New York, second edition, 1998.
+[Per16]
+Ekaterina Pervova. Diffeological vector pseudo-bundles. Topology Appl., 202:269–300,
+2016.
+[Ros84]
+J. Rosick´y. Abstract tangent functors. Diagrammes, 12:JR1–JR11, 1984.
+[Vin08]
+Martin
+Vincent.
+Diffeological
+differential
+geometry.
+Master’s
+thesis,
+Depart-
+ment of Mathematical Sciences,
+University of Copenhagen,
+2008. Available at
+https://www.math.ku.dk/english/research/tfa/top/paststudents/ms-theses/martinvincent.msthesis.pdf,
+downloaded on 4/15/2019.
+Max-Planck-Institut f¨ur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany
+Email address: blohmann@mpim-bonn.mpg.de
+
diff --git a/wdE0T4oBgHgl3EQfswF7/content/tmp_files/load_file.txt b/wdE0T4oBgHgl3EQfswF7/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf,len=1143
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='02583v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='DG]  6 Jan 2023  ELASTIC DIFFEOLOGICAL SPACES  CHRISTIAN BLOHMANN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We introduce a class of diffeological spaces, called elastic, on which  the left Kan extension of the tangent functor of smooth manifolds defines an  abstract tangent functor in the sense of Rosick´y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  On elastic spaces there is a  natural Cartan calculus, consisting of vector fields and differential forms, together  with the Lie bracket, de Rham differential, inner derivative, and Lie derivative,  satisfying the usual graded commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Elastic spaces are closed under  arbitrary coproducts, finite products, and retracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Examples include manifolds  with corners and cusps, diffeological groups and diffeological vector spaces with  a mild extra condition, mapping spaces between smooth manifolds, and spaces of  sections of smooth fiber bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The quest for a Cartan calculus on diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A category that  contains smooth manifolds as a full subcategory but has better properties, such as  having all limits, colimits, and exponential objects, is often called a convenient set-  ting for differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The price to be paid for this convenience is that such  categories are usually too large as to allow for strong geometric results that hold  for all its objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A typical example is the category of diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' It is a  quasi-topos with all its good categorical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' But since it contains arbitrary  quotients, arbitrary subsets, and arbitrary intersections of smooth manifolds, topo-  logical spaces, vector spaces of arbitrary cardinality, and much more, a theorem that  holds for all diffeological spaces must hold in all these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' So instead of trying to  prove statements that would have to cover this impossible generality of situations,  the task is often to identify conditions that are strong enough to prove a desired  result, but weak enough to allow for a wide range of examples and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  A considerable part of the infinitesimal differential geometric computations on  a smooth manifold M can be carried out in its Cartan calculus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' which consists of  the tangent bundle TM → M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' the Lie bracket of vector fields,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' the graded algebra  of differential forms Ω(M),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' together with the de Rham differential d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' the inner  derivative ιv and the Lie derivative Lv for every vector field v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' which satisfy the  relations  [d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' d] = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  [ιv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' ιw] = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  [ιv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' d] = Lv ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  [Lv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' ιw] = ι[v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='w] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  [Lv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' d] = 0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  [Lv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Lw] = L[v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='w] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  where the bracket is the graded commutator of graded derivations of Ω(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For  example, local definitons and calculations of symplectic geometry can typically be  worked out in the Cartan calculus, such as hamiltonian vector fields, Poisson brack-  ets, hamiltonian actions, Dirac structures, generalized complex geometry, contact  structures, the L∞-algebra of a multisymplectic structure, homotopy momentum  Date: January 9, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' 58A40 (58A03, 18F15, 18F40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Diffeological space, tangent structure, Cartan calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  1  2  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  maps, infinitesimal models for equivariant cohomology, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Another example is lo-  cal Lagrangian Field Theory, where the derivation of the Euler-Lagrange equations,  local symmetries, Noether’s theorems, the theory of Jacobi fields, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' take place in  the Cartan calculus of the infinite jet bundle, also known as the variational bicom-  plex [DF99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In fact, the question we will address in this paper was motivated by  current developments in Lagrangian Field Theory and geometric deformation the-  ory, areas where the basic geometric objects, spaces of fields and paths in a moduli  space of structures, are naturally equipped with diffeologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' What are the conditions a diffeological space must satisfy so that it  is equipped with a natural Cartan calculus?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Of course, there are always the tautological conditions which promote the desired  outcome to axioms, in our case the existence of a Cartan calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The task is to  identify a set of conditions that is minimal or at least so small that it can be verified  in a wide range of cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The basic structures of the Cartan calculus on a smooth manifold M, the differ-  ential graded algebra of differential forms Ω(M) and the tangent bundle TM → M,  are local, that is, they can be defined first on the open subsets U ⊂ Rn of a chart  U ⊂ M and then glued together on an atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' More precisely, the functor U �→ Ω(U)  is a sheaf and the tangent functor U �→ TU a cosheaf, so that  Ω(M) ∼= lim  U→M Ω(U)  TM ∼= colim  U→M TU ,  where the limit in differential graded algebras and the colimit in manifolds are taken  over the category of charts of a maximal atlas of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For a diffeological space X  we simply replace the category of charts by the category of plots and obtain the  definitions  Ω(X) := lim  U→X Ω(U)  TX := colim  U→X TU  of the de Rham complex and the tangent diffeological space of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The maps X �→ Ω(X) and X �→ TX are the pointwise left Kan extensions from  smooth manifolds to diffeological spaces, which shows that they are functorial in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The left Kan extension also applies to natural transformations, such as the bundle  projection πU : TU → U and the zero section 0U : U → TU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This suggests, that  the left Kan extension is the natural method to generalize the Cartan calculus of  smooth manifolds to diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  How do we define the Lie bracket of vector fields on a diffeological space?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The  first guess is to start from the Lie algebras X(U) = Γ(U, TU) of vector fields on all  plots U → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' However, U �→ X(U) is not a functor, so that the left Kan extension  cannot be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We could map the vector fields to the space of derivations of  C∞(X) = Ω0(X), which is equipped with the commutator bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' However, this  map is generally not injective, and even if it is, its image may not be closed under  the bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Worse, the map X �→ Der(C∞(X)) is still not a functor, so that this  does not solve the problem of naturality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The conclusion is that the spaces of vector  fields on plots are not a good starting point for the construction of a natural Cartan  calculus on diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  ELASTIC DIFFEOLOGICAL SPACES  3  Fortunately, the situation has been analyzed carefully by Rosick´y who has iden-  tified the natural structure of the tangent functor that is needed to define the Lie  bracket of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' He defines an abstract tangent structure on a category C  to be an endofunctor T : C → C together with the natural transformations of the  bundle projection πX : TX → X, zero section 0X : X → TX, fiberwise addition  +X : TX ×X TX → TX, exchange of order of differentiation τX : T 2X → T 2X,  and inclusion of the tangent fibers into the vertical tangent space λX : TX → T 2X,  which have to satisfy a rather long list of axioms (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' It is instructive  to see how all these structures come together to define the Lie bracket (8) of vec-  tor fields, avoiding any reference to the commutator bracket of derivations of some  structure ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  For us, the main advantage of Rosick´y’s approach is that all the structure is given  by functors and natural transformations, to which we can apply the left Kan exten-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' However, this does not yield an abstract tangent structure on all diffeological  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The main issue is that the pointwise left Kan extension, which is given by a  colimit, does not preserve limits, in particular the pullback on which the fiberwise  addition of tangent vectors is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' More precisely, the natural morphism  (1)  colim  U→X TU ×U TU −→ TX ×X TX ,  is not an isomorphism for all diffeological spaces X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In fact, this map is generally  neither surjective nor injective, as the following two examples show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2 (Axis cross of the plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Consider the subset {(x, y) ∈ R2 | xy =  0} ⊂ R2 with the subspace diffeology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The two tangent vectors at the orgin in  the direction of the x-axis and the y-axis cannot be represented on the same plot  (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' It follows that (1) is not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='3 (Folded line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Consider the diffeological quotient space of the action  Z2 × R → R, (k, x) �→ kx, where Z2 = {1, −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The quotient map R → R/Z2  is a plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The tangent vectors (0, 1) and (0, −1) on its domain represent the  same tangent vector on R/Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This implies that the pairs ζ =  �  (0, 1), (0, 1)  �  and  η =  �  (0, 1), (0, −1)  �  in TR ×R TR represent the same pair of tangent vectors in  T(R/Z2) ×R/Z2 T(R/Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since the tangent morphism of every morphism of plots  preserves the sum of a pair of tangent vectors at a point and since the sum of ζ is zero  but that of η is not, the two pairs cannot be equivalent in colimU→R/Z2 TU ×U TU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  We conclude that (1) is not injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The axiom of elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Only if (1) is an isomorphism, the left Kan extension  of the addition +U on on plots is a morphism +X : TX ×X TX → TX that can  be viewed as a fiberwise addition of tangent vectors on the diffeological space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Therefore, requiring (1) to be an isomorphism is the first condition we have to impose  for a diffeological space to have a natural Cartan calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  A k-form in Ω(X) is a family of k-forms on all plots U → X that are compatible  with the pullbacks along morphisms of plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A vector field, however, is not repre-  sented by a family of vector fields on the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For this reason, there is no natural  operation of inner derivative on Ω(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For the inner derivative, we have to define a  k-form as a fiberwise multilinear and antisymmetric morphism  α : TX ×X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' ×X TX  �  ��  �  =:TkX  −→ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  4  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  (We avoid defining a tensor product, which would entail the usual technical issues  of completion when the fibers are infinite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=') The notation TkX for the  k-fold fiber product is standard in the literature on abstract tangent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The  inner derivative of α with respect to a vector field v : X → TX is then given by  precomposition  ιvα : Tk−1X  ∼  =  −−→ X ×X Tk−1X  v×Xid  −−−−−→ TkX  α  −−→ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  If we define forms as maps TkX → R, how can we define the differential?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The  differential of a function f : TX → X is given by the tangent map,  df : TX  Tf  −−−→ TR  ∼  =  −−→ R × R  pr2  −−−→ R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  However, the functions and exact 1-forms do not generate the ring of forms, so that  this construction cannot be extended to higher forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  We are now in the following dilemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Either we define differential forms as  families of forms on the plots, in which case we have a differential but no inner  derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Or we define them as fiberwise multilinear and antisymmetric morphisms  TkX → R, in which case we have an inner derivative, but no differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The way  out is to require that the two notions of differential forms coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  We have already imposed the condition that (1) is an isomorphism, which induces  an isomorphism  (2)  Hom(TX ×X TX, R)  ∼  =  −−→ lim  U→X Hom(TU ×U TU, R) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  It is easy to see that this isomorphism is equivariant with respect to the exchange of  the two fractors for the fiber product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Moreover, the maps are fiberwise multilinear  on TkX if and only if they are on all TkU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This shows that the isomorphism (2)  induces an isomorphism from fiberwise multilinear and antisymmetric morphisms  on TX ×X TX to Ω2(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since we need such an isomorphism for forms of arbitrary  degree k, we have to impose the following axiom:  Axiom (E1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The natural morphisms  θk,X : colim  U→X TkU −→ TkX ,  are isomorphisms for all k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  This axiom has the following geometric interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Every tangent vector  vx ∈ TxX is represented by a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' One can picture this by stretching out x in  the direction of vx to a smooth path γ : (−ε, ε) → X of short but non-zero length  through γ(0) = x, such that the coordinate tangent vector  ∂  ∂t at the origin of the  interval is mapped by T0γ to vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In this sense, every point of a diffeological space  has some elasticity in a single infinitesimal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  However, we generally cannot simultaneously stretch out x in the directions of  several tangent vectors v1  x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' , vk  x ∈ TXx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' That is, we cannot always find a plot  p : U → X with p(0) = x such that (T0p) ∂  ∂ti = vi  x, where (t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' , tk) are the  canonical coordinates of U ⊂ Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  And even if we can find such a plot, it may  happen that the tangent map Tp is not injective at 0, so that we cannot identify  the tangent vectors on X with the coordinate vectors on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This identification is  possible at every point x ∈ X if and only if the morphism θk,X is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If in  addition we want this condition to be compatible with the smooth structure, then  we have to make the stronger assumption that θk,X is an isomorphism of diffeological  ELASTIC DIFFEOLOGICAL SPACES  5  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Diffeological subspaces of R2 with non-elastic points  marked in red, at which two tangent directions cannot be represented  on the same plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In this sense, Axiom (E1) captures the geometric idea of the “elasticity” of  a diffeological space in which any finite set of tangent directions can be streched out  to a smooth “membrane” given by the image of a plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='4 (Pasta diffeologies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We can equip a smooth manifold M with an  alternative diffeology by defining the plots be all smooth maps p : U → M such  that the rank of Tp : TU → TM is everywhere less than or equal to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since (i) the  precomposition of p with a smooth function f does not increase the rank, (ii) the  rank is a local property, and (iii) the rank of constant maps is zero, this defines a  diffeology, which we call the rank-r-restricted diffeology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  For r = 0 we obtain the discrete diffeology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If r = 1, then every plot factors  through R, so that we obtain the Spaghetti diffeology [IZ13, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='10, footnote  1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The case r = 2 might then be called the Fettuccine diffeology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  It was  suggested by the participants of the AMS-EMS-SMF meeting 2022 in Grenoble that  the case r = 3 should be called the Gnocchi diffeology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For the rank-r-restricted  diffeology the morphism θk,M of Axiom (E1) is an isomorphism for all k ≤ r but not  for r < k < dim M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The additional axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' So far we have the Axiom (E1) that ensures that we  have a fiberwise addtion on TX and an inner derivative on differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For  the definition of the Lie bracket we need more structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In particular, we need a  natural morphism τX : T 2X → T 2X that exchanges the order of differentiation when  we apply the tangent functor twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' On a euclidean space U ⊂ Rn, every tangent  vector is represented by a path R → U on some plot, so that a tangent vector on the  manifold of tangent vectors is represented by a smooth path of smooth paths, which  is the same as a smooth map R2 → U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Exchanging the order of differentiation is  achieved by exchanging the parameters,  τ1↔2 : R2 −→ R2  (t1, t2) �−→ (t2, t1) ,  which descends by the commutative diagram  (3)  C∞(R2, U)  C∞(R2, U)  T 2U  T 2U  τ ∗  1↔2  τU  to an endomorphism of T 2U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  6  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  When we extend τU to diffeological spaces, the problem arises that the left Kan  extension does not preserve the product of endofunctors, that is, the natural mor-  phism  θ2  X : colim  U→X T 2U −→ T 2X  is generally not an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  We could impose the condition that θ2  X is an  isomorphism, but this would be unnecessarily strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' It suffices to require the left  Kan extension of τU to descend to a morphism τX : T 2X → T 2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since θ2  X is a  subduction for all X (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='26), such a τX is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This condition can  be expressed more intuitively in terms of the smooth families in the same way as  for euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We can show that we can represent elements in T 2X by plots  R2 → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' More precisely, we have a subduction  Dflg(R2, X) −→ T 2X ,  where Dflg denotes the inner hom of diffeological spaces, that is, the set of morphisms  equipped with the functional diffeology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The second axiom can now be expressed in  a way that is completely analogous to diagram (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Axiom (E2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' There is a natural morphism τX : T 2X → T 2X, such that the  diagram  Dflg(R2, X)  Dflg(R2, U)  T 2X  T 2X  τ ∗  1↔2  τX  commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Next, consider the natural morphism λX : TX → T 2X that maps v ∈ TX to  the vertical tangent vector on TX represented by the path t �→ tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' On a smooth  manifold, this morphism induces an isomorphism between every tangent space and  the tangent space of the tangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For diffeologial vector spaces this can fail,  as the following example shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Consider Rn equipped with k-times differentiable maps as plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  This is a diffeological vector space that we denote by Rn  Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Its tangent diffeological  space is given for k > 0 by  TRn  Ck ∼= Rn  Ck × Rn  Ck−1 ,  which shows that the vector space and its tangent fiber are not isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Assume  that k > 1, so that we can apply the tangent functor twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The vertical lift,  λRn  Ck : Rn  Ck × Rn  Ck−1 −→ Rn  Ck × Rn  Ck−1 × Rn  Ck−1 × Rn  Ck−2  (x, v) �−→ (x, 0, 0, v) ,  is not a subduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The definition of the Lie bracket in terms of the tangent structure yields a map  from X to the vertical subbundle of T 2X restricted to the zero section of TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  We have to be able to identify this bundle with TX for the bracket to be again a  vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This condition is not specific to diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A vector field on  a Ck-manifold is a Ck-map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The commutator of two such vector fields is a Ck−1-  map which is, therefore, not a vector field on the Ck-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' To exclude such  phenomena we have to impose the following axiom:  ELASTIC DIFFEOLOGICAL SPACES  7  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Elastic diffeological subspaces of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The tangent spaces  are 0 at the marked points, R at points on the black lines, and R2 at  gray points in the interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Axiom (E3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The vertical lift λX : TX → T 2X is an induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  There are two more axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For smooth manifolds the tangent functor commutes  with pullbacks over submersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  This follows from the local standard form of  submersions, which is proved using the implicit function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Such a genuinely  analytic result cannot hold for all diffeological spaces, which is why we need to  impose the following axiom:  Axiom (E4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The tangent functor commutes with fiber products of the tangent  bundle, TTkX ∼= TkTX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Finally, we want the diffeological spaces that satisfy our axioms to form a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  This requires the collection of diffeological spaces that satisfy the axioms to be closed  under the functors Tk, which leads to the following axiom:  Axiom (E5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For every finite set of positive integers k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' , kn the diffeological  space X′ := Tk1 · · · TknX satisfies axioms (E1) through (E4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  A diffeological space that satisfies Axioms (E1)-(E5) will be called elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If we  drop Axiom (E5), then we still have a natural Cartan calculus on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We call a  diffeological space that satisfies Axioms (E1)-(E4) weakly elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The category of  weakly elastic spaces is not closed under the functors Tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The main result about elastic diffeological spaces is the follow-  ing: The left Kan extension of the tangent structure on euclidean spaces defines  a tangent structure with scalar R-multiplication on the category of elastic spaces  (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The proof of this statement, which is very long and technical, is  carried out in detail in a much longer paper [Blo].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Here we will present an overview  of the conceptual framework, the main properties, and important examples of elastic  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  In Section 2 we review Rosick´y’s concept of abstract tangent structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  We  also spell out the conditions for a compatible scalar multiplication, which is only  mentioned briefly in the original paper [Ros84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For clarity, we spell out the tangent  structure of euclidean spaces explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  In Section 3 we state some new results about the left Kan extension from eul-  cidean spaces to diffeological spaces, that will be needed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We use the definition  of diffeological spaces as concrete sheaves on the site of euclidean spaces, which  the best approach for the categorical constructions we will study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We then state,  without proof, our results on the compatibility of the left Kan extension with prod-  ucts, coproducts, subductions, composition of endofunctors, and the D-topology of  diffeological spaces, which are all needed for the proof of the main Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  8  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  Section 4 contains the formal definition of elastic spaces an the statement, without  proof, of the main Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We discuss some alternative choices of axioms for  which the theorem remains true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' First, we observe that we can drop Axiom (E5)  from the defintion of elastic spaces and still obtain a Cartan calculus on the space  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  This category of weaker diffeological spaces will no longer be closed under  the tangent functor and its fiber products, so that we no longer have a category  with an abstract tangent structure in the sense of Rosick´y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We can also slightly  relax the Axiom (E1) of elasticity by requiring the the condition only holds for  k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This weaker condition will be sufficient to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' As explained  in the introduction, our choice of the stronger Axiom (E1) is motivated by our  wish to obtain not just a tangent structure, but a full-fledged Cartan calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In  an earlier version of the notion of elastic spaces presented in talks, Axioms (E1),  (E2), and (E4) were replaced by a single condition, which later turned out to be  unnecessarily strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Finally, we state, without proof, that the category of elastic  spaces is closed under restrictions to open subsets, coproducts, finite products, and  retracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  In Section 5 we will give a number of examples for elastic diffeological spaces that  show that, while the conditions of elasticity are quite strong, they still allow for an  interesting range of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The first main result, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2, shows that a  diffeological Lie group G is elastic if and only if the natural map g → T0g from the  vector space g = TeG to its tangent space at 0 is an induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This is a surprisingly  weak condition, which is not particular to diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For example, it is  not satisfied by differentiable group structure on a Ck-manifold, k < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The proof  of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2, which is long and techincal, will be given in [Blo].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The second main  result, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='11, states that the diffeological space of sections of a smooth fiber  bundle F → M is elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The proof of this theorem is again long and involved, so  that it will be given in [Blo].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' More examples of elastic spaces in this sections are:  manifolds with corners and cusps, diffeological vector spaces satisfying a mild extra  condition, manifolds modelled on elastic vector spaces, mapping spaces Dflg(X, A)  to diffeological vector spaces satisfying A ∼= T0A, the space of smooth R-valued  functions on a diffeological space, the mapping space of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' One of the motivations to develop the concept of elastic spaces came  from classical field theory, where the basic strucure, the “space” of fields is the set  of sections of a smooth fiber bundle F → M leaving it often unclear or implicit  what “space” means mathematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' It is often observed that F = Γ(M, F) is a  Fr´echet manifold, which is subsequently viewed as blanket license to treat F as if  it were an ordinary finite-dimensional smooth manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For example, it is often an  implicit assumption that there is a natural differential bigraded algebra of smooth  forms on F × M that restricts to the variational bicomplex on J∞F [DF99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In  the same vein, in the study of symmetries, such as Noether’s theorems or BV-  theory, the constructions are often explained rigorously only for finite Lie groups  acting on finite-dimensional manifolds and then generalized with a leap of faith to  group-valued functions or diffeomorphisms acting on the spaces of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A closer  analysis shows that only the diffeological structure that is being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For example,  a variation of a field is a smooth path of sections and a tangent vector to the space  of solutions of the field equations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' a generalized Jacobi field) is represented by a  ELASTIC DIFFEOLOGICAL SPACES  9  smooth path of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' All this suggests that there is a diffeological construction of  a Cartan calculus on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This approach is validated by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Another motivation comes from geometric deformation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Conceptually, a  deformation is a path in the moduli space of structures, such as the morphisms  of an algebraic structure, riemannian metrics, or complex structures, all of which  are equipped with a natural functional diffeology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This suggests that the geomet-  ric moduli spaces can be conceptualized by (higher) stacks that are presented by  (higher) diffeological groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The infinitesimal deformation theory should then  be given by the fibers of the tangent bundle of the moduli space, which should be  presented by the corresponding (higher) Lie algebroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In order to define this pro-  cedure rigorously, a Lie theory for diffeological groupoids has to be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This  has lead us to the conclusion that we need a tangent structure on the diffeological  spaces of the groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' It is encouraging that for diffeological Lie groups the con-  dition of elasticity is surprisingly weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Lie theory for diffeological groupoids and  their application to geometric deformation theory is work in progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Abstract tangent structures  It is fairly straightforward to generalize the de Rham complex Ω(M), which is  a contravariant functor from smooth manifolds to differential graded algebras, to  other categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' While vector fields are in some sense dual to differential forms,  their generalization is a much more difficult problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' An immediate obstacle is  that the map M → X(M) that sends a manifold to its Lie algebra of vector fields  is not a functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' It is the tangent bundle TM → M that is functorial in M and,  therefore, lends itself easily to generalizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Given a generalized tangent bundle  πX : TX → X in some category, the vector fields are naturally defined as the  sections of the morphism πX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The obvious question is now the following:  Question 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Given a morphism TX → X in some catgory, what is the natural  structure needed to equip its space of sections with the structure of a Lie algebra?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  This question turns out to be quite involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A vector field on a smooth manifold  can be identified with a derivation of its ring of smooth functions C∞(M), which  is closed under the commutator bracket of the ambient ring of endomorphisms of  C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' However, M �→ Der(C∞(M)) is still no functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Moreover, in a generalized  setting, the identification of sections of TX → X with derivations on some structure  ring on X seems to be an overly strong requirement that is extraneous to differential  geometric considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' So how can we define the Lie bracket of vector fields on  M directly in terms of the tangent functor using a categorical approach that lends  itself to generalizations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  This issue has been solved by Rosick´y in [Ros84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In a first step, we observe that on  manifolds the vector space structure on vector fields is induced by the vector space  structure on the tangent fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' To allow for categories that do not contain the real  numbers as object, we relax the structure of R-vector space to that of an abelian  group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The structure we then need on a generalized tangent bundle TX → X is  the natural transformatios of addition +X : TX ×X TX → TX and a zero section  0X : X → TX that equip TX → X with the structure of an abelian group over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  For the definition of the Lie bracket, we start from the coordinate formula  �  vi ∂  ∂xi wj ∂  ∂xj  �  =  �  vi∂wj  ∂xi − wi∂vj  ∂xi  � ∂  ∂xj  10  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  for vector fields on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The right hand side is given by the derivation of w with  respect to v minus the derivation of v with respect to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In order to generalize this  formula we must make sense of the differentiation of one vector field with respect to  another and the subtraction of the two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The derivation of w : X → TX with respect to v : X → TX is given by the  composition  X  v  −−→ TX  Tw  −−−→ T 2X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  However, we cannot yet subtract Tw(vx) and Tv(wx) since the basepoint of Tw(vx)  in πTX : T(TX) → TX is wx, whereas that of Tv(wx) is vx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  We first have to  exchange the order of differentiation of the twofold tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' That is, we  need a natural transformation τX : T 2X → T 2X that satisfies τX ◦ πTX = TπX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Then the basepoint of τX(Tv(wx)) is also wx, so that we can take the difference  (4)  Tw(vx) − τX  �  Tv(wx)  �  =  �  +TX ◦ (Tw ◦ v, −τX ◦ Tv ◦ w)  �  x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The result lies in the vertical tangent bundle of T 2X → TX, that is, the kernel of  TπTX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In the last step we have to be able to identify at every point wx ∈ TX the  vertical tangent space with TxX itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  In a smooth manifold, there is a morphism λ2,M : TM ×M TM → T 2M that maps  the pair (wx, ux) to the vertical tangent vector in T 2M that is represented by the  path t �→ wx + tux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This map induces an isomorphism TM ×M TM ∼= ker TπTM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In  the generalized setting this structure is promoted to an axiom: We require a natural  morphism λ2,X : TX ×X TX → T 2X that induces an isomorphism TX ×X TX ∼=  ker TπTX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Using this isomorphism, the expression (4) can be viewed as an element  in TX ×X TX that can be projected onto the second factor which produces an  element in TX, which is the value of the bracket [v, w] at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  For this bracket to satisfy the Jacobi identity, a number of compatibility relations  and properties of the various structures, bundle projection, zero section, fiberwise  addition, exchange of the order of differentiation, and the vertical lift have to be  required [Ros84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Rosicky’s axiomatization of all this structure is the basis of this  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Preliminary remarks on terminology and notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Terminology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let Wibble be an algebraic theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let X be an object in a  category C such that the overcategory C ↓ X has all finite products (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' pullbacks  over X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A Wibble object in C ↓ X will be called a bundle of Wibbles over X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  In this paper Wibble will be one of: monoid, group, abelian group, module, R-  vector space (for categories containing R as an object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If W → X is a bundle of  Wibbles and if the pullback Wx = ∗ ×X W over a point x : ∗ → X exists in C, then  Wx is a Wibble object in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In other words, every fiber of a bundle of Wibbles is a  Wibble object in C, which justifies the terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Note, that the notion of bundle  of Wibbles does not make any assumptions on local trivializations, whatsoever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' So  a bundle of vector spaces over a manifold M is more general than a vector bundle  over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The main purpose of Terminology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2 is to unify (for the purpose of  this paper) the varied terminology found in the literature and to use a term that  is self-explanatory for a category theorist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In [Ros84, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' 1] a bundle of (abelian)  groups over an endofunctor F : C → C is called an “natural (abelian) group bundle  ELASTIC DIFFEOLOGICAL SPACES  11  over F”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A bundle of vector spaces over a diffeological space X is called a “regular  vector bundle” in [Vin08], a “diffeological vector space over X” in [CW16], and a  “diffeological vector pseudo-bundle” in [Per16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We will follow [Ros84] for the notation of the compositions of func-  tors and natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The composition of functors G : A → B and  F : B → C will be denoted by juxtaposition FG : A → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Therefore, the horizontal  composition of natural transformations α : F → F ′ and β : G → G′ (the Godement  product) will also be denoted by juxtaposition αβ : FG → F ′G′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Its components  are given by the following commutative diagram:  FG(A)  FG′(A)  F ′G(A)  F ′G′(A)  αG(A)  F (βA)  (αβ)A  αG′(A)  F ′(βA)  The identity natural transformation F → F will be denoted by F, so that  (Fβ)A = F(βA)  (αG)A = αG(A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The vertical composition of α with a natural transformation α′ : F ′ → F ′′ will be  denoted by α′ ◦ α : F → F ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Its components are given by (α′ ◦ α)A = α′  A ◦ αA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The  monoidal category of endofunctors will be denoted by End(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Rosick´y’s axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In [Ros84], Rosick´y introduced the notion of abstract  tangent functor, which captures much of the categorical structure of the tangent  functor of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The following notion is implicit in Rosick´y’s definition:  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let F : C → C be a functor and τ : F 2 → F 2 a natural transfor-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let τ12 := τ F and τ23 := F τ be the two trivial extensions of τ to natural  transformations F 3 → F 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We call τ a braiding on F if it satisfies the braid rela-  tions τ12 ◦ τ23 ◦ τ12 = τ23 ◦ τ12 ◦ τ23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A braiding τ is called a symmetric structure  on F if it satisfies τ ◦ τ = F 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A symmetric structure on F defines an action of the symmetric group  Sn on F n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  A bundle of groups over X consists of a morphism π : A → X, the bundle  projection, together with the morphisms 0 : X → A and + : A ×X A → A of the  group structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let π′ : A′ → X′, 0′ : X′ → A′, +′ : A′ ×X′ A′ → A′ be another  bundle of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A morphism of bundles is a commutative diagram  A  A′  X  X′  ϕ  π  π′  ψ  12  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  There is a unique morphism ϕ ×ψ ϕ : A ×X A → A′ ×X′ A′ that makes the following  diagram commutative:  A ×X A  A ×X A  A × A  A′ × A′  ϕ×ψϕ  ϕ×ϕ  The pair (ϕ, ψ) is a morphism of bundles of groups if the diagram  A ×X A  A′ ×X A′  A  A′  ϕ×ψϕ  +  +′  ϕ  commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' An endofunctor F preserves the fiber product if the natural mor-  phism of bundles over FX,  (5)  νk,X : F(A ×π,π  X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' ×π,π  X A) −→ FA ×F π,F π  F X  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' ×F π,F π  F X  FA ,  where both sides have the same number k of factors, is an isomorphism for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='7 (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' 2 in [Ros84], Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='3 in [CC14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A tangent structure of a  category C consists of a functor T : C → C together with natural transformations  π : T → 1, 0 : 1 → T, + : T2 → T, λ : T → T 2, and τ : T 2 → T 2, such that the  following axioms hold:  Fiber products: The pullbacks  Tk := T ×1 T ×1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' ×1 T  �  ��  �  k factors  over T  π→ 1 exist for all k ≥ 1, are pointwise, and preserved by T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Bundle of abelian groups: T  π→ 1 with neutral element 0 and addition +  is a bundle of abelian groups over 1 (Terminology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Symmetric structure: τ : T 2 → T 2 is a symmetric structure on T (Def-  inition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Moreover, τ is a morphism of bundles of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' That is, the  diagrams  T 2  T 2  T  τ  Tπ  πT  and  TT2  T 2 ×Tπ,Tπ  T  T 2  T 2T  T 2  T 2  ν2  T+  τ×T τ  +T  τ  commute, where ν2 is morphism (5) for A = TX  π→ X, F = T, and k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  ELASTIC DIFFEOLOGICAL SPACES  13  Vertical lift: The diagrams  T  T 2  1  T  λ  π  πT  0  T  T 2  T 2  T 3  λ  λ  λT  Tλ  commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Moreover, the first diagram is a morphism of bundles of groups,  that is (+T) ◦ (λ ×0 λ) = λ ◦ +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Compatibility of vertical lift and symmetric structure: The diagrams  T  T 2  T 2  λ  λ  τ  T 2  T 3  T 3  T 2  T 3  Tλ  τ  τT  Tτ  λT  commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The vertical lift is a kernel: The diagram  (6)  T  T 2  1  T2  λ  π  (πT,Tπ)  0×10  is a pointwise pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Terminology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In [CC14] and subsequent work, Rosick´y’s original condition  that T → 1 be a bundle of abelian groups was relaxed to a bundle of abelian  monoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In this terminology, Rosicky’s stronger notion is called a tangent structure  with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' All tangent structures in this paper will be with negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Diagram (6) is a pointwise pullback if and only if  T  T 2  T  λ  πT  Tπ  0◦π◦πT  is a pointwise triple equalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This condition is the original axiom in [Ros84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The vertical lift can be extended by the additive bundle structure  to the map  (7)  λ2 : T2  T0×0λ  −−−−−→ T2T  +T  −−−→ T 2  τ  −−→ T 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  It was shown in [CC14, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='10], assuming all other axioms of a tangent structure  (with negatives), that axiom (6) is satisfied if and only if  T2  T 2  1  T  λ2  π◦pr1  Tπ  0  is a pointwise pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  14  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Scalar multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let R be a ring object in the category C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This gives  rise to an endofunctor R × 1 : C → C, X �→ R × X, which is equipped with the  projection pr1 : R × 1 → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The ring structure of R equips R × 1 → 1 with the  structure of a ring internal to endofunctors over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let π : T → 1 be an abelian group  object in the category of endofunctors over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' An (R × 1 → 1)-module structure on  T → 1 is given explicitly by a natural transformation  κX : R × TX −→ TX ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  such that the following diagrams commute for all X ∈ C:  (i) Morphism of bundles:  R × TX  TX  X  κX  πX◦pr2  πX  (ii) Associativity:  R × R × TX  R × TX  R × TX  TX  idR×κX  ×idR×T X  κX  κX  (iii) Unitality:  {1} × TX  R × TX  TX  ∼  =  κX  (iv) Linearity in R:  R × R × TX  R × TX  (R × TX) ×X (R × TX)  TX ×X TX  TX  +×idT X  idR2×∆T X  κX  κX×XκX  +X  Here ∆TX : TX → TX ×X TX is the diagonal morphism and the factors of  the codomain are reordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  (v) Linearity in TX:  R × TX ×X TX  R × TX  (R × TX) ×X (R × TX)  TX ×X TX  TX  id×+X  ∆R×idT2X  κX  κX×XκX  +X  ELASTIC DIFFEOLOGICAL SPACES  15  Here ∆R : R → R × R is the diagonal morphisms and the factors of the  codomain are reordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  We will call this structure more succinctly an R-module structure on T → 1  and T → 1 a bundle of R-modules (Terminology 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If T is part of a tangent  structure on C, then we also have to require the compatibility with the symmetric  structure and the vertical lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let R be a ring internal to a category C with a tangent structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  An R-module structure κX : R×TX → TX will be called a scalar multiplication  of the tangent structure if the following diagrams commute for all X ∈ C:  (vi) Compatibility with the symmetric structure:  R × T 2X  R × T 2X  T 2X  idR×τX  κT X  TκX  (vii) Compatibilty with the vertical lift:  R × TX  R × T 2X  TX  T 2X  idR×λX  κX  κT X  λX  The tangent structures we consider here will all be equipped with an R-scalar  multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' As is the case for any module structure, the commutative dia-  gram (v) implies that the scalar multiplication by 0 ∈ R sends TX to the zero  section, that is, the diagram  {0} × TX  R × TX  X  TX  πX◦pr2  κX  0X  is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If κ is such that this diagram and diagrams (i)-(iii) are commutative,  then κ will be called an R-cone structure and T → 1 a bundle of R-cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The Lie bracket of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let C be a category with a tangent structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A vector field on  X ∈ C is a section of πX : TX → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The bracket of two vector fields v, w : X → TX is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The  composition of v and Tw : TX → T 2X satisfies  πTX ◦ Tw ◦ v = w ◦ πX ◦ v = w ◦ idX  = w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  When we exchange v and w, we have πX ◦Tv◦w = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In order to be able to subtract  the two terms in the fiber product T 2X ×TX T 2X, we have two apply the symmetric  16  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  structure on T 2, so that we obtain  πTX ◦ τX ◦ Tv ◦ w = TπX ◦ Tv ◦ w = TidX ◦ w = idTX ◦ w  = w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  This shows that Tw◦v and τX ◦Tv◦w project to the same fiber of πTX : T 2X → TX,  so that we can take the difference  δ(v, w)(x) := (Tw ◦ v)(x) − (τX ◦ Tv ◦ w)(x) ,  where the minus denotes the difference in the bundle of abelian groups πTX : T 2X →  TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We have  πTX ◦ δ(v, w) = 0 = TπX ◦ δ(v, w) ,  so that the map δ(v, w) : X → T 2X takes values in the kernel of TπX : T 2X → TX,  which is isomorphic to TX ×X TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' By projecting on the second factor we thus  obtain the vector field [v, w] : X → TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This construction can be summarized by  the following commutative diagram:  (8)  X  X × X  TX × TX  T 2X × T 2X  TX  TX ×X TX  T 2X  T 2X ×TX T 2X  T 2X × T 2X  X  TX  TX  TX × TX  ∆X  [v,w]  ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  ∃!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  v×w  Tw×Tv  idX×τX  ⌟  pr2  λ2,X  πX  TπX  ⌟  −T X  πT X×πT X  0X  ∆T X  This shows that all of the tangent structure is needed for the definition of the bracket  of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' It was announced in [Ros84] and proved in [CC15] with the input  of Rosick´y that [v, w] satisfies the Jacobi relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The set of vector fields Γ(X, TX) has the structure of an abelian group with  addition  v + w := +X ◦ (v × w) ◦ ∆X ,  where ∆X : X → X × X is the diagonal morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' When the tangent structure  has a scalar multiplication by R, then Γ(X, TX) is a module over the ring C(X, R)  of R-valued functions, given by  κ(f, v) = κX ◦ (f × v) ◦ ∆X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The tangent structure of euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The eponymous example for  tangent structures is the tangent functor of open subsets of real vector spaces, which  is the local model for the tangent functor of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let Eucl denote the  category which has open subsets of Rn, n ≥ 0 as objects and smooth maps as  morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Eucl will be called the category of euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Its tangent  functor will be denoted by  T : Eucl −→ Eucl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  ELASTIC DIFFEOLOGICAL SPACES  17  On an open subset U ⊂ Rn, the functors that appear in the definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='7 of a  tangent category are given explicitly by  TU = U × Rn  T 2U = U × Rn × Rn × Rn  T kU = U × (Rn)2k−1  T2U = U × Rn × Rn  TkU = U × (Rn)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  On a smooth map f : U → V ⊂ Rm the functors are given by  Tf : (u, ui  0) �−→  �  f(u), ∂f a  ∂xi ui  0  �  T 2f : (u, ui  0, ui  1, ui  01) �−→  �  f(u), ∂f a  ∂xi ui  0, ∂f a  ∂xi ui  1, ∂f a  ∂xi ui  01 + ∂2f a  ∂xi∂xj ui  0uj  1  �  T2f : (u, ui  0, ui  1) �−→  �  f(u), ∂f a  ∂xi ui  0, ∂f a  ∂xi ui  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The formulas for T k and Tk are analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The natural transformations of the  tangent category structure are given by  πU : (u, u0) �−→ u  0U : u �−→ (u, 0)  +U : (u, u0, v0) �−→ (u, u0 + v0)  λU : (u, u0) �−→ (u, 0, 0, u0)  τU : (u, u0, u1, u01) �−→ (u, u1, u0, u01) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The commutativity of T2 and T is given by the isomorphism  T(T2U) −→ T2(TU)  �  (u, u0, v0), (u1, u01, v01)  �  �−→  �  (u, u1), (u0, u01), (v0, v01)  �  The bundle projection extends to T 2 as  (πT)U = πTU : (u, u0, u1, u01) �−→ (u, u0)  (Tπ)U = TπU : (u, u0, u1, u01) �−→ (u, u1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The other natural transformations that appear in the definition, +T, T+, λT, Tλ,  τT, and Tτ, are obtained in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The extension (7) of the vertical lift is  given by  λ2 : (u, u0, v0) �−→ (u, u0, 0, v0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The following propositions can be checked by explicit elementary calculation:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Eucl with the tangent functor T and the natural transformations  π, 0, +, τ, and λ is a tangent structure on Eucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The fiberwise multiplication by real numbers,  κU : R × TU −→ TU  �  r, (u, u0)  �  �−→ (u, ru0) ,  is a scalar multiplication of the tangent structure (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  18  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Left Kan extension to diffeological spaces  The structures on smooth manifolds that we wish to generalize to diffeological  spaces, such as the tangent bundle and the algebra of differential forms, are local  and universal in the sense that they are defined on all open subsets of Rn and then  glued together along an atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In categorial terms, the local structure is given by a  functor F : Eucl → C and the glueing operation by the colimit  FM := colim  U→M FU  over a maximal atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In categorical terms, FM is the pointwise left Kan extension  of F along the inclusion of euclidean spaces into smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For the gen-  eralization of this construction to diffeological spaces we replace the charts of the  smooth manifold M with the plots of the diffeological space X,  FX := colim  U→X FU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  This is the pointwise left Kan extension of F along the inclusion of euclidean spaces  into diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  In order to use the left Kan extension as universal tool to generalize differential  geometric structures to diffeological spaces, we have to study its categorical prop-  erties in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For this it is best to work with the definition of diffeological  spaces as concrete sheaves on euclidean spaces, which was elucidated in [BH11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The site Eucl of euclidean spaces is concrete, which means that there is a faith-  ful functor Eucl → Set, U → |U| that maps covers to surjective maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A sheaf  D : Euclop → Set is concrete if it is a subsheaf of U �→ Set(|U|, D(∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Explicitly,  this means that D(U) is a collection of maps of sets |U| → X := D(∗), called plots,  that satisfy the properties of a sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In this way, we recover the original definition  of diffeology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  As is the case for any category of concrete sheaves, the category of diffeological  spaces Dflg is a quasi-topos, a category with a classifier for strong subobjects, which  has a number of good properties (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Other good properties of Dflg  are inherited from the site Eucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For example, the category of plots of a diffeological  space, the index category used to compute the pointwise left Kan extension, is sifted,  which suggest a compatibility with finite products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Every representable presheaf is a concrete sheaf, so that the Yoneda embedding  factors through a full and faithful functor  y : Eucl −→ Dflg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  While y is simply the Yoneda embedding restricted on its codomain, the restriction  changes the computation of colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  A colimit in Dflg is given by first taking  the pointwise colimit in the functor category SetEuclop and then applying the left  adjoint of the inclusion Dflg → SetEuclop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This second step has consequences for the  properties of the left Kan extension, which is given by a colimit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Most structures that we will consider are given by endofunctors F : Eucl →  Eucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since Eucl is not cocomplete and since we want to obtain an endofunctor of  diffeological spaces, we have to place the codomain of F in diffeological spaces by  composing with y before taking the Kan extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The Kan extended endofunctor  will be denoted by  LF := Lany yF : Dflg −→ Dflg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The main question we will address in this section is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  ELASTIC DIFFEOLOGICAL SPACES  19  Question 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' What are the categorical properties of L : End(Eucl) → End(Eucl)  and which properties of F are preserved by L?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  It follows from the naturality of the Kan extension that L is a functor  L : End(Eucl) −→ End(Dflg)  of categories of endofunctors, which means that the vertical composition of natural  transformations and, therefore, commutative diagrams of natural transformations  are preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This is the most important property for our purposes, since it im-  plies that the commutative diagrams that appear in the Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='7 of tangent  structures carry over to the diffeological setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Unfortunately, this is where the good news about the left Kan extension to dif-  feological spaces end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Unlike the Kan extension to presheaves, the Kan extension  of a functor to concrete sheaves does generally not preserve colimits, not even finite  coproducts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Nor does L preserve the composition of endofunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Fortunately, the  functors F we want to extend have some good categorical properties, that entail  good properties of their Kan extensions LF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Using that the category of plots U → X is sifted, we show that if F preserves finite  products, then so does its Kan extension LF (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' It also implies that  L preserves finite products of endofunctors (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For the compatibility  with coproducts we have to assume that F is a cosheaf, that is, F : Euclop →  Euclop is a sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Then LF preserves coproducts (Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='23) and subductions  (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  While L does not preserve the composition of endofunctors, there is a natural  morphism  (9)  L(FG) −→ (LF)(LG) ,  which is generally not an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We can show that if F is a cosheaf, then (9)  applied to a diffeological space is a subduction (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The morphism  L(FGH) −→ L(F)L(G)L(H)  we obtain by applying (9) twice does not depend on whether we first apply it to  (FG)H or to F(GH) (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In this sense µ is associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' It is straight-  forward to see that (9) is natural in F and G (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Finally, we study the compatibility of the Kan extension with the D-topology  of diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We observe that many of the functors on euclidean spaces  we are interested in come with a natural transformation F → 1 such that the  pullback along any open embedding V → U satisfies FV ∼= V ×U FU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We call  such an F → 1 a local bundle, by default of a better term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We can show that if  F → 1 is local, then so is its Kan extension LF → 1 (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Then we  prove that a morphism FX → GX of local bundles is an induction, subduction,  epimorphism, or isomorphism if all restrictions to the open subsets of a cover of X  are (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Diffeological spaces as concrete sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Recall that Eucl denotes the  category which has all open subsets of euclidean spaces Rn, n ≥ 0 as objects and all  smooth maps as morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Open covers define a Grothendieck pretopology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The small category Eucl together with the Grothendieck topology  generated by the pretopology of open covers will be called the site of euclidean  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  20  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  The terminal object in Eucl is ∗ := R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The functor of points,  |  | : Eucl −→ Set  U �−→ Eucl(∗, U) ,  is faithful, so that it equips Eucl with the structure of a concrete category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' More-  over, every cover {Ui → U} is surjective on the underlying sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A site with these  properties is called concrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Let X : Euclop → Set be a presheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Then there is a morphism of presheaves  defined by  (10)  X(U) −→ Set  �  |U|, |X|  �  p �−→  �  (∗  u→ U) �→ Xu(p)  �  ,  where Xu ≡ X(u) : X(U) → X(∗) is the restriction of X(U) to the point u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A presheaf X : Euclop → Set on a concrete site is concrete if (10)  is a monomorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' if the maps defined in (10) are injective for all U ∈ Eucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  A sheaf is concrete if it is concrete as a presheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A morphism between concrete  sheaves is a morphism of the underlying presheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A concrete sheaf on the site of euclidean spaces is called a diffeo-  logical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A morphism of diffeological spaces is a morphism of sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The  category of diffeological spaces will be denoted by Dflg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='5 (Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='25 in [BH11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The category of diffeological spaces is a qua-  sitopos with small limits and small colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='5 implies that the category of diffeological spaces has a number of  convenient properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  For clarity and later reference, we will spell out some of  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The category Dflg of diffeological spaces has the following prop-  erties:  Dflg is locally cartesian closed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' for every object X in Dflg the overcate-  gory Dflg ↓ X is cartesian closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Strong monomorphisms and strong epimorphisms are effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  (Strong) monomorphisms and (strong) epimorphisms are stable under pull-  back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Dflg is quasiadhesive, that is, the pushout of a strong monomorphism is a  strong monomorphism and the pushout square is a pullback square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The initial object is strict, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' every morphism X → ∅ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Coproducts are disjoint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' X → X ⊔ Y ← Y are monomorphisms and  X ×X⊔Y Y ∼= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The functor of points Dflg → Set, X → Dflg(∗, X) is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' It has a left  and a right adjoint, so that it preserves limits and colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Terminology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The strong epimorphisms in Dflg are called subductions, the  strong monomorphisms inductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Here are some of the most basic examples for diffeologies:  ELASTIC DIFFEOLOGICAL SPACES  21  (a) The fine diffeology or discrete diffeology on a set S is the diffeology for  which the plots are the locally constant maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1  (b) The coarse diffeology, or indiscrete diffeologoy, or trivial diffeology  on a set S is given by U �→ Set  �  |U|, S  �  , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' all maps are plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  (c) Every topological space X is equipped with the continuous diffeology  given by U �→ Top(U, X), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' the plots are the continuous maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  (d) Every smooth finite-dimensional manifold M is equipped with the natural  diffeology given by U �→ Mfld(U, M), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' the plots are the infinitely often  differentiable maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  (e) We will denote the exponential objects in Dflg by  Dflg(X, Y ) ≡ Y X  and call them the diffeological mapping spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The diffeology, which is  given by the universal property  Dflg  �  U, Dflg(X, Y )  � ∼= Dflg(U × X, Y ) ,  is called the functional diffeology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='6, the forgetful functor Dflg → Set, X �→ |X| has a  left and a right adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The right adjoint equips a set S with the coarse diffeology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The left adjoint equips it with the fine diffeology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In other words, the fine diffeology  on a set is the free diffeology, the coarse diffeology is the cofree diffeology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The map that sends a smooth manifold M to its natural diffeology  U �→ Mfld(U, M) defines a full, faithful, and injective functor Mfld → Dflg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Every diffeological space X is naturally equipped with the finest  topology such that all plots are continuous, which is called the D-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This  topology is determined by the smooth curves only, so that many different diffeologies  induce the same topology [CSW14, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Mapping a diffeology on X to the  induced topology is left adjoint to mapping a topology to the continuous diffeology  [CSW14, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The topology induced by the discrete (trivial) diffeology is  the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In general, however, neither the unit nor the counit of the  adjunction is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The maps p ∈ X(U) ⊂ Set(|U|, |X|) of a diffeological space X are called plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  If we spell out the defining conditions of a concrete sheaf, we obtain the traditional  definition of diffeological spaces in terms of plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  For every open cover {Ui → U} in Eucl, U is the coequalizer of �  i,j Ui ∩ Uj ⇒  � Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In other words, the site of euclidean spaces is subcanonical, so that every  representable presheaf is a sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since every representable presheaf is concrete, it  follows that all representable presheaves on Eucl are concrete sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We conclude  that the Yoneda embedding Y : X �→ Dflg( , X) factors as  SetEuclop  Eucl  Dflg  Y  y  I  1In [BH11, Example (2), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' 5794] it is stated incorrectly that the discrete diffeology is given by  the constant maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  22  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  through a functor y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since Y and I are full and faithful, so is y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since I is full  and faithful, the Yoneda lemma implies that the evaluation of the concrete sheaf  X ∈ Dflg on U ∈ Eucl is given by  X(U) ∼= SetEuclop(Y U, IX) ∼= SetEuclop(IyU, IX)  ∼= Dflg(yU, X) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  It follows that limits in Dflg are computed pointwise and that I preserves limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  By the adjoint functor theorem, I has a left adjoint,  K : SetEuclop  Dflg : I ,  which was computed and studied in [BH11, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Explicitly, K is given by a  procedure called concretization followed by the Grothendieck plus construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The left adjoint K is a retract, KI ∼= idDflg, which implies that the colimit of a  diagram in X : I → Dflg can be computed as  colim  i∈I  Xi ∼= colim  i∈I  KIXi  ∼= K colim  i∈I  IXi ,  that is, by first computing the colimit in presheaves and then applying K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' As a  further consequence, it can be shown that y is dense:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='12 (Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' 51 in [BH11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Every X ∈ Dflg is the colimit of y ↓ X →  Eucl → Dflg, which we will write as  X ∼= colim  yU→X yU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Terminology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The comma category y ↓ X is called the category of plots of  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' It is customary and convenient to identify notationally the domain  of a plot U ∈ Eucl with the diffeological space yU ∈ Dflg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In this paper, however, we  deal with a number of subtleties of Kan extensions along y where this identification  would invite wrong proofs by notation (a trap the author has fallen into more than  once).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Therefore, we will always spell out the embedding y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Left Kan extension to diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Notation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let F : Eucl → Eucl be an endofunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The left Kan extension of  yF along the embedding y : Eucl ֒→ Dflg will be denoted by  (11)  LF := Lany yF ,  which is an endofunctor of Dflg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The left Kan extension will be our device to extend the tangent structure of  euclidean to diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' L is functorial, which means that L preserves the  vertical compostion of natural transformations ˆα : ˆF → ˆF ′ and ˆα′ : ˆF ′ → ˆF ′′  between endofunctors ˆF, ˆF ′, ˆF ′′ ∈ End(Eucl), that is,  L(α′ ◦ α) = (Lα′) ◦ (Lα) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  ELASTIC DIFFEOLOGICAL SPACES  23  Since Eucl is small and Dflg is cocomplete, LF exists and is pointwise, that is, it  can be computed by the colimit [ML98, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='3]  (LF)(X) ∼= colim(y ↓ X −→ Eucl  yF  −→ Dflg)  = colim  yU→X yFU ,  for all X ∈ Dflg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let α : F → G be a natural transformation of functors Eucl →  Eucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Then yα : yF → yG is a natural transformation of functors Eucl → Dflg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The left Kan extension Lany yF is functorial in yF, so that we have a natural  transformation  (12)  Lα := Lany yα : LF −→ LG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Together (11) and (12) define a functor  (13)  L : End(Eucl) −→ End(Dflg) ,  where End(C) denotes the category of endofunctors and natural transformations of  the category C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The diagram  Eucl  Eucl  Dflg  Dflg  y  F  y  LF  commutes for all endofunctors F, that is  (LF)yU ∼= yFU  for all U ∈ Eucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since y is full and faithful, the statement follows from [ML98, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' 3, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='3]  or from [Kel05, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  □  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The functor (13) is full and faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Compatibility with products, coproducts, and subductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since y :  Eucl → Dflg is full and faithful, a smooth map f : U → V of euclidean spaces is a  strong epimorphism if and only if yf : yU → yV is a strong epimorphism, which is  the same thing as a subduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For this reason we will call a strong epimorphism  f a subduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This is the case if every point v0 ∈ V has an open neighborhood  V0 ⊂ V such that there is a smooth map g0 : V0 → U satisfying f ◦ g0 = idV0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  In short, f is a subduction if it has local sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In particular, every surjective  submersion is a subduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let α : F → G be a natural transformation of endofunctors of  Eucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If αU : FU → GU is a subduction for all U ∈ Eucl, then (Lα)X : (LF)X →  (LG)X is a subduction for all X ∈ Dflg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since αU is a subduction, so is yαU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Subductions in Dflg are the same as  regular epimorphisms, so that yαU is a regular epimorphism for all U ∈ Eucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The  left Kan extension αX = (Lα)X is given by the colimit over the category of plots  y ↓ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since colimits preserve regular epimorphisms, αX is a regular epimorphism,  that is, a subduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  □  24  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If a functor F : Eucl → Eucl preserves finite products, then so  does LF : Dflg → Dflg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Let F : I → End(Eucl), i �→ Fi be a functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Due to the universal properties of  colimits and limits, we have for every X ∈ Dflg the natural morphism  colim  yU→X lim  i∈I yFiU −→ lim  i∈I colim  yU→X yFiU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Assuming that the limit limi∈I Fi exists in End(Eucl), it can be written as the natural  transformation  L lim  i Fi −→ lim  i LFi ,  where we have used that y preserves limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This is not an isomorphism unless the  colimit and the limit commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Let G : J → End(Eucl) be another diagram, such that limi Gi exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Any natural  transformation αi : Fi → Gi induces a commutative diagram  L limi Fi  L limi Gi  limi LFi  limi LGi  L(limi αi)  limi Lαi  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' , Fk ∈ End(Eucl) be a finite family of endofunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Then we have an isomorphism  L(F1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' × Fk) ∼= LF1 × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' × LFk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  It is well-known, that the left Kan extension of an arbitrary functor along the  Yoneda embedding Y : Eucl → SetEuclop preserves all colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This is not true for  Kan extensions along y : Eucl → Dflg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Already for the preservation of coproducts  we have to make additional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Recall that a functor F : Eucl → C is a  cosheaf if F op : Euclop → Cop is a sheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The following functors on Eucl are cosheaves:  (a) the tangent functor T : Eucl → Eucl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  (b) fiber products of the tangent functor Tk : Eucl → Eucl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  (c) if F : Eucl → C and G : Eucl → Eucl are cosheaves, then so is their  composition FG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  (d) the de Rham functor Ω : Eucl → dgAlgop, which maps U to the differential  graded algebra of differential forms and smooth maps to the pullback of  forms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  (d) if F : Mfldop → C is a sheaf on the big site of manifolds and open covers,  then the restriction F : Eucl ֒→ Mfld → Cop is a cosheaf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If F : Eucl → C is a cosheaf, then its left Kan extension along  y : Eucl → Dflg preserves coproducts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If F : Eucl → Eucl is a cosheaf, then LF preserves coproducts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If F : Eucl → Eucl is a cosheaf, then LF preserves subductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let X : I → Dflg be a functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If F : Eucl → Eucl is a cosheaf,  then the natural morphism  colim(LF)X −→ (LF) colim X  ELASTIC DIFFEOLOGICAL SPACES  25  is a subduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Compatibility with the composition of endofunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let F, G : Eucl →  Eucl be endofunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The left Kan extension of their product is given by the colimit  L(FG)X ∼= colim  yU→X yFGU  ∼= colim  yU→X (LF)yGU ,  (14)  where we have used Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The product of the Kan extensions can be  written as  (15)  (LF)(LG)X ∼= (LF) colim  yU→X yGU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Due to the universal property of the colimit, we have a natural morphism  (16)  colim  yU→X (LF)yGU −→ (LF) colim  yU→X yGU .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Composing this morphism with the isomorphism (14) on its domain and with iso-  morphism (15) on the codomain, we obtain a natural morphism  (17)  µF,G : L(FG)X −→ (LF)(LG)X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  This is an isomorphism if and only if (16) is, which is generally not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If F is a cosheaf, then (17) is a subduction for all X ∈ Dflg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since F is a cosheaf, it follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='25 that (16) is a subduction,  which implies that (17) is a subduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  □  The morphism (17) is associative in the followoing sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let F, G, H : Eucl → Eucl be endofunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Then the diagram  of natural transformations  L(FGH)  (LF)(LGH)  L(FG)(LH)  (LF)(LG)(LH)  µF,GH  µF G,H  (LF )µG,H  µF,G(LH)  is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let α : F → F ′ and β : G → G′ be natural transformations of  endofunctors of Eucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Then the diagram  L(FG)  L(F ′G′)  (LF)(LG)  (LF ′)(LG′)  L(αβ)  µF,G  µF ′,G′  (Lα)(Lβ)  is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  26  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Compatibility with the D-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='29 (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='8 in [IZ13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The D-topology on a diffeological space is  the finest topology (on the underlying set) such that every plot is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Explicitly, a subset Y ⊂ X is open in the D-topolgy if and only if for every plot  p : yU → X, the preimage p−1(Y ) ⊂ U is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Every morphism of diffeological  spaces is continuous with respect to the D-topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In the following “open” and  ”continuous” are always meant with respect to the D-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' An open subset  S ⊂ X of a diffeological space is naturally equipped with the subspace diffeology, so  that the inclusion i : S → X is an open induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Let {Si ⊂ X}i∈I be an open cover of a diffeological space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Then the diagram  (18)  �  i,j Sij  �  i Si  X ,  where Sij = Si ×X Sj, is a coequalizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A functor F : Dflg → Dflg is a cosheaf if  F preserves the coequalizer (18), that is, if  �  i,j FSij  �  i FSi  FX  is a coequalizer for every open cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A special kind of cosheaf is given by a natural  bundle that satisfies the following locality condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A bundle π : F → 1 of endofunctors of diffeological spaces will  be called local if for every open induction i : S → X the commutative diagram  FS  FX  S  X  πS  F i  πX  i  is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If a bundle π : F → 1 of endofunctors of diffeological spaces is  local, then F is a cosheaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If a bundle π : F → 1 of endofunctors of diffeological spaces is  local, then F preserves open inductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let π : F → 1 and ρ : G → 1 be bundles of endofunctors of  diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If both bundles are local, then the fiber product F ×1 G → 1 and  the composition FG → 1 is local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let π : F → 1 and ρ : G → 1 be bundles of endofunctors of  diffeological spaces and α : F → G a morphism of bundles, ρ◦α = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If both bundles  are local, then the following are equivalent:  (i) αX : FX → GX is an induction (subduction, epimorphism, isomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  (ii) There is an open cover {Si → X}, such that αSi : FSi → GSi is an induction  (subduction, epimorphism, isomorphism) for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='30 applies also to bundles of endomorphisms of euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Explicitly, a bundle π : F → 1 is local if π−1  U (V ) ∼= FV for every open subset  V ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The next proposition shows that this notion locality is preserved by the left  Kan extension to diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If a bundle π : F → 1 of endofunctors of euclidean spaces is  local, then so is its left Kan extension Lπ : LF → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  ELASTIC DIFFEOLOGICAL SPACES  27  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Elastic diffeological spaces  The main result for the category of elastic spaces is that the left Kan extension of  the tangent structure on euclidean spaces defines a tangent structure with scalar R-  multiplication (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The proof of this statement is quite long and will be  given in [Blo].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' It uses a variety of techniques, some categorical, some geometric, and  all of the results about the left Kan extension given in the preceeding Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The tangent structure of elastic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The tangent structure of eu-  clidean spaces consists of the tangent functor ˆT : Eucl → Eucl together with the nat-  ural transformations of the bundle projection ˆπ : ˆT → 1, the zero section ˆ0 : 1 → ˆT,  the addition ˆ+ : ˆT2 → ˆT, the symmetric structure ˆτ : ˆT 2 → ˆT 2, and the vertical  lift ˆλ : ˆT → ˆT 2, which we from now one decorate with hats in order to distinguish  them from their Kan extensions to diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The structure was spelled  out explicitly in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  We would like to extend this structure to diffeological spaces by applying the Kan  extension functor L = Lany y  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The Kan extension of the tangent functor will be  denoted by  T := L ˆT : Dflg → Dflg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A diffeological space X is called elastic if the following axioms  hold:  (E1) The natural morphisms  θk,X : (L ˆTk)X −→ TkX  are isomorphisms for all k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  (E2) There is a natural morphism τX : T 2X → T 2X, such that the diagram  (L ˆT 2)X  (L ˆT 2)X  T 2X  T 2X  (Lˆτ)X  θ2  X  θ2  X  τX  commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  (E3) The natural morphism  λX : TX  (Lˆλ)X  −−−−→ (L ˆT 2)X  θ2  X  −−−→ T 2X  is an induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  (E4) The natural morphisms  νk,X : TTkX −→ T 2X ×TπX,TπX  TX  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' ×TπX,TπX  TX  T 2X  are injective for all k > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  (E5) For every finite set of positive integers k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' , kn the diffeological space X′ :=  Tk1 · · · TknX satisfies axioms (E1) through (E4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The full subcategory of elastic diffeological spaces will be denoted by Elst ⊂ Dflg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The category of elastic diffeological spaces has a tangent structure  given by the Kan extended tangent functor T = L ˆT, bundle projection πX := (Lˆπ)X,  and zero section 0X := (Lˆ0)X, the addition  +X := θ−1  2,X ◦ (Lˆ+)X ◦ θ2,X ,  28  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  the symmetric structure τX of Axiom (E2) of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1, and the vertical lift  λX of Axiom (E3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Moreover, the Kan extension of the scalar R-multiplication is a  scalar R-multiplication in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Alternative axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The axioms for abstract tangent structures are to hold  for the entire category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  This is why the Axiom (E5) has to be included in the  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1 of elastic spaces, so that the application of Tk does not lead out of the  subcategory of elastic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' However, Rosick´y’s axioms still make sense pointwise  for a single object X that lies in an ambient category where Tk is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This is  our situation, which suggests the following more general concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A diffeological space X is called weakly elastic if the Axioms (E1)-  (E4) of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  On a weakly elastic space we still have most of the structure of differential calculus,  like a Lie algebra of vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We only have to observe that TX and its fiber  products may no longer share the same good properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2 still holds if Axiom (E1) of Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1 is replaced  with the following weaker version:  (E1’) The natural morphism  θk,X : (L ˆTk)X −→ TkX  is an isomorphism for k = 2 and an epimorphism for all k > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  As explained in the introduction, we need the stronger Axiom (E1) so that we obtain  a Cartan calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  In earlier versions of the definition of elastic space, we have used the following  axiom:  (E0) The natural morphisms  L( ˆTk ˆT)X −→ TkTX  are isomorphisms for all k ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The Axiom (E0) implies the Axioms (E1), (E2), and (E4) of  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  While this proposition shows that Axiom (E0) is logically stronger than (E1),  (E2), and (E4), we currently do not know an example of an elastic space that does  not satisfy the stronger Axiom (E0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Stability properties of elastic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let X be a diffeological space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The following are equivalent:  (i) X is elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  (ii) X has an open cover {Si → X} by elastic spaces Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let X be a diffeological manifold modelled on the diffeological vector  space A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Then X is elastic if and only if A is elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let {Xi}i∈I be a small family of diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The following  are equivalent:  (i) The coproduct ⊔iXi is elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  ELASTIC DIFFEOLOGICAL SPACES  29  (ii) Every Xi is elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Finite products of elastic spaces are elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Retracts of elastic spaces are elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Examples  The conditions for a diffeological space to be elastic are quite strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Limits,  colimits, subspaces, and mapping spaces of elastic spaces are generally no longer  elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If by considering elastic spaces we have given away many of the conventient  properties of the category of diffeological spaces, the question arises whether there  are enough interesting examples and applications for the concept to be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  We have already stated that finite products and small coproducts of elastic spaces  are elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We have also seen that elastic spaces are stable under retracts, which  allows for the construction of elastic spaces with rather benign singular behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Basic examples are manifolds with corners [Joy12] and cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The original example that motivated the concept of diffeology by Souriau is that  of diffeological groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Our main result states that a diffeolgical group is elastic if  and only if the vertical lift λG : TG → T 2G is an induction (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This  mild condition is needed to avoid that the bracket of vector fields takes values in a  tangent bundle with a weaker diffeology, as is the case for vector fields on groups on  a Ck-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This shows that almost all diffeological groups that come to mind  are elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Even a seemingly pathological group like the quotient R/Q is elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Every diffeological vector space has an underlying abelian group, so that we can  apply the characterization of diffeological groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  It follows that a diffeological  vector space A is elastic if the natural map T0A → T0T0A that maps va to the tangent  vector represented by the path t �→ tva is an induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A diffeological manifold  modelled on A is elastic if and only if A is elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  A stronger condition which  ensures elasticity is that A → T0A is an isomorphism, which yields a trivialization  TA ∼= A × A of the tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We call such vector spaces tangent stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' All  fine diffeological vector spaces have this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  If A is tangent stable then the diffeological mapping space Dflg(X, A) is elastic  for all X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' An important example is the algebra C∞(X) ∼= Dflg(X, R) of smooth  functions on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The diffeological space of sections Γ(M, F) of a smooth fiber bundle  ρ : F → M is elastic (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' As expected, its tangent space is given by the  sections of the vertical tangent bundle ker Tρ → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' As a corollary, the diffeological  space Dflg(M, N) of smooth maps of manifolds is elastic, the tangent space being  given by  T Dflg(M, N) ∼= Dflg(M, TN) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  More examples can be constructed by forming retracts, which allows for mildly  singular situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Manifolds with corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Consider the set [0, ∞) ⊂ R equipped with the  subspace diffeology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Every smooth map p ∈ U → R with image p(U) ⊂ [0, ∞) has  vanishing derivatives to all orders at every point u0 with p(u0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' It follows that  T0[0, ∞) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  The tangent space at an interior point x ∈ (0, ∞) is given by Tx[0, ∞) = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  30  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  We want to show that [0, ∞) is elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For this, we consider the maps  π : R −→ [0, ∞)  x �−→ x2  and  σ : [0, ∞) −→ R  x �−→ √x ,  which satisfy π ◦σ = id[0,∞) as maps of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We have to show that σ is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let  p : U �→ [0, ∞) be a plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since σ is smooth on the interior (0, ∞), σ ◦ p is smooth  at a all points in U that are mapped to the interior (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Assume that p(u0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Since all derivatives of a plot p vanish at u0, p vanishes to all orders at u0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  p(u)  ∥u − u0∥k  u→u0  −−−→ 0 ,  for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This implies that  �  p(u) = (σ ◦ p)(u) vanishes to all orders at u0 as  well, so that σ ◦ p is differentiable at u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We conclude that σ is smooth, so that  [0, ∞) is a smooth retract of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='10, we conclude that [0, ∞) is elastic and by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='9  that any finite product  Rn  k := [0, ∞)k × Rn−k  is elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since the diffeological tangent functor commutes with products, the tan-  gent spaces are given by  T(x1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=',xn)Rn  k = Tx1[0, ∞) × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Txk[0, ∞) × Rn−k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Finally, it follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='6 that every diffeological space modeled locally  on Rn  k is elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Such spaces are called manifolds with corners [Joy12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  We  conclude:  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Manifolds with corners are elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Manifolds with cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Consider the following subset of R2,  X :=  �  (x, y) | x ≥ 0 ∧ |y| ≤ x  �  ,  with the subspace diffeology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  We can squeeze or stretch the corner at (0, 0) by  multiplying the y coordinate by a smooth function f ∈ C∞�  [0, ∞)  �  (see Figure 3),  ϕ : X −→ Xf  (x, y) �−→  �  x, f(x) y  �  ,  where  Xf :=  �  (x, y) | x ≥ 0 ∧ |y| ≤ f(x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Assume that f(x) > 0 for x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Then ϕ has an inverse map given by ϕ−1(0, 0) =  (0, 0) and  ϕ−1(x, y) =  �  x,  y  f(x)  �  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let us equip Xf with the pullback diffeology of ϕ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since ϕ−1 is sur-  jective, ϕ is an isomorphism of diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' X is [0, ∞) × [0, ∞) rotated by  minus 45 degrees, so it is elastic by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Since ϕ is an isomorphism Xf  is elastic, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  ELASTIC DIFFEOLOGICAL SPACES  31  ∼=  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Squeezing a corner by multiplying the y-coordinate with  the function f(x) = x  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The boundary of the resulting diffeological  subspace of R2 is the curve x2 = y5 with a cusp at (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  We cannot describe here in general, what a manifold with cusps or similar defects  is, since any squeezing operation that is a smooth retract will produce a new model  for elastic subspaces of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In Figure 2 on page 7 we have given a few examples of  elastic subspaces of R2 that can be obtained in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Diffeological groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A diffeological group is a group object (G, m, e) in Dflg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  As for ordinary groups, the inverse i : G → G, g �→ g−1 is unique, so it is a property  rather than a structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' By applying an endofunctor F : Dflg → Dflg that preserves  products, we obtain a diffeological group (FG, Fm, Fe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Examples for F are T and  Tk (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' (TG, Tm, Te) is called the tangent group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  From the multiplication of the tangent group we obtain the left G-translation of  TG given by  L : G × TG  0G×idT G  −−−−−−→ TG × TG  Tm  −−−→ TG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  If we restrict the second argument to the tangent fiber at e, we obtain the map  (19)  ϕG : G × TeG −→ TG  (g, ve) �−→ Lgve ,  where TeG = {e} ×G TG is the tangent fiber at the group identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' It has an inverse  given by  ϕ−1  G : TG  (i◦πG,idT G)  −−−−−−−→ G × TG  L  −−→ TeG ,  which maps vg �→ Lg−1vg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We conclude that (19) is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' In analogy to  Lie groups, we will denote  g := TeG ,  so that we obtain the trivialization of the tangent bundle,  TG ∼= G × g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  TG is itself a diffeological group with neutral element 0e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Using the isomorphism  T(x,y)(X, Y ) ∼= {(x, y)} ×X×Y T(X × Y )  ∼= ({x} ×X TX) × ({y} ×Y TY )  ∼= TxX × TyY ,  we obtain the trivialization  T 2G ∼= TG × T(0e)TG  ∼= G × g × T(e,0)(G × g)  ∼= G × g × TeG × T0g  ∼= G × g × g × T0g .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  32  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  In this trivialization the vertical lift λG : TG → T 2G is given by  G × g −→ G × g × g × T0g  (g, a) �−→  �  g, 0, 0, λ⊥  g (a)  �  where  λ⊥  g : g −→ T0g ,  maps a ∈ g to the tangent vector represented by the path t �→ ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A diffeological group G is elastic if and only if λ⊥  g : g → T0g is an  induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  It follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2 that g and T0g are diffeological  vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' For an elastic group λ⊥  g is an isomorphism and g is equipped with  the Lie bracket of invariant vector fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Every Lie group G is elastic when G is a smooth manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' However,  when equipp G with the Ck-diffeology, then g = Rn  Ck and T0g = Rn  Ck−1, so that λ⊥  g  is no longer an induction (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The diffeomorphism group of a smooth manifold is elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Diffeological vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A diffeological vector space is an R-vector space  object in Dflg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Explicitly, this means that the addition and scalar multiplication are  morphisms of diffeological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Diffeological vector spaces are a rich and subtle  structure [CW19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A diffeological vector space A is elastic if and only if the natural  linear map T0A → T0T0A is an induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This is Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='2 for the additive diffeological group A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  □  In many cases, elastic diffeological vector spaces satisfy the stronger condition  A ∼= T0A, which is equivalent to  TA ∼= A × A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  We will call diffeological vector spaces with this property tangent stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Tangent  stable vector spaces are elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' All fine diffeological vector spaces are tangent stable (hence elas-  tic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Diffeological manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' We recall that a diffeological manifold modelled  on the diffeological vector space A is a diffeological space X such that every point  of X has an open neighborhood that is isomorphic as diffeological space to an open  subset of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' A diffeological manifold is elastic if and only if it is modelled on  an elastic diffeological vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' This follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  □  ELASTIC DIFFEOLOGICAL SPACES  33  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Mapping spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If A is diffeological vector space, then the mapping space  Dflg(X, A) is a diffeological vector space with pointwise addition and scalar multi-  plication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let X be a diffeological space and A a diffeological vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  If A is tangent stable, then so is Dflg(X, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' If A is a tangent stable diffeological vector space then we have a  natural isomorphisms  T Dflg(X, A) ∼= Dflg(X, A) × Dflg(X, A) ∼= Dflg(X, TA)  for all X ∈ Dflg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The diffeological space  C∞(X) := Dflg(X, R)  of smooth R-valued functions on a diffeological space X is elastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Let ρ : F → M be a fiber bundle of smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Then the  diffeological space of sections Γ(M, F) is elastic with tangent space  TΓ(M, F) ∼= Γ(M, V F) ,  the space of sections of the vertical tangent bundle V F := ker Tρ → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The diffeological mapping space Dflg(M, N) of smooth manifolds  M and N is elastic with tangent space  T Dflg(M, N) ∼= Dflg(M, TN) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' The diffeological vector space of sections Γ(M, A) of a smooth  vector bundle A → M is tangent stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  References  [BH11]  John C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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+page_content=' Hoffnung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Convenient categories of smooth spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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+page_content=' In preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Cruttwell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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+page_content=' Cah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' G´eom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Diff´er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Cat´eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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+page_content=' Pacific J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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+page_content=' Daniel Christensen and Enxin Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Tangent spaces and tangent bundles for diffeological  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Cah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' G´eom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Diff´er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Cat´eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=', 57(1):3–50, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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+page_content=' Daniel Christensen and Enxin Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Diffeological vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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+page_content=' Classical field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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+page_content=' American Mathematical Society, Providence, RI, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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+page_content=' Lect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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+page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Press, Somerville, MA, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Kelly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Basic concepts of enriched category theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Repr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Theory Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Categ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=',  (10):vi+137, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Reprint of the 1982 original [Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Press, Cambridge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  MR0651714].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  34  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' BLOHMANN  [ML98]  Saunders Mac Lane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Categories for the working mathematician, volume 5 of Graduate  Texts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Springer-Verlag, New York, second edition, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  [Per16]  Ekaterina Pervova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Diffeological vector pseudo-bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Topology Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=', 202:269–300,  2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  [Ros84]  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Rosick´y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Abstract tangent functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Diagrammes, 12:JR1–JR11, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  [Vin08]  Martin  Vincent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Diffeological  differential  geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content='  Master’s  thesis,  Depart-  ment of Mathematical Sciences,  University of Copenhagen,  2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
+page_content=' Available at  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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+page_content='  Max-Planck-Institut f¨ur Mathematik, Vivatsgasse 7, 53111 Bonn, Germany  Email address: blohmann@mpim-bonn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wdE0T4oBgHgl3EQfswF7/content/2301.02583v1.pdf'}
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